Comparison of model potentials for molecular dynamics ...

Comparison of model potentials for moleculardynamics simulation of crystalline silica

Dissertation

zur Erlangung des Grades,,Doktor der Naturwissenschaften“

am Fachbereich Physikder Johannes Gutenberg–Universitat

Mainz

vorgelegt von

Daniel Herzbach

geboren in Frankfurt (Main)

Mainz, im Juni 2004

Datum der mundlichen Prufung: 05.07.2004

Abstract

Simulating SiO2 with the two-body potential developed by van Beest, Kramers, and van Santen(BKS) produces many satisfactory results but also characteristic flaws. We investigate thesefailures of the BKS potential and compare the performance of the BKS potential with that oftwo recently suggested potential energy surfaces that effectively incorporate many-body inter-actions. One approach, which is called the fluctuating-charge model, allows the ionic chargesto adjust depending on the chemical environment. The other approach assumes fixed, effectivecharges but allows for inducible dipole moments on the oxygen atoms.The emphasis in this work is placed on situations where BKS fails. We show that an anomalyin the ratio of quartz’s two independent lattice constantsa andc, which is observed experimen-tally at the transition betweenα andβ-quartz, is missing with BKS. Cristobalite and tridymiteappear mechanically unstable with BKS when periodic boundary conditions are employedthat are compatible with competing high-density silica polymorphs. Lastly, the BKS densityof states (DOS) shows characteristic discrepancies from the true DOS.The fluctuating-charge model slightly improves the phononic density of states and correctlyproduces stable cristobalite, but it does fail to show the experimentally observedc/a anomalyat theα-β transition. Moreover, many properties are reproduced much less satisfactorily thanwith BKS.The fluctuating dipole model remedies all mentioned artifacts. We confirm the view that theproper behaviour in thec/a ratio is due to the distortion of SiO4 tetrahedra. These distortionscan in turn be shown to be due to the many-body effects incorporated in the fluctuating dipolepotential.In addition, the pressure-induced phase transition inα quartz is studied. All three models showa transition at similar pressures to the same crystalline phase, which is probably the same asthat found experimentally for the high-pressure polymorph called quartz II. On decompres-sion, the BKS potential predicts the formation of an unknown phase at ambient pressures,while in the two other approaches the quartz II phase reverts toα-quartz with an intermediatephase similar to quartz II.Furthermore we show that the fluctuating-charge potential is the only known model potentialto predict the right pressure for a phase transition between two sixfold coordinated stishovitepolymorphs.We suggest two different methods to calculate piezoelectric coefficients in classical moleculardynamics simulation in the constant stress ensemble. We find that BKS reproduces experi-mental data reasonably well. However, the results for the fluctuating dipole potential turn outto underestimate the experimental data by more than 50%, unless the coupling of the electricalfield to the dipoles is switched off. We also show that there is a strong correlation between themagnitude of the dipoles and the bond-bending angles on the oxygen atoms and speculate thatquantum chemical effects that are non-electrostatic in nature were (successfully) parametrizedas inducible dipoles. With this interpretation, the fluctuating dipole potential always appearsto be closest to available experimental data out of the three approaches.

i

Contents

Introduction 1

1. Simulation Techniques and Model Potentials 61.1. Molecular Dynamics Techniques. . . . . . . . . . . . . . . . . . . . 6

1.1.1. Parrinello-Rahman Barostat. . . . . . . . . . . . . . . . . . 71.1.2. Gear Predictor Corrector Integrator. . . . . . . . . . . . . . 81.1.3. Langevin Thermostat. . . . . . . . . . . . . . . . . . . . . . 91.1.4. Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . 101.1.5. Ewald Summation. . . . . . . . . . . . . . . . . . . . . . . 10

1.2. BKS Potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3. Fluctuating Charge Potential. . . . . . . . . . . . . . . . . . . . . . 12

1.3.1. Charge Equilibration Algorithm. . . . . . . . . . . . . . . . 121.3.2. Direct solution . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.3. Extended Lagrangian. . . . . . . . . . . . . . . . . . . . . . 151.3.4. Morse-Stretch QEq Potential. . . . . . . . . . . . . . . . . . 161.3.5. Alternative Morse-Stretch fluc-Q Potential. . . . . . . . . . 181.3.6. Kinetic Parameters for Extended Lagrangian. . . . . . . . . 19

1.4. Fluctuating Dipole Model. . . . . . . . . . . . . . . . . . . . . . . . 201.5. Comparison of Performance. . . . . . . . . . . . . . . . . . . . . . 22

2. The α-β Quartz Transition 242.1. Transition Temperature. . . . . . . . . . . . . . . . . . . . . . . . . 262.2. Hysteresis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3. Thermal Expansion and c/a Ratio. . . . . . . . . . . . . . . . . . . . 30

2.3.1. Influence of Specific Properties of the Potentials. . . . . . . 322.4. Distortion of Tetrahedra. . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1. Influence of Atomic Bond Character. . . . . . . . . . . . . . 372.5. Elastic Constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.6. Phonon Density of States. . . . . . . . . . . . . . . . . . . . . . . . 43

3. Pressure-Driven Transitions in α-Quartz 473.1. Quartz II Transition in Simulations. . . . . . . . . . . . . . . . . . . 48

3.1.1. Transition Path Ways. . . . . . . . . . . . . . . . . . . . . . 493.2. Quartz II b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3. X-Ray Diffraction Spectra. . . . . . . . . . . . . . . . . . . . . . . 52

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3.4. Other High-Pressure Phases. . . . . . . . . . . . . . . . . . . . . . 54

4. Silica Polymorphs Other Than Quartz 564.1. Cristobalite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2. Tridymite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3. Stishovite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.1. High-Pressure Behaviour. . . . . . . . . . . . . . . . . . . . 61

5. Electromechanical and Dielectric Properties 635.1. Theory and Methods. . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1.1. Linear Response. . . . . . . . . . . . . . . . . . . . . . . . 645.1.2. Ambiguity of the dipole and its fluctuation. . . . . . . . . . 675.1.3. Fluctuation estimators for dipoles. . . . . . . . . . . . . . . 685.1.4. Direct estimators with noise reduction. . . . . . . . . . . . . 69

5.2. Temperature Dependence of d11 in Quartz . . . . . . . . . . . . . . . 715.3. Pressure Dependence of d11 in α-Quartz. . . . . . . . . . . . . . . . 745.4. Bond-Angle Dependence of the Dipole Moment. . . . . . . . . . . . 75

6. Conclusion and Outlook 77

A. Calculations 81A.1. Calculations for Fluc-Q Potential. . . . . . . . . . . . . . . . . . . . 81

A.1.1. Direct Solution of Charge Equilibration Equations. . . . . . 81A.1.2. Two Body Slater Integral. . . . . . . . . . . . . . . . . . . . 82

B. Programs 85B.1. Molecular Dynamics Code. . . . . . . . . . . . . . . . . . . . . . . 85

B.1.1. Inputfiles . . . . . . . . . . . . . . . . . . . . . . . . . . . .147B.2. Analysis Programs. . . . . . . . . . . . . . . . . . . . . . . . . . .148

List of Figures 158

Bibliography 164

iv

Introduction

Almost a century ago, the discovery of quantum mechanics allowed one to describe thefundamental interactions of ions and electrons in matter. Even some decades earlier,the equations of statistical mechanics were formulated, which relate the microscopicinteractions of ions and electrons to the resulting macroscopic properties of a givenmaterial. However it is generally not possible to solve these equations analytically forcomplex materials in order to predict their properties. Dirac expressed the problemin 1929 as follows: “The fundamental laws necessary for the mathematical treatmentof large parts of physics and the whole of chemistry are thus fully known, and thedifficulty lies only in the fact that application of these laws leads to equations that aretoo complex to be solved.” This statement still holds despite the increasing power ofcomputers. Simplified models for the interactions need to be formulated for analyticaltheories. More approximations are required if one wants to predict the behaviour ofmaterials beyond the harmonic approximation. If theory and experiment disagree, it isoften difficult to say at what point theory failed.The use of computers made it possible to reduce this uncertainty, as computer simu-lations made it possible to solve the equations for microscopic empirical model po-tentials with (theoretically) arbitrary accuracy, thus allowing us to see how the macro-scopic system behaves as a whole without the need to rely on further assumptions.The numerical computation of macroscopic observables can be realized by samplingrandomly over phase space in a Monte-Carlo (MC) simulation [58] or by followingthe trajectory of Newton’s equations of motion, and thus sampling over phase space aswell, in a molecular dynamics (MD) simulation [28].It is nevertheless not yet possible to solve the quantum mechanical equations exactlyexcept in an extremely limited number of cases. In fact, one can show that it is beyondany conceivable computational capacity to exactly solve the Schrodinger equation inits standard form for a system containing only ten electrons [84]. However it becamepossible to handle quantum mechanics in a computer simulation with the developmentof density functional theory (DFT), which states that it suffices to know the averagenumber of electrons located at any one point in space instead of the motion of eachindividual electron [70]. The practical application of this theory relies on good ap-proximations for the functional as the true functional is not known in a solid. Herethe local density approximation (LDA) has produced very accurate results for a wide

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range of materials. In 1985 Car and Parrinello [11] developed a very efficient methodto use DFT in the force calculation of an MD simulation, which made this so calledCar-Parrinello molecular dynamics (CPMD) one of the most widely used techniquesfor the calculation of dynamic and electronic properties of condensed matter. Thiskind of simulations, where a real (but approximative) potential without adjustable pa-rameters is used, is called first-principle or ab-initio simulations.So these two options, the ab-initio approach and the empirical model potentials, canbe implemented in a computer simulation, which involves different possibilities andrestrictions respectively. Of course ab-initio techniques lead to much higher chemicalaccuracy, as the calculation of the interatomic stresses and forces used for propagationof the particles is close to reality. On the other hand it is limited due to its computa-tional cost and thus restricted to small length and time scales. This is a major problemas it severely limits the range of properties that one can calculate and the number ofphenomena that can be simulated.In contrast an empirical model is parametrized to certain experimental properties un-der certain thermodynamic conditions, and its reliability has to be questioned when itis used away from the conditions under which it was parametrized or if properties haveto be calculated that did not enter the parametrization of the adjustable model param-eters. But they provide much faster calculation, which is beneficial for the calculationof thermal properties if the relaxation times in the system are large so that thermalaverages require sampling over long periods of time. Moreover, in a wide range ofapplications model potentials have been employed very successfully.The crucial questions with an empirical potential are to incorporate the relevant physi-cal effects into the formulation of the functional form and to have a proper adjustmentof the free parameters. In this work different approaches of model potentials will betested in the simulation of silica (SiO2) polymorphs. (Polymorphic materials are ma-terials that possess different crystalline structures for a fixed stoichiometry.) The forcefields studied in this work are the BKS two body pair potential [101], a model thatallows for fluctuation of charge between the atoms [20] and a third that employs fixedcharges but allows for inducible dipole moments on the oxygen atoms [90]. For theproblem of the parametrization the authors of the last potential used a new approach,which does not fit the potential to macroscopic experimental values but aims to matchthe microscopic forces, stresses and energies with values obtained from DFT calcula-tions [25, 57].

Silica or silicon dioxide is one of the most widely and intensively studied of all ma-terials. There are many reasons for this. First of all, it is one of the most commonmaterials in nature. Silicates, which is a compound consisting of silicon and oxygen,one or more metals and possibly hydrogen, make up more than 90% of the mineralsin the earth’s mantle and crust. Additionally it is used extensively in industrial appli-

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cation.An obvious example is that it makes up approximately 75% of the compositionof the glass that is used for everything from window panes to optical fibres. It is alsoused as an insulator or substrate in the semiconductor electronics industry.On the other hand it is interesting to study as crystalline silica has an extremely richphase diagram with a large number of polymorphs, shown in Fig.0.1 and the fluidphase, which remains metastable in the form of silica glass for a long time when cooleddown to ambient temperatures. The best known polymorphs are quartz, cristobalite,

Figure 0.1.: Phase diagram for silica polymorphs. From [54]

tridymite, coesite, and stishovite. The crystalline phases make up a good test casefor empirical potentials, as due to the reduced symmetries there are more independentsusceptibilities, while in disordered systems effects can be averaged out. Isotropic ma-terials have for instance only two independent elastic constants, while in quartz thereare six independent coefficients.

All the low pressure crystal structures are composed of corner-sharing SiO4 tetrahe-dra [39] and even in the glass and the liquid, almost all Si atoms are tetrahedrallycoordinated. In the higher pressure phases six-fold coordination becomes more andmore favorable for the Si atoms, which lead to octahedral coordination in stishoviteand other high-pressure phases. The high-pressure phases are of particular interest tounderstand the earth’s interior, where silica is exposed to extreme conditions of temper-ature and pressure. The high-pressure behaviour of quartz is complicated to investigatein experiment as well as in theory, due to the large number of six-coordinated struc-tures that can possibly form out of four-coordinated structures [52], the slow kinetics,metastabilities and certainly experimental difficulties [52, 105]. As shown in Fig.0.1,

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the stable phases with increasing pressure on quartz would be coesite and stishovite,however these transitions are kinetically inhibited at room temperature. Consequently,quartz can be kept metastable up to 21 GPa. At this pressure, a phase transition oc-curs. For the new forming phase different crystal structures [38, 95, 96], amorphousstructures [40] or partly crystalline structures [52] were proposed experimentally andtheoretically. In computer simulations, this phase transition is a good candidate for areliable model potential, as the system size and the time scales needed to simulate thetransition are out of reach for an ab-initio approach. Still the high pressure is not whatthe model potentials were made for, so the comparison between different approachesincreases the credibility in this case.But also in the quartz phase, which is prevalent at ambient conditions, there are un-resolved issues. When heatingα-quartz (or low quartz), the crystal turns toβ-quartz(or high quartz) at around 573◦C [39]. The average structure of both quartz phaseshas been known for several decades, and the displacive transition between them canbe easily understood in terms of tilting of certain groups of SiO4 tetrahedra. How-ever the real picture of the transition is far more difficult, as is the real structure ofβ-quartz. It is clear that the oxygen atoms do no fluctuate harmonically around theiraverage positions, but the dynamical disorder that leads to the average structure is stillan unanswered problem. There are currently two competing interpretations. One pic-ture of theβ-quartz phase sees the oxygens oscillate between two potential minima atthe positions of the left-handed and the right-handedα-quartz [107, 87, 100], while theother picture supports an ordered structure that has the atoms vibrate in a librationalmotion around the mean positions [3, 92, 13, 24, 81, 49]. These two interpretations fa-vor different pictures of the transition, either a disorder-order type transition or a weakfirst-order transition with a soft mode initiating the transition.The described dynamical disorder makes theβ-quartz phase and obviously the transi-tion difficult to simulate with an ab-initio method. It is definitely not sufficient to findthe average structure and extract the relevant properties ofβ-quartz by harmonic ap-proximations. It was shown that for example the bulk modulus inβ-quartz turned outto be twice as large as in experiment with local density approximation and subsequentharmonic approximation [21, 65]. In contrast it is necessary to find the thermodynamicobservables by averaging over a long simulation run in order to sample the dynamicdisorder (which leads to a bulk modulus comparable to experiment). This makes itnecessary to use an empirical model potential. However the standard two-body poten-tial generally employed for the simulation of silica turned out to be flawed in differentaspects of the transition. Thus a new potential modelling different microscopic effectscould improve the two-body potential in the simulation of theα-β transition.

At the beginning of the thesis in chapter1 the three potentials used will be introducedand explained, as well as different MD algorithms and techniques used in the simula-

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Contents

tions.The following chapter2 will be concerned with theα-β transition and the performanceof the different potentials at this transition. After the determination of the transitiontemperature, the emphasis is on the course of the lattice constants with temperature.Due to the failure of the BKS potential in this respect, the distortion of the tetrahedrawill be investigated in more detail. The phonon density of states and the elastic con-stants are compared with experiment.The simulation of the high-pressure post quartz phases in chapter3 concentrates ofthe determination and comparison of the transition path ways produced by the differ-ent potentials. As available the phases will be compared with experiment in terms ofdiffraction spectra.Chapter4 focuses on an instability of cristobalite in the BKS potential and the com-parison to the other models.As the new potential approaches incorporate electrostatic properties, namely fluctuat-ing charges and fluctuating dipole moments, the investigation of dielectric propertiesin chapter5 should highlight the question if these are real electrostatic variables andthus improve the prediction for the dielectric properties of the system. A method isdeveloped to measure dielectric and piezoelectric properties efficiently.The conclusions at the end will give a final assessment of the suitability of the newapproaches for silica simulations.

5

1. Simulation Techniques and ModelPotentials

1.1. Molecular Dynamics Techniques

The simulation scheme being used, a classical molecular dynamics (MD) computersimulation, is a numerical method to calculate the trajectory of a many-atom system.The starting point of an MD simulation is Newton’s equations of motion for the systemunder consideration:

miRi = −∇iV ({Rj}) = F i (1.1)

whereRi are the coordinates of the atoms,mi their masses,V is the total potentialenergy andF i the force acting on atomi.These equations form a system ofN coupled partial differential equations. Their so-lution is the phase space trajectory of the many-particle system. The aim of an MDsimulation is to numerically solve the equations of motion and to extract macroscopicobservables from the resulting trajectory.To obtain meaningful time thermal averages for macroscopic quantities, it is impor-tant to start with an equilibrated configuration. This ensures that time averages donot change in the course of the simulation. In order to simulate a crystal in thermalequilibrium, one has to start with a configuration being close to the expected crystalstructure and to run the simulation until the macroscopic quantities are reasonably sta-ble. Two aspects are particularly important for the simulation of a system. One is themodel potentialV from which the forces follow, the other aspect is which algorithm isused to integrate the equation of motion and how external imposed conditions such astemperature, stress, or an electric field can be taken into consideration.A variety of standard methods is available for for numerical details such as the inte-gration scheme. In this work the so-called Gear predictor-corrector algorithm up tofifth order was used for the integration of the equations of motion. [33]. While Gearpredictor-corrector algorithm of order higher than two do not conserve energy, theydo produce more accurate trajectories than the second order Gear predictor-correctoralgorithm, which is also called velocity Verlet. The 2nd-order predictor-corrector isknown to be energy and phase space conserving. However in the context of this workthis attribute is relevant for testing purposes only, when the correct implementation of

6


the routines was checked by means of energy conservation within an unthermostatted,i.e. microcanonical ensemble. For succeeding production runs a Langevin thermo-stat was used, which enables simulations in the canonical ensemble. The thermostatconstantly adds and removes energy to the system in order to maintain a constanttemperature. This renders the energy conservation of the integrator an unnecessaryquality. Also higher-order integration schemes give more accurate averages at a giventime step, once the thermostat is introduced, which is why they were used for the pro-duction runs.To allow for simulations in the isothermal-isobaric (NpT) ensemble, the ParrinelloRahman method was used. This method as well as the integrator and thermostat algo-rithms will be described in detail in the following.For acceleration of the time consuming force calculation standard binning and Verletneighborhood list techniques were used [1, 28].

1.1.1. Parrinello-Rahman Barostat

The Parrinello-Rahman method of pressure and stress control allow one to simulatemolecular systems under externally applied stress. This is useful for studying thestress-strain relationship of materials and phase transitions with non-isotropic shapevariations. Both the shape and the volume of the cell can change, so that the inter-nal stress of the system can match the externally applied stress. In this work only thepresence of uniform pressure is needed, so it is not necessary to allow for an arbitrarystress tensor. It is yet important to allow for an arbitrary strain.The method was presented in detail by Parrinello and Rahman [71] and is only sum-marized here. The fluctuating shape of the cell is given by the cell matrixh containingthe cell vectors so that the real space coordinatesRj of a particlej are related to thereduced coordinatesrj by Rj = hrj. While the coordinatesRj represent the real posi-tions of the atoms, the reduced coordinatesrj only live in a unit cube.The Langrangian of the system is modified such that a term representing the kineticenergy of the cell depends on a user-defined massMbox. An elastic energy termp det his related to the pressurep and the volumedet h of the system. These two terms leadto an extended Lagrangian:

L =1

2

N∑i=1

mir′ih′ hri − Vinteraction+

1

2MboxTrh′ h− p det h (1.2)

The equations of motion for the reduced coordinates and the cell vectors can be derivedfrom this Lagrangian by using the regular Lagrange formalism. The motion of the cellvectors, which determine the cell shape and size, is driven by the difference betweenthe target and the internal stress.

7

1. Simulation Techniques and Model Potentials

The choice of the fictitious massMbox is governed by the following considerations.A large Mbox means a heavy, slow cell. In the limiting case, infiniteMbox revertsto constant-volume dynamics. A smallMbox means fast motion of the cell vectors.Although the target stress can be reached faster, a too small mass can result in artificalperiodic movements of the cell and requires a smaller time step. As theh matrix iskept symmetric, it can be written in Voigt notation [102], which is a standard to writesymmetric crystallographic3 × 3 tensors as a6 dimensional vector with entriesh =(hxx, hyy, hzz, hxz, hyz, hxy). Once all equations of motion are derived, a thermostatcan be added to them. This can be done for the cell vectors in a similar way as for thereduced atomic coordinates.

1.1.2. Gear Predictor Corrector Integrator

Let ri(t) be the position of particlei at timet. First, in the predictor step the positionsand derivatives of the particle positions are predicted,

(∆t)n

n!

∂n

∂tnri(t+∆t) =

(∆t)n

n!

∂n

∂tnri(t)+

Norder∑m=n+1

Cpn,m

(∆t)m

m!

∂m

∂tmri(t) , n = 0, . . . , Norder ,

(1.3)whereCp

n,m is them-th predictor coefficient ofn-th order. The predictor coefficientsare just the Taylor expansion coefficients

Cpn,m =

m!

n!(m− n)!. (1.4)

For efficiency, the powers of the time-step and faculties are saved on the arrays holdingthe derivatives of the positions. When configurations are written out, they are howeverdivided out, to obtain the time-derivatives of the positions independently of the time-step. They can be used to propagate saved configurations with another time-step∆t.The force calculation in the following MD step is now based on these predicted po-sitions. After this, the propagation step is completed by the correction step. In thisstep, positions and derivatives are corrected proportionally to the difference betweencomputed and predicted force (or equivalently acceleration),

(∆t)n

n!

∂n

∂tnri(t+∆t) :=

(∆t)n

n!

∂n

∂tnri(t+∆t)+Cc

n

(∆t)2

2

(fd

i(t +∆t)

m− ∂2

∂t2ri(t +∆t)

).

(1.5)Here, Cc

n is the n-th corrector coefficient. Gear devised the correction coefficientsto minimize errors such that the local truncation error is ofO(∆tNorder+1) for lineardifferential equations [33]. For second-order differential equations the global erroris thenO(∆tNorder−1). The corrector coefficients are listed in table1.1.2 followingRef. [36].

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Norder = 2 Norder = 3 Norder = 4 Norder = 5

Cc0 0 1/6 19/120 3/16

Cc1 1 5/6 3/4 251/360

Cc2 1 1 1 1

Cc3 – 1/3 1/2 11/18

Cc4 – – 1/12 1/6

Cc5 – – – 1/60

Table 1.1.: Corrector coefficients for the Gear predictor corrector algorithm.

1.1.3. Langevin Thermostat

The Langevin thermostatting routine couples the system via friction and noise to a heatbath. In doing so, the simulation, which was based on Newton’s equation up to now,now integrates a Langevin equation

miRi = F i −mγRi + F ri . (1.6)

Therefore this method is also referred to as “stochastic dynamics”. The friction con-stantγ controls how fast the system relaxes into equilibrium. The simulation schemenow produces states which are distributed according to the canonical ensemble, andtogether with the barostat even the isobaric-isothermal ensemble. The temperature re-sults as the ratio of noise strength to friction, via the fluctuation-dissipation theorem.The stochstic forceF r

i is a random variable with mean zero and a widthσr chosen tosatisfy the fluctuation-dissipation theorem [1]:

σr =√

3

√2TmkBγ

∆t(1.7)

The uniform distribution is prefered to the Gaussian distribution for efficiency and sta-bility reasons, as the details of the distribution function do not matter. The frictionconstantγ is set to 0.1 in almost all simulations unless noted otherwise.The thermostat was used also in production runs, even though an equilibrated con-figuration should not change in temperature, when an energy conserving integrationscheme is employed. Nevertheless the thermostat decorrelates the system and is evenmore stable than simple microcanonical MD with the Verlet algorithm, because thestochastic dynamics thermostats every degree of freedom individually. As the stochas-tic force individually heats or cools single particles, it can prevent particles with akinetic energy much higher or lower than the thermal average from causing instabili-ties. This is in contrast to the Nose-Hoover thermostat [69], in which only the overallsystem is thermostatted.

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1.1.4. Interactions

Interaction forces are calculated using the positions from the predictor step, includingthe random contribution. In all models being used the leading order in the forces aregiven by pair potentials. In the case of the fluctuating-charge potential these forces alsodepend, via the charges, on the positions of all the other atoms, and are therefore manybody interactions. In the case of the fluctuating-dipole potential the two-body forcesare accomplished by additional dipole interactions. The values of the dipoles dependon all the other dipoles, and therefore the interactions between the dipoles represent amany body contribution.

1.1.5. Ewald Summation

All the potentials dealt with in this work contain a Coulomb interaction part. As withall the other parts in the potentials, the calculation of the total energy of a three-dimensional system theoretically involves the evaluation of interactions between allspecies within the unit cell and their periodic replications. Even when the interactionwas evaluated for an infinite number of replications, the electrostatic interaction wouldnot be well defined as it still depends on how the limit is carried out. One conisitentway is to sum upexp(−r/a)/r to infinity before takinga → ∞. Nevertheless in nu-merical computations some finite cutoff must be placed on the interactions. For thoseinteractions decaying quickly with distance, the summation can normally be readilyconverged directly in real space. For long-range interactions however the convergencemight be not so quick, particularly since the number of interactions increases withr2.For the electrostatic energy in particular, the number of interactions increases morerapidly with distance than the potential, which is proportional to1/r, decays. An ac-curate evaluation can be achieved through the Ewald summation [26, 93] in which theinverse distance is rewritten as its Laplace transform and then split into two rapidlyconvergent series, one in reciprocal space and one in real space. The distribution ofthe summation between real and reciprocal space is controlled by a parameterα. Theresulting expression for the energy is:

V rec =∑j>i

Rij<rewcut

QiQje2

Rij

erfc(αRij)−α√π

∑j

q2j e

2 (1.8)

+2π

V

∑q=2πh−1(nx,nz,nz)

0<q≤qcut

exp(−(

q2α

)2)q2

∣∣∣∣∣∑j

Qje exp(2πiRj · q

)∣∣∣∣∣2

(1.9)

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1.2. BKS Potential

The cutoffsrewcut in real space andqcut in reciprocal space can be set in dependency ofα

in a way that the error is minimized:

rewcut = s/α, qcut = (2sα)2 (1.10)

A value ofs = 3 has proven to be a good compromise between accuracy and speed.For the distribution between calculations in real and reciprocal space, a good balancecan be found at aboutα = 0.3 for system sizes of around1000 particles which wereconsidered in the present work. Both of these values were used throughout the simu-lations.

1.2. BKS Potential

The BKS Potential was proposed in 1990 by van Beest, Kramer and van Santen [101]and has been succesfully applied to a range simulations involving crystalline and vit-reous silica since [16, 68, 46, 103, 48, 91]. It consists of pure pair interactions withdifferent strength for different pairs of atom types. By combining suitable attractiveand repulsive forces for the different atom types it is possible to reproduce effects thatare usually caused by many-body interactions. So the BKS potential causes the stableformation of SiO4 tetrahedra without explicitly caring for the effects of covalent bond-ing.The total potential energy is thus a sum of pure pair potentials

V ({Ri}) =∑i<j

φij(Rij). (1.11)

The pair potentialφij is composed out of a long-range Coulomb term and a short-rangeBuckingham term:

φαβ(R) =qαqβe2

R+ Aαβe−BαβR − Cαβ

R6. (1.12)

The charges are fixed on the valuesqSi = 2.4 andqO = −1.2, while the parametersA, B andC in the Buckingham interaction were optimized for the different pairs ofatom types with respect to ab intio calculations as well as to macroscopic experimentalquantities like the elastic constants ofα-quartz. The parameters given in ref. [101] arelisted in table1.2. The Si-Si interaction is assumed to be purely Coulombic in the BKSpotential, which is why the Buckingham parameters are all0 for Si-Si interaction.

11


Aαβ[eV] Bαβ[A−1

] Cαβ[eVA6]

O-O 1388.7730 2.76000 175.0000Si-O 18003.7572 4.87318 133.5381

Table 1.2.: BKS Parameters

1.3. Fluctuating Charge Potential

A fluctuating charge potential is conceived as a model potential that makes use of vari-able charges on the atomic sites when calculating the Coulomb interaction. As thetotal charge of the system needs to be conserved, this charge adjustment may be un-derstood as relocation of charges within the system and indeed it implies a change inthe polarisation of the simulation box. This explains the common notation “polariz-able potential”, which will not be used in the following for better distinction betweenfluctuating charge and fluctuating dipole potentials.The purpose of fluctuating point charges in the simulation is to implicitly account forelectronic degrees of freedom. The effective charge of an atom in a fixed charge sim-ulation can be chosen with respect to its type and ionization, but it cannot change inresponse to changing electrostatic fields which arise from movement of the atoms dur-ing the simulation. Thus the charges used in simulations based on fixed charge forcefields must reflect average charge values for the particular phase and are in generalnot transferable to different thermodynamic states. The fluctuating charge approach,in contrast, adjusts the charge according to the instantaneous configuration of the sur-rounding atoms. Therefore it should be more adaptable to different phases and espe-cially to varying coordination numbers than a fixed charge approach.

1.3.1. Charge Equilibration Algorithm

The basic concept used for the fluctuating charge approach is the concept of electroneg-ativity equalization[82] according to which charge is transferred between atomic sitesin such a way that electronegativities are equalized. In doing so, the electronegativ-ity is dependent on the atom type and charge as well as the electronegativities of theneighboring atoms.Following the Mulliken definition, the electronegativity of an isolated atomi is thenegative of the chemical potential (µi) of the electron gas surrounding the nucleus:

χi = −µi = − ∂U

∂Ni

= e∂U

∂Qi

(1.13)

12


In this equation, use had been made of the fact that the ground state energyE, thenumber of electronsN in the atom and the charge on the atomQ fulfill the equationQ = −e(N − Z), whereZ is the atomic number of the atom.The energy of an isolated atom dependent on the effective charge can be expandedaround a neutral reference point as

Ei(Q) = Ei0 + Qi

(∂E

∂Q

)i0

+1

2Q2

i

(∂2E

∂Q2

)i0

+ . . . (1.14)

Neglecting higher-order terms, one can ask for the energy needed to remove or add asingle electron. In units ofe and withEi0 = 0 one obtains

Ei(+1) = Ei0 +

(∂E

∂Q

)i0

+1

2

(∂2E

∂Q2

)i0

(1.15)

Ei(−1) = Ei0 −(

∂E

∂Q

)i0

+1

2

(∂2E

∂Q2

)i0

. (1.16)

As Ei(+1) is just the ionization potentialIE andEi(−1) is the electron affinityEA,one finds for the first coefficients exactly the Mulliken definition of electronegativity(

∂E

∂Q

)i0

=1

2(IE + EA) = χ0

i . (1.17)

The second coefficent turns out to be a quantity called idempotential, representing theself repulsion between two electrons:(

∂2E

∂Q2

)i0

= (IE − EA) = J0ii (1.18)

This leads to the following expression for the charge dependence of the atomic energy,where the parametersχi andJ0

ii can be derived from atomic data:

Ei(Q) = χiQi +1

2J0

iiQ2i (1.19)

The total energy in the system is obtained by adding this energy to the interactionenergy, which is given by a two body termΦ consisting of a Coulomb interaction partand a short range non-Coulombic interaction part.

U({Qi}, {Ri}) =∑

i

Ei(Qi) +∑i<j

Φij(Rij)

=∑

i

[χiQi +

1

2J0

iiQ2i

]+∑i<j

[Jij(Rij)QiQj + V (Rij)

]=

∑i

χ0i Qi +

1

2

∑i,j

Jij(Rij)QiQj +∑i<j

V (Rij) (1.20)

13


Here the Coulomb interaction was assumed to obeyJii(R) → J0ii asR → 01. This

means thatJij(R) does not represent a pure1/R Coulomb potential but has to be ashielded potential with a finite value atR = 0.After all, we end up with an equation that relates the total energy to the positions andcharges of all the atoms in the system. Equations (1.13) and (1.20) can be used in twodifferent ways to find the right charges:

• treat the charges as parameters[77]: directly solve for the exact charges given bythe spatial configuration in every MD step, or

• treat the charges as generalized coordinates[78]: extend the Lagrangian for thecharges, propagate them in the same way as spatial coordinates and thermostatthem at 0 K.

A third possibility is to solve the equations selfconsistently. This solution would givethe possibilty to control the characteristics of the charge trajectory between the onewhich is always in the energy minimum, like in the first case, up to a retarded be-haviour, like in the second case. Nevertheless it turned out that this solution wasdifficult to control numerically and converged very slowly. As there are no obviousdifferences in the trajectories of the two solutions introduced above, this third possi-bility was skipped.

1.3.2. Direct solution

A set of charges forms the ground state of a given spatial configuration when the chem-ical potentials on all atom sites are equal. This implies theN − 1 conditions

χ1 = χ2 = · · · = χN . (1.21)

Adding the condition on charge neutrality

N∑i=1

Qi = 0 (1.22)

leads to a total ofN equations for theN equilibrium charges. In order to expressthe electronegativity in terms of the charges, one has to insert expression (1.20) intoeq. (1.13):

χi =∂U

∂Qi

= χ0i + J0

iiQi +∑j 6=i

JijQj (1.23)

1In Ref. [77] one can actually recalculate analytically that this condition is not fulfilled. To obtainbetter fits to dipole moments, it was dropped by the authors[76].

14


Equations (1.21) and (1.22) can be merged into a matrix equation

C ·Q = −D, (1.24)

where

Di = χ0i − χ0

1 (1.25)

and

C1i = 1 (1.26)

Cij = Jij − J1j (1.27)

Solving these linear equations with standard procedures leads to the sought-after charges.A more elaborate version of this matrix including all terms from Ewald summation canbe found in appendixA.1.1

1.3.3. Extended Lagrangian

The idea for treating charges as generalized coordinates is based on an extended La-grangian approach. While a charge dependent potential energy is obviously available(eq. (1.20)), one has to define additionally a kinetic energy expression. This is done byassigning a fictitious mass to the charges. The choice of an appropriate value will bediscussed later on. Again the charge neutrality has to be obeyed, and the simplest wayto implement this is to treat the charges as independent and use a Lagrange multiplierto enforce the constraint. The Lagrangian is

L =∑

i

1

2miR

2i +

∑i

1

2MQQ2

i − U({Qi}, {Ri})− λ∑

i

Qi (1.28)

While the Euler Lagrange equation for the positions remains unchanged, one gets anadditional expression for the charges:

MQQi = − ∂U

∂Qi

− λ (1.29)

As the total charge is zero, the expression∑

i Qi also vanishes. Therefore∑i

MQQi = −∑

i

∂U

∂Qi

−Nλ = 0 =⇒ λ = − 1

N

∑i

∂U

∂Qi

, (1.30)

which defines the Lagrange parameter. Substitution into the equation of motion yields

MQQi = − ∂U

∂Qi

+1

N

∑j

∂U

∂Qj

= − 1

N

∑j

(∂U

∂Qi

− ∂U

∂Qj

)(1.31)

15


The descriptive interpretation of this equation is that the electrons are attracted byatoms with higher electronegativity, which complies with the expected behaviour.This equation of motion can now be propagated by the MD integrator, provided thatparametersMQ for the inertia of the charges andγQ for the friction in the thermostathave been defined.The choice of these parameters is detailed below for the present case. The generalconsideration is to separate the time scales of ionic motion and electronic motion inorder to suppress coupling between internal charge modes and ionic motion. This is asimilar idea as in the Car Parrinello approach, where as well a ficitious mass is assignedto the electronic orbitals, which should be small enough that the motion of the orbitalswill be very fast relative to the motion of the ions. This way it is ensured that thefrequency spectra of the electronic orbitals and the ions are well separated from oneanother.

1.3.4. Morse-Stretch QEq Potential

The potential that we used for MD simulations was parametrized for SiO2 by Demi-ralp et al.[20]. In this model the fluctuating charge was implemented through thecharge equilibration (QEq) procedure developed by Rappe and Goddard[77], whichfeatures a direct matrix diagonalization approach. In agreement with Rappe and God-dard the QEq parameters were choosen as shown in Table1.3. Following Ref. [77],

χ(eV) J(eV) RQEq(A)

O 8.741 13.364 0.669Si 4.168 6.974 1.176

Table 1.3.: QEq parameters for O and Si

the shielded Coulomb interactionJij(R) is taken to be the Coulomb overlap integralbetween Slater orbitals centered on each atomic site

Jij(R) =

∫dRi

∫dRj |φni

(Ri)|21∣∣Ri −Rj −R

∣∣ ∣∣φnj(Rj)

∣∣2 . (1.32)

The Slater orbitals are given by

φni(R) = AiR

ni−1e−ζiR (1.33)

and are determined by the principal quantum numberni and an exponentζi. Thisexponent is given byζi = 0.4913 ·(2ni+1)/(2RQEq

i ), with RQEq from the table above2.

2The differing formula forζ given in ref. [20] is not correct[19].

16


A detailed description of how to solve the two-center overlap integral can be found inappendixA.1.2.The J(R) interaction models a shielded Coulomb potential that is asymptotically a1/R point-charge Coulomb interaction, when the particles are at a sufficient distance,but approaches a finite value forR → 0.The non-electrostatic short-range interactions are modelled by a Morse-Stretch (MS)term

UMSij (Rij) = D0

[eγ(1−Rij/R0) − 2eγ/2(1−Rij/R0)

]. (1.34)

The short range interaction is truncated at9.0A. The MS parameters were optimizedby Demiralp et al. to describe properties like density, cohesive energy, elastic moduliof α quartz and stishovite. The values can be found in table1.4. It is of course always

R0(A) D0(kcal/mol) γ

O-O 3.7835 0.5363 10.4112Si-Si 3.4103 0.2956 11.7139Si-O 1.6148 45.9970 8.8022

Table 1.4.: Morse Stretch parameters

assumed implicitly that the potential energy surface used to find the optimum charges,by setting∂V/∂Qi = 0 at fixed values ofR with the constraint

∑i Qi = 0, should be

the same as the potential energy surface used to calculate the forces∂V/partialRi atfixed values ofQ.As the description of the energy surface in Ref. [20] does not state anything different,we naturally assumed the only mentioned electrostatic potential energy in this paper,which is theJ(R) given by (1.32), as the one which is consistently used. When weimplemented the potential as stated by the authors, none of their results could be re-produced, and even the nearest-neighbor peak of the Si-O pair correlation functionin theα-quartz structure deviated significantly from the experimentally known1.6A.This flawed behaviour was exactly reproduced when we employed the MD simulationprogram GULP[30, 29, 31] for comparison, which was provied free of charge by theauthor Julian D. Gale.The reason for these discrepancies finally turned out be an inconsistent treatment ofthe potential energy, which was disguised in the publication. Even though they nevercomment on this in their paper or in private communication, Demiralp et al. did notuse the shielded Coulomb potentialJij(Rij) for the ion interaction. Instead they useda pure1/Rij Coulomb potential. The only verification for this fact was that only whenwe tried this inconsistent treatment, we could reproduce the results for the equation ofstate inα-quartz as they published it in the paper.So effectively they used the shielded Coulomb potential to calculate the charges as

17


overlapping Slater orbitals and subsequently used these values as point charges witha pure Coulomb potential in the MD step. Obviously this was done because in thecommercial software package they used the correct electrostatic energy functional andits derivatives have not been impelemented, thus the software was only capable to usethe point-charge interaction.A hint to this inconsistency and an attempt to correct it was made recently in a laterpublication by Sefcik et al.[85], where the Morse Stretch parameters where optimizedfor a consistent potential energy function. Unfortunatelyα-quartz turned out to beunstable with this new parameter set in our simulations at ambient temperature. Thecrystal immediately transformed toβ-quartz at temperatures well within the stabilityrange ofα-quartz. Because of this flawed behaviour of the new parameter set, whichrenders it useless for our simulations, we do not consider it in this work, but only usethe inconsistent original potential, repectively the modification of it described in thenext section.

1.3.5. Alternative Morse-Stretch fluc-Q Potential

Using different potentials for charge equilibration and force calculations makes it im-possible to use the extended Lagrange method. Moreover, it is an unphysical treat-ment to treat these two potentials inconsistently. However, as the results presented inRef. [20] seemed to be close to experimental data, it appeared to be useful to alter thepresent potential such that experimental data was reproduced and that charge equili-bration and force calculations were based consistently on the same energy surface. Todo this, an energy functional was constructed in a way that its derivatives with respectto position and charge should be as close as possible to the derivatives of the two differ-ent energy functionals used in ref. [20]. For this purpuse, both fluctuating charges andfixed charges were included in this potential.Jij(Rij) was used as Coulomb potentialand a termqiqj[Jij(Rij) − 1/Rij], with q the average charge inα quartz, was addedto the MS potential term. Using this modified potential energy within the extendedLagrangian formalism, the derivation with respect to the dynamical charges will onlytake notice of theJij(Rij) part, but for the derivation with respect to the positions andtherefore the ion motion the1/R is dominant. This modification is of course purelyheuristic, but it reproduces nicely the structure of different SiO2 polymorphs, as shownfor thegSiO(R) in α-quartz in Fig.1.1and it reproduces the original data as well. Thus,while this procedure does not necessarily yield the optimum values for the interactionparameters, it seems as though it allows one to analyze at least qualitatively if notsemi-quantitatively what benefits one can expect from a fluctuating charge model inSiO2.

18


3 4 5 6r/Å

0

1

2

3

4

5

g SiO

(r)

BKSmodified fluc-Q potentialoriginal fluc-Q potential

Figure 1.1.: Pair correlation functiongSiO(R) of α-quartz at 300 K. The curves for the new consistentfluctuating charge potential and the original unconsistent potential basically lie on top ofeach other. The nearest neighbor peak, which is located at 1.6A, is not included in thegraph.

1.3.6. Kinetic Parameters for Extended Lagrangian

In the extended Lagrangian approach, the equilibration of the effective charges reflectsthe movement of the electrons in the system. To adiabatically decouple the electronicmovement from the ionic movement, it is therefore necessary to separate the relevanttime scales. Therefore in addition to the usual relation∆t � τion � TSim, the newelectronic timescale has to fit in:∆t � τQEq � τion � TSim, so that the chargeequilibration time is well below the equilibration time of the ionic movement. This wasachieved by choosing a fictious mass for the charges of0.01 u. The damping constantin the Langevin thermostat was set by finding the value where the charge equilibrationis critically damped, which is2.0. With these numbers the equilibration time of thecharges is about10 fs, which means a frequency of100 THz, well beyond the typicalphonon frequencies in SiO2. To cover these higher frequencies in the simulation, thetime step was shortened by a factor of10 to ∆tMD = 0.1 fs in this type of simulation.As the authors of the fluctuating charge potential mention that they used the directapproach to calculate the charges every25 to 100 timesteps, it is interesting to note thatthis corresponds to frequencies of10 to 40 THz. This is within the phononic frequencyrange and leads to the situation that the charges cannot adapt to the instantaneous ionicconfiguration.

19


1.4. Fluctuating Dipole Model

In the fluctuating charge procedure, charge can only fluctuate from one atom to theother. This can bring about polarizations within the crystal, but it does not accountfor polarizations within the electron hull of an atom. The fluctuating dipole proce-dure in contrast directly assigns a dipole moment on each atom site. In case of thequartz system to be considered, only the oxygen atoms are assigned a dipole moment.This is because the silicon atoms have a rather stable nearest neighbor environmentthat consists of symmetrically arranged oxygen atoms in the corners of a tetrahedron.Therefore polarization on these atoms is unlikely to play a major role. The polarizationof the oxygen atoms however is well established[60].The force field for SiO2 used in the following was parametrized by P. Tangney andS. Scandolo[90]. For the parameter fitting they used theoptimal potential routinede-veloped by Ercolessi[25] and in this form by Laio et al[57]. It is an iterative procedurethat obeys the following algorithm: Beginning with the best potential available, thesystem is equilibrated with an MD simulation. Once equilibrated, a number of atomicconfigurations are generated. On these configurations, density functional theory cal-culations of total energy, forces and stress are performed and these are used to performthe parametrization. With the new parameters, the scheme will be repeated until thecurrent parameter set fits the current and the previous ab initio data to the same degree.Finally the emerging parameter set is supposed to produce potential energy, forces andstress in agreement with the DFT method used for parametrization, but much fasteras the ab initio method. This statement obviously only holds in terms of a certainaccuracy and in the vicinity of the surrounding conditions that were employed in theparametrization procedure, like temperature and external pressure. When this temper-ature and pressure is exceeded significantly, the accuracy of the potential cannot betaken for granted but has to be questioned as it is the case with every model potential.One major advantage of this method is that it does not rely on any experimental inputand thus the parametrization can take place in principle in any possible thermodynamicconditions, even where experimental data is unavailable.In this work, we used the Fortran subroutine provided by Paul Tangney that calculatesforces, stress tensor and potential energy for a given configuration. The parametriza-tion was done by Tangney and Scandolo in the liquid phase at3000 K, in order tofind a parameter set which is not biased towards a particular crystal structure. In thefollowing the algorithm used to solve for the dipoles will be outlined.Calculation of the induced polarization is a many-body problem, where the induceddipoles all depend on each other. Ifαi is the polarizability of sitei, the induced dipole

20

1.4. Fluctuating Dipole Model

piat positionRi is given by

pi

= αiEi (1.35)

Ei = E0i +

∑j 6=i

Tijp

j(1.36)

The polarization is proportional to the electric field, which in turn depends on all thedipoles via the dipole-dipole interaction tensorT

ij. HereE0

i is the “fixed” part of theelectric field due to charges and a possibly applied external field. To solve this pair ofequations amounts to solving a set of3N × 3N linear equations. Thus the possibilityto solve them directly or in an iterative way can be ruled out directly, as it would befar too computationally expensive. Therefore these equations have to be solved eitherwith an extended Lagrangian scheme as done in [88, 106] or with a self consistentapproach as in [94].However, Wilson and Madden[106] found that this model for polarization had signifi-cant shortcomings. The ions tended to become over-polarized, and small anion-cationdistances resulted in very large induction forces which overcame the short-range Pauliexclusion repulsion. This is because the model neglects the dipoles induced by short-range overlap effects, which should oppose the Coulomb-field induced dipoles. On thebasis of electronic structure calculations[27], Wilson and Madden suggested an addi-tional term to incorporate these dipoles into the present model. The induced dipolefrom eq. (1.35) will now be:

pi= αiEi +

∑j 6=i

αigij(Rij)qjRij

R3ij

(1.37)

where

gij(Rij) = − exp(−bRij)4∑

k=0

(bRij)k

k!. (1.38)

Putting everything together, the algorithm basically works according to the followingprinciple:

1. Calculate the contribution from the charges to the energy, forces and stress

2. Calculate the short-range induced dipole moment on each ion

3. Formulate a guess for the electric field on each ion by extrapolation from the lastthree timesteps

4. Solve for the dipoles, until they are consistent with the resulting electric field upto a certain accuracy

21


5. Use the converged values to calculate the contribution from charge-dipole anddipole-dipole interaction to energy, forces and stress

6. Calculate the contribution of the short-range induction of dipole moments to theenergy, forces and stress

7. Add up all contributions to the energy, forces and stress

The pair interaction between the ions used in the force field by Tangney and Scandolois a point charge Coulomb interaction plus a pair potential of Morse-Stretch form tomodel the short-range interactions:

Uij =qiqj

Rij

+ Dij[eγij(1−rij/r0

ij) − 2eγij/2(1−rij/r0ij)]. (1.39)

The parameter set used is shown in Table1.5.

D γ r0

O-O 2.4748× 10−4 12.07092 7.17005

Si-O 1.9033× 10−3 11.1523 4.6371

Si-Si −2.0846× 10−3 10.45517 5.75038

qO qSi α b c-1.38257 2.76514 8.89378 2.02989 -1.50435

Table 1.5.: Fluc. Dipole Force Field parameters (atomic units)

1.5. Comparison of Performance

As far as the speed of calculations is concerned, the simplest potential, which is thetwo-body BKS potential, is obviously the fastest. The fluctuating-charge potential isslowed down in comparison to BKS by about a factor of10 due to the shorter time stepthat is needed with the extended Lagrangian approach to equilibrate the charges andto decouple charge motion from the ionic motion. Using the direct calculation of thecharges at every 25th to 100th time step as suggested by the original authors [20] leadsto a similar slowing down.The comparison of the performance of the fluctuating-dipole potential with the stan-dard two-body potential is difficult as our BKS routine is highly optimized while thefluctuating-dipole routine uses new routines for every part of the calculation. As these

22

1.5. Comparison of Performance

were not optimized at all, the comparison would be meaningless because of differentways to calculate the details as for instance the Ewald sum, especially its reciprocalpart, or the neighbor lists. The original authors [90] claim to see a slowing down bya factor of100. However in comparison to our BKS routine the slowing down is evenmore significant, which is why in many instances in the following work the systemsizes for the fluctuating-dipole simulations will be smaller than for the other potentials.In general the fluctuating dipole can be treated comparable to the fluctuating charges,and thus in an optimized routine the slowing down should have a similar value, whichmeans a factor of10.

23

2. The α-β Quartz Transition

The transition fromα- to β-quartz is a displacive phase transition, which occurs at atransition temperature ofTtr = 846 K. The atomic structure of theα andβ polymorphsof quartz is known since 1925 [9] and was refined in the following decades by numer-ous experimental studies [39]. The structure ofα-quartz is most easily envisioned as

Figure 2.1.: Sketch of the averageβ-Quartz structure along the [100] direction

a distortion of the high-temperatureβ modification. Whenβ-quartz is cooled belowthe transition temperature, the expandedβ-quartz framework collapses to the denserα-quartz configuration. For the average structure, the contraction of the tetrahedral

Figure 2.2.: The two orientations ofα-Quartz

24

network may be described geometrically as the rotation of rigid tetrahedra around the[100] axes by an angleΘ. This angle can be used as on order parameter, which is zeroin theβ polymorph, whereasΘ is 16.3 for α-quartz at room temperature. Dependingon the sign of the tilting angle, the transformation yields two distinctα1 andα2 twinorientations as shown in Fig.2.2. These are called the Dauphine twins ofα-quartz.However the real picture of the transition is far more complicated. Even though theα − β transition has been subject of many studies over the past 100 years [39], it issurprising that there is still no firm picture of what actually happens at the phase tran-sition. This uncertainty is connected with the discussion about the real structure ofβ-quartz. While figure2.1shows the average atomic positions for the oxygen atoms, itturns out that these average position are rarely occupied. It is not completely resolvedwhat the instantaneous structure looks like that leads to the symmetric average posi-tions. There are basically two competing ideas ofβ-quartz.One picture of theβ-quartz structure is based on the idea that each oxygen atom isassociated with not one but two potential minima. As a consequence the atom wouldoscillate between two positions that correspond to the Dauphine twin configurations ofα-quartz. This means that in the high symmetry phase the atoms are hopping betweenthe positions corresponding to the different domains of the low symmetry phase, givingthe structure shown in Fig.2.1on average. Upon cooling, the dynamic disorder in thehigh symmetry phase gets ordered when the atoms progressively occupy the positionscorresponding to one of the orientations in the low symmetry phase. Therefore this pic-ture ofβ-quartz would imply an order-disorder type transition. It has been supportedby results of Neutron diffraction[107], NMR studies[87] and MD simulations[100].The other picture supports an ordered structure ofβ-quartz. The idea is that theatoms in principle vibrate around mean positions given by the average structure, whichchange on cooling belowTtr through symmetry breaking displacements. This point ofview explains the observed existence of soft modes inα- andβ-quartz[3, 92, 13, 24]and the absence of symmetry forbidden phonons in theβ phase of quartz[81], whichshould be detected ifβ-quartz consisted of clusters ofα-quartz that persist over longertime scales. However it is clear that the oxygen atoms do not fluctuate harmonicallyaround their average positions. This is supported by an X-ray study[49] that showedthat the oxygen probability density functions deviate considerably from Gaussians.This outcome was not interpreted as disorder in the sense of clusters of microtwins,but rather as librational motion of the oxygen atoms around their averaged positions,which lie on the straight connection of two Si atoms.

25


2.1. Transition Temperature

In the following the transition temperatureTtr of theα-β transition will be determinedfor the different potentials. For the BKS potential this value was recently shown [66]to be 740 K with an uncertainty of about5 K. For the fluctuating charge and the fluctu-ating dipole potential the transition temperature is not known and will be determinedin this section.In general for the calculation of the transition temperature in a computer simulationone is confronted with the problem, that due to the finite size of the simulation boxall thermodynamic properties behave smoothly near the phase transition, and so doesthe order parameter. In order to determine the transition temperatureTtr neverthelessaccurately, it is possible to use the fourth-order cumulant [8], which is defined in thecase of a one-component order parameter as

g4(N, T ) =1

2

(3− 〈(φ)4〉N

〈(φ)2〉2N

), (2.1)

where⟨φk⟩

deontes the thermal average of thek’th moment of the order parameterfor an N -particle system. It has been shown [104], that g4(N, T ), aside from smallcorrection terms, has a size-independent crossing point at a first-order phase transi-

500 600 700 800 900 1000T [K]

0

0.2

0.4

0.6

0.8

1

g 4(T)

N=72N=324N=864

Figure 2.3.: Fourth-order cumulantg4 as a function ofT for the fluctuating dipole potential

tion. This is valid under the restriction that the geometry of the simulation cell remainssimilar when the particle number is increased. This condition can only be approxi-mately satisfied in the case of a crystal, as it can only be expanded in multiples of the

26

2.1. Transition Temperature

unit cell. The unit cell used in the simulations is even larger than the primitive cell,as we fit the trigonalα-quartz phase into a rectangular simulation box. So only an ap-proximately coinciding crossing point can be expected for the different system sizes.The system sizes chosen areN = 72, N = 324, andN = 864. As shown in Fig.2.3,there is not a perfect crossing of the cumulants for the different system sizes, but it stillallows one to locate the transition temperature atT ≈ 715K with an uncertainty ofabout5K. This value is comparable to the BKS transition temperature, but even lowerand120K too low compared to the experimental transition temperature.To gain a more accurate picture of the transition it can be described within Landau

200 400 600 800 1000 1200T [K]

0

5

10

15

<Φ>

[°]

N=72N=324N=864L. T. (N=72)L. T. (N=324)L. T. (N=864)Landau Theory

Figure 2.4.: Order parameter〈|Φ|〉 as a function of temperature. Shown are different system sizessimulated with the fluctuating dipole potential, fits obtained with Landau theory and finite-size Landau theory.

theory in the same way as it is done in Ref. [66]. It is well established that the macro-scopic evolution of quartz at the transition point can be described by the standardLandau expansion for a first-order transition [13, 35, 4, 5]. Odd order terms inΦ arenot permitted by symmetry and the expansion is written as

F (Φ, T ) =1

2a(T − Tc)Φ

2 +1

4bΦ4 +

1

6cΦ6. (2.2)

F is the free energy per particle as a function of temperatureT and order parameterΦ anda, b, c, andTc are parameters. The parameter b should be negative to have a(weak) first order transition.These parameters can be specified by fitting the absolute value of the order parameter〈|Φ|〉 as a function of temperature to the data obtained from simulation. In finite-size

27


Landau theory for a system withN particles the expectation value reads

〈|Φ|〉 =

∫∞−∞ dΦ|Φ| exp{−βNF (Φ, T )}∫∞−∞ dΦ exp{−βNF (Φ, T )}

. (2.3)

This function is fitted against the data from theN = 324 system and shown in Fig.2.4.For the two other system sizes also the fit parameters obtained fromN = 324 areshown. It can be seen the behaviour of〈|Φ|〉 is reasonably described by Landau the-ory. The fit parameters area = 7.41 × 10−3 J/(mol · K), b = −4.97 × 10−3 J/mol,c = 8.03× 10−5 J/mol, andTc = 704 K. From Fig.2.4 the transition temperature canbe read asTtr = 712 K. Similar to the experiment [13], b is found to be negative, whichcharacterizes a first-order transition, andTtr − Tc = 8 K. This is also in good agree-ment with different studies based on experimental data, which typically determined thevalue ofTtr − Tc in the range of4− 13 K [3, 45, 35, 4, 22, 50]. In total the features ofexperimental studies are certainly reproduced, even though the transition temperatureis about130 K smaller.The same analysis for the fluctuating-charge potential (Fig.2.5) is done for systems

200 400 600 800 1000 1200 1400T [K]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

g 4(T)

N=324N=864N=1800

Figure 2.5.: Fourth-order cumulantg4 as a function ofT for the fluctuating charge potential.

sizesN = 324, N = 864, andN = 1800. The system sizes are chosen somewhatlarger than before for the reasons mentioned in chapter1.5. The non-monotonic be-haviour of the cumulant, which was also observed in Ref. [104], does occur withinthe plotted error bars. The cumulant analysis yields a transition temperature of aboutT = 550K. The parameters of the Landau expansion (eq. (2.2)) are fitted as describedbefore to the data of theN = 864 system. The parameters obtained were used for allthree system sizes shown in Fig.2.6. The fit parameters area = 8.92×10−3 J/(mol·K),

28

2.2. Hysteresis

200 400 600 800 1000 1200T [K]

0

5

10

15

<Φ>

[°]

N=324N=864N=1800L. T. (N=324)L. T. (N=864)L. T. (N=1800)Landau Theory

Figure 2.6.: Order parameter〈|Φ|〉 as a function of temperature. Shown are different system sizes sim-ulated with the fluctuating charge potential, fits obtained with Landau theory and finite-size Landau theory.

b = −4.30 × 10−3 J/mol, c = 1.02 × 10−4 J/mol, andTc = 570 K. Again, b is nega-tive and the differenceTtr − Tc is positive and about10 K, as we can find a transitiontemperature ofTtr = 581 K in Fig. 2.6. Again the characterisation of the phase transi-tion seen in experiment is obtained quite similar, but the discrepancy in the transitiontemperature of more than250 K is tremendous.

2.2. Hysteresis

An interesting feature in theα-β transition was discovered in Ref. [66] that is linkedto hysteresis effects in the transition. It was shown that one can distinguish betweenthe two phases from the local structure, which offers the possibility to study hysteresiseffects directly. This statement holds for our simulations, as it is shown for the case ofthe fluctuating-dipole potential in Fig.2.7. The pair correlation functiong(r) does notchange significantly with temperature above and belowTtr, but makes a sudden changeat the transition. So one can observe the hysteresis effect in theg(r) function. Whilethe structure atT = 725 K that was started with anα-quartz configuration equilibrated25 K below, it remains stable in this phase. On the other hand it also remains stablein theβ-quartz phase, when the initial configuration was equilibrated25 K above. Ofcourse in a simulation close to the transition temperature the system undergoes morechanges between the two phase for a small box, only for a larger particle number the

29


4 4.5 5 5.5 6r/Å

0

0.5

1

1.5

2

2.5

3g S

iO(r

)T=750 KT=725 K (cooling)T=725 K (heating)T=700 K

Figure 2.7.: Radial distribution functiongSi−O(r) at various temperatures for system size N=864 withfluctuating-charge potential.

system stays in one phase.

2.3. Thermal Expansion and c/a Ratio

One check for the different models is the temperature dependence of the volume closeto the transition. As can be seen in fig.2.8 the BKS potential gives fairly good agree-ment with experiment, even though the volume is sligthly overestimated and the tran-sition temperature differs from the experimental transition temperature as described inthe previous section. The result of the fluc-Q potential is quite similar, even thoughthe volume of both quartz phases is even bigger for this model. The best quantitativeagreement with experiment can be seen for the fluc-µ potential. The rather smoothbehaviour at the transition is due to the fact that this potential allows for the existenceof both of the two phases in a quite broad temperature range around the transition.However the volume expansion is not isotropic, but the jump at the transition was ob-served in experiment [13] to be of different magnitude for different spatial directions.This can be seen in the ratio of the lattice constantsc anda, which shows a disconti-nuity at the transition. It was recently pointed out [66], that in a computer simulationusing the BKS potential the behaviour of thec/a ratio differs qualitatively from the ex-perimental result, as the simulation does not see any discontinuity or even any distinctfeature in thec/a ratio at the transition. This behaviour is shown in figure2.9. Shown

30


0 400 800 1200 1600T/K

112

114

116

118

120

122

V/Å

3

BKSexperimentfluc Qfluc-dipole

Figure 2.8.: Temperature dependence of the volume per unit cell at theα-β transition. All of thethree different model potentials show qualitative agreement with experiment, while thequantitative agreement is best for the fluc-µ potential.

-500 0 500T-Ttr [K]

0

0.004

0.008

[c/a

](T)-

[c/a

](Ttr)

ExperimentBKSfluc-Qfluc-dipole

Figure 2.9.: Temperature dependence of thec/a ratio at theα-β transition. For easier comparison thecurves are shifted to match at their respective transition temperature.

in this figure are also the curves for the fluc-Q and the fluc-µ potential. In the caseof the fluc-Q potential one can see a change in the slope at the transition temperature.Thus there is at least an effect of the transition in thec/a ratio, but it is still far awayfrom reproducing the height of the stroke that occurs in experiment. In contrast thefluc-µ approach matches the experimental course of thec/a ratio quite well even in

31


quantitative terms. The smooth behaviour at the transition is again owing to the broadrange of phase coexistence as discussed above.

2.3.1. Influence of Specific Properties of the Potentials

0 200 400 600 800 1000 1200 1400T [K]

1.088

1.09

1.092

1.094

c/a

standard fluc-Q potentialfluc-Q potential with fixed Q

Figure 2.10.: Temperature dependence of thec/a ratio at theα-β transition for the fluc-Q potentialwith fluctuating and fixed charges.

A closer look at the origin of these differences can be done by investigating the in-fluence of the fluctuating charges and the fluctuating dipoles on this behaviour. If weuse the fluctuating charge potential with the charges fixed on the average charges inα-quartz, the behaviour of thec/a ratio is only marginally different, and only in thehigh temperature regime. The kink atTtr is still of the same shape as with the fluc-tuating charges. In general there is not much difference between the average chargesin α-quartz, which are about< QSi >= 1.318e for Si, and inβ-quartz, where thevalue is< QSi >= 1.293e. So it is justified to state that the fluctuations of the chargesbarely affect the simulation, at least the kink in thec/a ratio is certainly not a result offluctuatingcharges but rather an effect of the pure two body forces in the potential.The same analysis for the fluctuating dipole potential results in a completely differ-ent picture. The simulation crucically depends on the fluctuating dipoles. When thedipoles are fixed on the average values found inα-quartz, the crystal would not trans-form to theβ phase even at temperatures as high as1300K. Similarly, with fixedaverageβ quartz dipoles the crystal would stay in aβ-quartz configuration even at300K. This can be seen in the temperature dependence of the order parameter〈|Φ|〉 asshown in fig.2.11. While in the first case the volume (Fig.2.12) matches the volume ofthe run that has the dipoles fluctuating at a temperature of300K, where the averagingwas performed, the volume in the latter case is much higher than with a fluctuatingdipole run. This is due to the fact that the average dipoles are not a reasonable quantityin theβ-quartz phase, as they represent the dipoles that result from thermal averages

32


0 500 1000T [K]

0

5

10

15

c/a

fluc-dipolefixed α dipolesfixed β dipoles

Figure 2.11.: Order parameter〈|Φ|〉 as a function of temperature for the unaltered fluc-µ potential andfor the same potential with dipoles fixed on their average values inα-quartz orβ-quartz

0 400 800 1200T [K]

112

114

116

118

120

122

V [Å

3 ]

experimentfluc-dipolefixed β dipolesfixed α dipoles

Figure 2.12.: Volume per unit cell of quartz as a function of temperature for the unaltered fluc-µ

potential and for the same potential with dipoles fixed on their average values inα-quartz orβ-quartz

over the structure. So for example, for symmetry reasons, thez component of thedipole must be zero inβ-quartz but not inα-quartz. To constrain the dipoles to thevalues that they have inβ-quartz makes the crystal tend more to the average structurewith stretched bonds on the oxygen atoms, which explains the artificially increasedvolume that is observed when the constraint is applied in the simulations. Also thec/a ratio, shown in fig.2.13, is quite independent of temperature in this setting andshows an unphysical value for the fixedβ-quartz dipoles. To summmarize, one cansay that the dipoles play an important part in the model and are responsible for a bigpart of the forces that act on the atoms. While these simulations show the effect thatthe dipoles have on the structure and the phase transformation, the mechanism how thec/a anomaly at theα-β transition comes about is not yet resolved.

33


0 400 800 1200T [K]

1.085

1.09

1.095

1.1

c/a

experimentfluc-dipolefixed α dipolesfixed β dipoles

Figure 2.13.: c/a ratio of quartz as a function of temperature for the unaltered fluc-µ potential and forthe same potential with dipoles fixed on their average values inα-quartz orβ-quartz

2.4. Distortion of Tetrahedra

The reason for the behaviour of thec/a ratio in quartz was investigated by Grimm andDorner [35]. They isolated three tendencies that go along with increasing temperature.

• increasing Si-O-Si angle

• decreasing mean Si-O distance

• contraction of SiO4 groups in c direction

While it is possible to describe the change in the volume of quartz by rotation of rigidtetrahedra by leaving all nearest neighbor Si-O bond lengths constant, the behaviourof the c/a ratio cannot be reproduced by such a model. It was shown [86] by purelygeometrical arguments that a value ofc/a > 1.0981 can only be achieved in a quartzcrystal if the tetrahedra are deformed. The model employing only tilting of rigid tetra-hedral SiO4 units only can not account for a lower value in this ratio, even though itcan explain all volume effects. Given this argument and the experimentally knownlattice constants a deformation of the SiO4 tetrahedra must be present for tempera-tures lower than about 400 K. The shearing of the tetrahedra in the quartz phase isalso well established experimentally [35, 49, 13]. To test if such a deformation ispresent in the simulations, one can look at the tetraeder angleϑOSiO, which is shown infig. 2.14. However the fluctuating dipole potential, which definitely shows the samec/a ratio behaviour as in experiment, does not exhibit a dramatic change in the overallaverage of the OSiO angle when going to small temperatures (see fig.2.14) and noobvious difference compared to the BKS potential. On the other hand the average ofall tetrahedron angles is not sensitive to shearing, as the different angles in a deformedtetrahedron are likely to cancel out on average. The second moment of the distribution

34


500 1000T [K]

109.2

109.25

109.3

109.35

109.4

109.45

109.5

<ϑO

SiO

>

<δϑ><δϑ>

500 1000T [K]

3

4

5

6

7

8

<δϑ2 O

SiO

>

<δϑ2><δϑ2>

BKSfluc-dipole

Figure 2.14.: Average O-Si-O bond angle and its second moment as a function of temperature

could give more information, but the value obtained with fluc-µ is comparable to thatin BKS.A more sensitive analysis of the angle distribution can be done with the help of the

500 1000T [K]

-0.08

-0.06

-0.04

-0.02

0

g 4

fluc-dipoleBKS

Figure 2.15.: Average fourth order cumulant of the O-Si-O bond angle as a function of temperature

fourth-order cumulant shown in fig.2.15. As different mean values for the differentO-Si-O bond angles would result in a superposition of Gauss distributions with differ-ent mean, the overall average should deviate from a plain gaussian. A measurementfor the gaussian behaviour of a distribution is the fourth-order cumulant, defined as

g4 =1

2

(3− 〈(δϑ)4〉

〈(δϑ)2〉2

), (2.4)

which is zero for a gaussian distribution. One can see that the angle distribution gen-erated by the BKS potential deviates even more from gaussian behaviour than thefluc-dipole bond-angle distribution. This means that the tetrahedra in theα phase ofthe BKS potential are even more deformed than in the fluc-dipole potential. This canalso be seen quite obvious by an exemplary look at the average angles making up the

35


Figure 2.16.: [001] projection of theα1, α2 andβ configuration. Heights along [001] are shown inparantheses.z is 1/6 for β-quartz and slightly below or above for the two Dauphinetwins ofα-quartz. Picture taken from Kihara.

tetrahedron around one specific silicon atom. In order to enumerate the atoms in awell-defined way, the definition by Kihara [49] is used, which is shown in fig.2.16.The comparison of the angles around the Si1 position forα-quartz at298K is shownin the following table with experimental values by Kihara.

Experiment BKS fluc-dipoleO1-Si-O2 110.5 107.5 110.1O1-Si-O6 108.9 114.8 108.9O1-Si-O5 108.8 108.2 109.1O2-Si-O5 109.3 110.5 109.5

The table shows that the deviation between the ideal tetrahedral angle of 109.47 de-grees and the calculated angles has the correct sign and amplitude for the fluctuating-dipole potential as compared with experiment, in contrast to the BKS result. One cansee that the large deformation, that was also seen in the figures above, is actually wayto large compared to experiment and moreover has the wrong sign. In three out of thefour angles shown, the abberation from the tetraeder angle that is predicted by BKSgoes into the wrong direction. The explanation follows from the geometrical argumentgiven by Smith [86] when we look at the absolute value of thec/a ratio produced byBKS. It is shown in fig.2.17, that thec/a ratio does not exceed the value of1.0981 forthe case of the fluctuating dipole potential, in contrast to the BKS potential, which isbelow this value only for high temperature in theβ-quartz. So the BKS quartz has tobe strongly deformed even in theβ phase because of geometrical reasons, so it is clear

36


112 114 116 118 120 122 124V [Å3]

1.08

1.085

1.09

1.095

1.1

1.105

1.0981

c/a

ExperimentBKSfluc-Qfluc-dipole

Figure 2.17.: c/a ratio in quartz as a function of the volume per unit cell.

that the subtle shear effect in the tetrahedra that would lead to the correct behaviourof the lattice constants is completely suppressed by a large deformation having theincorrect sign.

2.4.1. Influence of Atomic Bond Character

The origin of the distortion of the tetrahedra that leads to a contraction of thez axisin quartz, was related by Hill and Gibbs [44] to an increasingπ bond order, i.e. anincreasing fraction ofπ-bonded valence electrons. At ambient conditions, Hill andGibbs have observed that the mean Si-O bond length is inversely proportional to thesize of the Si-O-Si angle. Plots ofRSiO versus the function−1/ cos(Si-O-Si) for awide variety of silicate structures are linear with the shorter bonds associated with thewider angles. On the basis of molecular orbital calculations [59, 34] a linear relationbetweenπ-bond order and both, the mean Si-O distance and−1/ cos(Si-O-Si), wasproposed. This was based on theπ-bonding model of Cruickshank [18], which is il-lustrated in Fig.2.18. The two pictures show how the p-orbitals of the oxygen atomscan be combined with thedz2 anddx2−y2 orbitals of the Si atom to formπ-bondingmolecular orbitals (the colours correspond to the signs of the wave function). It wasshown for SiO4 type tetrahedra that the decreasing Si-O distance is accompanied byan increasingπ-bond order. [18]. Furthermore thed-orbitals single out thez′ axis andat least for thedx2−y2 orbital it is obvious that an increasingπ-bond order will tend topull the oxygen atoms towards thex′y′ plane.To demonstrate the validity of this model for quartz, Grimm and Dorner showed thecorrelation between the mean Si-O distance and the Si-O-Si angle as mentioned above.To compare their conclusions to the simulations one has to determine the Si-O bond

37


Figure 2.18.: Models showing the formation of bindingπ-orbitals for XO4 groups. Pictures takenfrom Cruickshank.

11.051.11.151.21.251.3-1/cos(ϑSiOSi)

1.58

1.59

1.6

1.61

RS

iO[Å

]

Exp. (Grimm and Dorner)fluc-dipoleBKS

Figure 2.19.: Correlation between the mean Si-O distanceRSiO and the Si-O-Si angleθSiOSi

length in the average configuration rather than the average of the instantaneous bondlengths. The outcome in Fig.2.19shows clearly, that the angular dependence of theSi-O bond length is very similar to the experiment. The absolute numbers are compa-rable and the dependence is linear as predicted by the Hill-Gibbs relation. The figureshows that the effects of theπ bonds on the Si-O distance and the Si-O-Si angle, whichcontrols most of the volume expansion, are obviously included in the BKS potentialfairly well as well. Only the third effect, the contraction of the tetrahedra inz directionthat leads to the anomaly in thec/a ratio, is obviously not correctly modelled by theBKS potential as shown before, in contrast to the fluctuating dipole potential.Keeping in mind the Hill and Gibbs discussion ofπ-bond order and the results pre-sented above raises the question whether the effect ofπ-bonds has been parametrizedimplicitly as (fluctuating) dipoles in the potential by Tangney and Scandolo. It is shown

38

2.5. Elastic Constants

in Fig. 2.20, that the symmetries of the average dipoles in this potential very much re-

Figure 2.20.: Average dipoles in theα- (left) and theβ-quartz phase (right). Projection along the [100]direction. Obvious is the symmetry forbiddenz component (vertical) inβ quartz as wellas the dipole orientation orthogonal to the Si-O-Si connection line inα-quartz.

semble to the symmetries in theπ bond model. Their orientation around a tetrahedronis the same as for the oxygenπ orbitals and they single out thez direction. Of coursethis only constitutes an analogy as the quantum chemical interaction of the orbitals inFig. 2.18and the classical dipole interaction are of different nature. But it underlinesthe suspicion that the dipoles introduce the right effects into the system, but maybe doso by incorporating physics which is usually not to be described by classical dipolesalone. But it is not too surprising that this actually works, as already a simple classicaltwo body potential like BKS can model the tetrahedra quite well, even though theirformation is a quantum chemical effect of thesp3 hybrid orbitals.

In a later chapter on electromechanical coefficients, it will be shown that the piezoelec-trical response of quartz using the fluctuating charge potential reproduces available ex-perimental data much better if the external electric field is coupled to the charges onlybut not to the dipoles themselves.


There are two reasons why the calculation of elastic constants is crucial to test whetherthe model describes theα-β transition well. The first is that the elastic constants are

39


very sensitive to small changes in the model potential energy surface and second theyshow distinct features at theα-β transition in quartz, as discussed recently by Carpen-ter et al. [13].The calculation of the elastic constants in our simulations was done in the NpT ensem-ble using the fluctuations of the strain tensoru [72]. The strain tensor can be writtenin terms of the simulation boxh as

u =1

2

(h′−1

0h′hh−1

0− 1)

=1

2

(δh h−1

0+ h′−1

0δh′)

+O(δh2)

(2.5)

Now the elastic constants can be evaluated with the help of the fluctuations of thisquantity:

Cαβγδ =V

kBT〈δuαβδuγδ〉−1 (2.6)

As theh matrix is symmetric in the Parrinello Rahman method, derived tensors likethe strain tensor and the elastic constants will also be symmetric and can be written inVoigt notation (see chapter1.1.1). This reduces the 4th rank tensor of dimension 3 toa square matrix of dimension 6. In a crystal the number of independent coefficientsreduces even more, related to the crystal symmetry class. Forα-quartz, which is atrigonal holoaxial system, the elastic constants have the following symmetry:

C =

C11 C12 C13 C14 0 0C12 C11 C13 −C14 0 0C13 C13 C33 0 0 0C14 −C14 0 C44 0 00 0 0 0 C44 C14

0 0 0 0 C1412(C11 − C12)

(2.7)

With ascending symmetry the number of independent elastic constants decreases. Thuswhen going to theβ-quartz configuration, which is a tetragonal holoaxial system, thecrystals gains symmetry and therefore loses independent entries in the matrix, in thiscaseC14 will vanish. Note thatC14 can only be non-zero for systems that lack inversionsymmetry, as in tensor notationC14 becomesC1123 in which the indices 2 and 3 onlyoccur once. This is a reflection of quartz lacking inversion symmetry, see Fig.2.20.Consequently, the sign ofC14 depends on the chirality in quartz, i.e., whether we haveleft-handed or right-handed quartz. The elastic constants tensor ofβ-quartz looks like

C =

C11 C12 C13 0 0 0C12 C11 C13 0 0 0C13 C13 C33 0 0 00 0 0 C44 0 00 0 0 0 C44 00 0 0 0 0 1

2(C11 − C12)

(2.8)

40


40

60

80

100

120

140

Cii [G

Pa]

C33

C11

C44

C66

C11

C33

C11

C33

C11

C33

C33

C11

C44

C66

C33

C11

C44

C66

Exp.C11 (BKS)C33 (BKS)C44 (BKS)C66 (BKS)

40

60

80

100

120

140

Cii [G

Pa]

Exp.C11 (fluc. charge)C33 (fluc. charge)C44 (fluc. charge)C66 (fluc. charge)

-600 -400 -200 0 200 400T-Ttr [K]

40

60

80

100

120

140

Cii [G

Pa]

Exp.C11 (fluc. dipole)C33 (fluc. dipole)C44 (fluc. dipole)C66 (fluc. dipole)

Figure 2.21.: Elastic Constants

41


-20

0

20

40

60

Cij [G

Pa]

C14C12

C13

C14C13C12

C14C13C12

C14

C12

C13

C14

C12

C13

C14

C12

C13

Exp.C12 (BKS)C13 (BKS)C14 (BKS)

-20

0

20

40

60

Cij [G

Pa]

Exp.C12 (fluc. charge)C13 (fluc. charge)C14 (fluc. charge)

-600 -400 -200 0 200 400T-Ttr [K]

-20

0

20

40

60

Cij [G

Pa]

Exp.C12 (fluc. dipole)C13 (fluc. dipole)C14 (fluc. dipole)

Figure 2.22.: Elastic Constants

42

2.6. Phonon Density of States

The simulation results for the elastic constants are shown in Fig.2.21for the relevantdiagonal elementsC11, C33, C44, C66 and in Fig.2.22for the relevant non-diagonal el-ementsC12, C13, C14. The curves are shifted in temperature to match the respectivetransition temperatures of the models.All models show the general behaviour that there is a marked softening inC11, C12, C13,andC33 asT → Ttr. Only theC44 andC66 do not vary so much with temperature andnot at all in the stability field ofβ-quartz, whileC14 is quite constant inα-quartz andvanishes in theβ-quartz phase due to symmetry reasons.As already shown in Ref. [66], BKS shows good qualitative agreement with experi-ment, also for the strong temperature dependence nearTtr. Also most experimentalfeatures are well reflected, e.g.C11 andC33 cross between high and low temperaturessimilar toC44 andC66 (both shown in Fig.2.21), while C12 andC13 do not cross. Thisgood agreement is not too surprising considering the way in which the BKS parame-ters where determined: ab initio data along with elastic moduli were used to constructthe potential energy surface.In comparison to BKS the agreement of the fluctuating charge potential with exper-iment is not too convincing. While the qualitative trends are reproduced quite well,almost every value is significantly lower than in experiment.The fluctuating dipole potential is in better agreement, but does not improve on theBKS behaviour. While the agreement inα-quartz is better than BKS forC44, it is lessgood forC11. However the values forβ-quartz seem to have similar deviations forthe experimental values. Close to the transition temperature the softening is generallymore pronounced in all simulations due the finite system size, but at higher tempera-tures BKS and the fluctuating dipole potential agree on higher values forC12 andC13

than the experiment.


A quantity of interest in computer simulations of amorphous as well as crystalline sys-tems is the vibrational density of states (DOS). In general it is a measurement for thelattice dynamics, as it shows the distribution of vibrational states over frequency. Thismeansg(ν)dν give the number of eigenstates in the frequency rangeν andν + dν.While it can also be measured experimentally by neutron scattering, the simulationgives insight to where the microscopic origin of the distribution comes from. For ex-ample the partial distribution functions are often shown, which take into account onlythe contributions from a certain atom type, which is a quantity not possible to be mea-sured experimentally in general. However it was shown recently [7] for the example ofBKS in the simulation of amorphous silica, that the density of states is not well repro-duced by this model. Even though the structure of vitreous SiO2 was shown to change

43


only weakly by a subsequent first-principles calculation, the DOS was strongly mod-ified by using an ab initio treatment of the forces, and that this lead to a much betteragreement with experimental results. Moreover, in a large frequency range the natureof the excitations as determined from the effective potential differed significantly fromthe one determined from the ab initio force. This calls into question the explanatorypower of an analysis of the nature of the vibrational excitations determined from theBKS force field. On the other hand it gives rise to the question if other effective forcefields can do better, as ab-initio calculations suffer from their heavy computational costwhich restricts this type of calculations to the study of very small systems with ratherlow statistical accuracy.The definition of the vibrational density of states reads

g(ν) =1

3N

3N∑i=1

δ(ν − νi), (2.9)

with N being the particle number andνi, i = 1, . . . , 3N the eigenfrequencies of thedynamical matrixD. A less laborious way is the calculation in harmonical approx-imation, which only requires to calculate the Fourier transform of the velocity autocorrelation function:

g(ν) =1

NkBT

∑j

mj

∞∫−∞

dt⟨vj(t).vj(0)

⟩ei2πνt. (2.10)

This was calcluated in the simulations by averaging the velocity auto correlation func-tion over time intervals of2048 steps, which corresponds to real time intervals of2 ps,allowing for a resolution of1/2ps= 0.5 THz in the vDOS.The most obvious disagreement between the lattice dynamics of a BKS simulationand an ab initio simulation can already be seen directly in the vibrational DOS inFig. 2.23. The figure shows a comparison ofα quartz at300 K with ab initio datafrom Roma et al. [79], as no experimental measurement of the DOS in quartz couldbe found. The figure shows that the inter tetrahedral motions, the so called rigid unitmodes which account for the lower frequency range, show a significant discrepancy, asit is already known for amorphous systems. In the higher frequency range the doublepeak structure, which is available in both of the curves, comes about due to intratetra-hedral stretching vibrations [32, 73]. The four oxygen atoms in an SiO4 tetrahedronare moving at the higher frequency (“breathing mode”) in phase in relation to the cen-tral Si atom, at the lower frequency two O-Atoms oscillate in anti-phase to each other.However the frequency of these oscillations seems to be shifted from32.1 THz for thelow frequency peak, according to neutron scattering in vitreous silica [75], to around30 THz in α quartz, according to the ab initio data at hand. The lower peak of theBKS potential, which coincides with the experimental frequency quite well, is how-ever shifted rather to a higher frequency of about33 THz in α quartz.

44


0 10 20 30 40ω [THz]

0.00

0.02

0.04

0.06

0.08

G(ω

) [TH

z-1]

ab initioBKS

Figure 2.23.: Phonon Density of States inα-quartz at300K. Comparison of BKS and ab initio data.

The fluctuating charge model shows a slightly better agreement with the ab initio data

0 10 20 30 40ω [THz]

0.00

0.02

0.04

0.06

0.08

G(ω

) [TH

z-1]

ab initiofluc. charge

Figure 2.24.: Phonon Density of States inα-quartz at300K. Comparison of the fluctuating charepotential and ab initio data.

where the lower frequencies are concerned. The two peaks at11 THz and22 THz atleast exist close by and the structure is approximately maintained. The two peaks ofthe intra tetrahedral modes collapse on just a single peak, but exist at roughly the rightfrequency.The best overall agreement for both the rigid unit modes and the double peak isclearly shown by the fluctuating dipole potential. This is not too surprising as it wasparametrized to match ab initio forces and stresses, but also the BKS potential was

45


0 10 20 30 40ω [THz]

0.00

0.02

0.04

0.06

0.08G

(ω) [

THz-1

]ab initiofluc. dipole

Figure 2.25.: Phonon Density of States inα-quartz at300K. Comparison of the fluctuating dipolepotential and ab initio data.

parametrized to quartz data, even though the DOS looks better for vitreous silica. Thefluctuating dipole potential is almost congruent with the ab initio result in this case,only the tetrahedral breathing mode is slightly shifted in frequency. This convincingagreement suggests a good credibility for the lattice dynamics in this force field.

46

3. Pressure-Driven Transitions inα-Quartz

The high-pressure behaviour ofα-quartz is not yet fully understood neither on the theo-retical and nor on the experimental side. The majority of polymorphs that are observedat ambient to moderate pressures, such as cristobalite and quartz as well as other poly-morphs including tridymites and coesite, are built up of SiO4 tetrahedra [39]. At higherpressures dense forms containing SiO6 octahedra are observed such as stishovite above9 GPa, which has a rutile type structure. Because of the relatively strong Si-O bondingin silica, there are high kinetic barriers associated with the transitions to stable phasescontaining octahedrally coordinated silica at high pressure.As the pressure is increased, the four-coordinated structures become increasingly un-favorable and eventually the polymorph become unstable. New phases are formed thatcan be either crystalline or amorphous. The newly formed phases can often be reachedrelatively easily by local rearrangement of the atoms, however these phases are oftenonly thermodynamically metastable [52, 53, 105, 97, 99]. The transition path wayfrom quartz to stishovite, which is the proper high-temperature polymorph at elevatedpressure, would be highly reconstructive and is thus kinetically suppressed. Becauseof slow kinetics, the large number of possibilities to form six-coordinated structuresfrom four-coordinated structures, the metastability, but also experimental difficultiessuch as low x-ray scattering factors, the phase behaviour of high pressure quartz is stillnot completely resolved.The high-pressure behaviour ofα-quartz is particularly complex [52]. The stablephases of SiO2 in the order of increasing pressure are quartz, coesite (from about 3GPa), and stishovite (from about 7 GPa), as shown in the phase diagram in Fig.0.1.However, at room temperature the transformations are kinetically inhibited. Conse-quently, quartz can be kept metastable up to 21 GPa. Initially a pressure inducedamorphization was observed inα-quartz [40], but soon it was discovered, that someof these new materials formed at high pressure were found to exhibit unusual prop-erties for amorphous solids such as elastic anisotropy [64] and memory effects [55].The experimental observations at this point are somewhat complex, but it is clear thataround 21 GPa a transformation to a quartz II phase of unknown structure occurs.The amorphous phase develops out of this new phase and grows with pressure untila fully amorphous structure is reached above 30 GPa. Kingma et al. [52] report thatsamples recovered from above 30 GPa appeared to be entirely amorphous, while those

47

3. Pressure-Driven Transitions in α-Quartz

recovered from below this pressure contained crystalline regions. This explains theunexpected anisotropic properties of the under X-ray apparently amorphous phase. Adifferent result was obtained by Haines et al. [38], who saw a quartz II structure onlyas an intermediate phase from 19 to 26 GPa, which reverted for higher pressures to amonoclinic P21/c phase.

3.1. Quartz II Transition in Simulations

For molecular dynamics simulations employing the BKS potential it was shown that aphase transition occurs, whenα-quartz is compressed to 22 GPa [95, 96]. This tran-sition turned out to be not reversible [83]. It is crucial for the transition to occur that

Figure 3.1.: Snapshot of a Quartz II configuration with 2160 atoms. The configuration is rotated toalign the crystal axis horizontally, but it can still be seen that the box is sheared ratherthan orthorombic.

48

3.1. Quartz II Transition in Simulations

a constant pressure regime with a fluctuating box geometry is used, as the new phasecan only be established from theα-quartz configuration when the box is allowed toshear, which is shown in Fig.3.1. The new phase is perfectly crystalline, which is ob-vious from the structure snapshot, but it is subtle to achieve. In order to end up in thiscrystalline phase and not in an amorphous phase, the fictious mass of the box has to besufficiently small, and the particle number has to be large. For smaller particle numberthe transition would occur at a pressure of about 28 to 30 GPa. This is because atpressures such highα quartz is far beyond its stability range anyway and so it dependson various conditions if the metastability holds or if the crystal converts to a differentphase. However the pressure of 22 GPa is the smallest that could be found and is cer-tainly very comparable to experimental findings, which makes it a likely candidate forthe unknown Quartz II structure.From the pair correlation function or actually by viewing a snapshot of the system itcan be seen that only a third of the silicon atoms is four coordinated, but two thirds

Figure 3.2.: Silicon Coordination in Quartz II. Four coordinated silicon is printed in blue while the sixcoordinated silicon atoms are shown in yellow.

of silicon atoms have six neighbours, as it is common for the crystalline high pressurephases of SiO2. This can be seen in Fig.3.2, where in a snapshot of the configurationthe silicon atoms are coloured according to their coordination number. The four coor-dinated Si atoms are in the centres of SiO4 tetrahedra, while the six coordinated atomsare surrounded by oxygen atoms in the corners of an octahedron.

3.1.1. Transition Path Ways

The density changes in a compression/decompression cycle are shown in the equa-tion of state in Fig.3.3. Starting with theα-quartz configuration at zero pressure, thepressure is increased up to 21 GPa, where the system collapses from theα-quartz con-figuration into the quartz II phase. The transition is discontinous in the volume and

49


15 20V [cm3mol-1]

0

10

20

30P

[GP

a]

Quartz II

α-Quartz

δ-QuartzQ

uartz II b

experimentBKSfluc. charge

Figure 3.3.: Equation of State forα-quartz under pressure for BKS and fluctuating charge potential.

irreversible: in the simulation the new phase remains stable upon further compressionand even on decompression below the transition pressure. Even at small pressures thecyrstal does not revert to theα-quartz structure, but transforms into a newδ-quartzphase. This phase transition is reversible with a hysteresis, but is also non-displacive.The δ-quartz phase is stable within the BKS potential even at ambient pressure andwould not revert toα-quartz.For the fluctuating charge potential the equation of state for different quartz phases isalso shown in Fig.3.3. The phase transition occurs at a slightly higher pressure of 25GPa, and the volume of both theα quartz and the quartz II phase agree very well withthe BKS results. Nevertheles, the quartz II phase does not remain stable on decompres-sion, but the system undergoes a reversible phase transition at 21 GPa to a structurallysimilar phase named quartz II b in the figure. This phase has a smaller density than theBKS quartz II phase at the same pressure and reverts to theα quartz phase at 4 GPa.Theδ-quartz phase of BKS is not produced. Moreover theδ-quartz phase is not evenstable for the fluctuating charge potential but quickly transforms toα quartz.These findings for the fluctuating charge potential are also valid for the fluctuatingdipole potential. As shown in Fig.3.4, the variation of the volume with pressure isidentical to the fluctuating charge potential with respect to its qualitative features. Thetransition temperature to quartz II is 27 GPa now, and it has to be noted that the tran-sition only occurs for small systems. For larger systems theα-quartz configuration isstabilized for even much higher pressures. The transition to quartz II b occurs also at21 GPa and the next transition toα-quartz will take place not until 2 GPa. Also in thispotential the BKSδ-quartz phase is not stable and thus seems to remain an artificial

50

3.2. Quartz II b

15 20V [cm3mol-1]

0

10

20

30

P

[G

Pa]

Quartz II

α-Quartz

δ-QuartzQ

uartz II b

experimentBKSfluc. dipole

Figure 3.4.: Equation of State forα-quartz under pressure for BKS and fluctuating dipole potential.

polymorph of the BKS potential.

3.2. Quartz II b

The quartz II b phase, which does not form in a BKS simulation out of quartz II,emerges from quartz II phase with decreasing pressure at around 20 GPa for both thefluctuating dipole and the fluctuating charge potential. The connectivity of the atomsremains the same as in quartz II during the transition and the structure change is notvisible in a snapshot nor in the scattering spectra, but the change in the pair correlationfunction is obvious as shown exemplary in the Si-O pair correlation for the fluc. dipolepotential in Fig.3.5. From 23 to 20 GPa the cyrstal is in the quartz II phase and thepair correlation function changes only gradually. At the transition to the quartz II bphase at 19 GPa the pair correlation function exhibits a discontinuous jump, and ischanging only slightly on further decompression down to 17 GPa within the quartz IIb phase. The phase transition is reversible when the pressure is increased again, whilethere is a hysteresis effect that implies a higher transition pressure of about 2 GPaon recompression. These findings indicate a (quasi) displacive crystalline-crystallinefirst-order transition.

51


2 2.5 3 3.5 4r [Å]

0.05

0.1

0.15

0.2

g SiO

(r)

Q II

Q II b

17 GPa18 GPa19 GPa20 GPa21 GPa23 GPa

1.5 20

0.2

0.4

Figure 3.5.: Si-O Pair correlation for the fluctuating dipole potential for different pressures around theQuartz II - Quartz II b transition. The pressure was decreased subsequently starting fromquartz II.

3.3. X-Ray Diffraction Spectra

The similar behaviour of the three model potentials at the quartz II transition andthe apparent similarity of the crystalline structure emerging from these simulationsgives the simulation results a certain credibility for predictions towards the experi-ment. In experiments the structure of a crystal can be characterized by its diffractionpattern, such as the energy dispersive x-ray diffraction spectra measured for quartz IIby Kingma et al. [52]. To measure these data in a simulation one has to calculate thestructure factor

S(q) =

∣∣∣∣∣∑j,k

fjfk exp{iq(Rj −Rk)}

∣∣∣∣∣2

=

∣∣∣∣∣∑j

fj exp{iqRj}

∣∣∣∣∣2

(3.1)

with the appropriate atomic form factorsfi. These are tabulated as a function ofsin(Θ)/λ in Ref. [61], pp. 202, 203. A diffraction angle of 5◦ and a Gaussian broaden-ing with a 0.5 eV FWHM were used to generate the spectra. The wave vector is relatedto sin(Θ)/λ via the Bragg conditionλ = 2d sin Θ asq = 4πsin(Θ)/λ.The reciprocal lattice vectors have to fit in the simulation box:q = 2πh−1(nx, ny, nz),

which simplifies eq. (3.1) to

S(q) =

∣∣∣∣∣∑j

fj exp{2πi(nx, ny, nz)rj}

∣∣∣∣∣2

. (3.2)

52

3.3. X-Ray Diffraction Spectra

15 20 25 30 35Energy [keV]

BKSfluc. chargefluc. dipole

4.74 3.56 2.85 2.37 2.03d [Å]

Figure 3.6.: Energy dispersive x-ray diffraction patterns of Quartz II at 27 GPa. The experimentalx-ray diffraction spectrum shown in black is taken from Kingma et al. [52]. The Q’symbols represent shifted quartzlike peaks while the * depict new nonquartz peaks. Nedenotes diffraction from neon, which is used as pressure medium.

The diffraction spectra for the different potentials are compared to the data by Kingmaet al. in Fig.3.6. The three potentials are in good agreement with each other concern-ing the structure of quartz II and can be related quite reasonably to the features of theexperimental spectrum.An overview comparison for the spectra of the three potentials is given in Fig.3.7,where one can see the course of the peaks in theα quartz phase and the change to thequartz II peaks at the different transition temperatures. The position of the peaks canbe reexpressed as a Bragg distance between diffraction planes. The Bragg distancesas a function of pressure can then be compared to the data given by Kingma et al. InFig. 3.8 the interplanar spacings are shown in comparison with the experimental data.From this figure and fig.3.6it can be summarized that the features of the experimentalspectrum agree quite well with the simulations. The apparent absence of the interpla-nar spacing around 3.6A in all simulations might be explained with the fact that thepeaks at 3.8A in the simulation consists of two very close peaks which are not resolvedin the plot. On the other hand the apparent absence of the 2.8A Bragg distance in exper-iment might arise out of a misinterpretation of the experimentally complicated energyrange around 27 keV. In this energy range one peak arises from Neon that was used aspressure medium in experiment, which might cover the additional interplanar spacing

53


15 20 25 30 35Energy [keV]

4.74 3.56 2.85 2.37 2.03 d [Å]

15

18

21

25

27

28

29

30

p [G

Pa]

15 20 25 30 35Energy [keV]

4.74 3.56 2.85 2.37 2.03d [Å]

15 20 25 30 35Energy [keV]

4.74 3.56 2.85 2.37 2.03d [Å]

BKS Fluc. Charge Fluc. Dipole

Figure 3.7.: Comparison of X-Ray spectra at various pressures for the BKS potential (left), the fluc.charge potential (middle), and the fluc. dipole potential (right)

which is predicted by all of the simulations. In total it seems that all the three modelpotentials support the structure seen by Kingma. Still these results are questionable asthe models were parametrized for ambient conditions and are not verified to reproducethe real behaviour under such extreme pressures, but their stunningly similar findingsadds to the credibility of the results.

3.4. Other High-Pressure Phases

Wentzcovitch et al. [105] found a high pressure phase with symmetry P32 in simu-lations at 40 GPa. These simulation were done with an ab-initio MD approach with

54

3.4. Other High-Pressure Phases

10 15 20 25 30p [GPa]

2

2.5

3

3.5

4

d [Å

]

ExperimentBKSfluc. chargefluc. dipole

Figure 3.8.: Interplanar spacing for the compression ofα-quartz in experiment and simulation. Linesrepresent the different transition pressures. Experimental data by Kingma et al. [52].

variable cell shape and a box of 27 particles. This high-pressure phase phase did notemerge in our simulations fromα-quartz, however when using the crystal structurereported by Wentzcovitch as initial configuration, it proved to be stable under highpressures between 30 and 40 GPa with the fluctuating dipole potential and from 40GPa down to 15 GPa with the BKS potential. In the fluctuating charge approach thephase is only stable at around 30 GPa.At lower pressures, and even down to zero pressure, structures are forming in all po-tentials that are not entirely crystalline and not related to any known quartz phases.The density of the crystal at 40 GPa can be calculated from the lattice constants givenin Ref. [105] to be 4.60 g/cm3, while the BKS potenial estimates a density of 4.53g/cm3 and the fluctuating dipole potential finds 4.56 g/cm3. However the polymorphproposed by Wentzcovitch et al. seems to be a valid option for a high-pressure phaseof silica in the BKS and fluctuating dipole potential as well, however it is not nearly asstable as the Quartz II phase in the three potentials under consideration.

55

4. Silica Polymorphs Other Than Quartz

4.1. Cristobalite

Quartz is not the only polymorph that is (meta) stable at ambient conditions. Cristo-balite, a cubic polymorph, and tridymite, which has hexagonal symmetry, are foundto be stable as well during accessible experimental time scales [47, 23]. A modelpotential that is used to mimic the behaviour of silica should be able to reflect theexperimentally observed stabilities. In some cases, the observation of stability in asimulation might be fortuitous as it could be the consequence of periodic boundaryconditions and/or the number of atoms used. For example, the phase into which thesystem would like to convert is not compatible with the number of atoms used in thesimulation or it cannot be accessed without a major reconstruction of the cell shape.This issue will be discussed in more detail in this section with a focus on cristobalitethat shows subtle effects depending on the choice of the periodic boundary conditions.For the simulation of a crystalline system, the crystal structure has to be placed insidethe box in a way that the periodic images fit to each other at the box ends. For theinitial configuration of a simulation the atom positions are chosen to be identical withexperimental crystallographic data such that the atoms were set onto ideal lattice posi-tions and are equilibrated. For many systems and also for the case of cristobalite thereare different ways to arrange such an initial configuration.The most natural configuration is according to the usual description of cristobalite asa diamond structure for the silicon atoms with connecting oxygen atoms between eachneighboring pair of silicon atoms. This would place the [100] crystal direction parallelto the z axis of the simulation box, and the box geometry would be cubic.On the other hand there is a nice ABCABC layering of sheets of hexagonal rings inthe cristobalite structure, which is also a very common representation as it leads totridymite when the layers are stacked in an ABAB order. This layering is seen alongthe [111] direction in the crystal, and accordingly would place this direction parallelto the z axis of the simulation box when the layers are stacked in the xy plane.However both geometries are completed by the periodic boundary conditions to forman infinite cristobalite crystal, and therefore they are equivalent regarding the staticstructure. On the other hand the connectivity of the atoms at the boundary is differentand therefore the two geometries are not equivalent regarding phonons. When a lattice

56

4.2. Tridymite

vibration leaves a periodic image it would reenter the box at a different position forboth setups.This difference becomes obvious in the simulation ofβ-cristobalite. When relaxing

0 1 2 3 4 5t [ps]

40

42

44

46

48

50

VS

iO [Å

3 ]

001001001

0 1 2 3 4 5t [ps]

40

42

44

46

48

50

VS

iO [Å

3 ]

111111111

BKSfluc. chargefluc. dipole

Figure 4.1.: Time evolution per SiO2 unit VSiO2 for the two geometries with three different potentials.

the crystal atT = 1273 K and zero external pressure, where cristobalite is consideredto be thermodynamically stable [39], the volume of the simulation quickly stabilizeson a value close to the experimental value ofVSiO2 = 45.25 A3. In contrast with thesecond configuration, which had the [111] direction parallel to the z axis, the observedvoulme is, if at all, metastable only for a short period of time, before densification ofthe system takes place. After 2 ps the system is stable in a phase which is much denserthan cristobalite. It was tested that this state remains stable for more than a nanosec-ond, which is not shown in the picture.Obviously in contrast to this both the fluctuating charge and the fluctuating dipolepotential do not exhibit such a behaviour, but remain stable in the cristobalite phaseregardless of the initial condition.Apparently the cristobalite phase is not universally stable in the BKS potential, butonly if certain phonons are prohibited by special periodic boundary conditions.

4.2. Tridymite

Another high temperature polymorph of quartz is tridymite, which was accepted to bea stable phase of silica as late as in the 1960’s. There is a broad range of structuralmodifications upon heating, while the structures increase in complexity with decreas-

57


ing temperature. The ideal high-temperature tridymite, the HP-tridymite, is stable over650 K. Its structure consists of an AB layering in c direction of the same tetrahedralsheets that also form the cristobalite phase when stacked in a tripled ABC repeat along[111].We found that the HP-tridymite structure is not stable in BKS at 1000 K, which is well

0 5 10 15 20 25t [ps]

40

42

44

46

48

50

VS

iO [Å

3 ]

Exp.BKSfluc. charge

Figure 4.2.: Time evolution per SiO2 unit VSiO2 of tridymite for the three different potentials atT=1000 K.

in the experimental stability range. Surprisingly, also the fluctuating charge potential,in which cristobalite turned out to be stable, suggested in contradiction to experimentthat the HP-tridymite configuration is unstable. Shown in Fig.4.2is the time evolutionof the volume in a simulation run. While the crystal simulated with the BKS poten-tials collapses into a denser phase about as quickly as the cristobalite collapses, with

a) BKS b) Fluc. Charge c) Fluc. DipoleFigure 4.3.: Snapshot of the different phases that emerged after 20 ps at 1000 K. In all cases the initial

structure was HP-tridymite. Only the fluctuating dipole potential remained in the initialphase, which is stable experimentally.

58

4.3. Stishovite

the fluctuating charge potential the HP-tridymite is metastable for about 10 ps in thesimulation, only then it collapses slowly afterwards. The fluctuating dipole potentialis the only one that predicts that the HP-tridymite phase is stable.In Fig. 4.3 this is shown in terms of the averaged configuration. One can see thatthe collapsed structure of BKS and the fluctuating charge potential looks quite simi-lar. The densification is mainly achieved by a contraction in thea direction (vertical).The framework collapse induces distortions of hexagonal rings to oval configurations.Additionally the tetrahedrons are tilted relative to the stacking directionc and thus ad-jacent sheets are displaced alongb. While the later is known to occur in the tridymitephases at intermediate temperatures, these phases all have a superstructure of charac-teristically alternating oval and ditrigonal configurations. The occurence of only ovalrings in the present simulations may be caused by the finite box size, which does notallow the superstructure to emerge.

4.3. Stishovite

The quartz, cristobalite and tridymite phases studied in this work were composed outof corner-sharing SiO4 tetrahedra, which is the usual configuration in the low pres-sure phases. Stishovite in contrast is a polymorph that often serves as a prototypephase having octahedrally coordinated silicon. It is often assumed [20, 90] that withthe fluctuating charge approach the ability of the charges to adapt to the configurationis especially helpful for the transferability between differently coordinated structures.

0 10 20 30 40 50 60 70 80p [GPa]

40

42

44

46

48

V [Å

3 ]

BKSfluc. chargefluc. dipole Exp. (Ross, Hemley, Andrault)Exp. (Mao)

Figure 4.4.: Stishovite Equation of State. Solid circles are the data of Ross et al. [80], Hemleyet al. [43], and Andrault et al. [2]; open squares are the data of Mao et al. [62]

59


Also it is very likely that the degree of ionicity changes with pressure. Therefore, onemight expect that the fluctuating charge potential comes closest to the available exper-imental data out of the three models.Stishovite is usually stable at high pressures, above 8 GPa at room temperature, but itwas also discovered to be metastable at ambient pressures. It was first synthesized inthe laboratory [89] and later was discovered in association with coesite [14]. Stishovitehas the rutile structure, which consists of infinite chains of edge-shared SiO6 octahedraparallel to thec-axis. Each oxygen atom is coordinated by three silicon atoms [43].The equation of state for stishovite under pressure is shown in Fig.4.4. The agreementin the low pressure range is relatively good for the BKS potential, while the fluctuatingdipole and the fluctuating charge potential both do not see a stable stishovite phase atzero pressure. When increasing the pressure, these two potentials constantly underes-timate the density of stishovite, while the volume of the BKS simulation deviates morefor higher pressures.One feature of stishovite under pressure is a different compressibility in thea andcdirection, which is shown in Fig.4.5. This plot shows again a very good agreement of

0 20 40 60 80p [GPa]

0.61

0.62

0.63

0.64

0.65

c/a

fluc. chargeBKSfluc. dipoleExp. (Ross, Hemley, Andrault)Exp. (Mao)

Figure 4.5.: c/a ratio in stishovite against pressure. Experimental data as in Fig.4.4. Please note thatthe phase transition discussed in the next section does not have much influence on thec/a

ratio and the volume.

the BKS data with the experimental results. The effect of the different compressibilityin the fluc. charge and fluc. dipole potential is rather overestimated, while the fluctuat-ing dipole potential again improves in the quantitative prediction for higher pressures.The result of the fluctuating charge potential shows no improvement over neither ofthe two other potentials.

60

4.3. Stishovite

4.3.1. High-Pressure Behaviour

Shortly after the discovery of stishovite, a major question has evolved around possibletransformations to a denser structure at high pressures. By crystal chemical argu-ments [67], first-principle calculations [42, 15], and experimentally [41, 98, 51, 12] apressure-induced phase transition at 50 GPa was confirmed. At the transition the SiO6

octahedra are only slightly tilted, whith the result that the lattice constantsa andb,which are identical in the low pressure rutile-type, tetragonal phase, are of differentsize in the CaCl2-type orthorombic stishovite phase. This tilting has the same symme-try as the stishovite B1g vibrational mode, which was identified as the pressure-inducedsoft mode that drives the transition [67]. By means of a Landau expansion Carpenter etal. [12] found a classical second-order character. The transition has almost no impacton the volume or thec/a ratio, which is shown in the paper by Carpenter.However molecular dynamics models were not able to reproduce the tranistion pres-sure but only found the transition at pressures in the megabar region [42, 97, 56].As a comparison between molecular dynamics and experiment the elastic constantscan be analysed. Carpenter et al. [12] used a Landau expansion to generate expres-sions for the elastic constants of stishovite with pressure. Fig.4.6 shows the relevant

0 20 40 60 80 100 120 140p [GPa]

0

100

200

300

400

B11

-B12

[GPa

]

ExperimentBKSfluc. chargefluc. dipole

Figure 4.6.: Birch coefficients B11-B12 as a function of pressure. Simulation data is compared to theresult of a Landau expansion by Carpenter et al. [12]

shear modulus B11-B12, which softens on approach of the transition, as the B1g modedirectly contributes to it. The Birch coefficents Bij have to be considered for a stabilityanalysis rather than the elastic constants Cij, when the system is observed under ex-ternal pressure. In the case of the two coefficents needed here, they are related to eachother asB11 = C11 − p andB12 = C12 + p.

61


One can see that the BKS potential and the fluctuating dipole potential show a soften-ing towards pressures of over 100 GPa, which is in agreement to the molecular dynam-ics simulations cited above, and far away of the 50 GPa measured experimentally. Incontrast the fluctuating dipole potential predicts the transition in almost perfect agree-ment with the analysis by Carpenter at 50 GPa.This prediction can be verified by a plot of the lattice constantsa andb, which coincidein a tetragonal lattice and differ in an orthorombic lattice. In Fig.4.7 the course of

0 20 40 60 80 100 120p [GPa]

3.7

3.8

3.9

4

4.1

4.2

4.3

a, b

[Å]

fluc-dipoleExp. (Ross, Hemley, Andrault)Exp. (Mao)

tetragonal orthorombic

Figure 4.7.: Lattice constantsa andb against pressure. Experimental data as in Fig.4.4.

the lattice constants with pressure is compared to experimental data. The transition isclearly visible at a pressure of around 50 GPa in both the fluctuating-dipole simulationand experiment. The length of the lattice constant is overestimated constantly in thesimulation, which was already obvious in the volume plotted in Fig.4.4.However the agreement with experiment in this phase transition is far better than inthe other model potentials. This is rather surprising, as it was parametrized for tetra-hedrally coordinated silicon only.

62

5. Electromechanical and DielectricProperties

Dielectric and electromechanical properties of materials play an important role inmany technological applications, as for instance piezoelectric resonators which areused as frequency generators in computers. Furthermore it is interesting to check howthe results of the fluctuating charge and dipole potential compare to the standard two-body force field, as these new approaches claim to account for electronic propertiesand therefore should offer a better basis for the investigation of dielectric properties.Due to these reasons it is desirable to have methods at hand that allow one to computedielectric, piezoelectric and related tensors in atomistic simulations. Dielectric proper-ties are more challenging to calculate than thermal and mechanical properties, becausethe definition of polarization in periodically repeated cells is less obvious than the for-mulation of thermo-mechanical variables. The reason is that an electrical dipole, andhence the polarization, lives on the surface of a sample. One of the consequences forcomputer simulations is that the value of the dipole depends upon where we chose the’central image’ of the periodically repeated cell to be. Conversely, internal energy,lattice parameters, strain, etc. are independent of the central image’s position. As out-lined further below, the ambiguity in defining the polarization becomes particularlydelicate even in classical molecular dynamics simulations when one treats a systemcontaining free charges at finite temperatures and constant stress, which implies fluc-tuating box shapes.Despite the mentioned ambiguity in the definition of polarization in periodic cells, ithas been shown that there is a bulk polarization that is intrinsic to the crystal and onlydepends on variations in charge density induced by atomic displacements. It is there-fore independent of the surfaces and thus of the choice of the central image [63].In the following chapter different methods are established for the calculation of dielec-tric and piezoelectric properties. These are in particular the dielectric tensor, definedas

εαβ =1

ε0

∂Pα

∂Eβ

, (5.1)

and the piezoelectric coefficients. The latter represent the mechanical reaction of thesystem to an applied field, and can therefore be defined either in terms of the strain or

63

5. Electromechanical and Dielectric Properties

the stress. In the following the focus will be on the piezoelectric strain coefficients:

dαβ,γ =∂uαβ

∂Eγ

. (5.2)

Another quantity that will be used in the following derivation is the piezoelectric stresscoefficients at constant strain:

γαβ,γ =∂σαβ

∂Dγ

. (5.3)

Two different approaches for the calculation of these quantities will be employed. Thefirst is the direct calculation of the slope of the polarization and the strain respectivelyagainst an electric field applied on the simulation box in different spatial directions.The second method is an evaluation of fluctuation relations within a usual simulationrun in zero external field.

5.1. Theory and Methods

In order to develop the fluctuation relations, we will introduce different generalizedvectors and matrices. In the following, a vectorv denotes a column of numbers, whilea transposed vectorvT represents a row of numbers. This means that the scalar productof two vectorsu andv will be written asuT v anduvT is a matrix of rank two. WithinVoigt notation, all matrices except for the piezoelectric tensors are square matrices.The piecoelectric coefficients are represented by a6 × 3 matrix in Voigt notation. Asbefore, spatial coordinates are written in capital letters to denote real space, whilelowercase letters represent reduced space:

Ri = h ri.

Additionally a vectorsi is introduced, which represents the reduced coordinates ofthe central periodic image of the particlei. When the particle moves over the imageboundaries, the coordinate vectorri behaves smoothly while we have to add or subtractunity to si. This means that the components ofsi always lie within0 and1, whereasri can exceed these boundaries. This will prove to be a relevant property as the polar-ization must not depend on the boundary conditions.

5.1.1. Linear Response

Starting point is the isothermal partition functionZh,D(N, β) at fixed dielectric dis-placementD and constant box geometry as defined by a symmetric matrixh. Accord-

64


ing to the regular rules of statistical mechanics,Zh,D(N, β) reads

Zh,D(N, β) =

∫dΓδ(h(Γ)− h)δ(D(Γ)−D) exp (−βΦ(Γ)) . (5.4)

Here∫

dΓ is an integral over phase space, theδ(·)’s are the delta functions singlingout proper geometry and dieelectric displacement, andΦ(Γ) is the potential energy ofthe system as a function of phase space, this means as a function ofh and{s}. Fromthe partition function, the free energyF is defined via the equation

Fh,D(N, β) = −kBT ln Zh,D(N, β). (5.5)

As outlined in more detail below, these equations allow one to connect phenomenogi-cal materials parameters with ensemble averages over phase space. With the assump-tion of ergodicity the integration over phase space is carried out by averaging alongthe trajectory of the molecular dynamics simulation.In the theory of elasticity it is helpful to express the partition function and free en-ergy as a function of strain rather thanh. Unlike the box geometry, the definitionof the strain does not need the reference geometryh

0of the system, which is typi-

cally chosen to reflect the expectation value ofh at given temperature and stress, asan additional information. As used before in the calculation of elastic constants, theLagrangian strain with respect toh

0can be written as

uαβ =1

2

{(h−1

0

)αγ

hγδ hδε

(h−1

0

)εβ− δαβ

}, (5.6)

where tensor notation was used instead of Voigt notation. In Eq. (5.6), δαβ denotesthe Kronecker symbol. Since the relation betweenh andu is well defined for a givenreferenceh

0, it is possible to calculate the free energy as a function ofu in terms of a

Taylor series expansion inu away from a phase transition point, i.e.,

Fu,D = Fh0,D +∂Fh,D

∂h

∂h

∂uu + ... (5.7)

The new free energy depends on both, strainsu and displacementD. In the following,for convenience of calculations these variables will be grouped together into a gener-alized strainu = (u1, · · · , u6, D1, · · · , D3)

t, with (u1, · · · , u6) being the strain tensorin Voigt notation. The generalized stressesσ that are the conjugate thermodynamicvariables tou can be calculated asσ = (1/V ) ∂F(N, β)/∂u. Away from a phasetransition, it is possible to expandFu into a power series around a reference strainu0

(which one can usually set to zero ifu0 at the reference geometry), thus

V0σ =∂Fh0,D(N, β)

∂u+

∂2Fh,D(N, β)

∂ut ∂uδu + ... , (5.8)

65


where the derivatives are evaluated atu0, the relative generalized strain isδu = u− u0,andV0 denotes the volume of the system at the reference geometry. The (expectationvalue of the) generalized stress atδu = 0 will be calledσ0 andδσ = σ − σ0 denotesan (average) stress variation. One can then interpret the second-order derivative ofFu with respect tou as generalized elastic constantsC, that connectδu andδσ via

δσ = C δu, or if we representu andσ explicitly

(δσ

δE

)=

CD

γt

γ(ε0 ε

r,u

)−1

( δu

δD

). (5.9)

Here,CD

denotes ‘real’ elastic constants at constant dielectric displacement,γu

are

the piezoelectrical stress coefficients at constant strain, andεr,u

is the (isothermal)dielectric tensor at constant strain. Eq. (5.9) is a linear equation that connectsδu withδσ. It is therefore possible to chose up to nine linearly independent thermodynamicvariables, i.e., we can ‘fix’ theδσ and ‘measure’ the responseδu. This can be done bymultiplying both sides of Eq. (5.9) with the inverse ofC, which can be written as:(

δu

δD

)=

[C−1

Edt

d ε0 εr,σ

](δσ

δE

). (5.10)

HereCE

have to be interpreted as elastic constants at constant (external) electric fieldandd are the piezoelectrical strain coefficients. Eq. (5.9) uses the more natural rep-resentation of the linear-response coefficients than Eq. (5.10), as stress and (external)electrical field are typically the experimentally controlled parameters, while strain andpolarization and thus dielectric displacement are the measured responses.

Within linear-response theory,δu andδσ are conjugate to each other. In the constant-uensemble, we can therefore attribute an excess free energy∆F due to the fluctuationsin u, which is given by

∆F =V0

2utC u. (5.11)

Thermal fluctuations ofu at fixed σ will thus be related toC, which for harmonicapproximations, results in⟨[

δuδut δuδDt

δDδut δDδDt

]⟩=

kBT

V0

[C−1

Edt

d ε0 εr,σ

]. (5.12)

In this section, we have measured the free energy density by dividing the actual valueof F by the volumeV0 of the reference strain. This convention, which is typically used

66


in the theory of elasticity. is also beneficial when evaluating Eq. (5.12): The fluctua-tion of the electric displacement can be rewritten as fluctuation of the polarizationP .Introducing the dielectric susceptibility,χ = ε

r− 1, we can rewrite Eq. (5.10) as:(

δu

δP

)=

[C−1

Edt

d ε0 χσ

](δσ

δE

). (5.13)

and conclude with the same procedure the flucuation relation⟨[δuδut δuδP t

δPδut δPδP t

]⟩=

kBT

V0

[C−1

Edt

d ε0 χσ

]. (5.14)

Relations such as Eqs. (5.8) and (5.14) can be exploited in atomistic calculations todetermine the susceptibilities of interest. However, it will first be necessary to state theproper estimators for the polarization fluctuation.

5.1.2. Ambiguity of the dipole and its fluctuation

To understand the difficulties connected with the definition of bulk polarization and itsfluctuations in an infinitely replicated system it is helpful to consider Fig.5.1. Part (a)of Fig. 5.1 shows that the definition of the dipole momentµ in the cell (which leadsto the polarizationP through division by the volume of the simulation box), dependsupon the choice of the boundary of the central cell. By displacing the boundary, noneof the interatomic distances is altered, yet the dipole changes its value. As long asthe shape (here simply the length) of the cell is maintained, this ambiguity causesno complications for the response functions, because the only relevant quantity is thedifference between polarizations. A polarization change is independent of the choicefor the boundary - provided the ‘true’ trajectory of the atoms is measured, i.e., when acharged particle moves across a boundary the scaled coordinate for the calculation ofthe dipole is not increased or decreased by unity. Thus evaluating fluctuations relatedto dipoles (from which the constant-strain dielectric and piezoelectric constants couldbe calculated), do not depend on the choice for the position of the central image.When the cell fluctuates, even a polarization difference will depend upon where wechose the central image to be. This is illustrated in part (b) of Fig.5.1, where in onecase, the reduced coordinates of a charged particle is constrained to zero (bottom) andin the other case it is set constant to one half (top). Both times, we assume the presenceof an opposite background charge of constant density ensuring charge neutrality. Inthe upper part of Fig.5.1 (b), the fluctuation of the dipole vanishes exactly if we keepthe reduced positions of the chargeQ fixed in the center of mass. Therefore, thefluctuation of the dipole will be zero in the upper half. In the bottom part, we would

67


−

−

−

− −

−

−

−Q

Q

−Q

−Q(a)

(b)µµ

Figure 5.1.: Illustration of the ambiguity for (a) the definition of a dipoleµ and for (b) the fluctuationof the dipole. The boxes framed by a solid line present the position of the central image,while boxes framed with a broken line are periodic images. The lower left corner of thecentral image is defined to the origin of the coordinate system. In part (b), a positive back-ground neutralizes the point chargeQ and the arrows with the two arrow heads representfluctuation of the box geometry.

attribute a dipole to the cell, which would fluctuate with changing box size. Thus thedipole fluctuations are affected by the choice of the location for the central image. Theabove mentioned difficulties do not arise if one deals with neutral molecules whoseconstituents have charges that add up to zero. In that case it is easy to remove theambiguity due to surface effects or periodic-boundary conditions. One would onlyhave to suppress any minimum image convention within the molecules, i.e., for thecalculation of the dipole, theri have been to be defined such that two covalently bondedatoms are separated by the proper distance. In principle, it would be possible to definesimilar molecules in an ideal crystalline network such as quartz, i.e., by evaluating thedipole over entities that consist of a central silicon atom and its four oxygen neighbors,where the charges of the O atoms would only count half, in order to avoid doublecounting. However, such ‘tricks’ cannot be generally applied, for instance, if any typeof disorder or even impurities are present in the system, or if a non-rigid charge modelis employed.

5.1.3. Fluctuation estimators for dipoles

The problem of defining bulk dipoles can be overcome by considering the small wavelength limit of the dipole’s spatial Fourier transformµ. If the box cell is fluctuating, it isappropriate to work with scaled reciprocal vectorsk = 2π(mx, my, mz) where themα

are integer numbers that are related to the true reciprocal vectors throughK = k h−1.Since the box is fluctuating, it is more meaningful to work with reduced coordinates in

68


both real and reciprocal space. Let us first consider the Fourier transform of the chargedistribution ˜ρ(k)

ρ(k) =∑

i

Qieik si . (5.15)

It remains unchanged if we add or subtract unity from a particularsi or if the boxgeometry is rescaled. Yet, the formal derivativeρ(k) with respect tok (which onecannot do in practice as thekα’s are discrete) does not show the same invariance. Thederivative ofρ(k) evaluated atk = 0 corresponds to the contribution of the reduceddipole due to the point charges, which is defined as

µred

=∑

i

Qiri + h−1µi, (5.16)

If, however, one defines a reference configuration, which can be an initial configura-tion at timet = 0 that does not have to be equilibrated, then thedifferencebetween areduced dipoleµred(t) evaluated at timet and the reference dipoleµred(0) is invariantagainst the transformations discussed in Fig.5.1, as long as the transformation is per-formed on the configuration of interest and the reference configuration. For the finalevaluation of the dipole, the valueµred(0) has to be multiplied with the expectationvalue of the box shape. Thus our estimator becomes

µestimator = 〈h〉 µred (5.17)

and consequently that for the polarizationP estimator = µestimator/〈V 〉. These estima-tors do not enable one to calculate absolute dipoles or polarizations but only relativequantities, i.e., those to be used in Eq. (5.14) for the calculation of fluctuation relations.If absolute values for the polarization were needed, the dipole moment of the initialconfiguration should vanish. This could be accomplished by chosing a configurationwith vanishing dipole moment due to an underlying symmetry.

5.1.4. Direct estimators with noise reduction

An alternative method to evaluate piezoelectric coefficients is to apply an external elec-tric field E to the system. The appropriate force that acts on the reduced coordinatesrelated to a point chargeQi would readh−1EQi. It is yet not meaningful to simplyadd a term−

∑i QiE

tRi to the Hamiltonian. Such a perturbation would require tohave a force act onto the shape of the simulation cell and this force would depend onwhether a particle is counted within the central image or within a periodic image. Wethus suppress the force from the external field onto theh matrix, also because it wouldbe absent in a system without charges and dipoles.

69


If the shape of the system is known forE = 0 and the initial configuration is equi-librated, then one may switch on the field adiabatically and monitor the box shapevariation from which the strain can be calculated. This procedure will give correctresults, in particular if one averages over an appropriate number of independent, ini-tial configurations. However, this algorithm may produce large stochastic scatter inparticular at elevated temperatures.

The noise can be reduced and the necessity of having to know the average structure canbe omitted if one performs a reference simulation in which the electric field remainsswitched off. The reference simulation should be based on the same initial conditionand an identical initialization of the thermostat. Any instantaneous configuration atfinite field can be compared directly to the instantaneous configuration at zero field.Within a few molecular dynamics steps, it is then possible to obtain quite reasonableestimates for the piezoelectrical and other dielectric coefficients, even if the systemis rather anharmonic and contains slow modes, as is the case in quartz close to theα − β transition. Fig.5.2 confirms this expectation. In Fig.5.2, we increased the

0 2 4 6 8 10E [10-16 V/m]

-2

0

2

4

6 uE11(t)

uE11(t)-u

011(t)

1.826 m/V · E

0 2 4 6 8t [ps]

-2

0

2

4

6

u 11 [1

0-3]

Figure 5.2.: Response of the strainu11 as a function of time (below) or electrical field (above) for anN = 1080 particle system (α-quartz) at temperatureT = 300 K. The long dashed line isthe result for the piezo-electrical coefficient based on evaluating the fluctuation relationin Eq. (5.14).

external field slowly with time and also chose box inertia sufficiently small to have thecell shape adapt quickly to the new field. The box geometry was thermostatted with aLangevin thermostat to decrease the (stochastic) correlation time. Excellent estimatesfor the piezoelectrical coefficients can be obtained already after one hundred MD stepsby evaluating the slope∂u11/∂Ex. In this way, four simulations have to be run to

70

5.2. Temperature Dependence of d11 in Quartz

determine all generalized elastic constants; three in which the electric field is parallelto one of the three coordinate axis and one in which the electric field remains zero.As said above it is necessary that in all four runs the same initial conditions (one largesample in thermal equilibrium or an average over some small equilibrated samples)must be used as an input into the simulation and that all runs require the same randomnumber sequence.For the various silica polymorphs investigated here, we obtain a statistical accuracy ofbetter than 5% forγ if we first equilibrate the material for about 10,000 MD steps toproduce an equilibrated start configuration. Note that even though quartz is crystalline,it does takes relatively long to equilibrate, because it has an unusually large phonondensity of states at small frequencies. The start configuration is then used for fourindependent runs, namely one without electric field plus three runs with electric fieldin one of the three spatial dimensions. Each of these runs is about 2,500 MD steps longand produces a graph similar to that shown in Fig.5.2. This length of the simulationrun is by far long enough to obtain a small error in the linear regression, while in alonger run the abberations in the four trajectories get too large. Also non linear effectswould begin to dominate, at least for the set up and the slope of electric field againsttime used above. Thus, the required numerical effort to obtain the data is relativelysmall.

Using Nose Hoover thermostats or related (less sophisticated) rescaling methods doprobably not lead to similarly reliable results. In particular for large systems and/orsmall temperatures, the motion of the simulation cell will be quite harmonic and thusthe crucial exchange of (kinetic) energy between the simulation cell and the internaldegrees of freedom will be inefficient. This inefficiency will prevent the cell fromquickly finding its new preferred shape or from fluctuating around it.

5.2. Temperature Dependence of d 11 in Quartz

The temperature dependence of piezoelectrical constants is of technological relevancebecause one is often interested in having pressure sensors and pressure transducers atvarying temperatures. One of the few disadvantages ofα-quartz is the relatively lowtransition temperature toβ-quartz. Due to the increase in symmetry from the trig-onal holoaxialα-quartz structure to the tetragonal holoaxialβ-quartz structure, thepiezoelectricity decreases such that thed11 component becomes symmetry-forbidden.This means that quartz cannot be used as an effective piezoelectrical material at hightemperature. Fig.5.3 shows the temperature dependence of the piezoelectrical straincoefficientd11. One can see that both potentials underestimate the value ofd11, thefluctuating dipole potential underestimates its value in the low-temperature phase by

71


more than 50%. The discrepancy between experiment and the fluctuating dipole

400 600 800 1000 1200T [K]

0

0.5

1

1.5

2

2.5

3

d 11 [1

0-12 m

/V]

ExperimentBKSfluc. dipolefluc. dipole without dipoles

Figure 5.3.: Comparison of the temperature dependence of the piezoelectrical strain coefficientd11

between experiment and simulation. Different potentials were used, the BKS and thefluctuating dipole potential. In the fluc. potential, two set of runs were performed, one inwhich the electric field was coupled to the dipoles, and one in which this coupling wassuppressed for the reasons outlined in the text. Experimental data was taken from Cookand Weissler [17]

potential is suprising having in mind the otherwise good performance of this poten-tial. As already mentioned in chapter2.4.1, it is possible that Tangney and Scan-dolo parametrized quantum-chemical effects into the dipoles, so that their dielectricproperties should possibly not be treated as such. Re-running the simulations withoutcoupling the dipoles to the electric field, so that only the bare charges coupled toE,showed that the discrepancy between the fluctuating dipole potential and the experi-ment is significantly reduced and quantitative agreement is almost achieved.Of course in this comparison one has to take into account that the transition tempera-ture is about100 K underestimated by this potential.The high temperature behaviour of quartz was recently studied by Haines et al. [37],who observed a loss of piezoelectric properties inα-quartz still below the transitiontemperature. It had been known for many decades that the piezoelectric coefficientd11 of quartz decreases gradually above room temperature [10, 17, 6] as shown inFig. 5.3. Haines et al. measured the quality factor of piezoelectric resonators andfound that it decreases significantly already at550 K and reaches values characteristicof a poor resonator above750 K, which is well below the transition temperature. Thetime-averaged structure ofα quartz does not provide an explanation for this behaviour,as the order parameter decreases only gradually in this temperature range. Therefore

72

5.2. Temperature Dependence of d11 in Quartz

Haines et al. suspected that this deterioration of the piezoelectric response arises fromlocal instantaneous disorder. This can be characterized by the tilt angleΦ. In Fig. 5.4,the distribution of the tilt angles in the crystal is shown for different temperatures.

-40 -30 -20 -10 0 10 20 30 40Tilt angle Φ [º]

Rel

ativ

e pr

obab

ility

Experimentfluc. dipoleBKS

T=293 K

T=473 K

T=673 K

T=863 K

T=973 K

Figure 5.4.: Distribution functions of the tilt angleΦ in quartz as a function of temperature. Note thatat temperatures of863 K and973 K, the crystal is in theβ-quartz form. Experimental datataken from Haines et al. [37].

The comparison of the simulation data with experiment shows that this explanationalso holds in computer simulations. While the average tilt angle hardly changes forthe three temperatures in theα-quartz stability range, the distribution of tilt anglesis significantly broadened. The shift towards smaller tilt angles in the simulation ascompared to experiment atT = 673 K arises from the lower transition temperature inboth potentials. This increasing disorder leads to the early decline of the quality factor,while quantities that can be correlated to the static average value, like the piezoelectriccoupling constant, remain constant towards higher temperatures.In the α-quartz phase, where this phenomenon occurs, both potentials are in goodagreement with experiment, which is reflected in Fig.5.3 in the quite similar decreaseof d11 with temperature. However it is interesting to note that the tilt angle distribu-tion in β-quartz is obviously more accurate with the fluctuating dipole potential, whichmeans that the disorder in theβ-quartz phase is modelled closer to reality.

73


5.3. Pressure Dependence of d 11 in α-Quartz

In Fig. 5.5, we show the pressure dependence of quartz with the same approaches as inFig. 5.3. While probably the most accurate data is obtained with the fluctuating dipoleapproach by suppressing the coupling betweenµ andE, the “full” fluc.-dipole poten-tial and the BKS potential are included for comparison. The fluc.-dipole potential with

0 5 10 15 20 25p [GPa]

0

5

10

15

20

d 11 [1

0-12 m

/V]

BKSfluc. dipole without dipolesfluc. dipole

Figure 5.5.: Comparison of the pressure dependence of the piezoelectrical strain coefficientd11 be-tween experiment and simulation. Different potentials were used, the BKS and the fluctu-ating dipole potential. In the fluc. potential, two set of runs were performed, one in whichthe electric field was coupled to the dipoles, and one in which this couplig was suppressedfor the reasons outlined in the text.

suppressedµ-E coupling shows a surprisingly strong similarity to the BKS potentialup to 12 GPa. Above this pressure, the results for the piezoelectrical coefficients arestarting to differ. Keeping in mind that the transition pressure for theα → II transi-tion is 21 GPa in BKS compared to 28 GPa in TS, it is obvious that the results start todiffer on approach to the transition pressure. However the absolute values comparedat the respective transition temperature are again rather comparable. The increase inthe piezoelectrical activity originates in the softening of internal modes inα-quartz, inwhich the oxygen atoms are displaced with respect to the oppositely charged siliconatoms.

74

5.4. Bond-Angle Dependence of the Dipole Moment

5.4. Bond-Angle Dependence of the DipoleMoment

In chapter5.2 above the observation was made, that the predictions of piezoelectricproperties approve, when the dipoles are not fully treated as dipoles in the calculation,but when the dipoles do not couple to the electric field and do not contribute to thepolarization of the material. It was already suspected before in the discussion of thetetraeder distortion in chapter2.4.1that the value of the dipole moments calculated inthe potential by Tangney and Scandolo modelled effects caused by the quantum chemi-cal bonds. It was shown that the distortion of the SiO4 tetraehedron, which is modelledextremely accurately by this potential, was related by Grimm and Dorner [35] to theπ-bonds between Si and O atoms. Additionally the symmetry of theπ orbitals resem-bles closely to the symmetry of the dipoles.We want to investigate this possibility by having a closer look on the relation betweenthe dipole moment and its nearest surrounding. In Fig.5.6the distribution of the dipolemoments with respect to the Si-O-Si bond angle around the respective oxygen atom is

ϑ[◦] ϑ[◦]a) α-quartz b) β-quartz

µ[eA]

120 130 140 150 160 170 180

h

1.91.71.51.31.10.90.70.50.30.1

120 130 140 150 160 170 1800

0.2

0.4

Figure 5.6.: Distribution of the absolute value of the dipole momentµ against the Si-O-Si bond angleϑ on the adjacent oxygen atom. The probability density h is plotted in units of probabilityper degree andeA.

shown. One can see that the dipoles depend almost linearly of the oxygen bond an-gle ϑ. There is some statistical spread, which increases with higher temperature. Butapparently the absolute value of the dipoles depends mainly on the bond angle. Thismeans that they depend largely on local effects of the bonding to the closest neighbors.To measure the orientation of the dipole moments, we use the labels introduced inFig. 5.7. The orientation angleϕ is the angle between the direction of the dipole mo-ment and the direction given by the bisecting line of the Si-O-Si angleϑ.

75


Figure 5.7.: Sketch of bond angleϑ, dipole orientation angleϕ and absolute value of dipoleµ, as usedin Fig. 5.6and5.6.

One can see in the distribution of the orientation angle in Fig.5.8 that the dipole mo-ment points most probable in the direction of the bisecting line, which is within thelayer defined by the oxygen and the two neighboring silicon atoms and represents themost symmetric orientation at this point. In theβ-quartz phase the orientation mightdeviate more than 10◦ with a certain probability. But in these cases the dipoles have asmall absolute value, as shown in Fig.5.8, and therefore they have only little influenceon the forces.

ϕ[◦] ϕ[◦]a) α-quartz b) β-quartz

µ[eA]

0 10 20 30 40

h0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 10 20 30 40

0

0.2

0.4

Figure 5.8.: Distribution of the absolute value of the dipole momentµ against the deviation angleϕ. The deviation angle is measured as explained in the text and shown in Fig.5.7. Theprobability density h is plotted in units of probability per degree andeA.

As the orientation of the four dipoles sitting on the oxygen atoms around one siliconatom are pairwise opposed (see Fig.2.20), one can conclude that the contribution ofdistant dipoles on the force acting on an atom dissappear faster than the usual charge-dipole interaction.Summing up these observations it is likely that there was some contribution of thequantum chemistry to the forces, stresses and energy parametrized into the dipoles.

76

6. Conclusion and Outlook

In the present work, different model potentials were used for the simulation of silicapolymorphs, in order to study the influence of different enhancements of the standardtwo-body potential. The potentials used for comparison were the frequently used BKStwo body pair potential [101], a model that had the atomic charges dynamically cal-culated from the spatial configuration [20] and a third that employs fixed charges butallows for inducible dipole moments on the oxygen atoms [90]. In the case of thefluctuating charge potential some modifications had to be implemented as the originalparametrization was flawed.

For the testing of the different potentials we concentrated on crystalline phases, as indisordered structures effects are more likely to average out and therefore in crystalsmore independent susceptibilities can be observed. An example are the piezoelectricconstants or the elastic constantC14, which are forbidden for symmetry reasons in theglass phase, but are present in certain crystalline phases. The observables taken intoaccount are predominantly of static nature and sometimes dynamic.

We analysed several cases in which the BKS potential showed quantitatively or quali-tatively incorrect results. So theα-β quartz transition is located at a transition temper-ature that is 100 K smaller than in experiment, and the density of both theα and theβ quartz phase are underestimated by about 4%, but the global picture with respect tothe volume change, the weak first-order nature of the transition and the hysteresis ef-fects is certainly well reproduced. However one important detail, the ratio of the latticeconstantsc anda, shows a qualitative difference to the experimental behaviour at thetransition temperature. This discrepancy turned out to be originated in a distortion ofthe SiO4 tetrahedra that deviates strongly in direction and size from the true distortion.Another problematic quantity is the phonon density of states, which only reproducesbasic aspects of the lattice dynamics, namely the intra-tedrahedral modes, but even inthese modes the frequency differs from ab-initio data. The low frequency range cannotbe brought into agreement with ab-initio data at all.In all of these difficulties, the allowance for fluctuating charges could not be of signif-icant help. While the aberration in the transition temperature and the density was evenstronger in the same direction, the predictions for thec/a ratio and the phonon densityof states basically shows the same shortcomings as seen in the BKS potential.

77


The introduction of additional inducible dipole moments on the oxygen atoms was afar more successful supplement to the two-body forces. Even though the transitiontemperature deviates slightly more from experiment than in the BKS potential, thisremained the only drawback in the results of the fluctuating dipole potential. For theestimation of the density the agreement with experiment is very satisfactory, and mostimportant the qualitative failures of the BKS potentials in the vibrational density ofstates and thec/a ratio do not occur with the fluctuating dipoles. It could be shownthat the distortion of the tetrahedra, which causes the behaviour of thec/a ratio, is inperfect agreement with experiment. The resemblance of the dipole orientation to thesymmetry of theπ-bonds, which were hold responsible for the tetrahedra deformationin a work by Grimm and Dorner [35], gives rise to the suspicion that some of thequantum chemistry in the atomic bonding was incorporated into the dipole interactionduring the parametrization of this potential.The elastic constants were in generally good agreement for the BKS potential, andwere equally good reproduced by the fluctuating dipole potential. Only the fluctuatingcharge potential showed strong deviations. The good results of BKS in this respect arenot surprising given that the BKS parameters were adjusted to the experimental elasticconstants ofα-quartz.

To test the new potentials at extreme conditions, the post-quartz phases under highpressure were studied. Even though there is a rich variety of different high pressurepolymorphs and different suggestions for possible high-pressure transition path waysfrom experiment and theory, it is remarkable that all three potentials showed a tran-sition in a pressure range of 21 to 27 GPa to the same crystalline structure, whichis supposedly the structure of the still unclear quartz II phase. However theδ-quartzphase seen by the BKS potential on decompression to ambient pressures is not repro-duced by any of the two new potentials. Therefore the formation of this phase seemsto be an artifact of the BKS potential. In contrast, the two new potentials do see thequartz II phase revert toα-quartz, while at intermediate pressures a phase emerges thatis very similar to the quartz II phase at higher pressures.

As the new potential approaches incorporate electrostatic properties, namely fluctuat-ing charges and fluctuating dipole moments, the investigation of dielectric properties inchapter5 should highlight the question if these are real electrostatic variables and thusimprove the prediction for the dielectric properties of the system. As there was no ap-propriate method for the calculation of piezoelectric properties available in literature,we developed a method to measure these by fluctuation relations within the simulationrun. In order to achieve this the definition of the polarisation in an infinitely replicatedsystem had to be clarified, as the definition is ambigious due to surface effects and tobox fluctuations, and the fluctuation relation was established by linear response theory.

78

To test this method the direct impact of an external field was measured, where the noisewas reduced by subtracting the observables measured in a zero field run with identicalstart configuration and random number generator.We found that the BKS potential gives a quite reasonable estimate for the piezoelec-tric constants ofα-quartz against temperature. The fluctuating charge potential wasagain slightly more away from experiment than BKS. For the fluctuating dipole po-tential the observation was made that the result depends on the question whether thedipoles are taken seriously, this means if the dipoles are coupled to the electric fieldand contribute to the polarization in the material, or if they are ignored with respectto electrostatic quantities. The results clearly improved in the later case, again givingrise to the suspicion, that the dipoles that are induced on the oxygen atoms are not pureelectric dipole moments. This question was further investigated by an analysis of therelation between the dipole moment and its direct surrounding. It turned out that thestrength of the dipole depends linearly on the adjacent oxygen bond angle and is mostlikely in the direction of the bisecting line of this angle.

Putting these observations together, we can compare the potentials in the followingway:

BKS Fl-Q Fl-µDOS inα-quartz - 0 +Elastic constants inα andβ quartz + 0 +c/a anomaly at theα-β quartz transition - -/0 +Equation of state inα andβ-quarty 0 - +Stability of cristobalite and tridymite - 0 +Piezoelectric coefficients 0 0 (+)Pressure induced transition + + +Stishovite c/a and elasticity 0 - 0

The results shown above make it clear that the fluctuating charge approach is not ofuse for the simulation of silica. Although it is very likely that the effective charges,which represent the degree of ionicity, change with pressure and for different poly-morphs, there is no evidence to suggest that this happens dynamically at a given pres-sure or temperature in the real system. In the simulations it was shown that keepingthe charges fixed on their average values does not influence the results of the simu-lations. The fluctuation in the charges is quite small, and one can even see that theoriginal authors update the charges on a time scale which is far beyond the time scalesof ionic motion. In an equilibrated system the charges can therefore not react on thepresent spatial configuration. The overall conclusion is that the dynamic adaption ofthe charges within an equilibrated system does not influence the ionic motion notably.However this approach could be probably more useful for systems where the ionicity

79


changes dynamically, as for instance at a surface.

In contrast, the effects contained in the fluctuating dipoles have a significant influ-ence on the system. Their contribution is sufficiently large that theα-β transition issuppressed when the dipoles are constrained to their average values in the respectivephases. In summary, the results achieved with this potential turn out to be in very closeagreement with experiment with the exception of six-coordinated silicon at small pres-sures. It is the only potential that accurately models the tetrahedral distortion and thelattice dynamics in quartz.Possibly the quantum effects that are apparently parametrized in terms of dipoles couldbe expressed in a newly parametrized three body term and thus reduce the computa-tional expense.

All properties of four-coordinated silica could be described very precisely with thefluctuating dipole potential, even though the potential was not explicitly tested nor op-timized for these experimental results, but only parametrized to (microscopic) ab-initiodata. The parametrization was done in the predominantly tetrahedrally coordinated liq-uid phase and therefore slightly biased in favor of a tetrahedral crystal structure, whichexplains the weakness in the six coordinated stishovite phase at low pressure. How-ever it is even more surprising that it is the only model that predicts the experimentallymeasured transition pressure in stishovite. The overall success clearly approves thefunctional form of this model potential as well as the parametrization procedure, andgives rise to the hope that this procedure can find similarily good potentials for othermaterials as well.

80

A. Calculations

A.1. Calculations for Fluc-Q Potential

A.1.1. Direct Solution of Charge Equilibration Equations

Equation (1.24) and its following explanations only take into account a nonperiodicsystem. Regarding a system with periodic boundary conditions, the Coulomb interac-tion has to be calculated by making use of the Ewald summation1.1.5. With summa-tion over all periodic images, eq. (1.23) reads

χi = χ0i + J0

iiQi +∑

j

′JijQj (A.1)

The notation∑

j′ meaning the sum over all periodic pictures andi 6= j in the original

picture can be rewritten as∑j

′JijQj =

∑j

′(

1

Rij

−(

1

Rij

− Jij

))Qj (A.2)

=∑

j

′ 1

Rij

Qj −∑j 6=i

(1

Rij

− Jij

)Qj, (A.3)

becauseJij → 1/Rij for Rij → 0 and therefore the difference is short range. Theremaining pure Coulomb term can be evaluated using standard Ewald summation:∑

j

′ 1

Rij

Qj =∑j 6=i

1

Rij

erfc(√

αRij)Qj −2α√

πQi +

∑j

CijQj (A.4)

with Cij =4π

V

∑k 6=0

exp(− k2

4α)

k2exp(ikRi) exp(−ikRj) (A.5)

Inserted back to eq. (A.1) this gives

χi = χ0i + AijQi +

∑j 6=i

BijQj (A.6)

81

A. Calculations

with Ai = J0ii −

2α√π

+ Cii (A.7)

and Bij =1

Rij

erfc(√

αRij)−(

1

Rij

− Jij

)+ Cij (A.8)

Now the conditionχi = χ1 reads

−A1Q1 + AiQi +∑j 6=i

BijQj −∑j 6=1

B1jQj = −(χ0i − χ0

1) (A.9)

Together with the charge neutrality constraint this leads to the set of equations

C ·Q = −D, (A.10)

whereDi = χ0

i − χ01 (A.11)

and

C =

1 1 1 · · · 1

−A1 + B21 A2 −B12 B23 −B13 · · · B2N −B1N

−A1 + B31 B32 −B12 A3 −B13 · · · B3N −B1N...

......

......

−A1 + BN1 BN2 −B12 BN3 −B13 · · · AN −B1N

(A.12)

with Ai andBij as above.

A.1.2. Two Body Slater Integral

The integration over the overlap of two slater orbitals with two different centers canbe sketched as in FigureA.1. The value of the integral depends on the distancerbetween the centersa andb. The integral to be evaluated (see eq.1.32) looks in thisnomenclature like

J(r) =

∫∫φ2

i (ri1)1

r12

φ2j(rj2)dri1drj2 (A.13)

=

∫φ2

i (ri1)Uj(ri1)dri1 (A.14)

with

Uj(ri1) =

∫1

r12

φ2j(rj2)drj2 (A.15)

= A2j

∫1

r12

r2nj−2j2 e−2ζjrj2drj2 (A.16)

82

A.1. Calculations for Fluc-Q Potential

j i

1

2

ri1 rj2

r

r12

rj1

ri2

Figure A.1.: The i and j points are the centers of the orbitals, theri1 and rj2 are the integrationvariables.

TheAj is the normalization factor to ensure∫

φ2dV = 1 and can be calculated as

A =

√1

4π

(2ζ)2n+1

(2n)!(A.17)

Solving at first theUj(ri1) integral, one has to expand1/r12 in spherical harmonics:

1

r12

=∞∑

λ=0

4π

2λ + 1

r<

rλ> + 1

+λ∑κ=−λ

Y κλ (θ1, ϕ1)

∗ Y κλ (θ2, ϕ2) (A.18)

Herer< is the smaller andr> is the bigger ofrj1 andrj2. In the angular integrationonly λ = 0 delivers a non-vanishing contribution. So the radial integration is left over:

Uj(ri1) = A2j4π

∞∫0

1

r>

r2nj−2j2 e−2ζjrj2drj2 (A.19)

= A2j4π

1

rj1

rj1∫0

r2nj

j2 e−2ζjrj2drj2 +

∞∫rj1

r2nj−1j2 e−2ζjrj2drj2

(A.20)

= A2j4πr

2nj

j1

1∫

0

y2nj e−2xydy +

∞∫1

y2nj−1 e−2xydy

(A.21)

In the last line, the substitutiony = rj2/rj1 andx = rj1/ζj was applied. After evalua-tion of the two integrals, substitution ofAj from eq. (A.17) and further simplification,one obtains

Uj(ri1) =1

rj1

− e−2ζjrj1

2nj∑ν=1

ν(2ζj)2nj−ν

(2nj − ν)!2nj

r2nj−ν−1j1 (A.22)

83

A. Calculations

Substitution back into eq. (A.13) yields

J(r) = A2i

{∫r2n1−2i1

1

rj1

e−2ζ1ri1dri1 (A.23)

−2nj∑ν=1

ν(2ζj)2nj−ν

(2nj − ν)!2nj

∫r2n1−2i1 r

2nj−ν−1j1 e−2ζ1ri1e−2ζjrj1dri1

}.

Due to their symmetry these two integrals can be evaluated best in elliptical coordi-nates(ξ, η, ϕ). The transformation isri1 = (r/2)(ξ + η), rj1 = (r/2)(ξ − η) anddri1 = (r/2)3(ξ2 − η2)dξdηdϕ. After some simplifications, the final result for theoverlap integral now reads

J(r) =(2ζi)

2ni+1

2 (2ni)!

{(r

2

)2ni2ni−1∑ν=0

(2ni − 1

ν

)A2ni−1−ν(ζir)Bν(ζir) (A.24)

−2nj∑ν=1

ν(2ζj)2nj−ν

(2nj − ν)!2nj

(r

2

)2ni+2nj−ν2ni−1∑k=0

2nj−ν∑l=0

(2ni − 1

k

)(2nj − ν

l

)

×(−1)lA2(ni+nj)−k−ν−l−1(r(ζi + ζj))Bk+l(r(ζi − ζj))

}

with Ak(x) =

∞∫1

ξke−xξdξ, and Bk(x) =

+1∫−1

ηke−xηdη (A.25)

The auxiliary integralAk(x) can be evaluated by

Ak(x) = e−x k!

xk+1

k∑ν=0

xν

ν!, (A.26)

whereas a similar representation ofBk(x) turned out to be numerically unstable. There-foreBk(x) is evaluated by the following formula:

Bk(x) =∞∑

ν=0

1− (−1)k+ν+1

ν!(k + ν + 1)(−x)ν (A.27)

84

B. Programs

B.1. Molecular Dynamics Code

This is the main simulation code. Input parameters is handed over in a file calledpimd.inp, the start configuration in a fileconf.sta . If no start configuration ispresent, this file has to exist anyway with only the number 0 as content.The program consists of the filepimd.f , and the filepimd.com.f with declara-tions of common variables. The force routine for the fluctuating dipole potential is notprinted but can be obtained from Paul Tangney. The interface for this routine with theconversion of units is within the subroutine scandolopotential.The following routines from “Numerical Recipes in Fortran” [74] need to be includedin the code: ludcmp, lubksb, erfcc, four1. All variables needed by this force routineare in atomic units.The units used in the main program are as follows:

[l] = 1A [E] = 1kBK [t] = 1~/(kBK)

[m] = 1mp [Q] = 1e [p] = 1kBK/A3

The output files generated have the following content:

fort.60 Different observables with time, see code for columns.fort.61 Order parameter with timepimd.g r Pair correlations functionspimd.bon g(r) for Si-O at nearest neighbor distance, bond angle distributionsfort.62 Average, minimum and maximum charge with timeconf.end End configurationconf.xmo End configuration in xyz formatconfavg.xmo Averaged configuration in xyz formatconf.pol End dipole configurationconfavg.pol Averaged dipole configurationpimd.out Averaged observables and input configuration

85

B. Programspi

md.

fp

rog

ram

pim

dim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

na

vg_

cha

rge

(n_

typ

e),

ma

x_ch

arg

e(n

_ty

pe

),&

min

_ch

arg

e(n

_ty

pe

)in

teg

er

i_te

std

ou

ble

pre

cisi

on

qe

qre

c2(n

_p

art

,n_

pa

rt)

com

mo

n/q

eq

rect

est

/q

eq

rec2

do

ub

lep

reci

sio

np

mix

com

mo

n/p

mis

ch/

pm

ixd

ou

ble

pre

cisi

on

ers

tes_

v,zw

eite

s_v

do

ub

lep

reci

sio

nq

eq

rec2

1(n

_p

art

)

do

ub

lep

reci

sio

nv_

qa

bh

ae

ng

ig_

ete

stco

mm

on

/ext

lag

ete

st/

v_q

ab

ha

en

gig

_e

test

inte

ge

rco

unt,

c1,

c2,

cra

te,

cma

xre

al

pe

rfo

rma

nce

n_

run

=1

do

i_ru

n=

1,

n_

run

write

(*,*

)"i

_ru

n",

i_ru

nca

llp

ara

_in

itif

(i_

run

.eq.1

)ca

llco

nf_

init

call

ob

se_

init

call

che

ck_

skin

if(f

_sk

in.e

q.1

)th

en

call

bin

nin

ge

nd

if

if((

f_ch

arg

ea

dju

st.e

q.3

).o

r.(f

_ch

arg

ea

dju

st.e

q.4

))th

en

call

ma

trix

inve

rsio

ne

nd

if

r_re

al_

avg

=0

.

do

i_tim

e=

1,

n_

rela

x+

n_

ob

ser

r_tim

e=

r_tim

e+

dt

!---

pro

pa

ga

teco

ord

ina

tes

an

dits

de

riva

tive

so

ne

time

unit

call

pre

dic

tif

(f_

cha

rge

adju

st.e

q.3

)ca

lle

xtd

lag

ran

ge

_p

red

icto

r

!---

che

ckw

he

the

rn

eig

hb

or

lists

ne

ed

tob

eu

pd

ate

dif

(f_

po

ten

tial.n

e.1

)th

en

call

che

ck_

skin

if(f

_sk

in.e

q.1

)th

en

call

bin

nin

ge

nd

ife

nd

if

!---

tra

nsf

orm

pre

dic

ted

coo

rdin

ate

sin

tore

al

spa

ceca

lltr

an

s_re

al

!---

en

sure

ha

rmo

nic

bo

un

da

ryco

nd

itio

ns

if(f

_p

ote

ntia

l.ne

.1)

call

pb

_co

nd

it

!---

calc

ula

tein

tera

ctio

nif

(f_

po

ten

tial.e

q.1

)th

en

call

ha

rm_

osc

els

eif

(f_

po

ten

tial.e

q.2

)th

en

call

len

na

rd_

jon

es

els

eif

(f_

po

ten

tial.e

q.3

)th

en

call

bks

_p

ote

ntia

le

lse

if(f

_po

ten

tial.e

q.4

)th

en

call

bks

_p

ote

ntia

le

lse

if(f

_po

ten

tial.e

q.5

)th

en

call

go

dd

ard

_po

ten

tial

els

eif

(f_

po

ten

tial.e

q.6

)th

en

call

go

dd

ard

_po

ten

tial

els

eif

(f_

po

ten

tial.e

q.7

)th

en

call

go

dd

ard

_po

ten

tial

els

eif

(f_

po

ten

tial.e

q.8

)th

en

call

vash

ish

ta_

po

ten

tial

els

eif

(f_

po

ten

tial.e

q.9

)th

en

call

sca

nd

olo

_p

ote

ntia

le

nd

if

if(f

_e

_a

pp

lied

.gt.0

)ca

lla

pp

lied

_fie

ld

!---

calc

ula

ten

ew

cha

rge

sif

(f_

cha

rgea

dju

st.e

q.1

)th

en

!le

ave

cha

rge

su

nch

an

ge

de

lse

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

!d

on

oth

ing

(in

clu

de

din

go

dda

rd_

po

ten

tial)

els

eif

(f_

cha

rge

ad

just

.eq

.4)

the

nif

(i_

time

.le.n

_re

lax)

the

nif

(mo

d(i_

time

,25

).e

q.0

)ca

llm

atr

ixin

vers

ion

els

e if(m

od

(i_

time

,10

0).

eq

.0)

call

ma

trix

inve

rsio

ne

nd

ife

nd

if

if(i_

time

.gt.n

_re

lax)

the

nca

llo

bse

rva

tion

en

dif

!---

tra

nsf

orm

inte

ract

ion

sin

tore

cip

roca

lsp

ace

call

tra

ns_

reci

pif

(f_

po

ten

tial.n

e.1

)ca

llb

ox_

cha

in

!---

ad

dra

nd

om

forc

es

call

the

rmo

sta

t

!---

corr

ect

tra

ject

orie

sca

llco

rre

ctif

(f_

cha

rgea

dju

st.e

q.3

)ca

lle

xtd

lagra

ng

e_

corr

ect

or

r_re

al_

avg

=r_

rea

l_a

vg+

r_re

al

if(m

od

(i_

time

,n_

safe

).e

q.0

)ca

llco

nf_

save

en

dd

o

call

ob

ser_

ou

t

en

dd

o

en

dp

rog

ram

86


!!

!!

!!

!!

!!

!

sub

rou

tine

pa

ra_

init

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

inte

ge

rn

_o

rde

r_m

ax

pa

ram

ete

r(n

_o

rde

r_m

ax=

5)

do

ub

lep

reci

sion

pre

d_

coe

f(0

:n_

ord

er_

max-

1,0

:n_

ord

er_

ma

x),

&co

rr_

coe

f(0

:n_

ord

er_

ma

x)d

ou

ble

pre

cisi

on

sla

terc

uto

ff,

sla

terijr

do

ub

lep

reci

sion

sla

terin

t,sl

ate

rin

t_d

inte

ge

rjs

ee

dco

mm

on

/cjs

ee

d/

jse

ed

do

ub

lep

reci

sion

r_1

,r_

2,r

_6

,exp

_p

ot,v_

cou

lom

b,p

ot_

loc

do

ub

lep

reci

sion

d_

du

mm

1,

d_

du

mm

2

inte

ge

rm

sid

um

my1

,m

sid

um

my2

log

ica

le

xist

do

ub

lep

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sion

po

tch

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(n_

typ

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pe)

!**

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****

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po

ten

tial

1if

(f_

po

ten

tial.e

q.1

)th

en

if(i_

run

.eq

.1)

the

no

pe

n(1

0,

file

="p

imd

.inp

",st

atu

s="o

ld")

rea

d(1

0,2

10

)i_

du

mm

y,ch

ar_

du

mm

y

if(i_

du

mm

y.e

q.0

)th

en

!---

-ta

kede

fau

ltp

ara

me

ters

n_

rela

x=

50

n_

ob

ser

=5

0n

_sa

fe=

10

0is

ee

d=

47

59

1+

2*i

nt(

n_

pa

rt/2

)+2

*in

t(10

*tk)

+n

_tr

ot

jse

ed

=is

ee

d

dt

=0

.03

5d

0tk

=0

.27

5m

ass

(0,1

)=

1.d

0m

ass

(0,2

)=

ma

ss(0

,1)

if(n

_o

rde

r.eq

.2)

the

n!-

---

che

cke

ne

rgy

con

serv

atio

nfr

ictio

n(1

)=

0.

fric

_b

ox

=0

.d

t=

dt/2

els

efr

ictio

n(1

)=

0.0

1/

dt

fric

_b

ox

=0

.01

/d

te

nd

iffr

ictio

n(2

)=

fric

tion

(1)

!---

de

fine

ine

rtia

for

bo

xd

efo

rma

tion

ma

ss_

bo

x=

(n_

silic

*ma

ss(0

,1)

+n

_o

xy*m

ass

(0,2

))/

12

els

e!

i_du

mm

yif

(i_

du

mm

y.ne

.f_

po

ten

tial)

&st

op

"f_

po

ten

tial

an

dp

imd

.inp

mis

ma

tch

"re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.ne

.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n

_re

lax

rea

d(1

0,2

10

)n

_o

bse

rre

ad

(10

,21

0)

n_sa

fere

ad

(10

,21

0)

ise

ed

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

dt

rea

d(1

0,2

01

)tk

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

2)

(ma

ss(0

,i_ty

pe

),i_

typ

e=

1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

mass

_b

ox,

fric

_b

ox

en

dif

!i_

du

mm

ycl

ose

(10

)e

lse

!i_

run

tk=

tk-

0.2

5e

nd

if!

i_ru

n

!**

****

****

****

*f_

po

ten

tial

2e

lse

if(f

_po

ten

tial.e

q.2

)th

en

if(i_

run

.eq.1

)th

en

op

en

(10

,fil

e=

"pim

d.in

p",

sta

tus=

"old

")re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.e

q.0

)th

en

!---

-ta

ked

efa

ult

pa

ram

ete

rsn

_re

lax

=5

0n

_o

bse

r=

50

n_

safe

=1

00

ise

ed

=4

75

91

+2*i

nt(

n_

pa

rt/2

)+2

*in

t(1

0*t

k)+

n_

tro

tjs

ee

d=

ise

ed

for_

sca

_m

fa_

cor

=0

.

!---

-L

en

na

rdJo

ne

sp

ote

ntia

l;L

Ju

nits

!m

ass

(0,1

)=

10

0.d

0!

epsi

l(1

,1)

=1

.d0

!si

gm

a(1

,1)

=1

.d0

!---

-L

Jp

ote

ntia

ls;

my

un

its\h

ba

r=1

,k_

b=

1,A

=1

,e=

1!-

---

Arg

on

ma

ss(0

,1)

=4

0*m

ass

_a

mu

ep

sil(1

,1)

=1

.67

e-2

1*j

ou

lesi

gm

a(1

,1)

=3

.40

5!-

---

Heliu

m:

Azi

zp

ote

ntia

lm

ore

acc

ura

te!

mass

(0,1

)=

4*m

ass

_a

mu

!e

psi

l(1

,1)

=1

0.2

2!

sig

ma

(1,1

)=

2.5

56

!---

-N

eo

n!

mass

(0,1

)=

18

*ma

ss_

am

u!

epsi

l(1

,1)

=0

.49

4e

-21

*jo

ule

!si

gm

a(1

,1)

=2

.75

0d

0!-

---

sym

me

triz

em

ass

es

an

dL

en

nard

Jon

es

coe

ffic

ien

tsm

ass

(0,2

)=

ma

ss(0

,1)

ep

sil(2

,2)

=e

psi

l(1

,1)

ep

sil(1

,2)

=sq

rt(e

psi

l(1

,1)*

ep

sil(2

,2))

ep

sil(2

,1)

=e

psi

l(1

,2)

sig

ma

(2,2

)=

sig

ma

(1,1

)si

gm

a(1

,2)

=(s

igm

a(1

,1)+

sig

ma

(2,2

))/2

sig

ma

(2,1

)=

sig

ma

(1,2

)!-

---

cho

sea

pp

rop

ria

tecu

toff

and

skin

87

B. Programsd

oi_

typ

e=

1,n

_ty

pe

do

j_ty

pe

=1

,n_

typ

er_

cuto

ff(i_

typ

e,j_

typ

e)

=1

.7*s

igm

a(i_

typ

e,j_

typ

e)

!r_

cuto

ff(i_

typ

e,j_

typ

e)

=3

.00

*sig

ma

(i_

typ

e,j_

typ

e)

r_sk

in(i_

typ

e,j_

typ

e)

=0

.18

*r_

cuto

ff(i_

typ

e,j_

typ

e)

en

dd

oe

nd

do

!---

-ch

ose

ap

pro

pria

test

art

para

me

ters

tk=

0.4

*ep

sil(1

,1)

dt

=0

.01

25

*sq

rt(s

igm

a(1

,1)*

*2*m

ass

(0,1

)/e

psi

l(1

,1))

f_p

ress

ure

=1

pre

ss=

.00

1*e

psi

l(1

,1)/

sig

ma

(1,1

)**3

if(n

_o

rde

r.e

q.2

)th

en

!---

-ch

eck

en

erg

yco

nse

rva

tion

fric

tion

(1)

=0

.fr

ic_

bo

x=

0.

dt

=d

t/2

els

efr

ictio

n(1

)=

0.0

1/

dt

fric

_b

ox

=0.0

1/

dt

en

dif

fric

tion

(2)

=fr

ictio

n(1

)!-

--d

efin

ein

ert

iafo

rb

ox

de

form

atio

nm

ass

_b

ox

=(n

_si

lic*m

ass

(0,1

)+

n_

oxy

*ma

ss(0

,2))

/1

2

ma

ss_

bo

x=

ma

ss_

bo

x/

2

els

e!

i_d

um

my

if(i_

du

mm

y.n

e.f_

po

ten

tial)

&st

op

"f_p

ote

ntia

la

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)i_

du

mm

y,ch

ar_

du

mm

yif

(i_

du

mm

y.n

e.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n_

rela

xre

ad

(10

,21

0)

n_

ob

ser

rea

d(1

0,2

10

)n_

safe

rea

d(1

0,2

10

)is

ee

dre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

01

)dt

rea

d(1

0,2

01

)tk

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

2)

(ma

ss(0

,i_ty

pe

),i_

typ

e=

1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

ma

ss_

bo

x,fr

ic_

bo

x

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

pre

ssre

ad

(10

,21

0)

f_p

ress

ure

rea

d(1

0,2

00

)ch

ar_

du

mm

y

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(ep

sil(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

ore

ad

(10

,20

0)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(sig

ma

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00

)ch

ar_

du

mm

y

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(r_

cuto

ff(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00

)ch

ar_

du

mm

yd

oi_

typ

e=

1,n

_ty

pe

rea

d(1

0,2

02

)(r

_sk

in(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

en

dif

clo

se(1

0)

els

etk

=tk

-0

.25

en

dif

if(i_

run

.eq

.1)

the

n!-

--a

bso

rbfa

cto

r4

into

ep

silo

nan

din

itia

lize

sig

ma

_2

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

sig

ma

_2

(i_

typ

e,j_

typ

e)

=si

gm

a(i_

typ

e,j_

typ

e)*

*2e

psi

l(i_

typ

e,j_

typ

e)

=4

*ep

sil(i_

typ

e,j_

typ

e)

en

dd

oe

nd

do

en

dif

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_cu

toff_

2(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_ty

pe

,j_ty

pe

)**2

r_ra

ng

e(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

type

,j_ty

pe

)+

&r_

skin

(i_

typ

e,j_

typ

e)

r_ra

ng

e_

2(i_

typ

e,j_

typ

e)

=r_

ran

ge

(i_

typ

e,j_

typ

e)*

*2e

nd

do

en

dd

o

!---

de

fine

shift

sin

en

erg

y

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_d

um

my

=(s

igm

a_

2(j_

typ

e,i_

typ

e)/

r_cu

toff_

2(j_

typ

e,i_

typ

e))

**3

e_

shift

(j_

typ

e,i_

typ

e)

=r_

du

mm

y*(r

_du

mm

y-1

.)e

nd

do

en

dd

o

!**

****

****

****

f_p

ote

ntia

l3

els

eif

(f_

po

ten

tial.e

q.3

)th

en

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

sto

p"B

KS

pote

ntia

ld

oe

sn

ot

wo

rkw

ithva

ria

ble

cha

rge

"e

nd

if

if(i_

run

.eq

.1)

the

no

pe

n(1

0,

file

="p

imd

.inp

",st

atu

s="o

ld")

rea

d(1

0,2

10

)i_

du

mm

y,ch

ar_

du

mm

y

if(i_

du

mm

y.e

q.0

)th

en

!---

-ta

kede

fau

ltp

ara

me

ters

n_

rela

x=

50

n_

ob

ser

=5

0n

_sa

fe=

100

ise

ed

=4

75

91

+2

*in

t(n

_p

art

/2)+

2*i

nt(

10

*tk)

+n

_tr

ot

88


jse

ed

=is

ee

d

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

qe

qe

lect

rone

g(1

)=

4.1

68

*11

60

4.5

04

9!

eV

->kB

Kq

eq

ele

ctro

ne

g(2

)=

8.7

41

*11

60

4.5

04

9q

eq

ha

rdn

ess

(1)

=6

.97

4*1

16

04

.50

49

qe

qh

ard

ne

ss(2

)=

13

.36

4*1

16

04

.50

49

en

dif

n_

rela

x=

30

n_

ob

ser

=3

0n

_sa

fe=

30

!---

-si

liciu

mm

ass

(0,1

)=

28

.1*m

ass

_a

mu

!---

-o

xyge

nm

ass

(0,2

)=

16

.0*m

ass

_a

mu

ep

sil(1

,1)

=0

.*e

ve

psi

l(2

,2)

=1

38

8.7

73

*ev

ep

sil(1

,2)

=1

80

03

.75

72

*ev

ep

sil(2

,1)

=e

psi

l(1

,2)

sig

ma

(1,1

)=

0.

sig

ma

(2,2

)=

2.7

6si

gm

a(1

,2)

=4

.87

31

8si

gm

a(2

,1)

=si

gm

a(1

,2)

c_va

nd

wa

als

(1,1

)=

0.

c_va

nd

wa

als

(2,2

)=

17

5.0

*ev

c_va

nd

wa

als

(1,2

)=

13

3.5

38

1*e

vc_

van

dw

aa

ls(2

,1)

=c_

van

dw

aa

ls(1

,2)

alp

ha

=0

.3*

(10

02

./n

_p

art

)**(

1./6

)a

lph

a=

0.3

v_cu

t=

0.0

1*e

lec_

pre

fac*

(1.2

*2.4

)/1

.6/s

qrt

(1.*

n_

pa

rt)

f_o

pt_

rec

=0

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_sk

in(i_

typ

e,j_

typ

e)

=0

.45

en

dd

oe

nd

do

tk=

36

00

./n

_tr

ot

!--

--tim

est

ep

=1

fsd

t=

1.3

09

1e

-4f_

pre

ssu

re=

1p

ress

=.0

if(n

_o

rde

r.eq

.2)

the

n!-

---

che

cke

ne

rgy

con

serv

atio

nfr

ictio

n(1

)=

0.

fric

_b

ox

=0

.d

t=

dt/2

els

efr

ictio

n(1

)=

0.0

1/

dt

fric

_b

ox

=0

.01

/d

te

nd

iffr

ictio

n(2

)=

fric

tion

(1)

!---

de

fine

ine

rtia

for

bo

xd

efo

rma

tion

ma

ss_

bo

x=

(n_

silic

*ma

ss(0

,1)

+n

_o

xy*m

ass

(0,2

))/

12

ma

ss_

bo

x=

ma

ss_

bo

x/

8e

lse

!i_

dum

my

if(i_

du

mm

y.ne

.f_

po

ten

tial)

&st

op

"f_

po

ten

tial

an

dp

imd

.inp

mis

ma

tch

"re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.ne

.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n

_re

lax

rea

d(1

0,2

10

)n

_o

bse

rre

ad

(10

,21

0)

n_sa

fere

ad

(10

,21

0)

ise

ed

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

dt

rea

d(1

0,2

01

)tk

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

2)

(ma

ss(0

,i_ty

pe

),i_

typ

e=

1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

mass

_b

ox,

fric

_b

ox

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

pre

ssre

ad

(10

,21

0)

f_p

ress

ure

rea

d(1

0,2

10

)f_

op

t_re

cre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

00

)ch

ar_

du

mm

y

if(f

_ch

arg

ead

just

.ne

.1)

the

nre

ad

(10

,20

2)

(qe

qe

lect

ron

eg

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

02

)(q

eq

ha

rdn

ess

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

en

dif

rea

d(1

0,2

00

)ch

ar_

du

mm

y

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(ep

sil(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

ore

ad

(10

,20

0)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(sig

ma

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00

)ch

ar_

du

mm

yd

oi_

typ

e=

1,n

_ty

pe

rea

d(1

0,2

02)

(c_

van

dw

aa

ls(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00

)ch

ar_

du

mm

y

rea

d(1

0,2

02)

alp

ha

,v_

cut

rea

d(1

0,2

00

)ch

ar_

du

mm

yd

oi_

typ

e=

1,n

_ty

pe

rea

d(1

0,2

02)

(r_

skin

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

en

dif

!i_

du

mm

ycl

ose

(10

)e

lse

!i_ru

ntk

=tk

-0

.25

en

dif

!i_ru

n

alp

ha

_p

refa

c=

2*a

lph

a/s

qrt

(pi)

r_cu

toff(1

,1)

=3

./a

lph

ar_

cuto

ff(2

,2)

=3

./a

lph

ar_

cuto

ff(2

,1)

=3

./a

lph

ar_

cuto

ff(1

,2)

=r_

cuto

ff(2

,1)

!if

(r_

cuto

ff(1

,1).

lt.r_

cut_

sho

rt)

!&

stop

"cu

to

ffto

osm

all

for

sho

rtra

ng

e"

89

B. Programsq

_cu

t_2

=(6

.*a

lph

a)*

*2!-

--a

ny

pe

ak

|f(k

|ˆ2

ino

rde

r1

/sq

rt(N

)g

ets

cut

off

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

sig

ma

_2

(i_

typ

e,j_

typ

e)

=si

gm

a(i_

typ

e,j_

typ

e)*

*2e

nd

do

en

dd

or_

cuto

ff_

ma

x=

0.0

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_cu

toff_

ma

x=

ma

x(r_

cuto

ff_

ma

x,r_

cuto

ff(i_

typ

e,j_

typ

e))

en

dd

oe

nd

do

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_cu

toff_

2(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)*

*2r_

ran

ge

(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)

+&

r_sk

in(i_

typ

e,j_

typ

e)

r_ra

ng

e_

2(i_ty

pe

,j_ty

pe

)=

r_ra

ng

e(i_ty

pe

,j_ty

pe

)**2

en

dd

oe

nd

do

!---

de

fine

shift

sin

en

erg

y

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

!---

-e

lect

rost

atic

e_

shift

(i_

typ

e,j_

typ

e)

=e

rfcc

(alp

ha

*r_cu

toff(i_

typ

e,j_

typ

e))

*&

ele

c_p

refa

c/r_

cuto

ff(i_

typ

e,j_

typ

e)

!---

-sh

ort

ran

ge

e_

shift

_sh

ort

(i_

typ

e,j_

typ

e)

=e

psi

l(i_

typ

e,j_

typ

e)*

&e

xp(-

sig

ma(i_

typ

e,j_

typ

e)*

r_cu

t_sh

ort

)-&

c_va

nd

wa

als

(i_

typ

e,j_

typ

e)/

r_cu

t_sh

ort

**6

en

dd

oe

nd

do

!---

-ta

bu

late

po

ten

tial

v(r)

an

dd

v/d

rd

oi_

typ

e=

1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

do

i_p

ote

ntia

l=

0,n

_p

ote

ntia

lr_

1=

i_p

ote

ntia

l*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial

r_2

=r_

1**

2!-

----

-cu

toff

at

5.5

\AA

sho

rtra

nge

inte

ract

ion

(SiO

_2

)if(

r_1

.le.r

_cu

t_sh

ort

)th

en

exp

_p

ot

=e

psi

l(i_

typ

e,j_

typ

e)*

exp

(-si

gm

a(i_

typ

e,j_

typ

e)*

r_1

)e

lse

exp

_p

ot

=0

.0e

nd

ifif(

r_2

.lt.2

.0)

the

nif(

i_ty

pe

.eq

.j_ty

pe

)th

en

r_2

=2

.0r_

1=

sqrt

2e

lse

if(r_

2.lt

.1.2

1)

the

nr_

2=

1.2

1r_

1=

1.1

en

dif

en

dif

!---

---

cuto

ffa

tsh

ort

ran

ge

inte

ract

ion

(SiO

_2

)if(

r_1

.le.r

_cu

t_sh

ort

)th

en

r_6

=c_

van

dw

aa

ls(i_

typ

e,j_

typ

e)*

(1./r_

2)*

*3e

lse

r_6

=0

.e

nd

ifv_

cou

lom

b=

erf

cc(a

lph

a*r

_1

)*e

lec_

pre

fac/

r_1

po

t_lo

c=

exp

_p

ot

-r_

6if(

r_1

.le.r

_cu

t_sh

ort

)&

po

t_lo

c=

po

t_lo

c-

e_

shift

_sh

ort

(i_

typ

e,j_

typ

e)

!---

-r_

dum

my

={\

pa

rtia

lV

\ove

r\p

art

ial

r}/

rr_

du

mm

y=

-sig

ma

(i_

typ

e,j_

typ

e)*

exp

_po

t/r_

1+

6*r

_6

/r_

2r_

po

t_1

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=p

ot_

loc

r_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=-e

_sh

ift(i_

typ

e,j_

typ

e)

&+

v_co

ulo

mb

r_p

ot_

d_

1(i_ty

pe

,j_ty

pe

,i_p

ote

ntia

l)=

r_d

um

my

r_p

ot_

d_

2(i_ty

pe

,j_ty

pe

,i_p

ote

ntia

l)=

-(1

./r_

2)

*(v

_co

ulo

mb+

&e

lec_

pre

fac*

alp

ha

_p

refa

c*e

xp(-

(alp

ha

*r_

1)*

*2))

en

dd

oe

nd

do

en

dd

od

oi_

po

ten

tial

=-n

_p

ote

ntia

l,n_

po

ten

tial

r_d

um

my

=i_

po

ten

tial*

2*p

i/n_

po

ten

tial

c_d

um

my

=r_

du

mm

y*(

0.,1

.)ta

b_

exp

_im

a(i_

po

ten

tial)

=e

xp(c

_d

um

my)

en

dd

o

!**

****

****

****

*f_

po

ten

tial

4e

lse

if(f

_p

ote

ntia

l.eq

.4)

the

nif

(i_

run

.eq

.1)

the

no

pe

n(1

0,

file

="p

imd

.inp

",st

atu

s="o

ld")

rea

d(1

0,2

10

)i_

du

mm

y,ch

ar_

du

mm

y

if(i_

du

mm

y.e

q.0

)th

en

!---

-ta

kede

fau

ltp

ara

me

ters

n_

rela

x=

50

n_

ob

ser

=5

0n

_sa

fe=

100

ise

ed

=4

75

91

+2

*in

t(n

_p

art

/2)+

2*i

nt(

10

*tk)

+n

_tr

ot

jse

ed

=is

eed

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

qe

qe

lect

ron

eg

(1)

=4

.16

8*1

16

04

.50

49

!e

V->

kBK

qe

qe

lect

ron

eg

(2)

=8

.74

1*1

16

04

.50

49

qe

qh

ard

ne

ss(1

)=

6.9

74

*11

60

4.5

04

9q

eq

ha

rdn

ess

(2)

=1

3.3

64

*11

60

4.5

04

9e

nd

if

n_

rela

x=

50

n_

ob

ser

=5

0n

_sa

fe=

10

0m

ass

(0,1

)=

4*m

ass

_a

mu

ma

ss(0

,2)

=m

ass

(0,1

)r_

cuto

ff(1

,1)

=1

0.

r_sk

in(1

,1)

=1

.1tk

=4

.5d

t=

0.5

e-2

f_p

ress

ure

=1

!--

-cr

it.is

ob

ar

of

He

3?

pre

ss=

.72

43

e-2

!--

-cr

it.is

ob

ar

of

He

lium

4p

ress

=.1

647

50

e-1

!--

-in

itia

liza

tion

of

fo_

sca

...c

or

ina

ziz_

loca

l

if(n

_o

rde

r.e

q.2

)th

en

!---

-ch

eck

en

erg

yco

nse

rva

tion

90


fric

tion

(1)

=0

.fr

ic_

bo

x=

0.

dt

=d

t/2

els

efr

ictio

n(1

)=

0.0

1/

dt

fric

_b

ox

=0

.01

/d

te

nd

iffr

ictio

n(2

)=

fric

tion

(1)

!---

de

fine

ine

rtia

for

bo

xd

efo

rma

tion

ma

ss_

bo

x=

(n_

silic

*ma

ss(0

,1)

+n

_o

xy*m

ass

(0,2

))/

12

ma

ss_

bo

x=

mass

_b

ox

/5

els

e!

i_d

um

my

if(i_

du

mm

y.ne

.f_

po

ten

tial)

&st

op

"f_

po

ten

tial

an

dp

imd

.inp

mis

ma

tch

"re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.ne

.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n

_re

lax

rea

d(1

0,2

10

)n

_o

bse

rre

ad

(10

,21

0)

n_

safe

rea

d(1

0,2

10

)is

ee

dre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

01

)d

tre

ad

(10

,20

1)

tkre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

02

)(m

ass

(0,i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe=

1,n

_ty

pe

)re

ad

(10

,20

2)

ma

ss_

bo

x,fr

ic_

bo

x

rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

01)

pre

ssre

ad

(10

,21

0)

f_p

ress

ure

rea

d(1

0,2

00)

cha

r_d

um

my

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

rea

d(1

0,2

02)

(qe

qe

lect

ron

eg

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

02)

(qe

qh

ard

ne

ss(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

en

dif

rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

00)

cha

r_d

um

my

do

i_ty

pe

=1,n

_ty

pe

rea

d(1

0,2

02)

(r_

cuto

ff(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(r_

skin

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

en

dif

!i_d

um

my

clo

se(1

0)

els

e!

i_ru

ntk

=tk

-0

.25

en

dif

!i_

run

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_cu

toff_

2(i_ty

pe

,j_ty

pe

)=

r_cu

toff(i_

typ

e,j_

typ

e)*

*2r_

ran

ge

(i_

type

,j_ty

pe

)=

r_cu

toff(i_

typ

e,j_

typ

e)

+&

r_sk

in(i_

typ

e,j_

typ

e)

r_ra

ng

e_

2(i_

typ

e,j_

typ

e)

=r_

ran

ge

(i_ty

pe

,j_ty

pe

)**2

en

dd

oe

nd

do

!---

de

fine

shift

sin

en

erg

y

i_ty

pe

=1

r_1

=r_

cuto

ff(1

,1)

call

azi

z_lo

cal(r_

1,d

_d

um

m1

)e

_sh

ift_

sho

rt(1

,1)

=d

_d

um

m1

do

i_p

ote

ntia

l=

5,

n_

po

ten

tial

r_1

=i_

po

ten

tial

*r_

ran

ge

(1,1

)/

n_

po

ten

tial

call

azi

z_lo

cal(r_

1,d

_d

um

m1

)r_

po

t(1

,1,i_

po

ten

tial)

=d

_d

um

m1

-e

_sh

ift_

sho

rt(1

,1)

r_1

=(i_

pote

ntia

l-1

)*

r_ra

ng

e(1

,1)

/n

_p

ote

ntia

lr_

2=

r_1

**2

call

azi

z_lo

cal(r_

1,d

_d

um

m1

)r_

1=

(i_

pote

ntia

l+1

)*

r_ra

ng

e(1

,1)

/n

_p

ote

ntia

lr_

2=

r_1

**2

call

azi

z_lo

cal(r_

1,d

_d

um

m2

)r_

po

t_d

(1,1

,i_p

ote

ntia

l)=

(d_

du

mm

2-d

_d

um

m1

)&

/(r

_ra

ng

e(1

,1)

/n

_p

ote

ntia

l)**

2/

(2*i

_p

ote

ntia

l)e

nd

do

!**

****

****

****

*f_

po

ten

tial

5+

6+

7

els

eif

((f_

po

ten

tial.g

e.5

).a

nd

.(f_

pote

ntia

l.le

.7))

the

n

if((

f_p

ote

ntia

l.eq

.5).

an

d.(

f_ch

arg

ead

just

.eq

.3))

the

nst

op

"MS

-Qp

ote

ntia

ld

oe

sn

ot

wo

rkw

ithE

xtL

ag

ran

ge

"e

nd

if

if((

f_p

ote

ntia

l.eq

.6).

an

d.(

f_ch

arg

ead

just

.eq

.4))

the

nst

op

"Ms-

fluc

Qp

ote

ntia

ld

oe

sn

ot

wo

rkw

ithe

xact

&d

iag

on

alis

atio

n"

en

dif

con

sta

nt_

cha

rge

(1)

=1

.3D

0co

nst

an

t_ch

arg

e(2

)=

-0.6

5D

0

if(i_

run

.eq

.1)

the

no

pe

n(1

0,

file=

"pim

d.in

p",

sta

tus=

"old

")re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.eq

.0)

the

n!-

---

take

de

fau

ltp

ara

me

ters

n_

rela

x=

50

n_

ob

ser

=5

0n

_sa

fe=

10

0is

ee

d=

47

59

1+

2*i

nt(

n_

pa

rt/2

)+2

*in

t(1

0*t

k)+

n_

tro

tjs

ee

d=

ise

ed

n_

rela

x=

30

n_

ob

ser

=30

n_

safe

=30

!---

-si

liciu

mm

ass

(0,1

)=

28

.1*m

ass

_a

mu

91

B. Programs!-

---

oxy

ge

nm

ass

(0,2

)=

16

.0*m

ass

_a

mu

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

qe

qe

lect

ron

eg(1

)=

4.1

68

*11

60

4.5

04

9!

eV

->kB

Kq

eq

ele

ctro

ne

g(2

)=

8.7

41

*11

60

4.5

04

9q

eq

ha

rdn

ess

(1)

=6

.97

4*1

16

04

.50

49

qe

qh

ard

ne

ss(2

)=

13

.36

4*1

16

04

.50

49

en

dif

d0

(1,1

)=

0.2

956

d0

(2,2

)=

0.5

363

d0

(1,2

)=

45

.997

!26

.37

65

d0

(2,1

)=

d0

(1,2

)m

sr0

(1,1

)=

3.4

10

3m

sr0

(2,2

)=

3.7

83

5m

sr0

(1,2

)=

1.6

14

8!1

.83

77

9m

sr0

(2,1

)=

msr

0(1

,2)

ga

mm

a(1

,1)

=1

1.7

13

9g

am

ma

(2,2

)=

10.4

11

2g

am

ma

(1,2

)=

8.8

02

2!8

.1g

am

ma

(2,1

)=

gam

ma

(1,2

)

msn

(1)=

3m

sn(2

)=2

msz

eta

(1,1

)=0

.14

88

10

e1

msz

eta

(1,2

)=0

.14

88

10

e1

msz

eta

(2,1

)=0

.18

68

46

e1

msz

eta

(2,2

)=0

.18

68

46

e1

alp

ha

=0

.3*

(10

02

./n

_p

art

)**(

1./6

)a

lph

a=

0.4

v_cu

t=

0.0

1*e

lec_

pre

fac*

(1.2

*2.4

)/1

.6/s

qrt

(1.*

n_

pa

rt)

v_cu

t=

0.

f_o

pt_

rec

=0

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_sk

in(i_

typ

e,j_

typ

e)

=0

.45

en

dd

oe

nd

do

tk=

36

00

./n

_tr

ot

!--

--tim

est

ep

=1

fsd

t=

1.3

09

1e

-4f_

pre

ssu

re=

1p

ress

=.0

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

qe

qm

ass

(1)

=0

.04

do

i_ty

pe

=2,

n_

typ

eq

eq

ma

ss(i_

typ

e)=

&q

eq

ma

ss(1

)*

sqrt

(qe

qh

ard

ne

ss(i_

typ

e)/

qe

qh

ard

ne

ss(1

))e

nd

do

qe

qfr

ictio

n=

0.1

/d

te

nd

if

if(n

_o

rde

r.e

q.2

)th

en

!---

-ch

eck

en

erg

yco

nse

rva

tion

fric

tion

(1)

=0

.fr

ic_

bo

x=

0.

dt

=d

t/2

els

e

fric

tion

(1)

=0

.01

/d

tfr

ic_

bo

x=

0.0

1/

dt

en

dif

fric

tion

(2)

=fr

ictio

n(1

)!-

--d

efin

ein

ert

iafo

rb

ox

de

form

atio

nm

ass

_b

ox

=(n

_si

lic*m

ass

(0,1

)+

n_

oxy

*ma

ss(0

,2))

/1

2m

ass

_b

ox

=m

ass

_b

ox

/8

els

e!

i_d

um

my

if(i_

du

mm

y.n

e.f_

po

ten

tial)

&st

op

"f_

po

ten

tial

an

dp

imd

.inp

mis

ma

tch

"re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.n

e.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n

_re

lax

rea

d(1

0,2

10

)n

_o

bse

rre

ad

(10

,21

0)

n_

safe

rea

d(1

0,2

10

)is

ee

dre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

01

)d

tre

ad

(10

,20

1)

tkre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

02

)(m

ass

(0,i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe=

1,n

_ty

pe

)re

ad

(10

,20

2)

ma

ss_

bo

x,fr

ic_

bo

x

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

pre

ssre

ad

(10

,21

0)

f_p

ress

ure

rea

d(1

0,2

10

)f_

op

t_re

cre

ad

(10

,20

0)

cha

r_d

um

my

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

rea

d(1

0,2

02

)(q

eq

ele

ctro

ne

g(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

02

)(q

eq

ha

rdn

ess

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

en

dif

rea

d(1

0,2

00

)ch

ar_

du

mm

y

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

rea

d(1

0,2

02

)(q

eq

ma

ss(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

01

)qe

qfr

ictio

ne

nd

ifre

ad

(10

,20

0)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(d0

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

ore

ad

(10

,20

0)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(msr

0(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00

)ch

ar_

du

mm

yd

oi_

typ

e=

1,n

_ty

pe

rea

d(1

0,2

02

)(g

am

ma

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00

)ch

ar_

du

mm

yd

oi_

typ

e=

1,n

_ty

pe

rea

d(1

0,2

02

)(m

sze

ta(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

92


rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

20)

(msn

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

02

)a

lph

a,

v_cu

t

rea

d(1

0,2

00)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(r_

skin

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

en

dif

!i_

du

mm

ycl

ose

(10

)e

lse

!i_ru

ntk

=tk

-0

.25

en

dif

!i_ru

n

!a

lph

a=

0.0

!clu

ste

ra

lph

a_

pre

fac

=2

*alp

ha

/sq

rt(p

i)r_

cuto

ff(1

,1)

=3

./a

lph

ar_

cuto

ff(2

,2)

=r_

cuto

ff(1

,1)

r_cu

toff(2

,1)

=r_

cuto

ff(1

,1)

r_cu

toff(1

,2)

=r_

cuto

ff(1

,1)

!if

(r_

cuto

ff(1

,1).

lt.r_

cut_

short

)!

&st

op

"cu

to

ffto

osm

all

for

sho

rtra

ng

e"

q_

cut_

2=

(6.*

alp

ha

)**2

!clu

ste

r!-

--a

ny

pea

k|f(k

|ˆ2

ino

rde

r1

/sq

rt(N

)g

ets

cut

off

r_cu

toff_

ma

x=

0.0

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_cu

toff_

max

=m

ax(

r_cu

toff_

ma

x,r_

cuto

ff(i_

typ

e,j_

typ

e))

en

dd

oe

nd

do

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_cu

toff_

2(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)*

*2r_

ran

ge

(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)

+&

r_sk

in(i_ty

pe

,j_ty

pe

)r_

ran

ge

_2

(i_

typ

e,j_

typ

e)

=r_

ran

ge

(i_

typ

e,j_

typ

e)*

*2e

nd

do

en

dd

o

!ch

arg

e(1

)=

0.

!1

.3fu

er

de

nn

eu

en

Ha

milt

on

ian

,w

ird

!ch

arg

e(2

)=

0.

!-0

.65

na

chh

er

wie

de

ru

eb

ers

chrieb

en

!a

dd

mo

rse

-str

etc

hte

rma

nd

cou

lom

bte

rmd

oi_

typ

e=

1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

sla

terc

uto

ff=

sla

terin

t(i_

typ

e,j_

typ

e,r

_cu

toff(i_

typ

e,j_

typ

e))

do

i_p

ote

ntia

l=1

,n

_p

ote

ntia

lr_

1=

i_p

ote

ntia

l*r_

ran

ge

(i_

typ

e,j_

type

)/n

_p

ote

ntia

l

sla

terijr

=sl

ate

rin

t(i_

typ

e,j_

typ

e,r

_1

)

!ta

bu

late

pote

ntia

l

!m

ors

est

retc

hte

rm

if(r

_1

.lt.r

_cu

t_sh

ort

)th

en

po

t_lo

c=

tra

ns_

un

its*d

0(i_

typ

e,j_

typ

e)*

&(e

xp(g

am

ma

(i_

typ

e,j_

typ

e)*

(1-r

_cu

t_sh

ort

/msr

0(i_

typ

e,j_

type

)))

&-2

*exp

(ga

mm

a(i_

typ

e,j_

typ

e)/

2&

*(1-r

_cu

t_sh

ort

/msr

0(i_

typ

e,j_

typ

e))

))r_

po

t_1

(i_

type

,j_ty

pe

,i_p

ote

ntia

l)=

-p

ot_

loc

&+

tra

ns_

un

its*d

0(i_

typ

e,j_

typ

e)*

&(e

xp(g

am

ma

(i_

typ

e,j_

typ

e)*

(1-r

_1

/msr

0(i_

typ

e,j_

typ

e))

)&

-2*e

xp(g

am

ma

(i_

typ

e,j_

typ

e)/

2&

*(1-r

_1

/msr

0(i_

typ

e,j_

typ

e))

))

if(f

_p

ote

ntia

l.eq

.6)

the

nr_

po

t_1

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=&

r_p

ot_

1(i_

typ

e,j_

typ

e,i_

po

ten

tial)

+&

con

sta

nt_

cha

rge

(i_

typ

e)*

con

sta

nt_

cha

rge

(j_

typ

e)

&*e

lec_

pre

fac*

((1

./r_

1-s

late

rijr)

&-(

1./r_

cuto

ff(i_

typ

e,j_

typ

e)-

sla

terc

uto

ff))

en

dif

els

e r_p

ot_

1(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=0.0

en

dif

!S

late

ro

rbita

lo

verla

p

r_p

ot_

2q

(i_ty

pe

,j_ty

pe

,i_p

ote

ntia

l)=

&e

lec_

pre

fac*

(sla

terijr-s

late

rcu

toff)

&-

ele

c_p

refa

c*(1

./r_

1-1

./r_

cuto

ff(i_

typ

e,j_

typ

e))

&+

erf

cc(a

lph

a*r

_1

)*!c

lust

er

&e

lec_

pre

fac/

r_1

-&

erf

cc(a

lph

a*r

_cu

toff(i_

typ

e,j_

typ

e))

*!c

lust

er

&e

lec_

pre

fac/

r_cu

toff(i_

typ

e,j_

typ

e)

if((

f_p

ote

ntia

l.eq

.6).

or.

(f_

po

ten

tial.e

q.7

))th

en

r_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=&

r_po

t_2

q(i_

typ

e,j_

typ

e,i_

pote

ntia

l)e

nd

if

if(f

_p

ote

ntia

l.eq

.5)

the

nr_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=&

erf

cc(a

lph

a*r

_1

)*!c

lust

er

&e

lec_

pre

fac/

r_1

-&

erf

cc(a

lph

a*r

_cu

toff(i_

typ

e,j_

typ

e))

*!c

lust

er

&e

lec_

pre

fac/

r_cu

toff(i_

typ

e,j_

typ

e)

en

dif

if(r

_1

.ge

.r_

cuto

ff(i_

typ

e,j_

typ

e))

the

nr_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)=

0.0

r_p

ot_

2q

(i_

type

,j_ty

pe

,i_p

ote

ntia

l)=

0.0

en

dif

en

dd

oe

nd

do

en

dd

o

!n

um

erica

lde

riva

tive

for

the

two

pote

ntia

lte

rms

do

i_p

ote

ntia

l=

2,

n_

po

ten

tial-1

do

i_ty

pe

=1

,n

_ty

pe

do

j_ty

pe

=1,

n_

typ

er_

1=

i_p

ote

ntia

l*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial

93

B. Programsr_

po

t_d

_1

(i_

typ

e,j_

typ

e,i_

po

ten

tial)=

1./r_

1*

&(r

_p

ot_

1(i_

typ

e,j_

typ

e,i_

po

ten

tial+

1)-

&r_

po

t_1

(i_

typ

e,j_

typ

e,i_

po

ten

tial-1

))/

&(2

.*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial)

r_p

ot_

d_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial)=

1./r_

1*

&(r

_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial+

1)-

&r_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial-1

))/

&(2

.*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial)

r_p

ot_

d_

2q

(i_

typ

e,j_

typ

e,i_

po

ten

tial)=

1./r_

1*

&(r

_p

ot_

2q

(i_

typ

e,j_

type

,i_p

ote

ntia

l+1

)-&

r_p

ot_

2q

(i_

typ

e,j_

typ

e,i_

po

ten

tial-1

))/

&(2

.*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial)

en

dd

oe

nd

do

en

dd

o

do

i_p

ote

ntia

l=

-n_

po

ten

tial,n

_p

ote

ntia

lr_

du

mm

y=

i_p

ote

ntia

l*2

*pi/n

_p

ote

ntia

lc_

du

mm

y=

r_d

um

my

*(0

.,1

.)ta

b_

exp

_im

a(i_p

ote

ntia

l)=

exp

(c_

du

mm

y)e

nd

do

!**

****

****

****

f_p

ote

ntia

l8

els

eif

(f_

po

ten

tial.e

q.8

)th

en

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

sto

p"V

ash

ish

tap

ote

ntia

ld

oe

sn

ot

wo

rkw

ithva

ria

ble

cha

rge

"e

nd

if

n_

an

g(1

)=n

_an

gu

lar

n_

an

g(2

)=1

if(i_

run

.eq.1

)th

en

op

en

(10

,fil

e=

"pim

d.in

p",

sta

tus=

"old

")re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

con

sta

nt_

charg

e(1

)=

1.6

D0

con

sta

nt_

charg

e(2

)=

-0.8

D0

if(i_

du

mm

y.e

q.0

)th

en

!---

-ta

ked

efa

ult

pa

ram

ete

rsn

_re

lax

=5

0n

_o

bse

r=

50

n_

safe

=1

00

ise

ed

=4

75

91

+2

*in

t(n

_p

art

/2)+

2*i

nt(

10*t

k)+

n_

tro

tjs

ee

d=

ise

ed

n_

rela

x=

30

n_

ob

ser

=3

0n

_sa

fe=

30

!---

-si

liciu

mm

ass

(0,1

)=

28.1

*ma

ss_

am

u!-

---

oxy

gen

ma

ss(0

,2)

=1

6.0

*ma

ss_

am

uva

sh_

h(1

,1)

=0

.05

7*e

lec_

pre

fac

vash

_h

(2,2

)=

51

.69

2*e

lec_

pre

fac

vash

_h

(2,1

)=

11

.38

7*e

lec_

pre

fac

vash

_h

(1,2

)=

vash

_h

(2,1

)va

sh_

eta

(1,1

)=

11

.d0

vash

_e

ta(2

,2)

=7

.d0

vash

_e

ta(2

,1)

=9

.d0

vash

_e

ta(1

,2)

=va

sh_

eta

(2,1

)

vash

_a

lph

a(1

)=

0.d

0*e

lec_

pre

fac

vash

_a

lph

a(2

)=

2.4

d0

*ele

c_p

refa

cva

sh_

B(1

)=

0.3

5va

sh_

B(2

)=

1.4

vash

_l(1

)=

1.0

vash

_l(2

)=

1.0

vash

_ct

he

ta(1

)=

cos(

pi*

10

9.4

7/1

80

.)va

sh_

cth

eta

(2)

=co

s(p

i*1

41

.00

/18

0.)

vash

_r0

(1)

=2

.6va

sh_

r0(2

)=

2.6

alp

ha

=0

.3*

(10

02

./n

_p

art

)**(

1./6

)a

lph

a=

0.3

v_cu

t=

0.0

1*e

lec_

pre

fac*

(1.2

*2.4

)/1

.6/s

qrt

(1.*

n_

pa

rt)

f_o

pt_

rec

=0

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_sk

in(i_

typ

e,j_

typ

e)

=0

.45

en

dd

oe

nd

do

tk=

36

00

./n_

tro

t!

----

time

ste

p=

1fs

dt

=1

.30

91

e-4

f_p

ress

ure

=1

pre

ss=

.0

if(n

_o

rde

r.e

q.2

)th

en

!---

-ch

eck

en

erg

yco

nse

rva

tion

fric

tion

(1)

=0

.fr

ic_

bo

x=

0.

dt

=d

t/2

els

efr

ictio

n(1

)=

0.0

1/

dt

fric

_b

ox

=0

.01

/d

te

nd

iffr

ictio

n(2

)=

fric

tion

(1)

!---

de

fine

ine

rtia

for

bo

xd

efo

rmatio

nm

ass

_b

ox

=(n

_si

lic*m

ass

(0,1

)+

n_

oxy

*ma

ss(0

,2))

/1

2m

ass

_b

ox

=m

ass

_b

ox

/8

els

e!

i_d

um

my

if(i_

du

mm

y.n

e.f_

po

ten

tial)

&st

op

"f_

po

ten

tial

an

dp

imd

.inp

mis

ma

tch

"re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.n

e.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n

_re

lax

rea

d(1

0,2

10

)n

_o

bse

rre

ad

(10

,21

0)

n_

safe

rea

d(1

0,2

10

)is

ee

dre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

01

)d

tre

ad

(10

,20

1)

tkre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

02

)(m

ass

(0,i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe=

1,n

_ty

pe

)re

ad

(10

,20

2)

ma

ss_

bo

x,fr

ic_

bo

x

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

pre

ssre

ad

(10

,21

0)

f_p

ress

ure

rea

d(1

0,2

10

)f_

op

t_re

c

94


rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

00)

cha

r_d

um

my

do

i_ty

pe

=1,n

_ty

pe

rea

d(1

0,2

02)

(va

sh_

h(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

rea

d(1

0,2

00)

cha

r_d

um

my

do

i_ty

pe

=1,n

_ty

pe

rea

d(1

0,2

02)

(va

sh_

eta

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

ore

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

02

)(v

ash

_a

lph

a(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

02

)(v

ash

_B

(i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

(va

sh_

l(i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

(va

sh_

cth

eta

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

02

)(v

ash

_r0

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e)

rea

d(1

0,2

00)

cha

r_d

um

my

rea

d(1

0,2

02

)a

lph

a,

v_cu

t

rea

d(1

0,2

00)

cha

r_d

um

my

do

i_ty

pe

=1

,n_

typ

ere

ad

(10

,20

2)

(r_

skin

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

en

dif

!i_

du

mm

ycl

ose

(10

)e

lse

!i_ru

ntk

=tk

-0

.25

en

dif

!i_ru

n

alp

ha

_p

refa

c=

2*a

lph

a/s

qrt

(pi)

r_cu

toff(1

,1)

=3

./a

lph

ar_

cuto

ff(2

,2)

=3

./a

lph

ar_

cuto

ff(2

,1)

=3

./a

lph

ar_

cuto

ff(1

,2)

=r_

cuto

ff(2

,1)

!if

(r_

cuto

ff(1

,1).

lt.r_

cut_

short

)!

&st

op

"cu

to

ffto

osm

all

for

sho

rtra

ng

e"

q_

cut_

2=

(6.*

alp

ha

)**2

!---

an

yp

ea

k|f(k

|ˆ2

ino

rde

r1

/sq

rt(N

)g

ets

cut

off

r_cu

toff_

ma

x=

0.0

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_cu

toff_

max

=m

ax(

r_cu

toff_

ma

x,r_

cuto

ff(i_

typ

e,j_

typ

e))

en

dd

oe

nd

do

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_cu

toff_

2(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)*

*2r_

ran

ge

(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)

+&

r_sk

in(i_ty

pe

,j_ty

pe

)r_

ran

ge

_2

(i_

typ

e,j_

typ

e)

=r_

ran

ge

(i_

typ

e,j_

typ

e)*

*2e

nd

do

en

dd

o

!---

de

fine

shift

sin

en

erg

y

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

!---

-e

lect

rost

atic

e_

shift

(i_

typ

e,j_

typ

e)

=e

rfcc

(alp

ha

*r_

cuto

ff(i_

typ

e,j_

typ

e))

*&

ele

c_p

refa

c/r_

cuto

ff(i_

typ

e,j_

typ

e)

po

t_lo

c=

r_cu

t_sh

ort

**(v

ash

_e

ta(i_ty

pe

,j_ty

pe

))p

ot_

loc

=va

sh_

h(i_

typ

e,j_

typ

e)/

po

t_lo

ce

xp_

po

t=

0.5

*(va

sh_

alp

ha

(i_

typ

e)*

con

sta

nt_

cha

rge

(j_

typ

e)*

*2&

+va

sh_

alp

ha

(j_

typ

e)*

con

sta

nt_

cha

rge

(i_

typ

e)*

*2)

exp

_p

ot

=exp

_p

ot/(r

_cu

t_sh

ort

**4

)e

xp_

po

t=

exp

_p

ot

*e

xp(-

r_cu

t_sh

ort

/4.4

3d

0)

e_

shift

_sh

ort

(i_

typ

e,j_

typ

e)

=p

ot_

loc

-e

xp_

po

te

nd

do

en

dd

o!-

---

tabu

late

po

ten

tial

v(r)

an

dd

v/d

rd

oi_

typ

e=

1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

do

i_p

ote

ntia

l=

1,n

_p

ote

ntia

lr_

1=

i_p

ote

ntia

l*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial

r_2

=r_

1**

2!-

----

-cu

toff

at

sho

rtra

ng

ein

tera

ctio

nif(

r_1

.le.r

_cu

t_sh

ort

)th

en

po

t_lo

c=

r_1

**(v

ash

_e

ta(i_

typ

e,j_

typ

e))

po

t_lo

c=

vash

_h

(i_

typ

e,j_

typ

e)/

po

t_lo

ce

xp_

po

t=

0.5

*(va

sh_

alp

ha

(i_

typ

e)*

con

sta

nt_

cha

rge

(j_

typ

e)*

*2&

+va

sh_

alp

ha

(j_

typ

e)*

con

sta

nt_

cha

rge

(i_

typ

e)*

*2)

exp

_p

ot

=e

xp_

po

t/(r

_1

**4

)e

xp_

po

t=

exp

_p

ot

*e

xp(-

r_1

/4.4

3d

0)

r_p

ot_

1(i_

type

,j_ty

pe

,i_p

ote

ntia

l)=

po

t_lo

c-

exp

_p

ot

&-e

_sh

ift_

sho

rt(i_

typ

e,j_

typ

e)

els

e r_p

ot_

1(i_

type

,j_ty

pe

,i_p

ote

ntia

l)=

0.d

0e

nd

if

if(r

_1

.le.r

_cu

toff(i_

typ

e,j_

typ

e))

the

nr_

po

t_2

(i_

type

,j_ty

pe

,i_p

ote

ntia

l)=

-e_

shift

(i_

typ

e,j_

typ

e)

&+

erf

cc(a

lph

a*r

_1

)*e

lec_

pre

fac/

r_1

els

e r_p

ot_

2(i_

type

,j_ty

pe

,i_p

ote

ntia

l)=

0.d

0e

nd

if

!r_

po

t_d

_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

=-(

1./r_

2)

*(v

_co

ulo

mb

+!

&e

lec_

pre

fac*

alp

ha

_p

refa

c*e

xp(-

(alp

ha

*r_

1)*

*2))

en

dd

oe

nd

do

en

dd

o

do

i_p

ote

ntia

l=

2,

n_

po

ten

tial-1

do

i_ty

pe

=1

,n

_ty

pe

do

j_ty

pe

=1

,n

_ty

pe

r_1

=i_

po

ten

tial*

r_ra

ng

e(i_

typ

e,j_

typ

e)/

n_

po

ten

tial

r_p

ot_

d_

1(i_

typ

e,j_

typ

e,i_

po

ten

tial)=

1./r_

1*

&(r

_p

ot_

1(i_

typ

e,j_

typ

e,i_

po

ten

tial+

1)-

&r_

po

t_1

(i_

typ

e,j_

typ

e,i_

po

ten

tial-1

))/

&(2

.*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial)

r_p

ot_

d_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial)=

1./r_

1*

&(r

_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial+

1)-

&r_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial-1

))/

&(2

.*r_

ran

ge

(i_

typ

e,j_

typ

e)/

n_

po

ten

tial)

en

dd

o

95

B. Programse

nd

do

en

dd

o

do

i_p

ote

ntia

l=

-n_

po

ten

tial,n

_p

ote

ntia

lr_

du

mm

y=

i_p

ote

ntia

l*2

*pi/n

_p

ote

ntia

lc_

du

mm

y=

r_d

um

my

*(0

.,1

.)ta

b_

exp

_im

a(i_

po

ten

tial)

=e

xp(c

_d

um

my)

en

dd

o

!**

****

****

****

f_p

ote

ntia

l9

(Sca

nd

olo

)

els

eif

(f_

po

ten

tial.e

q.9

)th

en

if(f

_ch

arg

ea

dju

st.n

e.1

)th

en

sto

p"S

can

do

lop

ote

ntia

ld

oe

sn

ot

wo

rkw

ithva

ria

ble

cha

rge

"e

nd

if

if(n

_tr

ot.ne

.1)

the

nst

op

"Sca

nd

olo

po

ten

tial

do

es

no

tw

ork

with

n_

tro

tte

r>

1"

en

dif

if(i_

run

.eq.1

)th

en

op

en

(10

,fil

e=

"pim

d.in

p",

sta

tus=

"old

")re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.e

q.0

)th

en

!---

-ta

ked

efa

ult

pa

ram

ete

rsn

_re

lax

=5

0n

_o

bse

r=

50

n_

safe

=1

00

ise

ed

=4

75

91

+2

*in

t(n

_p

art

/2)+

2*i

nt(

10

*tk)

+n

_tr

ot

jse

ed

=is

ee

d

n_

rela

x=

0n

_o

bse

r=

0n

_sa

fe=

0!-

---

silic

ium

ma

ss(0

,1)

=2

8.1

*ma

ss_

am

u!-

---

oxy

ge

nm

ass

(0,2

)=

16

.0*m

ass

_a

mu

alp

ha

=0

.3*

(10

02

./n

_p

art

)**(

1./6

)a

lph

a=

0.3

v_cu

t=

0.0

1*e

lec_

pre

fac*

(1.2

*2.4

)/1

.6/s

qrt

(1.*

n_

pa

rt)

f_o

pt_

rec

=0

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_sk

in(i_

typ

e,j_

typ

e)

=0

.45

en

dd

oe

nd

do

tk=

36

00

./n

_tr

ot

!--

--tim

est

ep

=1

fsd

t=

1.3

09

1e

-4f_

pre

ssu

re=

1p

ress

=.0

if(n

_o

rde

r.e

q.2

)th

en

!---

-ch

eck

en

erg

yco

nse

rva

tion

fric

tion

(1)

=0

.fr

ic_

bo

x=

0.

dt

=d

t/2

els

e

fric

tion

(1)

=0

.01

/d

tfr

ic_

bo

x=

0.0

1/

dt

en

dif

fric

tion

(2)

=fr

ictio

n(1

)!-

--d

efin

ein

ert

iafo

rb

ox

de

form

atio

nm

ass

_b

ox

=(n

_si

lic*m

ass

(0,1

)+

n_

oxy

*ma

ss(0

,2))

/1

2m

ass

_b

ox

=m

ass

_b

ox

/8

els

e!

i_d

um

my

if(i_

du

mm

y.n

e.f_

po

ten

tial)

&st

op

"f_

po

ten

tial

an

dp

imd

.inp

mis

ma

tch

"re

ad

(10

,21

0)

i_d

um

my,

cha

r_d

um

my

if(i_

du

mm

y.n

e.f_

cha

rge

ad

just

)&

sto

p"f

_ch

arg

ea

dju

sta

nd

pim

d.in

pm

ism

atc

h"

rea

d(1

0,2

10

)n

_re

lax

rea

d(1

0,2

10

)n

_o

bse

rre

ad

(10

,21

0)

n_

safe

rea

d(1

0,2

10

)is

ee

dre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

01

)d

tre

ad

(10

,20

1)

tkre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

02

)(m

ass

(0,i_

typ

e),

i_ty

pe

=1,n

_ty

pe

)re

ad

(10

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe=

1,n

_ty

pe

)re

ad

(10

,20

2)

ma

ss_

bo

x,fr

ic_

bo

x

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

1)

pre

ssre

ad

(10

,21

0)

f_p

ress

ure

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

00

)ch

ar_

du

mm

yre

ad

(10

,20

0)

cha

r_d

um

my

rea

d(1

0,2

02

)a

lph

a,

v_cu

t

rea

d(1

0,2

00

)ch

ar_

du

mm

yd

oi_

typ

e=

1,n

_ty

pe

rea

d(1

0,2

02

)(r

_sk

in(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e)

en

dd

o

en

dif

!i_

du

mm

ycl

ose

(10

)e

lse

!i_ru

ntk

=tk

-0

.25

en

dif

!i_ru

n

alp

ha

_p

refa

c=

2*a

lph

a/s

qrt

(pi)

r_cu

toff(1

,1)

=3

./a

lph

ar_

cuto

ff(2

,2)

=3

./a

lph

ar_

cuto

ff(2

,1)

=3

./a

lph

ar_

cuto

ff(1

,2)

=r_

cuto

ff(2

,1)

!if

(r_

cuto

ff(1

,1).

lt.r_

cut_

sho

rt)

!&

sto

p"c

ut

off

too

sma

llfo

rsh

ort

ran

ge

"q

_cu

t_2

=(6

.*a

lph

a)*

*2!-

--a

ny

pea

k|f(k

|ˆ2

ino

rde

r1

/sqrt

(N)

ge

tscu

to

ff

r_cu

toff_

ma

x=

0.0

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

r_cu

toff_

ma

x=

ma

x(r_

cuto

ff_

ma

x,r_

cuto

ff(i_

typ

e,j_

typ

e))

96


en

dd

oe

nd

do

do

i_ty

pe

=1,n

_ty

pe

do

j_ty

pe

=1,n

_ty

pe

r_cu

toff_

2(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_

typ

e,j_

typ

e)*

*2r_

ran

ge

(i_

typ

e,j_

typ

e)

=r_

cuto

ff(i_ty

pe

,j_ty

pe

)+

&r_

skin

(i_ty

pe

,j_ty

pe

)r_

ran

ge

_2

(i_

typ

e,j_

typ

e)

=r_

ran

ge

(i_

typ

e,j_

typ

e)*

*2e

nd

do

en

dd

o

n_

pb

c=1

do

i_d

im=

1,

n_

dim

if((

int(

r_cu

toff_

ma

x/sc

al_

0(i_

dim

))+

1).

gt.n

_p

bc)

the

nn

_p

bc=

(in

t(r_

cuto

ff_

ma

x/sc

al_

0(i_

dim

))+

1)

en

dif

en

dd

o

!p

ote

ntia

lis

calc

ula

ted

inp

au

lta

ngn

eys

rou

tine

els

e sto

P"f

_p

ote

ntia

lfla

gg

ma

kes

no

sen

se"

en

dif

!**

****

****

****

all

valu

es

f_p

ote

ntia

l

if(i_

run

.eq

.1)

ma

ss_

bo

x=

n_

tro

t*

ma

ss_

bo

x

jse

ed

=is

ee

dif

(f_

po

ten

tial.e

q.1

)f_

pre

ssu

re=

0

dt

=d

t/n

_tim

e_sc

al

dt_

2=

dt*

*2tk

p=

tk*n

_tr

ot

sqrt

p=

sqrt

(rea

l(n

_tr

ot)

)b

eta

=1

.d0

/tk

be

tap

=1

.d0

/tkp

do

i_ty

pe

=1

,n_

typ

ek_

qu

an

t(i_

type

)=

ma

ss(0

,i_ty

pe

)/b

eta

p**

2!-

---

sprin

gco

nst

an

tin

reci

pro

cal

spa

ced

oi_

tro

t=

0,n

_tr

ot-

1k_

eig

en

(i_

tro

t,i_

typ

e)

=4

*k_

qu

an

t(i_

typ

e)

&*s

in(i_

tro

t*p

i/n_

tro

t)**

2e

nd

do

v_th

erm

(0,i_

typ

e)

=sq

rt(t

kp/m

ass

(0,i_

typ

e))

w_

the

rm=

sqrt

(tkp

/ma

ss_

bo

x)!-

---

fictic

iuo

sm

ass

es

do

i_tr

ot

=1,n

_tr

ot-

1if(

f_p

ote

ntia

l.eq

.1)

the

nm

ass

(i_

tro

t,i_

typ

e)

=m

ass

(0,i_

typ

e)*

&(1

.+k_

eig

en

(i_

tro

t,i_

typ

e))

ma

ss(i_

tro

t,i_

typ

e)

=m

ass

(i_

tro

t,i_

typ

e)

/n

_tim

e_

sca

l**2

els

eif

(f_p

ote

ntia

l.eq

.2)

the

nm

ass

(i_

tro

t,i_

typ

e)

=m

ass

(0,i_

typ

e)*

&(2

56

*ep

sil(i_

typ

e,i_

typ

e)/

sig

ma

(i_

typ

e,i_

typ

e)*

*2&

+k_

eig

en(i_

tro

t,i_

typ

e))

/&

(25

6*e

psi

l(i_

typ

e,i_

typ

e)/

sig

ma

(i_

typ

e,i_

typ

e)*

*2)

ma

ss(i_

tro

t,i_

typ

e)

=m

ass

(i_

tro

t,i_

typ

e)

/n

_tim

e_

sca

l**2

els

eif

(f_

pote

ntia

l.eq

.3)

the

nm

ass

(i_

tro

t,i_

typ

e)

=m

ass

(0,i_

typ

e)*

&(1

.25

e6

+k_

eig

en

(i_

tro

t,i_

typ

e))

/1.2

5e

6m

ass

(i_

tro

t,i_

typ

e)

=m

ass

(i_

tro

t,i_

type

)/

n_

time

_sc

al*

*2e

lse

if(f

_p

ote

ntia

l.eq

.4)

the

nm

ass

(i_

tro

t,i_

typ

e)

=m

ass

(0,i_

typ

e)*

&(4

25

.73

3+

k_e

ige

n(i_

tro

t,i_

typ

e))

/4

25

.73

3m

ass

(i_

tro

t,i_

typ

e)

=m

ass

(i_

tro

t,i_

type

)/

n_

time

_sc

al*

*2e

lse

if((

f_po

ten

tial.g

e.5

).a

nd

.(f_

pote

ntia

l.le

.8))

the

nm

ass

(i_

tro

t,i_

typ

e)

=m

ass

(0,i_

typ

e)*

&(1

.25

e6

+k_

eig

en

(i_

tro

t,i_

typ

e))

/1

.25

e6

ma

ss(i_

tro

t,i_

typ

e)

=m

ass

(i_

tro

t,i_

type

)/

n_

time

_sc

al*

*2e

nd

ifv_

the

rm(i_

tro

t,i_

typ

e)

=sq

rt(t

kp/m

ass

(i_

tro

t,i_

typ

e))

en

dd

o!-

---

wid

tho

fG

au

ssia

nd

istr

ibu

ted

no

ise

do

i_tr

ot

=0

,n_

tro

t-1

rf_

wid

th(i_

tro

t,i_

typ

e)

=&

sqrt

(2*t

kp*m

ass

(i_

tro

t,i_

typ

e)*

fric

tion

(i_

typ

e)/

dt)

!---

-lin

ea

rb

lock

dis

trib

utio

nrf

_w

idth

(i_

tro

t,i_

typ

e)

=sq

rt3

*rf_

wid

th(i_

tro

t,i_

typ

e)

if(m

od

(2*i

_tr

ot,n

_tr

ot)

.ne

.0)

the

nrf

_w

idth

(i_

tro

t,i_

typ

e)

=rf

_w

idth

(i_

tro

t,i_

typ

e)/

sqrt

2e

nd

ife

nd

do

en

dd

orf

b_

wid

th=

sqrt

(2

*tkp

*ma

ss_

bo

x*fr

ic_

bo

x/d

t)

rfb

_w

idth

=sq

rt3

*rf

b_

wid

th

!---

initi

aliz

ep

red

icto

rco

effic

ien

tsd

oi_

ord

er

=0

,n_

ord

er_

ma

x-1

do

j_o

rde

r=

0,n

_o

rde

r_m

ax

pre

d_

coe

f(i_

ord

er,

j_o

rde

r)=

0.

en

dd

op

red

_co

ef(

0,i_

ord

er+

1)

=1

.if(

i_o

rde

r.g

e.1

)p

red

_co

ef(

1,i_

ord

er+

1)

=i_

ord

er+

1p

red

_co

ef(

i_o

rde

r,i_

ord

er)

=1

.e

nd

do

pre

d_

coe

f(2

,3)

=3

.p

red

_co

ef(

2,4

)=

6.

pre

d_

coe

f(2

,5)

=1

0.

pre

d_

coe

f(3

,4)

=4

.p

red

_co

ef(

3,5

)=

10

.p

red

_co

ef(

4,5

)=

5.

do

i_o

rde

r=

0,n

_o

rde

r-1

do

j_o

rde

r=

0,n

_o

rde

rp

red

ict_

coe

f(i_

ord

er,

j_o

rde

r)=

0.

en

dd

od

oj_

ord

er

=i_

ord

er,

n_

ord

er

pre

dic

t_co

ef(

i_o

rde

r,j_

ord

er)

=p

red

_co

ef(

i_o

rde

r,j_

ord

er)

en

dd

oe

nd

do

!---

initi

aliz

eco

rre

cto

rd

oi_

ord

er

=1

,n_

ord

er

corr

_co

ef(

i_o

rde

r)=

0.

en

dd

oi_

ord

er

=n

_o

rde

rif(

i_o

rde

r.eq

.2)

the

n!-

---

velo

city

Ve

rle

ta

lgo

rith

mco

rr_

coe

f(0

)=

0.

corr

_co

ef(

1)

=1

.

97

B. Programse

lse

if(i_

ord

er.

eq

.3)

the

nco

rr_

coe

f(0

)=

1./6

corr

_co

ef(

1)

=5

./6

corr

_co

ef(

3)

=1

./3

els

eif

(i_

ord

er.

eq

.4)

the

nco

rr_

coe

f(0

)=

19

./9

0!

corr

_co

ef(

0)

=1

9./1

20

on

lyif

the

rmo

sta

ta

bse

nt

corr

_co

ef(

1)

=3

./4

corr

_co

ef(

3)

=1

./2

corr

_co

ef(

4)

=1

./1

2e

lse

if(i_

ord

er.

eq

.5)

the

nco

rr_

coe

f(0

)=

3./1

6!

corr

_co

ef(

0)

=3

./2

0o

nly

ifth

erm

ost

at

ab

sen

tco

rr_

coe

f(1

)=

25

1./3

60

corr

_co

ef(

3)

=1

1./1

8co

rr_

coe

f(4

)=

1./6

corr

_co

ef(

5)

=1

./6

0e

lse

sto

P’O

rde

rfo

rp

red

icto

rco

rre

cto

ru

nava

ilab

le’

en

dif

corr

_co

ef(

2)

=1

.d

oi_

ord

er

=0

,n_

ord

er

corr

ect

_co

ef(

i_o

rde

r)=

corr

_co

ef(

i_o

rde

r)e

nd

do

if(i_

run

.eq

.1)

the

n!-

---

initi

aliz

era

nd

om

nu

mb

er

ge

nera

tor

kptr

=1

call

inr2

50

(mz,

ise

ed

,kp

tr)

if(n

_ru

n.g

t.1

)th

en

n_

rela

x=

1*n

_re

lax

en

dif

els

eif(

n_

rela

x.e

q.2

)th

en

n_

rela

x=

n_

rela

x/1

en

dif

20

0fo

rma

t(a

)2

10

form

at(

1i1

4,a

)2

20

form

at(

2i1

4,a

)2

01

form

at(

1e

14

.6,a

)2

02

form

at(

2e

14

.6,a

)2

03

form

at(

3e

14

.6,a

)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

do

ub

lep

reci

sio

nfu

nct

ion

sla

terin

t(i_

typ

e_

sl,j_

typ

e_

sl,r

_1

_sl

)im

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nr_

1_

slin

teg

er

i_ty

pe

_sl

,j_

typ

e_

sld

ou

ble

pre

cisi

on

ate

mp

,b

tem

pin

teg

er

bin

om

ial,

fact

oria

lin

teg

er

v,k,

ld

ou

ble

pre

cisi

on

sum

1,

sum

2,

sum

21

,su

m2

2

if(r

_1

_sl

.gt.1

e-8

)th

en

sum

1=

0d

ov

=0

,2

*msn

(i_

typ

e_

sl)-

1su

m1

=su

m1

+b

ino

mia

l(2

*msn

(i_

typ

e_

sl)-

1,v

)*&

ate

mp(2

*msn

(i_

typ

e_

sl)-

1-v

,1.d

0*

&m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)*

r_1

_sl

)*&

bte

mp

(v,1

.d0

*msz

eta

(i_

typ

e_sl

,j_ty

pe

_sl

)*r_

1_

sl)

en

dd

osu

m1

=(r

_1

_sl

/2)*

*(2

*msn

(i_

typ

e_

sl))

*sum

1su

m2

=0

do

v=

1,

2*m

sn(j_

typ

e_

sl)

sum

22

=0

do

k=

0,

2*m

sn(i_

typ

e_

sl)-

1su

m2

1=

0d

ol

=0

,2*m

sn(j_

typ

e_

sl)-

vsu

m2

1=

sum

21

+bin

om

ial(2

*msn

(j_

typ

e_

sl)-

v,l)*(

-1)*

*l*

&a

tem

p(2

*msn

(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

k-v-

l-1

,&

1.d

0*(

msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)

&+

msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

*r_

1_

sl)*

&b

tem

p(k

+l,1

.d0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&-m

sze

ta(j_

typ

e_

sl,i_

typ

e_

sl))

*r_

1_

sl)

en

dd

osu

m2

2=

sum

22+

sum

21

*bin

om

ial(2

*msn

(i_ty

pe

_sl

)-1

,k)

en

dd

osu

m2

2=

sum

22

r_d

um

my=

(r_

1_

sl/2

)**(

2*m

sn(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

v)r_

du

mm

y=r_

dum

my*

v*(2

*&

msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

**(2

*msn

(j_

typ

e_

sl)-

v)r_

du

mm

y=r_

du

mm

y/(f

act

oria

l(2

*msn

(j_

typ

e_

sl)-

v)*2

*msn

(j_

typ

e_

sl))

sum

2=

sum

2+

r_d

um

my*

sum

22

en

dd

o

r_d

um

my=

0.5

*(2*m

sze

ta(i_

typ

e_

sl,j_

typ

e_sl

))**

(2*m

sn(i_

typ

e_

sl)+

1)

r_d

um

my=

r_d

um

my/

fact

oria

l(2

*msn

(i_

typ

e_sl

))sl

ate

rin

t=r_

dum

my*

(su

m1

-su

m2

)

els

e!

spe

cia

lca

ser=

0

sum

2=

0

do

v=

1,

2*m

sn(j_

typ

e_

sl)

r_d

um

my=

(2.0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&+

msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

)**

&(2

*msn

(i_

type

_sl

)+2

*msn

(j_

typ

e_

sl)-

v)su

m2

1=

1.0

*fa

cto

ria

l(2

*msn

(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

v-1

)/&

r_d

um

my

sum

21

=su

m2

1*v

*((2

*msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

**&

(2*m

sn(j_

typ

e_

sl)-

v))/

&(f

act

oria

l(2

*msn

(j_

typ

e_

sl)-

v)*2

*msn

(j_

typ

e_

sl))

sum

2=

sum

2+

sum

21

en

dd

o

r_d

um

my=

(2.0

*msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl))

**(2

*msn

(i_

typ

e_

sl)+

1)

r_d

um

my=

r_d

um

my*

1.0

/(fa

cto

ria

l(2

*msn

(i_ty

pe

_sl

)))

sla

terin

t=-r

_du

mm

y*su

m2

+m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)/

msn

(i_

typ

e_sl

)

en

dif

en

dfu

nct

ion

!---

----

----

----

----

----

----

----

----

----

-

do

ub

lep

reci

sio

nfu

nct

ion

sla

terin

t_d

(i_

typ

e_

sl,j_

typ

e_

sl,r

_1

_sl

)im

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

98


do

ub

lep

reci

sion

r_1

_sl

inte

ge

ri_

typ

e_sl

,j_

typ

e_

sld

ou

ble

pre

cisi

on

ate

mp

,b

tem

pin

teg

er

bin

om

ial,

fact

oria

lin

teg

er

v,k,

ld

ou

ble

pre

cisi

on

sum

1,

sum

2,

sum

21

,su

m2

2,

sum

21

1

sum

1=

0d

ov

=0

,2

*msn

(i_

typ

e_

sl)-

1su

m1

=su

m1

+b

ino

mia

l(2

*msn

(i_

typ

e_

sl)-

1,v

)*&

(m

sn(i_

typ

e_

sl)*

&a

tem

p(2

*msn

(i_

typ

e_

sl)-

1-v

,1.d

0&

*msz

eta

(i_

typ

e_

sl,j_

typ

e_sl

)*r_

1_

sl)*

&b

tem

p(v

,1.d

0*m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)*

r_1

_sl

)&

-r_

1_

sl/2

*msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)*

&a

tem

p(2

*msn

(i_

typ

e_

sl)-

v,1

.d0

&*m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)*

r_1

_sl

)*&

bte

mp

(v,1

.d0

*msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)*

r_1

_sl

)&

-r_

1_

sl/2

*msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)*

&a

tem

p(2

*msn

(i_

typ

e_

sl)-

1-v

,&

1.d

0*m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)*

r_1

_sl

)*&

bte

mp

(v+

1,1

.d0

*msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)*

r_1

_sl

))

en

dd

osu

m1

=(r

_1

_sl

/2)*

*(2

*msn

(i_

typ

e_

sl)-

1)*

sum

1su

m2

=0

do

v=

1,

2*m

sn(j_

typ

e_

sl)

sum

22

=0

do

k=

0,

2*m

sn(i_

typ

e_

sl)-

1su

m2

1=

0d

ol

=0

,2

*msn

(j_

typ

e_

sl)-

vsu

m2

11

=(m

sn(i_

typ

e_

sl)+

msn

(j_

typ

e_

sl)-

0.5

*v)*

&a

tem

p(2

*msn

(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

k-v-

l-1,

&1

.d0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&+

msz

eta

(j_ty

pe

_sl

,i_ty

pe

_sl

))*r

_1

_sl

)*&

bte

mp

(k+

l,&

1.d

0*(

msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)

&-m

sze

ta(j_

typ

e_

sl,i_

typ

e_

sl))

*r_

1_

sl)

sum

21

1=

sum

211

-.5

d0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&+

msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

*r_

1_

sl*

&a

tem

p(2

*msn

(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

k-v-

l,&

1.d

0*(

msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)

&+

msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

*r_

1_

sl)*

&b

tem

p(k

+l,

&1

.d0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&-m

sze

ta(j_

typ

e_

sl,i_

typ

e_

sl))

*r_1

_sl

)su

m2

11

=su

m2

11

-.5

d0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&-m

sze

ta(j_

typ

e_

sl,i_

typ

e_

sl))

*r_

1_

sl*

&a

tem

p(2

*msn

(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

k-v-

l-1,

&1

.d0

*(m

sze

ta(i_

typ

e_

sl,j_

typ

e_

sl)

&+

msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

*r_1

_sl

)*&

bte

mp

(k+

l+1

,&

1.d

0*(

msz

eta

(i_

typ

e_

sl,j_

typ

e_

sl)

&-m

sze

ta(j_

typ

e_

sl,i_

typ

e_

sl))

*r_1

_sl

)su

m2

1=

sum

21

+b

ino

mia

l(2

*msn

(j_

typ

e_

sl)-

v,l)*(

-1)*

*l&

*su

m2

11

en

dd

osu

m2

2=

sum

22

+su

m2

1*b

ino

mia

l(2

*msn

(i_

typ

e_

sl)-

1,k

)e

nd

do

sum

22

=su

m2

2r_

du

mm

y=(r

_1

_sl

/2)*

*(2

*msn

(i_

typ

e_

sl)+

2*m

sn(j_

typ

e_

sl)-

v-1

)r_

du

mm

y=r_

du

mm

y*v*

&(2

*msz

eta

(j_

typ

e_

sl,i_

typ

e_

sl))

**(2

*msn

(j_

typ

e_sl

)-v)

r_d

um

my=

r_d

um

my/

(fa

cto

ria

l(2

*msn

(j_

typ

e_

sl)-

v)*2

*msn

(j_

typ

e_

sl))

sum

2=

sum

2+

r_d

um

my*

sum

22

en

dd

o

r_d

um

my=

0.5

*(2

*msz

eta

(i_

typ

e_

sl,j_

type

_sl

))**

(2*m

sn(i_

typ

e_

sl)+

1)

r_d

um

my=

r_d

um

my/

fact

oria

l(2

*msn

(i_

type

_sl

))sl

ate

rin

t_d

=r_

du

mm

y*(s

um

1-s

um

2)

en

dfu

nct

ion

!---

----

----

----

----

----

----

----

----

----

-

inte

ge

rfu

nct

ion

fact

oria

l(k)

imp

licit

no

ne

inte

ge

rk

inte

ge

ri,f

ac

fac=

1d

oi=

1,

kfa

c=fa

c*i

en

dd

ofa

cto

ria

l=fa

ce

nd

fun

ctio

n

!---

----

----

----

----

----

----

----

----

----

-

inte

ge

rfu

nct

ion

bin

om

ial(n

,l)im

plic

itn

on

ein

teg

er

fact

oria

l

inte

ge

rn

,lin

teg

er

de

no

m,n

um

,id

en

om

=1

nu

m=

1d

oi=

1,l

de

no

m=

de

no

m*i

nu

m=

nu

m*(

n-i+

1)

en

dd

ob

ino

mia

l=n

um

/de

no

me

nd

fun

ctio

n

!---

----

----

----

----

----

----

----

----

----

-

do

ub

lep

reci

sio

nfu

nct

ion

ate

mp

(k,x

)im

plic

itn

on

ein

teg

er

fact

oria

l

inte

ge

rk

do

ub

lep

reci

sio

nx

do

ub

lep

reci

sio

nsu

m1

inte

ge

rv

sum

1=

0d

ov=

0,

ksu

m1

=su

m1

+x*

*v/fa

cto

ria

l(v)

en

dd

oa

tem

p=

exp

(-x)

*fa

cto

ria

l(k)

/(x*

*(k+

1))

*su

m1

en

dfu

nct

ion

!---

----

----

----

----

----

----

----

----

----

-

do

ub

lep

reci

sio

nfu

nct

ion

bte

mp

(k,x

)im

plic

itn

on

ein

teg

er

fact

oria

l,k,

vd

ou

ble

pre

cisi

on

ate

mp

,x,

sum

1,

sum

1b

,a

t,fr

ac

99

B. Programsif

(ab

s(x)

.gt.1

e-9

)th

en

sum

1=

0.d

0if

(mo

d(k

,2).

eq.0

)th

en

fra

c=1

.d0

do

v=0

,1

00

00

sum

1b

=2

.d0

*fra

c/(k

+(2

.d0

*v)+

1.d

0)

fra

c=fr

ac*

(x**

2)/

((2

.d0

*v+

1.d

0)*

(2.d

0*v

+2

.d0

))su

m1

=su

m1

+su

m1

be

nd

do

els

e fra

c=-x

do

v=0

,1

00

00

sum

1b

=2

.d0

*fra

c/(k

+(2

.d0

*v+

1.d

0)+

1.d

0)

fra

c=fr

ac*

(x**

2)/

((2

.d0

*v+

2.d

0)*

(2.d

0*v

+3

.d0

))su

m1

=su

m1

+su

m1

be

nd

do

en

dif

bte

mp

=su

m1

els

e bte

mp

=1

.d0

*(1

-(-1

)**(

k+1

))/(

k+1

.d0

)e

nd

if

en

dfu

nct

ion

!!

!!

!!

!!

!!

!

sub

rou

tine

azi

z_lo

cal(r,

azi

z)im

plic

itn

one

!---

Le

nn

ard

Jon

es

as

che

ck!

rea

lr

!re

al

ep

s,si

g!

do

ub

lep

reci

sio

nv_

po

t!

ep

s=

10

.22

!si

g=

2.5

56

!v_

po

t=

4*e

ps

*(s

ig/r

)**6

*((

sig

/r)*

*6-1

.)!-

--tr

ue

azi

zp

ote

ntia

ld

ou

ble

pre

cisi

on

azi

zd

ou

ble

pre

cisi

on

r,x,

ep

s,rm

,d

,pre

fac

inte

ge

ri_

coe

f,n

_co

ef

pa

ram

ete

r(n

_co

ef=

3)

do

ub

lep

reci

sio

na

,a

lph

a,

be

ta,

c(0

:n_

coe

f-1

)e

ps

=1

0.9

48

rm=

2.9

63

d=

1.4

82

6a

=1

.84

43

101

e5

alp

ha

=1

0.4

33

29

5b

eta

=-2

.279

65

1c(

0)

=1

.36

74

45

21

c(1

)=

0.4

212

38

07

c(2

)=

0.1

747

33

18

x=

r/

rma

ziz

=0

.d

oi_

coe

f=

0,

n_

coe

f-1

azi

z=

azi

z+

c(i_

coe

f)/

x**(

2*i

_co

ef+

6)

en

dd

oif

(x.lt

.d)

the

np

refa

c=

exp

(-(d

/x-1

.)**

2)

els

ep

refa

c=

1.

en

dif

azi

z=

-p

refa

c*

azi

za

ziz

=a

ziz

+a

*e

xp(-

alp

ha

*x+

be

ta*x

**2

)a

ziz

=e

ps

*azi

z!

66

6fo

r_sc

a_

mfa

_co

r=

4*p

i*

ep

s*c(

0)

*rm

**6

/3

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

conf_

init

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

inte

ge

rd

i_tr

ot

do

ub

lep

reci

sio

nx_

qu

an

t(2

*n_

tro

t),f_

qu

an

t(2

*n_

tro

t)co

mm

on

/xf_

qu

an

t/x_

qu

an

t,f_

qu

an

t

op

en

(10

,fil

e=

"co

nf.st

a",

sta

tus=

"old

")re

ad

(10

,*)

i_du

mm

yif

(i_

du

mm

y.e

q.0

)th

en

call

con

f_d

efa

ult

els

ere

ad

(10

,*)

i_tr

ot

if(i_

tro

t.eq

.n_

tro

t)th

en

di_

tro

t=

1e

lse

if(i_

tro

t.e

q.2

*n_

tro

t)th

en

di_

tro

t=

1e

lse

if(2

*i_

tro

t.e

q.n

_tr

ot)

the

nd

i_tr

ot

=2

els

ed

i_tr

ot

=1

sto

P"t

rotte

rn

um

be

rin

com

pa

btib

le"

en

dif

rea

d(1

0,1

03

)(s

cal_

0(i_

dim

),i_

dim

=1

,n_

dim

+3

)re

ad

(10

,10

3)

(sca

l_x(

1,i_

dim

),i_

dim

=1

,n_

dim

+3

)d

oi_

tro

t=

0,

n_

tro

t-1

,d

i_tr

ot

do

i_p

art

=1

,n_

pa

rtre

ad

(10

,14

4)

typ

e(i_

pa

rt),

&(p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt),

i_d

im=

1,n

_d

im),

&(r

0(i_

dim

,i_tr

ot,i_

pa

rt),

i_d

im=

1,n

_d

im),

&ch

arg

e(i_

pa

rt)

if(d

i_tr

ot.e

q.2

)th

en

do

i_d

im=

1,

n_

dim

pb

c0(i_

dim

,i_tr

ot+

1,i_

pa

rt)

=p

bc0

(i_d

im,i_

tro

t,i_

pa

rt)

r0(i_

dim

,i_tr

ot+

1,i_

pa

rt)

=r0

(i_

dim

,i_tr

ot,i_

pa

rt)

en

dd

oe

nd

ife

nd

do

en

dd

o!-

--se

tp

bc

ba

ckto

sma

lln

um

be

rsd

oi_

pa

rt=

1,n

_p

art

do

i_d

im=

1,

n_

dim

do

i_tr

ot

=n

_tr

ot-

1,

0,

-1!

pb

c0(i_

dim

,i_tr

ot,i_

pa

rt)

=p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt)

!&

-pb

c0(i_

dim

,0,i_

pa

rt)

!d

on

td

oth

at,

we

like

big

ge

rn

um

be

rse

nd

do

en

dd

oe

nd

do

do

i_d

im=

1,

n_

dim

+e

_d

imsc

al_

x(1

,i_d

im)

=sc

al_

x(1

,i_d

im)*

dt

en

dd

o

100


if(e

_d

im.e

q.0

)th

en

do

i_d

im=

1,

n_

dim

sca

l_x(

1,n

_d

im+

i_d

im)

=0

.e

nd

do

en

dif

en

dif

!---

-se

tB

KS

cha

rge

sb

ack

tod

efa

ult

if(f

_p

ote

ntia

l.eq

.3)

the

nd

oi_

pa

rt=

1,

n_

pa

rtif

(typ

e(i_

pa

rt).

eq

.1)

the

nch

arg

e(i_

pa

rt)

=2

.4e

lse ch

arg

e(i_

pa

rt)

=-1

.2e

nd

ife

nd

do

en

dif

!---

-se

tS

can

do

loch

arg

es

ba

ckto

de

fau

ltif

(f_

po

ten

tial.e

q.9

)th

en

do

i_p

art

=1

,n

_p

art

if(t

ype

(i_

pa

rt).

eq

.1)

the

nch

arg

e(i_

pa

rt)

=2

.76

51

4e

lse ch

arg

e(i_

pa

rt)

=-1

.38

25

7e

nd

ife

nd

do

en

dif

!---

-ca

lcu

late

de

fau

ltin

tera

ctio

nra

ng

ea

nd

skin

if(f

_p

ress

ure

.eq

.0)

the

nd

oi_

dim

=1

,n

_d

im+

e_

dim

sca

l_x(

1,i_

dim

)=

0.

en

dd

oe

nd

if

n_

ne

igh

_m

ax

=0

!---

de

fine

refe

ren

tp

oin

tsfo

rp

art

icle

sd

oi_

pa

rt=

1,n

_p

art

do

i_d

im=

1,n

_d

imr_

ref(

i_d

im,i_

pa

rt)

=r0

(i_

dim

,0,i_

part

)+&

1.

*p

bc0

(i_

dim

,0,i_

pa

rt)

en

dd

oe

nd

do

!--

initi

aliz

eu

0fie

ldd

oi_

pa

rt=

1,

n_

pa

rtd

oi_

dim

=1

,n

_d

imd

oi_

tro

t=

0,

n_

tro

t-1

x_q

ua

nt(

2*i

_tr

ot+

1)

=r0

(i_

dim

,i_tr

ot,i_

pa

rt)

+&

1.

*p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r_

ref(

i_d

im,i_

pa

rt)

x_q

ua

nt(

2*i

_tr

ot+

1)

=x_

qu

an

t(2

*i_

tro

t+1

)/

n_

tro

t*

sqrt

px_

qu

an

t(2

*i_

tro

t+2

)=

0.

en

dd

o!

---

tra

nsf

orm

x_q

ua

nt

into

rec.

space

call

fou

r1(x

_q

ua

nt,n

_tr

ot,1

)d

oi_

tro

t=

1,

2*n

_tr

ot

u0

(i_

dim

,i_tr

ot,i_

pa

rt)

=x_

qu

an

t(i_

tro

t)e

nd

do

en

dd

oe

nd

do

!---

-in

itia

lize

volu

me

an

dve

loci

ties

call

sca

l_0

_u

pd

ate

call

vel_

de

fau

lt(i_

du

mm

y)ca

llca

lc_

rv_

rea

l

if(f

_p

ote

ntia

l.ne

.1)

call

pb

_co

nd

itca

lltr

an

s_re

cip

if(f

_p

ote

ntia

l.ne

.1)

call

bin

nin

g

do

i_p

art

=1

,n_

pa

rtd

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

dim

=1,n

_d

imfo

rce

(i_

dim

,i_tr

ot,i_

pa

rt)

=0

.d0

en

dd

oe

nd

do

en

dd

o

if(f_

po

ten

tial.e

q.1

)th

en

call

ha

rm_

osc

els

eca

llsc

al_

0_

up

da

teif

(f_

po

ten

tial.e

q.2

)th

en

call

len

na

rd_jo

ne

se

lse

if(f

_p

ote

ntia

l.eq

.3)

the

nse

lf_e

ne

rgy

=0

.d

oi_

pa

rt=

1,

n_

pa

rtse

lf_e

ne

rgy

=se

lf_e

ne

rgy

+ch

arg

e(i_

pa

rt)*

*2e

nd

do

self_

en

erg

y=

-n_

tro

t*e

lec_

pre

fac*

alp

ha

*se

lf_e

ne

rgy/

sqrt

(pi)

call

bks

_p

ote

ntia

le

lse

if(f

_p

ote

ntia

l.eq

.4)

the

nca

llb

ks_

po

ten

tial

els

eif

((f_

po

ten

tial.g

e.5

).a

nd

.(f_

po

ten

tial.l

e.8

))th

en

!h

ier

ae

nde

rn?

self_

en

erg

y=

0.

do

i_p

art

=1,

n_

pa

rtse

lf_e

ne

rgy

=se

lf_e

ne

rgy

+ch

arg

e(i_

pa

rt)*

*2e

nd

do

self_

en

erg

y=

-n_

tro

t*e

lec_

pre

fac*

alp

ha

*se

lf_e

ne

rgy/

sqrt

(pi)

call

go

dd

ard

_p

ote

ntia

le

nd

ife

nd

ifca

lltr

an

s_re

cip

if(f

_p

ote

ntia

l.ne

.1)

call

bo

x_ch

ain

call

the

rmo

sta

t

!---

-co

nve

rtfo

rce

sin

toa

cce

lera

tion

sd

oi_

dim

=1

,n_

dim

do

i_tr

ot

=0

,n_

tro

t-1

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

ux(

2,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

=&

f0(i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)/

ma

ss(i_

tro

t,i_

typ

e)*

dt_

2u

x(2

,i_d

im,2

*i_tr

ot+

2,i_

pa

rt)

=&

f0(i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)/

ma

ss(i_

tro

t,i_

typ

e)*

dt_

2e

nd

do

en

dd

oe

nd

do

101

B. Programs!-

---

bo

xsh

ap

ed

oi_

dim

=1,n

_d

im+

e_

dim

sca

l_x(

2,i_

dim

)=

forc

e_

sca

l(i_

dim

)/

ma

ss_

bo

x*

dt_

2/

2e

nd

do

call

con

f_sa

veif

(f_

po

ten

tial.e

q.3

)ca

llin

it_b

on

ds

if((

f_p

ote

ntia

l.ge

.5).

an

d.(

f_p

ote

ntia

l.le

.9))

call

init_

bo

nds

!---

-co

un

ta

tom

sp

er

typ

ed

oi_

typ

e=

1,

n_

typ

en

_a

tom

typ

e(i_

typ

e)

=0

en

dd

od

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

n_

ato

mty

pe

(i_

typ

e)

=n

_a

tom

typ

e(i_

typ

e)

+1

en

dd

o

10

3fo

rma

t(3f1

3.5

)1

44

form

at(

4i5

,4f1

4.6

)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

init_

bo

nd

sim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nr_

2

r_n

eig

h_

min

=1

.0r_

ne

igh

_m

ax

=2

.3d

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

i_d

um

my

=1

,4

i_n

eig

h_

ori(i_

pa

rt,i_

du

mm

y)=

0e

nd

do

i_d

um

my

=0

do

j_p

art

=1

,n

_p

art

j_ty

pe

=ty

pe(j_

pa

rt)

if(i_

typ

e.n

e.j_

typ

e)

the

nca

llca

lc_

dis

tan

ce(r

_2

,0,i_

pa

rt,j_

pa

rt)

r_d

um

my

=sq

rt(r

_2

)if

(r_

du

mm

y.lt.

r_n

eig

h_

ma

x)th

en

!--

--ne

igh

bo

rfo

un

di_

du

mm

y=

i_du

mm

y+

1if

(i_

du

mm

y.le

.4)

the

ni_

ne

igh

_o

ri(i_

pa

rt,i_

du

mm

y)=

j_p

art

els

ew

rite

(*,*

)"m

ore

tha

n4

ne

are

stn

eig

hb

ors

"e

nd

ife

nd

ife

nd

ife

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

sca

l_0

_u

pd

ate

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nve

c_d

um

my(

n_

dim

+3

),m

at_

du

mm

y(n

_d

im,n

_d

im),

&q

_2

_m

ind

ou

ble

pre

cisi

on

r_m

at(

n_

dim

,n_

dim

)

call

tra

ns_

to_te

ns(

sca

l_0

,h_

ma

t)ca

llsq

ua

re_

mat(

h_

ma

t,h

_m

at_

2)

call

tra

ns_

to_vo

igt(

h_

ma

t_2

,me

t_te

n_

voi)

volu

me

=sc

al_

0(1

)*sc

al_

0(2

)*sc

al_

0(3

)if

(e_

dim

.eq

.3)

the

nvo

lum

e=

volu

me

+&

2*s

cal_

0(4

)*sc

al_

0(5

)*sc

al_

0(6

)-

sca

l_0

(4)*

*2*s

cal_

0(3

)&

-sc

al_

0(5

)**2

*sca

l_0

(1)

-sc

al_

0(6

)**2

*sca

l_0

(2)

en

dif

do

i_d

im=

1,

n_

dim

+3

vec_

du

mm

y(i_

dim

)=

sca

l_x(

1,i_

dim

)e

nd

do

call

tra

ns_

to_te

ns(

vec_

du

mm

y,h

_m

at_

do

t)ca

llm

at_

mu

l(h_

ma

t,h

_m

at_

do

t,m

at_

du

mm

y)d

oi_

dim

=1

,n

_d

imd

oj_

dim

=1

,n

_d

imh

_m

at_

2d

(i_

dim

,j_d

im)

=m

at_

du

mm

y(i_

dim

,j_d

im)

en

dd

oe

nd

do

call

ma

t_m

ul(h_

ma

t_d

ot,h

_m

at,m

at_

du

mm

y)d

oi_

dim

=1

,n

_d

imd

oj_

dim

=1

,n

_d

imh

_m

at_

2d

(i_

dim

,j_d

im)

=h

_m

at_

2d

(i_d

im,j_

dim

)+

&m

at_

du

mm

y(i_

dim

,j_d

im)

h_

ma

t_2

d(i_

dim

,j_d

im)

=h

_m

at_

2d

(i_

dim

,j_d

im)

/1

en

dd

oe

nd

do

call

inv_

ma

t(h_

ma

t_2

,h_

ma

t_2

_in

v)ca

llm

at_

mu

l(h_

ma

t_2

_in

v,h

_m

at_

2d

,g_

mat_

do

t)ca

lltr

an

s_to

_vo

igt(

g_

ma

t_d

ot,g

_vo

i_d

ot)

call

inv_

ma

t(h_

ma

t,h

_m

at_

inv)

if(f

_p

ote

ntia

l.eq

.3)

the

n!-

--ch

an

ge:

cuto

ffcr

iterio

nfo

rre

cve

cto

rd

oi_

dim

=1,

n_

dim

do

j_d

im=

1,

n_

dim

dq

(i_

dim

,j_dim

)=

two

pi

*h

_m

at_

inv(

i_d

im,j_

dim

)e

nd

do

en

dd

oca

llja

cob

i(n

_d

im,

dq

,ro

t_m

at,

eig

en

,1

00

)q

_2

_m

in=

eig

en

(1,1

)**2

do

i_d

im=

2,

n_

dim

r_d

um

my

=e

igen

(i_

dim

,i_d

im)*

*2if

(r_

du

mm

y.lt.

q_

2_

min

)q

_2

_m

in=

r_d

um

my

en

dd

od

oi_

dim

=1,

n_

dim

nq

_m

ax(

i_d

im)

=in

t(sq

rt(q

_cu

t_2

/q_

2_

min

))e

nd

do

en

dif

if((

f_p

ote

ntia

l.ge

.5).

an

d.(

f_p

ote

ntia

l.le

.8))

the

n!

hie

ra

en

de

rn?

!---

cha

ng

e:

cuto

ffcr

iterio

nfo

rre

cve

cto

rd

oi_

dim

=1,

n_

dim

102


do

j_d

im=

1,

n_

dim

dq

(i_

dim

,j_d

im)

=tw

op

i*

h_

ma

t_in

v(i_

dim

,j_d

im)

en

dd

oe

nd

do

call

jaco

bi(n_

dim

,d

q,

rot_

ma

t,e

ige

n,

10

0)

q_

2_

min

=e

ige

n(1

,1)*

*2d

oi_

dim

=2

,n

_d

imr_

du

mm

y=

eig

en

(i_

dim

,i_d

im)*

*2if

(r_

du

mm

y.lt.

q_

2_

min

)q

_2

_m

in=

r_d

um

my

en

dd

od

oi_

dim

=1

,n

_d

imn

q_

ma

x(i_

dim

)=

int(

sqrt

(q_

cut_

2/q

_2

_m

in))

en

dd

oe

nd

if

!---

see

wh

eth

er

min

ima

ge

con

v.h

old

sif

(f_

pre

ssu

re.e

q.1

)th

en

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

r_m

at(

i_d

im,j_

dim

)=

h_

ma

t(i_

dim

,j_d

im)

en

dd

oe

nd

do

if(f

_p

ote

ntia

l.gt.1

)th

en

call

jaco

bi(n

_d

im,

r_m

at,

rot_

ma

t,e

ige

n,

10

0)

do

i_d

im=

1,

n_

dim

!if(

ab

s(e

ige

n(i_

dim

,i_d

im))

.lt.

2*r

_cu

toff_

ma

x)!

&st

op

"cu

toff

rad

ius

gro

win

gto

ost

ron

gly

"e

nd

do

en

dif

en

dif

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

inv_

ma

t(m

at,m

at_

inv)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,si

gn

do

ub

lep

reci

sion

ma

t(3

,3),

ma

t_in

v(3

,3),

de

t_m

at

do

ub

lep

reci

sion

de

term

de

t_m

at

=d

ete

rm(m

at)

do

i_d

im=

1,

3d

oj_

dim

=1

,3

ma

t_in

v(i_

dim

,j_d

im)

=0

.d0

en

dd

oe

nd

do

ma

t_in

v(1

,1)

=m

at(

2,2

)*m

at(

3,3

)-

ma

t(2

,3)*

*2m

at_

inv(

2,2

)=

ma

t(3

,3)*

ma

t(1

,1)

-m

at(

3,1

)**2

ma

t_in

v(3

,3)

=m

at(

1,1

)*m

at(

2,2

)-

ma

t(1

,2)*

*2m

at_

inv(

1,2

)=

-(m

at(

2,1

)*m

at(

3,3

)-

ma

t(2

,3)*

ma

t(3

,1))

ma

t_in

v(2

,3)

=-

(ma

t(3

,2)*

ma

t(1

,1)

-m

at(

3,1

)*m

at(

1,2

))m

at_

inv(

1,3

)=

+(m

at(

1,2

)*m

at(

2,3

)-

ma

t(1

,3)*

ma

t(2

,2))

do

i_d

im=

1,

3d

oj_

dim

=i_

dim

,3

ma

t_in

v(i_

dim

,j_d

im)

=m

at_

inv(

i_d

im,j_

dim

)/

de

t_m

at

ma

t_in

v(j_

dim

,i_d

im)

=m

at_

inv(

i_d

im,j_

dim

)e

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

do

ub

lep

reci

sio

nfu

nct

ion

de

term

(ma

t)im

plic

itn

one

do

ub

lep

reci

sio

nm

at(

3,3

)d

ete

rm=

ma

t(1

,1)*

ma

t(2

,2)*

ma

t(3

,3)

+2

*ma

t(1

,2)*

ma

t(2

,3)*

ma

t(3

,1)

&-m

at(

1,1

)*m

at(

2,3

)**2

-ma

t(2

,2)*

ma

t(3,1

)**2

-ma

t(3

,3)*

ma

t(1

,2)*

*2e

nd

fun

ctio

n

!!

!!

!!

!!

!!

sub

rou

tine

ma

t_m

ul(m

at1

,ma

t2,m

at3

)im

plic

itn

one

inte

ge

ri_

dim

,j_

dim

,k_

dim

do

ub

lep

reci

sio

nm

at1

(3,3

),m

at2

(3,3

),m

at3

(3,3

)ca

llze

ro_

mat(

ma

t3)

do

i_d

im=

1,

3d

oj_

dim

=1,

3d

ok_

dim

=1,

3m

at3

(i_

dim

,j_d

im)

=m

at3

(i_

dim

,j_d

im)

+&

ma

t1(i_

dim

,k_

dim

)*

ma

t2(k

_d

im,j_

dim

)e

nd

do

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

zero

_m

at(

ma

trix

)im

plic

itn

one

inte

ge

ri_

dim

,j_d

imd

ou

ble

pre

cisi

on

ma

trix

(3,3

)

do

i_d

im=

1,3

do

j_d

im=

1,3

ma

trix

(j_

dim

,i_d

im)

=0

.e

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

to_

ten

s(ve

cto

r,m

atr

ix)

imp

licit

no

ne

do

ub

lep

reci

sio

nm

atr

ix(3

,3),

vect

or(

6)

ma

trix

(1,1

)=

vect

or(

1)

ma

trix

(2,2

)=

vect

or(

2)

ma

trix

(3,3

)=

vect

or(

3)

ma

trix

(1,2

)=

vect

or(

4)

ma

trix

(2,1

)=

vect

or(

4)

ma

trix

(2,3

)=

vect

or(

5)

ma

trix

(3,2

)=

vect

or(

5)

ma

trix

(3,1

)=

vect

or(

6)

ma

trix

(1,3

)=

vect

or(

6)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

103

B. Programssu

bro

utin

esq

ua

re_

ma

t(m

at,m

at_

2)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,k_

dim

do

ub

lep

reci

sio

nm

at(

3,3

),m

at_

2(3

,3)

do

i_d

im=

1,

3d

oj_

dim

=1,

3m

at_

2(i_

dim

,j_d

im)

=0

.e

nd

do

en

dd

od

oi_

dim

=1,

3d

oj_

dim

=1,

3d

ok_

dim

=1,

3m

at_

2(i_

dim

,j_d

im)

=m

at_

2(i_

dim

,j_d

im)

+&

ma

t(k_

dim

,i_d

im)

*m

at(

k_d

im,j_

dim

)e

nd

do

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

to_

voig

t(m

atr

ix,v

ect

or)

imp

licit

no

ne

do

ub

lep

reci

sio

nm

atr

ix(3

,3),

vect

or(

6)

vect

or(

1)

=m

atr

ix(1

,1)

vect

or(

2)

=m

atr

ix(2

,2)

vect

or(

3)

=m

atr

ix(3

,3)

vect

or(

4)

=2

*m

atr

ix(1

,2)

vect

or(

5)

=2

*m

atr

ix(2

,3)

vect

or(

6)

=2

*m

atr

ix(3

,1)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

con

f_d

efa

ult

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nr_

2d

ou

ble

pre

cisi

on

r_d

um

my_

1,r

_d

um

my_

2,r

_d

um

my_

3in

teg

er

i_b

in_

x,i_

bin

_y,

i_b

in_

z

do

i_d

im=

1,

n_

dim

+3

if(i_

dim

.le.n

_d

im)

the

nsc

al_

0(i_

dim

)=

1.

els

esc

al_

0(i_

dim

)=

0.

en

dif

do

i_o

rde

r=

1,n

_o

rde

rsc

al_

x(i_

ord

er,

i_d

im)

=0

.e

nd

do

en

dd

o

if(f

_p

ote

ntia

l.eq

.1)

the

n!-

---

ha

rmo

nic

osc

illa

tor

do

i_p

art

=1

,n_

pa

rtty

pe

(i_

pa

rt)

=1

do

i_d

im=

1,n

_d

imr0

(i_

dim

,0,i_

part

)=

0.

en

dd

oe

nd

do

els

eif(

f_sy

mm

etr

y.e

q.1

)th

en

!---

-fc

csy

mm

etr

y

if(f

_p

ote

ntia

l.eq

.2)

the

n!-

---

thre

ep

art

icle

sin

on

eb

ox

at

de

nsi

ty1

.05

(fcc

)r_

du

mm

y=

(4.d

0/1

.05

)**(

1.d

0/3

)!-

---

cla

sica

lg

rou

nd

sta

teif

LJ

ne

xtn

eig

hb

or

inte

ract

ion

!r_

dum

my

=1

.58

74

01

els

eif

(f_

po

ten

tial.e

q.3

)th

en

!---

-a

pp

rop

ria

ted

ista

nce

for

SiO

2"m

elt"

r_d

um

my

=(4

.d0

/0.0

69

)**(

1./3

)e

lse

if(f

_po

ten

tial.e

q.4

)th

en

!---

-H

eliu

m3

at

criti

cal

de

nsi

tyr_

du

mm

y=

(4.d

0*7

.26

)**(

1./3

)*

2.5

56

!---

-H

eliu

m4

at

crit.

de

nsi

tyr_

du

mm

y=

(4.d

0*9

6.1

4)*

*(1

./3

)e

lse

if((

f_p

ote

ntia

l.ge

.5).

an

d.(

f_po

ten

tial.l

e.9

))th

en

!h

ier

ae

nd

ern

?!-

---

ap

pro

pria

ted

ista

nce

for

SiO

2"m

elt"

r_d

um

my

=(4

.d0

/0.0

69

)**(

1./3

)e

nd

ifsc

al_

0(1

)=

n_

bin

_x*

r_d

um

my

sca

l_0

(2)

=n

_b

in_

y*r_

du

mm

ysc

al_

0(3

)=

n_

bin

_z*

r_d

um

my

!---

-sc

ale

bo

xle

ng

tho

nrig

ht

len

gth

sca

leif

(f_

po

ten

tial.e

q.2

)th

en

do

i_d

im=

1,n

_d

imsc

al_

0(i_

dim

)=

sig

ma

(1,1

)*

sca

l_0

(i_

dim

)e

nd

do

en

dif

i_p

art

=0

do

i_b

in_

x=

0,n

_b

in_

x-1

do

i_b

in_

y=

0,n

_b

in_

y-1

do

i_b

in_

z=

0,n

_b

in_

z-1

!---

-se

tfir

sta

tom

on

ed

ge

of

bo

xi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=(r

ea

l(i_

bin

_x)

+0

.0)/

n_b

in_

xr0

(2,0

,i_p

art

)=

(re

al(i_

bin

_y)

+0

.0)/

n_b

in_

yr0

(3,0

,i_p

art

)=

(re

al(i_

bin

_z)

+0

.0)/

n_b

in_

z

!---

-se

tne

xtth

ree

ato

ms

on

face

dia

go

na

li_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-1)

+0

.5/n

_b

in_

xr0

(2,0

,i_p

art

)=

r0(2

,0,i_

pa

rt-1

)+

0.5

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-1)

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-2)

+0

.5/n

_b

in_

xr0

(2,0

,i_p

art

)=

r0(2

,0,i_

pa

rt-2

)r0

(3,0

,i_p

art

)=

r0(3

,0,i_

pa

rt-2

)+

0.5

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-3)

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-3)

+0

.5/n

_b

in_

yr0

(3,0

,i_p

art

)=

r0(3

,0,i_

pa

rt-3

)+

0.5

/n_

bin

_z

en

dd

oe

nd

do

104


en

dd

o

!---

-m

ake

eve

ryth

ird

pa

rtic

leS

iif

BK

Sp

ote

ntia

lif

(f_

po

ten

tial.e

q.3

)th

en

do

i_p

art

=1

,n_

pa

rtif(

mo

d(i_

pa

rt,3

).e

q.1

)th

en

typ

e(i_

pa

rt)

=1

els

ety

pe

(i_

pa

rt)

=2

en

dif

en

dd

oe

nd

if

!h

ier

ae

nd

ern

?!-

---

ma

kee

very

third

pa

rtic

leS

iif

BK

Sp

ote

ntia

lif

((f_

po

ten

tial.g

e.5

).a

nd

.(f_

po

ten

tial.l

e.8

))th

en

do

i_p

art

=1

,n_

pa

rtif(

mo

d(i_

pa

rt,3

).e

q.1

)th

en

typ

e(i_

pa

rt)

=1

els

ety

pe

(i_

pa

rt)

=2

en

dif

en

dd

oe

nd

if

els

eif(

(f_

sym

me

try.

eq

.2).

or.

(f_

sym

me

try.

eq

.3))

the

n!-

---

trid

ymite

:f_

sym

me

try

=2

!---

-cr

isto

ba

lit:

f_sy

mm

etr

y=

3r_

du

mm

y=

3.1

4sc

al_

0(1

)=

r_d

um

my*

sqrt

(8./3

)*n

_b

in_x

sca

l_0

(2)

=r_

du

mm

y*sq

rt(8

.)*n

_b

in_

yif(

f_sy

mm

etr

y.e

q.2

)th

en

sca

l_0

(3)

=8

.*r_

du

mm

y/3

*n_

bin

_z

els

esc

al_

0(3

)=

12

.*r_

du

mm

y/3

*n_

bin

_z

en

dif

r_d

um

my_

1=

r_d

um

my/

sca

l_0

(1)

r_d

um

my_

2=

r_d

um

my/

sca

l_0

(2)

r_d

um

my_

3=

r_d

um

my/

sca

l_0

(3)

i_p

art

=0

!---

-se

tu

pS

ia

tom

so

nW

urt

zit

stru

ctu

re

do

i_b

in_

x=

0,n

_b

in_

x-1

do

i_b

in_

y=

0,n

_b

in_

y-1

do

i_b

in_

z=

0,n

_b

in_

z-1

!---

---

Ad

ou

ble

laye

r

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.0

01

r0(2

,0,i_

part

)=

i_b

in_

y*1

./n

_b

in_

y+

0.0

01

r0(3

,0,i_

part

)=

i_b

in_

z*1

./n

_b

in_

z+

0.0

01

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

part

)=

r0(1

,0,i_

pa

rt-1

)r0

(2,0

,i_p

art

)=

r0(2

,0,i_

pa

rt-1

)r0

(3,0

,i_p

art

)=

r0(3

,0,i_

pa

rt-1

)+r_

du

mm

y/sc

al_

0(3

)

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-2

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-2

)+sq

rt(3

./4

)*sq

rt(8

./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-2

)

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-2

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-2

)+sq

rt(3

./4

)*sq

rt(8

./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-2

)

!---

-B

do

ub

lela

yer

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-4

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-4

)+sq

rt(1

./1

2)*

sqrt

(8./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-4

)+(1

.+1./3

)*r_

du

mm

y_3

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-1)

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-1)

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-1)+

r_du

mm

y_3

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-2

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-2

)+sq

rt(3

./4

)*sq

rt(8

./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-2

)

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-2

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-2

)+sq

rt(3

./4

)*sq

rt(8

./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-2

)

!---

-C

do

ub

lela

yer

if(f_

sym

me

try.

eq

.3)

the

ni_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-4

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-4

)+sq

rt(1

./1

2)*

sqrt

(8./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-4

)+(1

.+1./3

)*r_

du

mm

y_3

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-1)

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-1)

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-1)+

r_du

mm

y_3

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-2

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

r0(2

,0,i_

pa

rt)=

r0(2

,0,i_

pa

rt-2

)+sq

rt(3

./4

)*sq

rt(8

./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-2

)

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)=

r0(1

,0,i_

pa

rt-2

)+0

.5*s

qrt

(8./3

)*r_

du

mm

y_1

105

B. Programsr0

(2,0

,i_p

art

)=r0

(2,0

,i_p

art

-2)+

sqrt

(3./4

)*sq

rt(8

./3

)*r_

du

mm

y_2

r0(3

,0,i_

pa

rt)=

r0(3

,0,i_

pa

rt-2

)e

nd

if

en

dd

oe

nd

do

en

dd

o

els

eif(

f_sy

mm

etr

y.e

q.4

)th

en

r_d

um

my

=3

.14

sca

l_0

(1)

=4

*r_

du

mm

y/sq

rt(3

.)*n

_b

in_

xsc

al_

0(2

)=

4*r

_d

um

my/

sqrt

(3.)

*n_

bin

_y

sca

l_0

(3)

=4

*r_

du

mm

y/sq

rt(3

.)*n

_b

in_

z

i_p

art

=0

.d

oi_

bin

_x

=0,n

_b

in_

x-1

do

i_b

in_

y=

0,n

_b

in_

y-1

do

i_b

in_

z=

0,n

_b

in_

z-1

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

zi_

pa

rt=

i_pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-1)

+0.5

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-1)

+0.5

/n_

bin

_x

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-1)

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-2)

+0.5

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-2)

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-2)

+0.5

/n_

bin

_x

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-3)

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-3)

+0.5

/n_

bin

_x

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-3)

+0.5

/n_

bin

_x

do

j_p

art

=1

,4i_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=r0

(1,0

,i_p

art

-4)

+0

.25

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=r0

(2,0

,i_p

art

-4)

+0

.25

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=r0

(3,0

,i_p

art

-4)

+0

.25

/n_

bin

_z

en

dd

oe

nd

do

en

dd

oe

nd

do

els

eif(

f_sy

mm

etr

y.e

q.6

)th

en

!---

-b

eta

qu

arz

sca

l_0

(1)

=4

.99

77

*n_

bin

_x*

1.0

24

39

sca

l_0

(2)

=8

.65

63

*n_

bin

_y*

1.0

24

39

sca

l_0

(3)

=5

.46

01

*n_

bin

_z*

1.0

24

39

i_p

art

=0

do

i_b

in_

x=

0,n

_b

in_

x-1

do

i_b

in_

y=

0,n

_b

in_

y-1

do

i_b

in_

z=

0,n

_b

in_

z-1

i_p

art

=i_

pa

rt+

1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

(1.*

0)/

(1*n

_b

in_

x)r0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

(1.*

0)/

(1*n

_b

in_

y)r0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

(1.*

0)/

(1*n

_b

in_

z)i_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

(1.*

1)/

(4*n

_b

in_

x)r0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

(1.*

3)/

(4*n

_b

in_

y)r0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

(1.*

1)/

(3*n

_b

in_

z)i_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

(1.*

1)/

(4*n

_b

in_

x)r0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

(1.*

1)/

(4*n

_b

in_

y)r0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

(1.*

2)/

(3*n

_b

in_

z)i_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

(1.*

1)/

(2*n

_b

in_

x)r0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

(1.*

1)/

(2*n

_b

in_

y)r0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

(1.*

0)/

(1*n

_b

in_

z)i_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

(1.*

3)/

(4*n

_b

in_

x)r0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

(1.*

3)/

(4*n

_b

in_

y)r0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

(1.*

2)/

(3*n

_b

in_

z)i_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

(1.*

3)/

(4*n

_b

in_

x)r0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

(1.*

1)/

(4*n

_b

in_

y)r0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

(1.*

1)/

(3*n

_b

in_

z)

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0./n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.7

07

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.5

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0./n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.2

92

8/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.5

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.1

89

22

65

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.8

96

4/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.1

66

66

67

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.1

89

22

65

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.1

03

6/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.8

33

33

33

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.3

10

84

35

4/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.6

03

6/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.1

66

66

66

7/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.3

10

84

35

4/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.3

96

4/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.8

33

33

33

3/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

106


r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.5

/n_

bin

_x

r0(2

,0,i_

part

)=

i_b

in_

y*1

./n

_b

in_

y+

0.7

92

8/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.5

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.5

/n_

bin

_x

r0(2

,0,i_

part

)=

i_b

in_

y*1

./n

_b

in_

y+

0.2

07

2/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.5

/n_

bin

_z

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.6

89

29

65

5/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.6

03

6/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.8

33

33

33

3/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.6

89

29

65

5/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.3

96

4/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.1

66

66

66

7/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.8

10

91

35

8/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.8

96

4/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.8

33

33

33

3/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.8

10

91

35

8/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.1

03

6/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.1

66

66

66

7/n

_b

in_

z

en

dd

oe

nd

do

en

dd

oe

lse

if(f_

sym

metr

y.e

q.7

)th

en

!---

-a

lph

aq

ua

rz

sca

l_0

(1)

=4.9

14

85

*n_

bin

_x

sca

l_0

(2)

=8.5

12

77

*n_

bin

_y

sca

l_0

(3)

=5.4

06

29

*n_

bin

_z

i_p

art

=0

do

i_b

in_

x=

0,n

_b

in_

x-1

do

i_b

in_

y=

0,n

_b

in_

y-1

do

i_b

in_

z=

0,n

_b

in_

z-1

i_p

art

=i_

pa

rt+

1ty

pe

(i_

pa

rt)

=1

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0./n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0./n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0./n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.2

97

7/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.7

65

9/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.6

66

66

7/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.2

97

7/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.2

34

1/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.3

33

33

3/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

part

)=

i_b

in_

x*1

./n

_b

in_

x+

0.5

/n_

bin

_x

r0(2

,0,i_

part

)=

i_b

in_

y*1

./n

_b

in_

y+

0.5

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0./n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.7

97

7/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.7

34

1/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.3

33

33

3/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=1

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.7

97

7/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.2

65

9/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.6

66

66

67

/n_

bin

_z

i_p

art

=i_

part

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.0

91

35

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.7

06

55

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.4

53

56

7/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.0

91

35

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.2

93

45

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.5

46

43

36

9/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.1

92

2/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.9

26

5/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.7

86

9/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.1

92

2/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.0

73

5/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.2

13

1/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.3

11

85

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.3

66

95

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.1

20

23

36

9/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.3

11

85

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.6

33

05

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.8

79

76

7/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.5

91

35

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.2

06

55

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.4

53

56

7/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.5

91

35

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.7

93

45

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.5

46

43

36

9/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.6

92

2/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.5

73

5/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.2

13

1/n

_b

in_

zi_

pa

rt=

i_p

art

+1

typ

e(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.6

92

2/n

_b

in_

xr0

(2,0

,i_p

art

)=

i_b

in_

y*1

./n

_b

in_

y+

0.4

26

5/n

_b

in_

yr0

(3,0

,i_p

art

)=

i_b

in_

z*1

./n

_b

in_

z+

0.7

86

9/n

_b

in_

z

107

B. Programsi_

pa

rt=

i_pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.8

11

85

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.8

66

95

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.1

20

23

36

9/n

_b

in_

zi_

pa

rt=

i_pa

rt+

1ty

pe

(i_

pa

rt)

=2

r0(1

,0,i_

pa

rt)

=i_

bin

_x*

1./n

_b

in_

x+

0.8

11

85

/n_

bin

_x

r0(2

,0,i_

pa

rt)

=i_

bin

_y*

1./n

_b

in_

y+

0.1

33

05

/n_

bin

_y

r0(3

,0,i_

pa

rt)

=i_

bin

_z*

1./n

_b

in_

z+

0.8

79

76

7/n

_b

in_

ze

nd

do

en

dd

oe

nd

do

en

dif

if((f

_sy

mm

etr

y.g

e.2

).a

nd

.(f_

sym

me

try.

lt.6

))th

en

call

sca

l_0

_u

pd

ate

call

pb

_co

nd

it!-

---

pu

to

xyg

en

so

nb

on

ds

i_d

um

my

=i_

part

do

i_p

art

=1

,n_

silic

-1d

oj_

pa

rt=

i_p

art

+1

,n_

silic

call

calc

_d

ista

nce

(r_

2,0

,i_p

art

,j_p

art

)if(

r_2

.le.(

1.0

01

*r_

du

mm

y)**

2)

the

n!-

---

oxy

ge

nfo

un

di_

du

mm

y=

i_d

um

my

+1

typ

e(i_

du

mm

y)=

2d

oi_

dim

=1

,n_d

imr0

(i_

dim

,0,i_

du

mm

y)=

r0(i_

dim

,0,i_

part

)-

&d

elta

_ce

ll(i_

dim

)/

2e

nd

do

en

dif

en

dd

oe

nd

do

en

dif

!---

-g

en

era

teco

llap

sed

po

lym

ers

and

set

pe

rio

dic

bo

un

da

ries

toze

rod

oi_

pa

rt=

1,n

_p

art

do

i_tr

ot

=0

,n_

tro

t-1

do

i_d

im=

1,n

_d

imr0

(i_

dim

,i_tr

ot,i_

pa

rt)

=r0

(i_

dim

,0,i_

pa

rt)

pb

c0(i_

dim

,i_tr

ot,i_

pa

rt)

=0

en

dd

oe

nd

do

en

dd

o

!---

-se

tall

velo

citie

sa

nd

hig

he

rd

eriva

tive

sto

zero

do

i_p

art

=1

,n_

pa

rtd

oi_

tro

t=

1,2

*n_

tro

td

oi_

dim

=1,n

_d

imd

oi_

ord

er

=1

,n_

ord

er

ux(

i_o

rde

r,i_

dim

,i_tr

ot,i_

pa

rt)

=0

.e

nd

do

en

dd

oe

nd

do

en

dd

o

!---

-se

tde

fau

ltch

arg

es

acc

ord

ing

toty

pe

if(f

_p

ote

ntia

l.eq

.3)

the

nd

oi_

pa

rt=

1,n

_p

art

if(ty

pe

(i_

pa

rt).

eq

.1)

the

nch

arg

e(i_

pa

rt)

=2

.4e

lse

if(ty

pe

(i_p

art

).e

q.2

)th

en

cha

rge

(i_

pa

rt)

=-1

.2e

nd

ife

nd

do

els

eif

((f_

pote

ntia

l.ge

.5).

an

d.(

f_p

ote

ntia

l.le

.7))

the

nd

oi_

pa

rt=

1,n

_p

art

if(ty

pe

(i_

pa

rt).

eq

.1)

the

nch

arg

e(i_

pa

rt)

=1

.3e

lse

if(ty

pe

(i_p

art

).e

q.2

)th

en

cha

rge

(i_

pa

rt)

=-1

.3/2

.e

nd

ife

nd

do

els

eif

(f_

pote

ntia

l.eq

.8)

the

nd

oi_

pa

rt=

1,n

_p

art

if(ty

pe

(i_

pa

rt).

eq

.1)

the

nch

arg

e(i_

pa

rt)

=1

.6e

lse

if(ty

pe

(i_

pa

rt).

eq

.2)

the

nch

arg

e(i_

pa

rt)

=-1

.6/2

.e

nd

ife

nd

do

els

e do

i_p

art

=1

,n_

pa

rtif(

typ

e(i_

pa

rt).

eq

.1)

the

nch

arg

e(i_

pa

rt)

=2

.4e

lse

if(ty

pe

(i_

pa

rt).

eq

.2)

the

nch

arg

e(i_

pa

rt)

=-1

.2e

nd

ife

nd

do

en

dif

call

calc

_rv

_re

al

call

con

f_sa

ve

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

pb_

con

dit

imp

licit

no

ne

incl

ud

e"p

imd.c

om

.f"

!---

-m

ove

pa

rtic

les

into

bo

xd

oi_

pa

rt=

1,n

_p

art

do

i_tr

ot

=0

,n_

tro

t-1

do

i_d

im=

1,n

_d

imif(

r0(i_

dim

,i_tr

ot,i_

pa

rt).

gt.1

.)th

en

r0(i_

dim

,i_tr

ot,i_

pa

rt)

=r0

(i_

dim

,i_tr

ot,i_

pa

rt)-

1.

pb

c0(i_

dim

,i_tr

ot,i_

pa

rt)

=p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt)

+1

els

eif(

r0(i_

dim

,i_tr

ot,i_

pa

rt).

lt.(0

.d0

))th

en

r0(i_

dim

,i_tr

ot,i_

pa

rt)

=r0

(i_

dim

,i_tr

ot,i_

pa

rt)+

1.

pb

c0(i_

dim

,i_tr

ot,i_

pa

rt)

=p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt)

-1

en

dif

en

dd

oe

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

108


sub

rou

tine

vel_

de

fau

lt(i_

du

mm

y)im

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

i_tr

ot

=0,n

_tr

ot-

1!-

---

Ga

uss

ian

wid

thr_

du

mm

y=

sqrt

(tkp

/ma

ss(i_

tro

t,i_

typ

e))

*dt

!---

-lin

ear

bo

xd

istr

ibu

tion

r_d

um

my

=sq

rt3

*r_

du

mm

y!-

---

take

twic

eki

ne

tice

ne

rgy

at

t=

0if

(i_

du

mm

y.e

q.0

)r_

du

mm

y=

sqrt

2*r

_d

um

my

if(m

od

(2*i

_tr

ot,n

_tr

ot)

.ne

.0)

r_d

um

my

=r_

du

mm

y/sq

rt2

call

r25

0(m

z,ra

nd

om

,n_

dim

,n_

dim

,kp

tr)

!---

-te

rmin

itve

loci

ties

do

i_d

im=

1,

n_

dim

ux(

1,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

=0

.d

oj_

dim

=1

,n

_d

imu

x(1

,i_d

im,2

*i_

tro

t+1

,i_p

art

)=

ux(

1,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

&+

h_

ma

t_in

v(i_

dim

,j_d

im)

*(2

*ran

do

m(j_

dim

)-1

.)*

r_d

um

my

en

dd

oe

nd

do

en

dd

o!-

---

sym

me

triz

ed

oi_

tro

t=

1,n

_tr

ot/2

-1d

oi_

dim

=1

,n_

dim

ux(

1,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

=&

ux(

1,i_

dim

,2*(

n_

tro

t-i_

tro

t)+

1,i_

pa

rt)

ux(

1,i_

dim

,2*(

n_

tro

t-i_

tro

t)+

1,i_

pa

rt)

=&

ux(

1,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

ux(

1,i_

dim

,2*(

n_

tro

t-i_

tro

t)+

2,i_

pa

rt)

=&

-u

x(1

,i_dim

,2*i

_tr

ot+

2,i_

pa

rt)

en

dd

oe

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

con

f_sa

veca

llco

nf_

save

_n

r(-1

)e

nd

sub

rou

tine

!!

!!

!!

!!

!!

!

sub

rou

tine

con

f_sa

ve_

nr(

file

nr)

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

cha

ract

er(

20

)fil

en

am

ein

teg

er

file

nr

if(f

ilen

r.g

e.0

)th

en

file

na

me

="c

onf."

call

ad

dn

um

tost

rin

g(f

ilen

am

e,file

nr)

els

e file

na

me

="c

onf.xm

o"

en

dif

op

en

(10

,file

="c

on

f.e

nd

",st

atu

s="u

nkn

ow

n")

write

(10

,’(i1

)’)

1w

rite

(10

,*)

n_

tro

tw

rite

(10

,10

3)

(sca

l_0

(i_

dim

),i_

dim

=1

,n_

dim

+3

)w

rite

(10

,10

3)

(sca

l_x(

1,i_

dim

)/d

t,i_

dim

=1

,n_

dim

+3

)d

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

pa

rt=

1,n

_p

art

write

(10

,14

4)

typ

e(i_

pa

rt),

&(p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt),

i_d

im=

1,n

_d

im),

&(r

0(i_

dim

,i_tr

ot,i_

pa

rt),

i_d

im=

1,n

_dim

),&

cha

rge

(i_

part

)e

nd

do

en

dd

ow

rite

(10

,*)

n_

ne

igh

_m

ax,

"m

ax.

nu

nb

er

of

ne

igh

bo

rsse

ein

inru

n"

write

(10

,*)

n_

ne

igh

,"d

ime

nsi

on

of

neig

hb

or

field

"cl

ose

(10

)

if(i_

run

.eq

.1)

op

en

(11

,file

=fil

en

am

e,s

tatu

s="u

nkn

ow

n")

write

(11

,’(i5

)’)

n_

pa

rtw

rite

(11

,*)

do

i_p

art

=1

,n_

pa

rtif(

typ

e(i_

pa

rt).

eq

.1)

the

nw

rite

(11

,20

4)

"Si

",(r

_re

al(i_

dim

,0,i_

pa

rt),

i_d

im=

1,n

_d

im),

&ch

arg

e(i_

pa

rt)

els

eif(

typ

e(i_

pa

rt).

eq

.2)

the

nw

rite

(11

,20

4)

"O",

(r_

rea

l(i_

dim

,0,i_

pa

rt),

i_d

im=

1,n

_d

im),

&ch

arg

e(i_

pa

rt)

en

dif

en

dd

oif(

i_ru

n.e

q.n

_ru

n)

clo

se(1

1)

op

en

(35

,file

="c

on

favg

.xm

o",

sta

tus=

"unkn

ow

n")

write

(35

,’(i5

)’)

n_

pa

rtw

rite

(35

,*)

do

i_p

art

=1

,n_

pa

rtif(

typ

e(i_

pa

rt).

eq

.1)

the

nw

rite

(35

,20

4)

"Si

",(r

_re

al_

avg

(i_

dim

,0,i_

pa

rt)/

i_tim

e,

&i_

dim

=1

,n_

dim

),&

cha

rge

(i_

pa

rt)

els

eif(

typ

e(i_

pa

rt).

eq

.2)

the

nw

rite

(35

,20

4)

"O",

(r_

rea

l_a

vg(i_

dim

,0,i_

pa

rt)/

i_tim

e,

&i_

dim

=1

,n_

dim

),&

charg

e(i_

pa

rt)

en

dif

en

dd

ocl

ose

(35

)

op

en

(34

,file

="c

on

favg

.po

l",s

tatu

s="u

nkn

ow

n")

do

i_p

art

=1

,n

_p

art

write

(34

,10

4)

(avg

_d

ipo

le_

pa

rt(i_

dim

,i_p

art

)/i_

time

,&

i_d

im=

0,n

_d

im)

en

dd

ocl

ose

(34

)

op

en

(36

,file

="c

on

f.p

ol"

,sta

tus=

"un

know

n")

do

i_p

art

=1

,n

_p

art

write

(36

,10

4)

(dip

ole

(i_

dim

,i_p

art

),&

i_d

im=

1,n

_d

im)

en

dd

ocl

ose

(36

)

10

3fo

rma

t(3

f13

.5)

109

B. Programs1

04

form

at(

4f1

3.5

)1

44

form

at(

4i5

,4f1

4.6

)

20

4fo

rma

t(a

,3e

13

.4,e

30

.21

)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

reci

pim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nx_

qu

an

t(2

*n_

tro

t),f_q

ua

nt(

2*n

_tr

ot)

com

mo

n/x

f_qu

an

t/x_

qu

an

t,f_

qu

an

t

do

i_p

art

=1

,n_

pa

rtd

oi_

dim

=1,n

_d

im

do

i_tr

ot

=0

,n_

tro

t-1

f_q

ua

nt(

2*i

_tr

ot+

1)

=fo

rce

(i_

dim

,i_tr

ot,i_

pa

rt)/

n_

tro

t*sq

rtp

!---

-a

bove

no

rma

liza

tion

sn

ece

ssary

be

cau

sefo

ur1

inco

rrect

f_q

ua

nt(

2*i

_tr

ot+

2)

=0

.e

nd

do

!---

-tr

ansf

orm

into

reci

pro

cal

space

call

fou

r1(f

_qu

an

t,n

_tr

ot,1

)

do

i_tr

ot

=1

,2*n

_tr

ot

f0(i_

dim

,i_tr

ot,i_

pa

rt)

=f_

qu

an

t(i_

tro

t)e

nd

do

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

rea

lim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nx_

qu

an

t(2

*n_

tro

t),

f_q

ua

nt(

2*n

_tr

ot)

,&

ma

t_d

um

my(

3,3

)d

ou

ble

pre

cisi

on

forc

e_

sca

l_id

ea

l(n

_dim

+3

),&

forc

e_

sca

l_co

nf(

n_

dim

+3

)co

mm

on

/xf_

qu

an

t/x_

qu

an

t,f_

qu

an

t

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

i_d

im=

1,n

_d

im

do

i_tr

ot

=1,2

*n_

tro

tx_

qu

an

t(i_

tro

t)=

u0

(i_

dim

,i_tr

ot,i_

pa

rt)/

sqrt

pe

nd

do

!---

--tr

an

sfo

rmin

tore

al

spa

ceca

llfo

ur1

(x_q

ua

nt,n

_tr

ot,-1

)

do

i_tr

ot

=0,n

_tr

ot-

1r0

(i_

dim

,i_tr

ot,i_

pa

rt)

=x_

qu

an

t(2*i

_tr

ot+

1)

-&

1.

*p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt)

+r_

ref(

i_d

im,i_

pa

rt)

en

dd

o

en

dd

oe

nd

do

call

calc

_rv

_re

al

v_p

ress

=0

.d

oi_

dim

=1

,n

_d

im+

e_

dim

forc

e_

sca

l_co

nf(

i_d

im)

=0

.e

nd

do

!---

-T

ER

Mq

ua

nt

do

i_tr

ot

=0

,n_

tro

t-1

j_tr

ot

=m

od(i_

tro

t+1

,n_

tro

t)d

oi_

pa

rt=

1,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

do

i_d

im=

1,n

_d

imfo

rce

_sc

al_

con

f(i_

dim

)=

forc

e_

sca

l_co

nf(

i_d

im)

-&

k_q

ua

nt(

i_ty

pe

)*

&(r

0(i_

dim

,i_tr

ot,i_

pa

rt)+

1.*

pb

c0(i_

dim

,i_tr

ot,i_

pa

rt)

&-r

0(i_

dim

,j_tr

ot,i_

pa

rt)-

1.*

pb

c0(i_

dim

,j_tr

ot,i_

pa

rt))

*&

(r_

rea

l(i_

dim

,i_tr

ot,i_

pa

rt)-

r_re

al(i_

dim

,j_tr

ot,i_

pa

rt))

en

dd

oe

nd

do

if(e

_d

im.e

q.3

)th

en

do

i_p

art

=1

,n

_p

art

forc

e_

sca

l_co

nf(

4)

=fo

rce

_sc

al_

con

f(4

)-

&k_

qu

an

t(i_

typ

e)

*(

&(

r0(1

,i_tr

ot,i_

pa

rt)

+1

.*

pb

c0(1

,i_tr

ot,i_

pa

rt)

&-

r0(1

,j_tr

ot,i_

pa

rt)

-1

.*

pb

c0(1

,j_tr

ot,i_

pa

rt)

)*

&(

r_re

al(2

,i_tr

ot,i_

pa

rt)

-r_

rea

l(2

,j_tr

ot,i_

pa

rt)

)+

&(

r0(2

,i_tr

ot,i_

pa

rt)

+1

.*

pb

c0(2

,i_tr

ot,i_

pa

rt)

&-

r0(2

,j_tr

ot,i_

pa

rt)

-1

.*

pb

c0(2

,j_tr

ot,i_

pa

rt)

)*

&(

r_re

al(1

,i_tr

ot,i_

pa

rt)

-r_

rea

l(1

,j_tr

ot,i_

pa

rt)

))

/2

forc

e_

sca

l_co

nf(

5)

=fo

rce

_sc

al_

con

f(5

)-

&k_

qu

an

t(i_

typ

e)

*(

&(

r0(2

,i_tr

ot,i_

pa

rt)

+1

.*

pb

c0(2

,i_tr

ot,i_

pa

rt)

&-

r0(2

,j_tr

ot,i_

pa

rt)

-1

.*

pb

c0(2

,j_tr

ot,i_

pa

rt)

)*

&(

r_re

al(3

,i_tr

ot,i_

pa

rt)

-r_

rea

l(3

,j_tr

ot,i_

pa

rt)

)+

&(

r0(3

,i_tr

ot,i_

pa

rt)

+1

.*

pb

c0(3

,i_tr

ot,i_

pa

rt)

&-

r0(3

,j_tr

ot,i_

pa

rt)

-1

.*

pb

c0(3

,j_tr

ot,i_

pa

rt)

)*

&(

r_re

al(2

,i_tr

ot,i_

pa

rt)

-r_

rea

l(2

,j_tr

ot,i_

pa

rt)

))

/2

forc

e_

sca

l_co

nf(

6)

=fo

rce

_sc

al_

con

f(6

)-

&k_

qu

an

t(i_

typ

e)

*(

&(

r0(3

,i_tr

ot,i_

pa

rt)

+1

.*

pb

c0(3

,i_tr

ot,i_

pa

rt)

&-

r0(3

,j_tr

ot,i_

pa

rt)

-1

.*

pb

c0(3

,j_tr

ot,i_

pa

rt)

)*

&(

r_re

al(1

,i_tr

ot,i_

pa

rt)

-r_

rea

l(1

,j_tr

ot,i_

pa

rt)

)+

&(

r0(1

,i_tr

ot,i_

pa

rt)

+1

.*

pb

c0(1

,i_tr

ot,i_

pa

rt)

&-

r0(1

,j_tr

ot,i_

pa

rt)

-1

.*

pb

c0(1

,j_tr

ot,i_

pa

rt)

)*

&(

r_re

al(3

,i_tr

ot,i_

pa

rt)

-r_

rea

l(3

,j_tr

ot,i_

pa

rt)

))

/2

en

dd

oe

nd

ife

nd

do

!---

term

iiid

oi_

dim

=1

,n

_d

im+

3fo

rce

_sc

al_

ide

al(i_

dim

)=

0.

en

dd

od

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

type

(i_

pa

rt)

do

i_tr

ot

=0

,n

_tr

ot-

1

110


!---

-fir

stp

art

do

i_d

im=

1,

n_

dim

forc

e_

sca

l_id

ea

l(i_

dim

)=

forc

e_

sca

l_id

ea

l(i_

dim

)+

&m

ass

(i_

tro

t,i_

typ

e)

*v_

rea

l(i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

*&

ux(

1,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

en

dd

oif

(e_

dim

.eq

.3)

the

n!-

--sy

mm

etr

ize

no

nd

iag

ele

me

nts

forc

e_

sca

l_id

ea

l(4

)=

forc

e_

sca

l_id

ea

l(4

)+

&m

ass

(i_

tro

t,i_

typ

e)

*&

(v_

rea

l(1

,2*i

_tr

ot+

1,i_

pa

rt)

*ux(

1,2

,2*i

_tr

ot+

1,i_

pa

rt)

&+

v_re

al(2

,2*i

_tr

ot+

1,i_

pa

rt)

*ux(

1,1

,2*i

_tr

ot+

1,i_

pa

rt)

)fo

rce

_sc

al_

ide

al(5

)=

forc

e_

sca

l_id

ea

l(5

)+

&m

ass

(i_

tro

t,i_

typ

e)

*&

(v_

rea

l(2

,2*i

_tr

ot+

1,i_

pa

rt)

*ux(

1,3

,2*i

_tr

ot+

1,i_

pa

rt)

&+

v_re

al(3

,2*i

_tr

ot+

1,i_

pa

rt)

*ux(

1,2

,2*i

_tr

ot+

1,i_

pa

rt)

)fo

rce

_sc

al_

ide

al(6

)=

forc

e_

sca

l_id

ea

l(6

)+

&m

ass

(i_

tro

t,i_

typ

e)

*&

(v_

rea

l(3

,2*i

_tr

ot+

1,i_

pa

rt)

*ux(

1,1

,2*i

_tr

ot+

1,i_

pa

rt)

&+

v_re

al(1

,2*i

_tr

ot+

1,i_

pa

rt)

*ux(

1,3

,2*i

_tr

ot+

1,i_

pa

rt)

)e

nd

if!-

---

seco

nd

pa

rtif

(n_

tro

t.g

t.1

)th

en

do

i_d

im=

1,

n_

dim

forc

e_

sca

l_id

ea

l(i_

dim

)=

forc

e_

sca

l_id

ea

l(i_

dim

)+

&m

ass

(i_tr

ot,i_

typ

e)

*v_

rea

l(i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

&*

ux(

1,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

en

dd

oif

(e_

dim

.eq

.3)

the

n!-

--sy

mm

etr

ize

no

nd

iag

ele

me

nts

forc

e_

sca

l_id

ea

l(4

)=

forc

e_

sca

l_id

ea

l(4

)+

&m

ass

(i_tr

ot,i_

typ

e)

*&

(v_

real(1

,2*i

_tr

ot+

2,i_

pa

rt)

*u

x(1

,2,2

*i_

tro

t+2

,i_p

art

)&

+v_

real(2

,2*i

_tr

ot+

2,i_

pa

rt)

*u

x(1

,1,2

*i_

tro

t+2

,i_p

art

))

forc

e_

sca

l_id

ea

l(5

)=

forc

e_

sca

l_id

ea

l(5

)+

&m

ass

(i_tr

ot,i_

typ

e)

*&

(v_

real(2

,2*i

_tr

ot+

2,i_

pa

rt)

*u

x(1

,3,2

*i_

tro

t+2

,i_p

art

)&

+v_

real(3

,2*i

_tr

ot+

2,i_

pa

rt)

*u

x(1

,2,2

*i_

tro

t+2

,i_p

art

))

forc

e_

sca

l_id

ea

l(6

)=

forc

e_

sca

l_id

ea

l(6

)+

&m

ass

(i_

tro

t,i_

typ

e)

*&

(v_

rea

l(3

,2*i

_tr

ot+

2,i_

pa

rt)

*u

x(1

,1,2

*i_

tro

t+2

,i_p

art

)&

+v_

rea

l(1

,2*i

_tr

ot+

2,i_

pa

rt)

*u

x(1

,3,2

*i_

tro

t+2

,i_p

art

))

en

dif

en

dif

en

dd

oe

nd

do

do

i_d

im=

1,

n_

dim

+e

_d

imfo

rce

_sc

al_

ide

al(i_

dim

)=

forc

e_

sca

l_id

ea

l(i_

dim

)/

dt_

2if

(i_

dim

.gt.n

_d

im)

&fo

rce

_sc

al_

ide

al(i_

dim

)=

forc

e_

scal_

ide

al(i_

dim

)/

2e

nd

do

!6

66

take

cuto

ffco

rre

ctio

nin

toca

lcu

latio

no

fe

latic

con

sta

nts

if(f

_p

ote

ntia

l.ne

.1)

the

nd

oi_

dim

=1

,n

_d

im+

e_

dim

forc

e_

sca

l(i_

dim

)=

forc

e_

sca

l_id

ea

l(i_

dim

)&

+fo

rce

_sc

al_

con

f(i_

dim

)e

nd

do

!---

corr

ect

ion

du

eto

cuto

ffo

fr*

*(-6

)p

ote

ntia

lv_

pre

ss=

v_pre

ss&

-n

_tr

ot

*n

_p

art

**2

*fo

r_sc

a_

mfa

_co

r/

volu

me

/2

do

i_d

im=

1,

n_

dim

if(i_

dim

.le.n

_d

im)

the

nr_

du

mm

y=

h_

ma

t_in

v(i_

dim

,i_d

im)

els

eif

(i_

dim

.eq

.4)

the

nr_

du

mm

y=

(h

_m

at_

inv(

1,2

)+

h_

ma

t_in

v(2

,1)

)/

2e

lse

if(i_

dim

.eq

.5)

the

nr_

du

mm

y=

(h

_m

at_

inv(

2,3

)+

h_

ma

t_in

v(3

,2)

)/

2e

lse

r_d

um

my

=(

h_

ma

t_in

v(3

,1)

+h

_m

at_

inv(

1,3

))

/2

en

dif

r_d

um

my

=-

n_

pa

rt**

2*

for_

sca

_m

fa_

cor

*r_

du

mm

y/

volu

me

forc

e_

sca

l(i_

dim

)=

forc

e_

sca

l(i_

dim

)+

r_d

um

my

en

dd

oe

nd

if

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_rv

_re

al

imp

licit

no

ne

incl

ud

e"p

imd.c

om

.f"

do

i_p

art

=1

,n_

pa

rtd

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

dim

=1

,n_

dim

r_re

al(i_

dim

,i_tr

ot,i_

pa

rt)

=sc

al_

0(i_

dim

)*

&(r

0(i_

dim

,i_tr

ot,i_

pa

rt)

+1

.d0

*pb

c0(i_

dim

,i_tr

ot,i_

pa

rt))

v_re

al(i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

=&

sca

l_0

(i_

dim

)*

ux(

1,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

if(n

_tr

ot.g

t.1

)v_

rea

l(i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

=&

sca

l_0

(i_

dim

)*

ux(

1,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

en

dd

oif

(e_

dim

.eq

.3)

the

nr_

rea

l(1

,i_tr

ot,i_

pa

rt)

=r_

rea

l(1

,i_tr

ot,i_

pa

rt)

+&

sca

l_0

(4)

*(r

0(2

,i_tr

ot,i_

pa

rt)

+1.d

0*p

bc0

(2,i_

tro

t,i_

pa

rt))

+&

sca

l_0

(6)

*(r

0(3

,i_tr

ot,i_

pa

rt)

+1.d

0*p

bc0

(3,i_

tro

t,i_

pa

rt))

r_re

al(2

,i_tr

ot,i_

pa

rt)

=r_

rea

l(2

,i_tr

ot,i_

pa

rt)

+&

sca

l_0

(4)

*(r

0(1

,i_tr

ot,i_

pa

rt)

+1.d

0*p

bc0

(1,i_

tro

t,i_

pa

rt))

+&

sca

l_0

(5)

*(r

0(3

,i_tr

ot,i_

pa

rt)

+1.d

0*p

bc0

(3,i_

tro

t,i_

pa

rt))

r_re

al(3

,i_tr

ot,i_

pa

rt)

=r_

rea

l(3

,i_tr

ot,i_

pa

rt)

+&

sca

l_0

(6)

*(r

0(1

,i_tr

ot,i_

pa

rt)

+1.d

0*p

bc0

(1,i_

tro

t,i_

pa

rt))

+&

sca

l_0

(5)

*(r

0(2

,i_tr

ot,i_

pa

rt)

+1.d

0*p

bc0

(2,i_

tro

t,i_

pa

rt))

v_re

al(1

,2*i

_tr

ot+

1,i_

pa

rt)

=v_

rea

l(1

,2*i

_tr

ot+

1,i_

pa

rt)

+&

sca

l_0

(4)

*u

x(1

,2,2

*i_

tro

t+1

,i_p

art

)+

&sc

al_

0(6

)*

ux(

1,3

,2*i

_tr

ot+

1,i_

pa

rt)

v_re

al(2

,2*i

_tr

ot+

1,i_

pa

rt)

=v_

rea

l(2

,2*i

_tr

ot+

1,i_

pa

rt)

+&

sca

l_0

(4)

*u

x(1

,1,2

*i_

tro

t+1

,i_p

art

)+

&sc

al_

0(5

)*

ux(

1,3

,2*i

_tr

ot+

1,i_

pa

rt)

v_re

al(3

,2*i

_tr

ot+

1,i_

pa

rt)

=v_

rea

l(3

,2*i

_tr

ot+

1,i_

pa

rt)

+&

sca

l_0

(6)

*u

x(1

,1,2

*i_

tro

t+1

,i_p

art

)+

&sc

al_

0(5

)*

ux(

1,2

,2*i

_tr

ot+

1,i_

pa

rt)

if(n

_tr

ot.g

t.1

)th

en

v_re

al(1

,2*i

_tr

ot+

2,i_

pa

rt)

=v_

real(1

,2*i

_tr

ot+

2,i_

pa

rt)

+&

sca

l_0

(4)

*u

x(1

,2,2

*i_

tro

t+2

,i_p

art

)+

&sc

al_

0(6

)*

ux(

1,3

,2*i

_tr

ot+

2,i_

pa

rt)

v_re

al(2

,2*i

_tr

ot+

2,i_

pa

rt)

=v_

real(2

,2*i

_tr

ot+

2,i_

pa

rt)

+&

sca

l_0

(4)

*u

x(1

,1,2

*i_

tro

t+2

,i_p

art

)+

&sc

al_

0(5

)*

ux(

1,3

,2*i

_tr

ot+

2,i_

pa

rt)

v_re

al(3

,2*i

_tr

ot+

2,i_

pa

rt)

=v_

real(3

,2*i

_tr

ot+

2,i_

pa

rt)

+&

sca

l_0

(6)

*u

x(1

,1,2

*i_

tro

t+2

,i_p

art

)+

&sc

al_

0(5

)*

ux(

1,2

,2*i

_tr

ot+

2,i_

pa

rt)

en

dif

111

B. Programse

nd

ife

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

bin

nin

gim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

inte

ge

ri_

bin

,j_b

in

inte

ge

ri_

bin

_p

art

,ni_

bin

_p

art

inte

ge

rj_

bin

_p

art

,nj_

bin

_p

art

inte

ge

ri_

bin

_x,

i_b

in_

y,i_

bin

_z

inte

ge

rj_

bin

_x,

j_b

in_

y,j_

bin

_z

inte

ge

rd

j_bin

_x,

dj_

bin

_y,

dj_

bin

_z

do

ub

lep

reci

sio

nr_

2

do

i_p

art

=1

,n_

pa

rtn

eig

hb

or(

0,i_

pa

rt)

=0

en

dd

o

do

i_p

art

=1

,n_

pa

rt-1

i_ty

pe

=ty

pe

(i_

pa

rt)

do

j_p

art

=i_

pa

rt+

1,n

_p

art

j_ty

pe

=ty

pe

(j_

pa

rt)

do

i_tr

ot

=0

,n_

tro

t-1

!---

-ca

lcu

late

dis

tan

cew

ithrig

ht

bo

un

da

rie

sca

llca

lc_

dis

tan

ce(r

_2

,i_tr

ot,i_

pa

rt,j_

pa

rt)

if(r_

2.le

.r_

rang

e_

2(i_

typ

e,j_

typ

e))

then

if(d

elta

_r(

1).

lt.0

.)th

en

ne

igh

bo

r(0

,i_p

art

)=

ne

igh

bo

r(0

,i_pa

rt)

+1

i_d

um

my

=ne

igh

bo

r(0

,i_p

art

)n

eig

hb

or(

i_d

um

my,

i_p

art

)=

j_p

art

els

en

eig

hb

or(

0,j_

pa

rt)

=n

eig

hb

or(

0,j_

pa

rt)

+1

if(n

eig

hb

or(

0,j_

pa

rt).

gt.n

_n

eig

h)

sto

p"n

eig

hb

or

field

"i_

du

mm

y=

ne

igh

bo

r(0

,j_p

art

)n

eig

hb

or(

i_d

um

my,

j_p

art

)=

i_p

art

en

dif

n_

ne

igh

_m

ax

=m

ax(

n_

ne

igh

_m

ax,

ne

igh

bor(

0,j_

pa

rt))

n_

ne

igh

_m

ax

=m

ax(

n_

ne

igh

_m

ax,

ne

igh

bor(

0,i_

pa

rt))

go

to1

50

en

dif

en

dd

o1

50

con

tinu

ee

nd

do

en

dd

o

if(n

_n

eig

h_

ma

x.g

t.n

_n

eig

h)

sto

p’m

ake

ne

igh

bo

rfie

ldla

rge

r’

f_sk

in=

0d

oi_

pa

rt=

1,n

_p

art

do

i_tr

ot

=0

,n_

tro

t-1

do

i_d

im=

1,n

_d

imr0

_o

ld(i_

dim

,i_tr

ot,i_

pa

rt)

=r_

rea

l(i_

dim

,i_tr

ot,i_

pa

rt)

en

dd

o

en

dd

oe

nd

do

!6

66

if(f

_p

ote

ntia

l.eq

.3)

call

rec_

bin

nin

gif

(f_

po

ten

tial.e

q.5

)ca

llre

c_b

inn

ing

if(f

_p

ote

ntia

l.eq

.6)

call

rec_

bin

nin

gif

(f_

po

ten

tial.e

q.7

)ca

llre

c_b

inn

ing

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

rec_

bin

nin

gim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sio

nq

_ve

c(n

_d

im),

q_

2,lo

c_p

refa

c,p

ot_

loc

com

ple

x*1

6f_

of_

k

rec_

po

t_o

ffse

t=

0.

do

i_d

im=

1,

n_

dim

+e

_d

imre

c_fo

rce

_o

ffse

t(i_

dim

)=

0.

en

dd

olo

c_p

refa

c=

ele

c_p

refa

c*tw

op

i/vo

lum

e

n_

rec_

vec

=0

do

k_d

im=

0,

nq

_m

ax(

3)

do

j_d

im=

-nq

_m

ax(

2),

nq

_m

ax(

2)

do

i_d

im=

-nq_

ma

x(1

),n

q_

ma

x(1

)q

_ve

c(1

)=

i_d

im*

dq

(1,1

)+

j_d

im*

dq

(1,2

)+

k_d

im*

dq

(1,3

)q

_ve

c(2

)=

i_d

im*

dq

(2,1

)+

j_d

im*

dq

(2,2

)+

k_d

im*

dq

(2,3

)q

_ve

c(3

)=

i_d

im*

dq

(3,1

)+

j_d

im*

dq

(3,2

)+

k_d

im*

dq

(3,3

)q

_2

=0

.d

ol_

dim

=1,n

_d

imq

_2

=q

_2

+q

_ve

c(l_

dim

)**2

en

dd

oif(

(q_

2.le

.q_

cut_

2).

an

d.(

q_

2.g

t.0

.))

the

n!-

----

calc

ula

teF

To

fch

arg

ed

en

sity

f_o

f_k

=(0

.,0.)

do

i_p

art

=1

,n_

pa

rt!-

--op

t r_d

um

my

=&

i_d

im*(

r_re

f(1

,i_p

art

)+u

0(1

,1,i_

pa

rt)/

sqrt

p)

+&

j_d

im*(

r_re

f(2

,i_p

art

)+u

0(2

,1,i_

pa

rt)/

sqrt

p)

+&

k_d

im*(

r_re

f(3

,i_p

art

)+u

0(3

,1,i_

pa

rt)/

sqrt

p)

r_d

um

my

=d

mod

(r_

du

mm

y,1

.d0

)i_

du

mm

y=

nin

t(r_

du

mm

y*n

_p

ote

ntia

l)f_

of_

k=

f_o

f_k

+ch

arg

e(i_

pa

rt)*

!&

tab

_e

xp_

ima

(i_

du

mm

y)&

exp

(2.*

pi*

(0.,1

.)*r

_d

um

my)

en

dd

or_

du

mm

y=

loc_

pre

fac*

exp

(-q

_2

/(2

*alp

ha

)**2

)/q

_2

po

t_lo

c=

r_d

um

my

*a

bs(

f_o

f_k)

**2

!w

rite

(30

+i_

time

,*)

po

t_lo

c,(q

_ve

c(l_

dim

),l_

dim

=1

,n_d

im)

if(p

ot_

loc.

ge

.v_

cut)

the

nn

_re

c_ve

c=

n_

rec_

vec

+1

if(n

_re

c_ve

c.g

t.n

_re

c_ve

c_m

ax)

sto

p"n

_re

c_ve

cto

ob

ig"

rec_

vec_

list(

n_

rec_

vec,

1)

=i_

dim

rec_

vec_

list(

n_

rec_

vec,

2)

=j_

dim

rec_

vec_

list(

n_

rec_

vec,

3)

=k_

dim

els

eif(

k_d

im.n

e.0

)r_

du

mm

y=

r_d

um

my*

2

112


po

t_lo

c=

r_du

mm

y*a

bs(

f_o

f_k)

**2

rec_

po

t_o

ffse

t=

rec_

po

t_o

ffse

t+

po

t_lo

cif(

f_p

ress

ure

.eq

.1)

the

nr_

du

mm

y=

(1./(2

*alp

ha

**2

)+2

./q

_2

)*po

t_lo

cd

ol_

dim

=1,n

_d

imre

c_fo

rce

_o

ffse

t(l_

dim

)=

rec_

forc

e_

offse

t(l_

dim

)+

&p

ot_

loc

-r_

du

mm

y*q

_ve

c(l_

dim

)**2

en

dd

oe

nd

ife

nd

ife

nd

ife

nd

do

en

dd

oe

nd

do

!---

-ch

an

ge

do

i_d

im=

1,n

_d

imre

c_fo

rce

_offse

t(i_

dim

)=

n_

tro

t*re

c_fo

rce

_o

ffse

t(i_

dim

)e

nd

do

rec_

po

t_o

ffse

t=

n_

tro

t*re

c_p

ot_

offse

t

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

rec_

ew

ald

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

inte

ge

ri_

q(n

_dim

),tr

ot_

ma

x,k_

tro

td

ou

ble

pre

cisi

on

q_

vec(

n_

dim

),q

_2

,loc_

pre

fac,

rec_

pre

fac

do

ub

lep

reci

sion

two

_re

c_p

refa

c,v_

mo

de

,fo

rce

_sc

al_

rec(

n_

dim

)co

mp

lex*

16

qe

qre

ck(n

_p

art

)co

mp

lex*

16

f_o

f_k

do

ub

lep

reci

sion

f_o

f_k_

rea

l,f_

of_

k_im

ag

do

i_p

art

=1

,n

_p

art

qe

qre

c(i_

pa

rt)=

0.

en

dd

o

v_e

wa

ld=

0.

do

i_d

im=

1,n

_d

imfo

rce

_sc

al_

rec(

i_d

im)

=0

.e

nd

do

loc_

pre

fac

=e

lec_

pre

fac*

two

pi/v

olu

me

j_tr

ot

=m

od

(int(

n_

tro

t*ra

nd

om

(1))

,n_

tro

t)d

oi_

rec_

vec

=1

,n_

rec_

vec

q_

2=

0.

do

i_d

im=

1,n

_d

imq

_ve

c(i_

dim

)=

0.

do

j_d

im=

1,

n_

dim

i_q

(j_

dim

)=

rec_

vec_

list(

i_re

c_ve

c,j_

dim

)q

_ve

c(i_

dim

)=

q_

vec(

i_d

im)

+i_

q(j_

dim

)*d

q(j_

dim

,i_d

im)

en

dd

oq

_2

=q

_2

+q

_ve

c(i_

dim

)**2

en

dd

ore

c_p

refa

c=

loc_

pre

fac*

exp

(-q

_2

/(2

*alp

ha

)**2

)/q

_2

if(re

c_ve

c_lis

t(i_

rec_

vec,

3).

ne

.0)

rec_

pre

fac

=2

*re

c_p

refa

ctw

o_

rec_

pre

fac

=2

*re

c_p

refa

ctr

ot_

ma

x=

n_tr

ot-

1if

(f_

op

t_re

c.e

q.1

)tr

ot_

ma

x=

0d

oi_

tro

t=

0,

tro

t_m

ax

if(f

_o

pt_

rec.

eq

.0)

j_tr

ot

=i_

tro

tf_

of_

k=

(0.,0

.)f_

of_

k_re

al

=0

.d0

f_o

f_k_

ima

g=

0.d

0d

oi_

pa

rt=

1,n

_p

art

r_d

um

my

=0.

if(f

_o

pt_

rec.

eq

.0)

the

nd

oi_

dim

=1

,n_

dim

r_d

um

my

=r_

du

mm

y+

i_q

(i_

dim

)*r0

(i_d

im,j_

tro

t,i_

pa

rt)

en

dd

oe

lse

do

i_d

im=

1,

n_

dim

r_d

um

my

=r_

du

mm

y+

i_q

(i_

dim

)*

&(

r_re

f(i_

dim

,i_p

art

)+

u0

(i_

dim

,1,i_

pa

rt)/

sqrt

p)

en

dd

oe

nd

ifr_

du

mm

y=

dm

od

(r_

du

mm

y,1

.d0

)i_

du

mm

y=

nin

t(r_

du

mm

y*n

_p

ote

ntia

l)f_

of_

k=

f_o

f_k

+ch

arg

e(i_

pa

rt)*

!&

tab

_e

xp_

ima

(i_

du

mm

y)&

exp

(2.*

pi*

(0.,1

.)*r

_d

um

my)

f_o

f_k_

rea

l=

f_o

f_k_

rea

l+ch

arg

e(i_

pa

rt)*

&co

s(2.*

pi*

r_d

um

my)

f_o

f_k_

ima

g=

f_o

f_k_

ima

g+

cha

rge

(i_

part

)*&

sin

(2.*

pi*

r_d

um

my)

en

dd

o

v_m

od

e=

rec_

pre

fac

*a

bs(

f_o

f_k)

**2

if(f

_o

pt_

rec.

eq

.1)

v_m

od

e=

n_

tro

t*

v_m

od

ev_

ew

ald

=v_

ew

ald

+v_

mo

de

!---

-ca

lcu

late

forc

efr

om

mo

de

"i_

rec_

vec"

on

ea

cha

tom

do

i_p

art

=1

,n_

pa

rtr_

du

mm

y=

0.

do

i_d

im=

1,n

_d

imr_

du

mm

y=

r_du

mm

y+

i_q

(i_

dim

)*r0

(i_

dim

,j_tr

ot,i_

pa

rt)

en

dd

or_

du

mm

y=

-dm

od(r

_d

um

my,

1.d

0)

i_d

um

my

=n

int(

r_d

um

my*

n_

po

ten

tial)

!c_

du

mm

y=

tab

_e

xp_

ima

(i_

du

mm

y)c_

du

mm

y=

exp

(2.*

pi*

(0.,1

.)*r

_d

um

my)

!qe

qre

c(i_

pa

rt)

=q

eq

rec(

i_p

art

)+

!&

two

_re

c_p

refa

c*d

ble

(c_

du

mm

y*f_

of_

k)q

eq

rec(

i_p

art

)=

qe

qre

c(i_

pa

rt)

+tw

o_

rec_

pre

fac

*&

(co

s(2

.*p

i*r_

du

mm

y)*f

_o

f_k_

rea

l&

-si

n(2

.*p

i*r_

du

mm

y)*f

_o

f_k_

ima

g)

r_d

um

my

=im

ag

(c_

du

mm

y*f_

of_

k)

if(f

_o

pt_

rec.

eq

.0)

the

nd

oi_

dim

=1

,n_

dim

forc

e(i_

dim

,i_tr

ot,i_

pa

rt)

=fo

rce

(i_

dim

,i_tr

ot,i_

pa

rt)

&-

two

_re

c_p

refa

c*ch

arg

e(i_

pa

rt)*

q_

vec(

i_d

im)*

r_d

um

my

&*

h_

ma

t_in

v(i_

dim

,i_d

im)

en

dd

oe

lse

do

k_tr

ot

=n_

tro

t-1

,0,-

1d

oi_

dim

=1

,n_

dim

forc

e(i_

dim

,k_

tro

t,i_

pa

rt)

=fo

rce

(i_

dim

,k_

tro

t,i_

pa

rt)

&-

two

_re

c_p

refa

c*ch

arg

e(i_

pa

rt)*

q_

vec(

i_d

im)*

r_d

um

my

&*

h_

ma

t_in

v(i_

dim

,i_d

im)

en

dd

o

113

B. Programse

nd

do

en

dif

en

dd

o!-

---

calc

ula

tefo

rce

fro

me

ach

mod

e"i

_re

c_ve

c"o

nb

ox

sha

pe

if(f_

pre

ssu

re.e

q.1

)th

en

r_d

um

my

=(1

./(2

*alp

ha

**2

)+2

./q

_2

)*v_

mod

ed

oi_

dim

=1

,n_d

imfo

rce

_sc

al_

rec(

i_d

im)

=fo

rce

_sc

al_

rec(

i_d

im)

+&

(v_

mo

de

-r_

du

mm

y*q

_ve

c(i_

dim

)**2

)e

nd

do

en

dif

en

dd

o

en

dd

o

v_e

wa

ld=

v_e

wa

ld+

rec_

po

t_o

ffse

t

if(f_

pre

ssu

re.e

q.1

)th

en

do

i_d

im=

1,n

_d

imfo

rce

_sc

al_

rec(

i_d

im)

=fo

rce

_sc

al_

rec(

i_d

im)

+&

rec_

forc

e_

offse

t(i_

dim

)d

oj_

dim

=1,

n_

dim

forc

e_

sca

l(i_

dim

)=

forc

e_

sca

l(i_

dim

)+

&h

_m

at_

inv(

i_d

im,j_

dim

)*

forc

e_

scal_

rec(

j_d

im)

en

dd

oe

nd

do

en

dif

v_in

ter

=v_

inte

r+

v_e

wa

ld

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_d

ista

nce

(r_

2,i_

tro

t,i_

pa

rt,j_

pa

rt)

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nr_

2

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_

dim

,i_tr

ot,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_d

im)

=d

elta

_ce

ll(i_

dim

)+

1.

en

dif

en

dd

o!-

---

term

squ

are

r_2

=0

.d

oi_

dim

=1,

n_

dim

r_2

=r_

2+

met_

ten

_vo

i(i_

dim

)*

de

lta_ce

ll(i_

dim

)**2

en

dd

oif

(e_

dim

.eq.3

)th

en

r_2

=r_

2&

+m

et_

ten

_vo

i(4

)*

de

lta_

cell(

1)

*de

lta_

cell(

2)

&+

me

t_te

n_

voi(5

)*

de

lta_

cell(

2)

*de

lta_

cell(

3)

&+

me

t_te

n_

voi(6

)*

de

lta_

cell(

3)

*de

lta_

cell(

1)

en

dif

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_d

ist_

pb

c(r_

2,i_

tro

t,i_

pa

rt,j_

pa

rt,

&p

bc_

x,p

bc_

y,p

bc_

z)im

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sio

nr_

2in

teg

er

pb

c_x,

pb

c_y,

pb

c_z,

pb

c(n

_d

im)

pb

c(1

)=p

bc_

xp

bc(

2)=

pb

c_y

pb

c(3

)=p

bc_

z

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_

dim

,i_tr

ot,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_dim

)=

de

lta_

cell(

i_d

im)

+1

.e

nd

ife

nd

do

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

+&

1.0

*pb

c(i_

dim

)e

nd

do

!---

-te

rmsq

ua

rer_

2=

0.

do

i_d

im=

1,

n_

dim

r_2

=r_

2+

me

t_te

n_

voi(i_

dim

)*

de

lta_

cell(

i_d

im)*

*2e

nd

do

if(e

_d

im.e

q.3

)th

en

r_2

=r_

2&

+m

et_

ten

_vo

i(4

)*

de

lta_

cell(

1)

*d

elta

_ce

ll(2

)&

+m

et_

ten

_vo

i(5

)*

de

lta_

cell(

2)

*d

elta

_ce

ll(3

)&

+m

et_

ten

_vo

i(6

)*

de

lta_

cell(

3)

*d

elta

_ce

ll(1

)e

nd

if

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_d

ist_

int(

r_2

)im

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sio

nr_

2!-

---

term

squ

are

do

i_d

im=

1,n

_d

imd

elta

_r(

i_d

im)

=sc

al_

0(i_

dim

)*

de

lta_

cell(

i_d

im)

r_2

=r_

2+

me

t_te

n_

voi(i_

dim

)*

de

lta_

cell(

i_d

im)*

*2e

nd

do

if(e

_d

im.e

q.3

)th

en

r_2

=r_

2&

+m

et_

ten

_vo

i(4

)*

de

lta_

cell(

1)*

de

lta_

cell(

2)

&+

me

t_te

n_vo

i(5

)*

de

lta_

cell(

2)*

de

lta_

cell(

3)

&+

me

t_te

n_vo

i(6

)*

de

lta_

cell(

3)*

de

lta_

cell(

1)

en

dif

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

114


sub

rou

tine

ha

rm_

osc

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sion

v_in

ter_

loc

do

ub

lep

reci

sion

r_1

,r_

2

v_in

ter

=0

.!-

---

ha

rmo

nic

osc

illa

tor

do

i_p

art

=1

,n_

pa

rtv_

inte

r_lo

c=

0.

do

i_d

im=

1,n

_d

imd

oi_

tro

t=

0,n

_tr

ot-

1r_

du

mm

y=

r0(i_

dim

,i_tr

ot,i_

pa

rt)

forc

e(i_

dim

,i_tr

ot,i_

pa

rt)

=fo

rce

(i_

dim

,i_tr

ot,i_

pa

rt)

-&

r_d

um

my

v_in

ter_

loc

=v_

inte

r_lo

c+

r_d

um

my*

*2e

nd

do

en

dd

ov_

inte

r=

v_in

ter

+v_

inte

r_lo

ce

nd

do

v_in

ter

=v_

inte

r/2

viri_

est

=v_

inte

r

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

bo

x_ch

ain

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

!---

term

coup

lh

_d

ot

an

dve

loci

ties

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

j_tr

ot

=0,n

_tr

ot-

1i_

tro

t=

j_tr

ot

+1

if(i_

tro

t.eq

.2)

i_tr

ot

=n

_tr

ot

+1

i_d

um

my

=(i_

tro

t-1

)/2

do

i_d

im=

1,n

_d

imf0

(i_

dim

,i_tr

ot,i_

pa

rt)

=f0

(i_

dim

,i_tr

ot,i_

pa

rt)

-&

g_

voi_

dot(

i_d

im)

*m

ass

(i_

du

mm

y,i_

typ

e)

*&

ux(

1,i_

dim

,i_tr

ot,i_

pa

rt)

/d

t_2

en

dd

oif

(e_

dim

.eq

.3)

the

nf0

(1,i_

tro

t,i_

pa

rt)

=f0

(1,i_

tro

t,i_

part

)-

(&

g_

voi_

dot(

4)*

ma

ss(i_

du

mm

y,i_

typ

e)*

ux(

1,2

,i_tr

ot,i_

pa

rt)

+&

g_

voi_

dot(

6)*

ma

ss(i_

du

mm

y,i_

typ

e)*

ux(

1,3

,i_tr

ot,i_

pa

rt)

)/d

t_2

&/

2f0

(2,i_

tro

t,i_

pa

rt)

=f0

(2,i_

tro

t,i_

part

)-

(&

g_

voi_

dot(

5)*

ma

ss(i_

du

mm

y,i_

typ

e)*

ux(

1,3

,i_tr

ot,i_

pa

rt)

+&

g_

voi_

dot(

4)*

ma

ss(i_

du

mm

y,i_

typ

e)*

ux(

1,1

,i_tr

ot,i_

pa

rt)

)/d

t_2

&/

2f0

(3,i_

tro

t,i_

pa

rt)

=f0

(3,i_

tro

t,i_

part

)-

(&

g_

voi_

dot(

6)*

ma

ss(i_

du

mm

y,i_

typ

e)*

ux(

1,1

,i_tr

ot,i_

pa

rt)

+&

g_

voi_

dot(

5)*

ma

ss(i_

du

mm

y,i_

typ

e)*

ux(

1,2

,i_tr

ot,i_

pa

rt)

)/d

t_2

&/

2e

nd

ife

nd

do

en

dd

o

!---

-sy

mm

etr

ize

do

i_tr

ot

=1

,n_

tro

t/2

-1d

oi_

dim

=1,n

_d

imd

oi_

pa

rt=

1,n

_p

art

f0(i_

dim

,2*(

n_

tro

t-i_

tro

t)+

1,i_

pa

rt)

=&

f0(i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

en

dd

oe

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

len

na

rd_

jon

es

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nv_

inte

r_lo

cd

ou

ble

pre

cisi

on

r_2

,r_

6,

r_1

2,

po

t_lo

c,r_

4_

inv

do

ub

lep

reci

sio

nd

yn_

ma

t(n

_d

im,n

_d

im),

loc_

dyn

_m

at(

n_

dim

,n_

dim

)

v_in

ter

=0

.d

oi_

dim

=1,

n_

dim

+3

forc

e_

sca

l_in

ter(

i_d

im)

=0

.e

nd

do

if(

(i_

time.g

t.n

_re

lax)

.an

d.

(mo

d(i_

time

,20

).e

q.0

))

the

nd

oi_

dim

=1

,n_

dim

do

j_d

im=

i_d

im,

n_

dim

do

k_d

im=

i_d

im,n

_d

imd

ol_

dim

=k_

dim

,n_

dim

if(i_

dim

+j_

dim

.le.k

_d

im+

l_d

im)

then

ela

_b

orn

(i_

dim

,j_d

im,k

_d

im,l_

dim

)=

0.

en

dif

en

dd

oe

nd

do

en

dd

oe

nd

do

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

stra

in(i_

dim

,j_d

im)

=0

.e

nd

do

en

dd

oe

nd

if

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

v_in

ter_

loc

=0.

do

i_n

eig

h=

1,n

eig

hb

or(

0,i_

pa

rt)

j_p

art

=n

eig

hb

or(

i_n

eig

h,i_

pa

rt)

j_ty

pe

=ty

pe(j_

pa

rt)

po

t_lo

c=

0.

do

i_tr

ot

=0,n

_tr

ot-

1

call

calc

_dis

tan

ce(r

_2

,i_tr

ot,i_

pa

rt,j_

pa

rt)

if(r_

2.lt

.r_

cuto

ff_

2(i_

typ

e,j_

typ

e))

the

nr_

6=

(sig

ma

_2

(i_

typ

e,j_

typ

e)/

r_2

)**3

r_1

2=

r_6

**2

po

t_lo

c=

po

t_lo

c+

(r_

12

-r_

6)

-e

_sh

ift(i_

typ

e,j_

typ

e)

!---

-r_

du

mm

y=

{\p

art

ial

V\o

ver

\pa

rtia

lr}

/r

115

B. Programsr_

du

mm

y=

ep

sil(i_

typ

e,j_

typ

e)*

(-1

2*r

_1

2+

6*r

_6

)/r_

2

call

calc

_fo

rce

_lo

c(r_

du

mm

y,i_

tro

t,i_

pa

rt,j_

pa

rt)

if(

(i_

time

.gt.n

_re

lax)

.an

d.

(mo

d(i_

time

,20

).e

q.0

))

the

n!-

---

calc

ula

teb

orn

ela

stic

con

sta

nts

r_4

_in

v=

1.

/r_

2**

2ca

lllo

c_d

ynm

at(

loc_

dyn

_m

at,

de

lta_

r,r_

2,

-12

.d0

)d

oi_

dim

=1

,n

_d

imd

oj_

dim

=1

,n

_d

imd

yn_

ma

t(i_

dim

,j_d

im)

=si

gm

a_

2(i_

typ

e,j_

typ

e)*

*6*

&lo

c_d

yn_

ma

t(i_

dim

,j_d

im)

en

dd

oe

nd

do

call

loc_

dyn

ma

t(lo

c_d

yn_

ma

t,d

elta

_r,

r_2

,-6

.d0

)d

oi_

dim

=1

,n

_d

imd

oj_

dim

=1

,n

_d

imd

yn_

ma

t(i_

dim

,j_d

im)

=e

psi

l(i_

typ

e,j_

typ

e)

*r_

4_

inv

*&

(d

yn_

ma

t(i_

dim

,j_d

im)

-&

sig

ma_

2(i_

typ

e,j_

typ

e)*

*3*

loc_

dyn

_m

at(

i_d

im,j_

dim

))

en

dd

oe

nd

do

do

i_d

im=

1,

n_

dim

do

j_d

im=

i_d

im,

n_

dim

do

k_d

im=

i_d

im,

n_

dim

do

l_d

im=

k_d

im,

n_

dim

if(i_

dim

+j_

dim

.le.k

_d

im+

l_d

im)

the

ne

la_

bo

rn(i_d

im,

j_d

im,

k_d

im,

l_d

im)

=&

ela

_b

orn

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)+

de

lta_

r(i_

dim

)*

&d

yn_

mat(

j_d

im,k

_d

im)

*d

elta

_r(

l_d

im)

en

dif

en

dd

oe

nd

do

en

dd

oe

nd

do

!---

-ca

lcu

late

stra

inte

nso

rd

oi_

dim

=1

,n

_d

imd

oj_

dim

=1

,n

_d

imst

rain

(i_

dim

,j_d

im)

=st

rain

(i_

dim

,j_dim

)+

&r_

du

mm

y*

de

lta_

r(i_

dim

)*

de

lta_

r(j_

dim

)e

nd

do

en

dd

oe

nd

if

en

dif

en

dd

ov_

inte

r_lo

c=

v_in

ter_

loc

+e

psi

l(i_

typ

e,j_

typ

e)*

po

t_lo

ce

nd

do

v_in

ter

=v_

inte

r+

v_in

ter_

loc

en

dd

o

call

forc

e_

on

_sc

al

if(

(i_

time.g

t.n

_re

lax)

.an

d.

(mo

d(i_

time

,20

).e

q.0

))

the

ntim

e_

bo

rn=

time

_b

orn

+d

td

oi_

dim

=1

,n

_d

imd

oj_

dim

=i_

dim

,n

_d

imd

ok_

dim

=i_

dim

,n

_d

imd

ol_

dim

=k_

dim

,n

_d

imif

(i_

dim

+j_

dim

.le.k

_d

im+

l_d

im)

the

n

ela

_b

orn

_1

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

&e

la_

born

_1

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)+

dt

*&

ela

_b

orn

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)/

(vo

lum

e*n

_tr

ot)

en

dif

en

dd

oe

nd

do

en

dd

oe

nd

do

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

stra

in_

1(i_

dim

,j_d

im)

=st

rain

_1

(i_

dim

,j_d

im)

&+

stra

in(i_

dim

,j_d

im)

*d

td

ok_

dim

=i_

dim

,n

_d

imd

ol_

dim

=k_

dim

,n

_d

imst

rain

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

&st

rain

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)+

&st

rain

(i_

dim

,j_

dim

)*

stra

in(k

_d

im,

l_d

im)

*d

te

nd

do

en

dd

oe

nd

do

en

dd

oe

nd

if

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

loc_

dyn

ma

t(d

yn_

ma

t,d

r,r_

2,

n)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,n

_d

imp

ara

me

ter

(n_

dim

=3

)d

ou

ble

pre

cisi

on

dr(

n_

dim

),r_

2,

r_n,

nd

ou

ble

pre

cisi

on

dyn

_m

at(

n_

dim

,n_

dim

),r_

du

mm

y

if(n

.ge

.0.)

the

nr_

n=

r_2

**(0

.5*n

)e

lse

r_n

=1

./

r_2**

(-0

.5*n

)e

nd

if

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

if(i_

dim

.eq

.j_d

im)

r_d

um

my

=n

*r_

2if(

i_d

im.n

e.j_

dim

)r_

du

mm

y=

0.

dyn

_m

at(

i_d

im,j_

dim

)=

r_d

um

my

&+

n*(

n-2

)*

dr(

i_d

im)

*d

r(j_

dim

)d

yn_

ma

t(i_

dim

,j_d

im)

=d

yn_

ma

t(i_

dim

,j_d

im)

*r_

ne

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_fo

rce

_lo

c(r_

du

mm

y,i_

tro

t,i_

pa

rt,j_

pa

rt)

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

i_d

im=

1,

n_

dim

de

lta_

r(i_

dim

)=

sca

l_0

(i_

dim

)*d

elta

_ce

ll(i_

dim

)e

nd

do

if(e

_d

im.e

q.3

)th

en

116


de

lta_

r(1

)=

de

lta_

r(1

)&

+sc

al_

0(4

)*d

elta

_ce

ll(2

)+

sca

l_0

(6)*

de

lta_

cell(

3)

de

lta_

r(2

)=

de

lta_

r(2

)&

+sc

al_

0(5

)*d

elta

_ce

ll(3

)+

sca

l_0

(4)*

de

lta_

cell(

1)

de

lta_

r(3

)=

de

lta_

r(3

)&

+sc

al_

0(6

)*d

elta

_ce

ll(1

)+

sca

l_0

(5)*

de

lta_

cell(

2)

en

dif

do

i_d

im=

1,

n_

dim

!---

-te

rmii

forc

e_

loc(

i_dim

)=

r_d

um

my

*d

elta

_ce

ll(i_

dim

)fo

rce

(i_

dim

,i_tr

ot,i_

pa

rt)

=-

forc

e_lo

c(i_

dim

)+

&fo

rce

(i_

dim

,i_tr

ot,i_

pa

rt)

forc

e(i_

dim

,i_tr

ot,j_

pa

rt)

=+

forc

e_lo

c(i_

dim

)+

&fo

rce

(i_

dim

,i_tr

ot,j_

pa

rt)

forc

e_

sca

l_in

ter(

i_d

im)

=fo

rce

_sc

al_

inte

r(i_

dim

)-

&fo

rce

_lo

c(i_

dim

)*

de

lta_

r(i_

dim

)e

nd

do

!---

-te

rmiv

if(e

_d

im.e

q.3

)th

en

!---

sym

me

triz

ed

"str

ess

"fie

ldfo

rce

_sc

al_

inte

r(4

)=

forc

e_

sca

l_in

ter(

4)

-&

(fo

rce

_lo

c(1

)*d

elta

_r(

2)

+fo

rce

_lo

c(2

)*d

elta

_r(

1)

)/

2fo

rce

_sc

al_

inte

r(5

)=

forc

e_

sca

l_in

ter(

5)

-&

(fo

rce

_lo

c(2

)*d

elta

_r(

3)

+fo

rce

_lo

c(3

)*d

elta

_r(

2)

)/

2fo

rce

_sc

al_

inte

r(6

)=

forc

e_

sca

l_in

ter(

6)

-&

(fo

rce

_lo

c(3

)*d

elta

_r(

1)

+fo

rce

_lo

c(1

)*d

elta

_r(

3)

)/

2e

nd

ife

nd

sub

rou

tine

!!

!!

!!

!!

!!

!

sub

rou

tine

forc

e_

on

_sc

al

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

!---

-ch

an

ge

tra

nsf

orm

do

i_d

im=

1,n

_d

im+

e_

dim

forc

e_

sca

l(i_

dim

)=

forc

e_

sca

l(i_

dim

)+

forc

e_

sca

l_in

ter(

i_d

im)

en

dd

o

!---

ad

de

xte

rna

lp

ress

ure

!---

term

pre

ssv_

pre

ss=

v_p

ress

+n

_tr

ot

*p

ress

*vo

lum

ed

oi_

dim

=1

,n

_d

imfo

rce

_sc

al(i_

dim

)=

forc

e_

sca

l(i_

dim

)-

&n

_tr

ot

*pre

ss*v

olu

me

*h

_m

at_

inv(

i_d

im,i_

dim

)e

nd

do

if(e

_d

im.e

q.3

)th

en

forc

e_

sca

l(4

)=

forc

e_

sca

l(4

)-

&n

_tr

ot

*pre

ss*v

olu

me

*h

_m

at_

inv(

1,2

)fo

rce

_sc

al(5

)=

forc

e_

sca

l(5

)-

&n

_tr

ot

*pre

ss*v

olu

me

*h

_m

at_

inv(

2,3

)fo

rce

_sc

al(6

)=

forc

e_

sca

l(6

)-

&n

_tr

ot

*pre

ss*v

olu

me

*h

_m

at_

inv(

3,1

)e

nd

if

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_3

bo

dy_

forc

e_

loc(

fi,fj,

fk,

&i_

tro

t,i_

pa

rt,j_

pa

rt,k

_p

art

)im

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nfi,

fj,fk

inte

ge

rk_

part

do

i_d

im=

1,

n_

dim

de

lta_

r(i_

dim

)=

sca

l_0

(i_

dim

)*d

elta

_ce

ll(i_

dim

)e

nd

do

if(e

_d

im.e

q.3

)th

en

de

lta_

r(1

)=

de

lta_

r(1

)&

+sc

al_

0(4

)*d

elta

_ce

ll(2

)+

sca

l_0

(6)*

de

lta_

cell(

3)

de

lta_

r(2

)=

de

lta_

r(2

)&

+sc

al_

0(5

)*d

elta

_ce

ll(3

)+

sca

l_0

(4)*

de

lta_

cell(

1)

de

lta_

r(3

)=

de

lta_

r(3

)&

+sc

al_

0(6

)*d

elta

_ce

ll(1

)+

sca

l_0

(5)*

de

lta_

cell(

2)

en

dif

do

i_d

im=

1,

n_

dim

!---

-te

rmii

forc

e(i_

dim

,i_tr

ot,i_

pa

rt)

=+

fi*

de

lta_

cell(

i_d

im)

+&

forc

e(i_

dim

,i_tr

ot,i_

pa

rt)

forc

e(i_

dim

,i_tr

ot,j_

pa

rt)

=+

fj*

de

lta_

cell(

i_d

im)

+&

forc

e(i_

dim

,i_tr

ot,j_

pa

rt)

forc

e(i_

dim

,i_tr

ot,k_

pa

rt)

=+

fk*

de

lta_

cell(

i_d

im)

+&

forc

e(i_

dim

,i_tr

ot,k_

pa

rt)

forc

e_

loc(

i_d

im)

=fj

*d

elta

_ce

ll(i_

dim

)fo

rce

_sc

al_

inte

r(i_

dim

)=

forc

e_

sca

l_in

ter(

i_d

im)

-&

forc

e_lo

c(i_

dim

)*

de

lta_

r(i_

dim

)e

nd

do

!---

-te

rmiv

if(e

_d

im.e

q.3

)th

en

!---

sym

me

triz

ed

"str

ess

"fie

ldfo

rce

_sc

al_

inte

r(4

)=

forc

e_

sca

l_in

ter(

4)

-&

(fo

rce

_lo

c(1

)*d

elta

_r(

2)

+fo

rce

_lo

c(2

)*d

elta

_r(

1)

)/

2fo

rce

_sc

al_

inte

r(5

)=

forc

e_

sca

l_in

ter(

5)

-&

(fo

rce

_lo

c(2

)*d

elta

_r(

3)

+fo

rce

_lo

c(3

)*d

elta

_r(

2)

)/

2fo

rce

_sc

al_

inte

r(6

)=

forc

e_

sca

l_in

ter(

6)

-&

(fo

rce

_lo

c(3

)*d

elta

_r(

1)

+fo

rce

_lo

c(1

)*d

elta

_r(

3)

)/

2e

nd

if

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

go

dd

ard

_p

ote

ntia

lim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

call

two

bo

dy_

po

ten

tial

call

forc

e_

on

_sc

al

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

vash

ish

ta_

po

ten

tial

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

inte

ge

ri_

ang

,k_

pa

rtd

ou

ble

pre

cisi

on

cosj

ik,

rij,

rik,

fij,

fik,

fjj,

fjk,

fkj,

fkk

do

ub

lep

reci

sio

nfa

cto

r1,

term

1,

term

j,te

rmk,

&r_

ij(n

_d

im),

r_ik

(n_

dim

),&

de

lta_

ij(n

_d

im),

de

lta_

ik(n

_d

im)

117

B. Programsca

lltw

ob

od

y_p

ote

ntia

l!

thre

eb

ody

con

trib

utio

ns:

j-i-k

call

bo

nd

_a

na

lysi

sv_

thre

eb

od

y=0

.d0

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

do

i_a

ng

=1

,n

_an

g(i_

typ

e)

j_p

art

=tr

iple

_p

art

j(i_

pa

rt,i_

an

g)

k_p

art

=tr

iple

_p

art

k(i_

pa

rt,i_

an

g)

cosj

ik=

trip

le_

cos(

i_p

art

,i_a

ng

)rij=

trip

le_

rij(i_

pa

rt,i_

an

g)

rik=

trip

le_

rik(

i_p

art

,i_a

ng

)

if((

rij.g

e.v

ash

_r0

(i_

typ

e))

.or.

(rik

.ge

.va

sh_

r0(i_

typ

e))

)&

the

n!

v_th

ree

bo

dy=

=0

:a

dd

no

forc

es,

no

en

erg

ies

els

e!

on

lyif

rij

an

drik

<r0

fact

or1

=1

/(rij-va

sh_

r0(i_

typ

e))

+1

/(rik-

vash

_r0

(i_

typ

e))

fact

or1

=e

xp(v

ash

_l(i_

typ

e)*

fact

or1

)fa

cto

r1=

fact

or1

*(co

sjik

-va

sh_

cth

eta

(i_ty

pe

))fa

cto

r1=

vash

_B

(i_

typ

e)*

fact

or1

fact

or1

=fa

ctor1

*ele

c_p

refa

c

term

j=va

sh_

l(i_

typ

e)*

(co

sjik

-va

sh_

cth

eta

(i_

typ

e))

term

k=te

rmj

term

j=te

rmj/(

rij*

(rij-

vash

_r0

(i_

typ

e))

**2

)te

rmj=

2*c

osj

ik/(

rij*

*2)+

term

j

term

k=te

rmk/

(rik

*(rik-

vash

_r0

(i_

typ

e))

**2

)te

rmk=

2*c

osj

ik/(

rik*

*2)+

term

k

term

1=

2/(

rik*

rij)

!w

rite

(*,*

)i_

pa

rt,i_

typ

e,i_

an

g,j_

pa

rt,k

_p

art

,!

&1

80

*aco

s(co

sjik

)/p

i,rij,

rik

!ca

lcu

late

forc

es

inij

dire

ctio

nd

oi_

dim

=1

,n_

dim

de

lta_

ij(i_

dim

)=tr

iple

_d

elta

ij(i_

pa

rt,i_

an

g,i_

dim

)e

nd

do

fij=

-fa

cto

r1*(

term

1-t

erm

j)fjj

=-f

act

or1

*te

rmj

fkj=

+fa

cto

r1*t

erm

1!

call

calc

_3

bo

dy_

forc

e_lo

c(fi,

fj,fk

,0,i_

pa

rt,j_

pa

rt,k

_p

art

)

!ca

lcu

late

forc

es

inik

dire

ctio

nd

oi_

dim

=1

,n_

dim

de

lta_

ik(i_

dim

)=tr

iple

_d

elta

ik(i_

pa

rt,i_

an

g,i_

dim

)e

nd

do

fik=

-fa

cto

r1*(

term

1-t

erm

k)fjk

=+

fact

or1

*te

rm1

fkk=

-fa

cto

r1*t

erm

k!

call

calc

_3

bo

dy_

forc

e_lo

c(fi,

fj,fk

,0,i_

pa

rt,j_

pa

rt,k

_p

art

)

do

i_d

im=

1,

n_

dim

r_ij(

i_d

im)

=sc

al_

0(i_

dim

)*d

elta

_ij(

i_d

im)

en

dd

oif

(e_

dim

.eq.3

)th

en

r_ij(

1)

=r_

ij(1

)&

+sc

al_

0(4

)*d

elta

_ij(

2)

+sc

al_

0(6

)*de

lta_

ij(3

)

r_ij(

2)

=r_

ij(2

)&

+sc

al_

0(5

)*d

elta

_ij(

3)

+sc

al_

0(4

)*d

elta

_ij(

1)

r_ij(

3)

=r_

ij(3

)&

+sc

al_

0(6

)*d

elta

_ij(

1)

+sc

al_

0(5

)*d

elta

_ij(

2)

en

dif

do

i_d

im=

1,

n_

dim

r_ik

(i_

dim

)=

sca

l_0

(i_

dim

)*d

elta

_ik

(i_

dim

)e

nd

do

if(e

_d

im.e

q.3

)th

en

r_ik

(1)

=r_

ik(1

)&

+sc

al_

0(4

)*d

elta

_ik

(2)

+sc

al_

0(6

)*d

elta

_ik

(3)

r_ik

(2)

=r_

ik(2

)&

+sc

al_

0(5

)*d

elta

_ik

(3)

+sc

al_

0(4

)*d

elta

_ik

(1)

r_ik

(3)

=r_

ik(3

)&

+sc

al_

0(6

)*d

elta

_ik

(1)

+sc

al_

0(5

)*d

elta

_ik

(2)

en

dif

i_tr

ot=

0

do

i_d

im=

1,

n_

dim

!---

-te

rmii

forc

e(i_

dim

,i_tr

ot,i_

pa

rt)

=+

fij*

de

lta_

ij(i_

dim

)+

&fik

*d

elta

_ik

(i_

dim

)+

&fo

rce(i_

dim

,i_tr

ot,i_

pa

rt)

forc

e(i_

dim

,i_tr

ot,j_

pa

rt)

=+

fjj*

de

lta_

ij(i_

dim

)+

&fjk

*d

elta

_ik

(i_

dim

)+

&fo

rce(i_

dim

,i_tr

ot,j_

pa

rt)

forc

e(i_

dim

,i_tr

ot,k_

pa

rt)

=+

fkj

*d

elta

_ij(

i_d

im)

+&

fkk

*d

elta

_ik

(i_

dim

)+

&fo

rce(i_

dim

,i_tr

ot,k_

pa

rt)

forc

e_

sca

l_in

ter(

i_d

im)

=fo

rce

_sc

al_

inte

r(i_

dim

)&

-fjj

*d

elta

_ij(

i_d

im)

*r_

ij(i_

dim

)&

-fjk

*d

elta

_ik

(i_

dim

)*

r_ij(

i_d

im)

&-

fkj

*d

elta

_ij(

i_d

im)

*r_

ik(i_

dim

)&

-fk

k*

de

lta_

ik(i_

dim

)*

r_ik

(i_

dim

)e

nd

do

!---

-te

rmiv

if(e

_d

im.e

q.3

)th

en

!---

sym

me

triz

ed

"str

ess

"fie

ldfo

rce

_sc

al_

inte

r(4

)=

forc

e_

sca

l_in

ter(

4)

-&

(fjj

*d

elta

_ij(

1)*

r_ij(

2)

+fjj

*d

elta

_ij(

2)*

r_ij(

1)

)/

2-

&(

fjk*

delta

_ik

(1)*

r_ij(

2)

+fjk

*d

elta

_ik

(2)*

r_ij(

1)

)/

2-

&(

fkj

*d

elta

_ij(

1)*

r_ik

(2)

+fk

j*

de

lta_

ij(2

)*r_

ik(1

))

/2

-&

(fk

k*

delta

_ik

(1)*

r_ik

(2)

+fk

k*

de

lta_

ik(2

)*r_

ik(1

))

/2

forc

e_

sca

l_in

ter(

5)

=fo

rce

_sc

al_

inte

r(5

)-

&(

fjj*

delta

_ij(

2)*

r_ij(

3)

+fjj

*d

elta

_ij(

3)*

r_ij(

2)

)/

2-

&(

fjk*

delta

_ik

(2)*

r_ij(

3)

+fjk

*d

elta

_ik

(3)*

r_ij(

2)

)/

2-

&(

fkj

*d

elta

_ij(

2)*

r_ik

(3)

+fk

j*

de

lta_

ij(3

)*r_

ik(2

))

/2

-&

(fk

k*

delta

_ik

(2)*

r_ik

(3)

+fk

k*

de

lta_

ik(3

)*r_

ik(2

))

/2

forc

e_

sca

l_in

ter(

6)

=fo

rce

_sc

al_

inte

r(6

)-

&(

fjj*

delta

_ij(

3)*

r_ij(

1)

+fjj

*d

elta

_ij(

1)*

r_ij(

3)

)/

2-

&(

fjk*

delta

_ik

(3)*

r_ij(

1)

+fjk

*d

elta

_ik

(1)*

r_ij(

3)

)/

2-

&(

fkj

*d

elta

_ij(

3)*

r_ik

(1)

+fk

j*

de

lta_

ij(1

)*r_

ik(3

))

/2

-&

(fk

k*

delta

_ik

(3)*

r_ik

(1)

+fk

k*

de

lta_

ik(1

)*r_

ik(3

))

/2

en

dif

v_th

ree

bo

dy

=v_

thre

eb

od

y&

+fa

cto

r1*(

cosj

ik-

vash

_ct

he

ta(i_

typ

e))

en

dif

!o

nly

ifrij

an

drik

<r0

118


en

dd

oe

nd

do

v_in

ter

=v_

inte

r+

v_th

ree

bo

dy

call

forc

e_

on

_sc

al

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!!

!!

sub

rou

tine

sca

nd

olo

_p

ote

ntia

lc

use

me

tric

use

ne

igh

bou

ru

sep

ola

r!

toa

cce

ssd

ipo

les

imp

licit

dou

ble

pre

cisi

on

(a-h

,o-z

)in

clu

de

"pim

d.c

om

2.f"

cch

ara

cte

r*2

5fil

ep

os,

file

cel

rea

l*8

,a

lloca

tab

le::

r(:,:)

rea

l*8

,a

lloca

tab

le::

sfo

rce

(:,:)

rea

l*8

stre

ss(3

,3),

htin

(3,3

)lo

gic

al

rest

art

,tce

lin

teg

er

n_

pa

rt_ty

pe

(n_

typ

e)

inte

ge

ri_

cla

ss

!a

ssig

nth

eva

ria

ble

sn

_re

lax_

sc=

n_re

lax

na

t=n

_p

art

nsp

=n

_ty

pe

A_

ew

=1

.0d

-8rc

ut=

18

.02

70

3n

na

tma

x=n

_n

eig

hie

sr=

n_

pb

c!

sea

rch

cell

an

dn

eig

hb

orin

gim

ag

ece

lls

ca

lloca

te(s

forc

e(3

,na

t))

allo

cate

(r(3

,na

t))

allo

cate

(sp

ind(n

at)

)c

do

i=1

,na

td

oj=

1,

3r(

j,i)

=r_

real(j,0

,i)sp

ind

(i)

=ty

pe

(i)

en

dd

oe

nd

do

call

tra

ns_

to_

ten

s(sc

al_

0,

ht)

cr

=r/

0.5

29

2d0

ht

=h

t/0

.52

92

d0

ca

lloca

te(n

nlis

t(n

at,n

na

tma

x))

allo

cate

(nn

at(

na

t))

allo

cate

(nn

_im

ag

(3,n

at,n

na

tma

x))

cca

llin

v3(h

t,h

tm1

,om

eg

a)

call

nb

rlis

t(r,

nn

atm

ax)

ch

tin=

ht

c !se

tre

start

tru

eif

first

MD

ste

pif

(i_

time

.eq

.1)

the

nre

sta

rt=

.tru

e.

els

e rest

art

=.fals

e.

en

dif

if(f

_p

ress

ure

.eq

.1)

the

ntc

el

=.tru

e.

els

e tce

l=

.fa

lse.

en

dif

cca

llfo

rce

_ft(r

,en

erg

y,sf

orc

e,s

tre

ss,a

_e

w,i_

time

,.fa

lse

.,&

.fa

lse

.,h

tin,r

est

art

,tce

l)c c !

con

vert

en

erg

yfr

om

Ha

rtre

eto

kBK

v_in

ter

=e

nerg

y*3

15

77

3.2

!co

nve

rtst

ress

:!

stre

ss(i,j)

*om

eg

ais

r_i*

F_

j/vo

lum

e*v

olu

me

inh

art

ree

!fo

rce

_sc

al_

inte

r()

isr_

i*

(hˆ-

1*

F)_

jin

kBK

/a

ng

stro

m!

htm

isin

a_

Bo

hr,

soh

tm1

*str

ess

*om

eg

ais

inh

art

ree

/a_

Bo

hr

forc

e_

sca

l_in

ter

=0

.d0

stre

ss=

+st

ress

do

j=1

,3d

oi_

dim

=1

,3fo

rce

_sc

al_

inte

r(i_

dim

)=

forc

e_

sca

l_in

ter(

i_d

im)

+&

htm

1(i_

dim

,j)*s

tre

ss(i_

dim

,j)*o

me

ga

*3

15

77

3.2

/0

.52

92

d0

en

dd

oif

(e_

dim

.eq

.3)

the

n!-

--sy

mm

etr

ize

d"s

tre

ss"

field

forc

e_

sca

l_in

ter(

4)

=fo

rce

_sc

al_

inte

r(4

)+

&(

htm

1(2

,j)*s

tre

ss(1

,j)*o

me

ga

*3

15

77

3.2

/0

.52

92

d0

+&

htm

1(1

,j)*s

tre

ss(2

,j)*o

me

ga

*3

15

77

3.2

/0

.52

92

d0

)*0

.5fo

rce

_sc

al_

inte

r(5

)=

forc

e_

sca

l_in

ter(

5)

+&

(h

tm1

(3,j)

*str

ess

(2,j)

*om

eg

a*

315

77

3.2

/0

.52

92

d0

+&

htm

1(2

,j)*s

tre

ss(3

,j)*o

me

ga

*3

15

77

3.2

/0

.52

92

d0

)*0

.5fo

rce

_sc

al_

inte

r(6

)=

forc

e_

sca

l_in

ter(

6)

+&

(h

tm1

(3,j)

*str

ess

(1,j)

*om

eg

a*

315

77

3.2

/0

.52

92

d0

+&

htm

1(1

,j)*s

tre

ss(3

,j)*o

me

ga

*3

15

77

3.2

/0

.52

92

d0

)*0

.5e

nd

ife

nd

do

!co

nve

rtfo

rce

:!

sfo

rce

()is

Fin

ha

rtre

e/

a_

Bo

hr

!fo

rce

()is

hˆ-

1*

Fin

kBK

/a

ng

stro

mˆ2

!h

tmis

ina

_B

oh

r,so

htm

1*s

forc

eis

inh

art

ree

/a_

Bo

hrˆ

2d

oi_

pa

rt=

1,n

at

do

j=1

,3fo

rce

(j,0

,i_p

art

)=0

do

i=1

,3fo

rce

(j,0

,i_p

art

)=fo

rce

(j,0

,i_p

art

)+

&h

tm1

(j,i)

*sfo

rce

(i,i_

pa

rt)*

&3

15

77

3.2

/(0

.52

92

d0

**2

)e

nd

do

en

dd

oe

nd

do

!co

nve

rtdip

ole

s

119

B. Programsd

oi_

pa

rt=

1,

n_

pa

rtd

oi_

dim

=1

,3

dip

ole

(i_

dim

,i_p

art

)=

dip

(i_

dim

,i_p

art

)*

0.5

29

2d

0e

nd

do

en

dd

oa

vg_

dip

ole

=0

.d0

n_

pa

rt_

typ

e=

0

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

avg

_d

ipo

le(0

,i_ty

pe

)=

avg

_d

ipo

le(0

,i_ty

pe

)+

&sq

rt(

dip

ole

(1,i_

pa

rt)*

*2&

+d

ipo

le(2

,i_p

art

)**2

+d

ipo

le(3

,i_p

art

)**2

)d

oi_

dim

=1

,n

_d

ima

vg_

dip

ole

(i_d

im,i_

typ

e)

=a

vg_

dip

ole

(i_

dim

,i_ty

pe

)&

+d

ipo

le(i_

dim

,i_p

art

)e

nd

do

n_

pa

rt_

typ

e(i_ty

pe

)=

n_

pa

rt_

typ

e(i_

type

)+

1e

nd

do

do

i_ty

pe

=1

,n

_ty

pe

do

i_d

im=

0,

n_

dim

avg

_d

ipo

le(i_d

im,i_

typ

e)

=a

vg_

dip

ole

(i_

dim

,i_ty

pe

)/

&n_

pa

rt_

typ

e(i_

typ

e)

en

dd

oe

nd

do

if(i_

time

.eq

.1)

the

na

vg_

dip

ole

_p

art

=0

.d0

en

dif

do

i_p

art

=1

,n

_p

art

avg

_d

ipo

le_

pa

rt(0

,i_p

art

)=

avg

_d

ipo

le_p

art

(0,i_

pa

rt)

+&

sqrt

(d

ipo

le(1

,i_p

art

)**2

&+

dip

ole

(2,i_

pa

rt)*

*2+

dip

ole

(3,i_

pa

rt)*

*2)

do

i_d

im=

1,

n_

dim

avg

_d

ipo

le_

part

(i_

dim

,i_p

art

)=

&a

vg_

dip

ole

_p

art

(i_

dim

,i_p

art

)+

dip

ole

(i_

dim

,i_p

art

)e

nd

do

en

dd

o

dip

ole

_cl

ass

=0

.d0

do

i_p

art

=1

,n

_p

art

if(t

ype

(i_

pa

rt).

eq

.2)

the

ni_

cla

ss=

mo

d(i_

pa

rt-1

,18

)-5

dip

ole

_cl

ass

(0,i_

cla

ss)

=&

dip

ole

_cl

ass

(0,i_

cla

ss)

&+

sqrt

(dip

ole

(1,i_

pa

rt)*

*2+

dip

ole

(2,i_

pa

rt)*

*2+

&dip

ole

(3,i_

pa

rt)*

*2)/

(n_

oxy

/12

.)d

oi_

dim

=1

,3d

ipo

le_

cla

ss(i_

dim

,i_cl

ass

)=

&d

ipo

le_

cla

ss(i_

dim

,i_cl

ass

)&

+d

ipo

le(i_

dim

,i_p

art

)/(n

_o

xy/1

2.)

!if

(i_

dim

.eq

.3)

write

(37

,*)

i_p

art

,i_

cla

ss,

!&

dip

ole

(i_

dim

,i_p

art

)e

nd

do

en

dif

en

dd

o!

write

(37

,*)

"==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

==

"

do

i_cl

ass

=1,1

2d

oi_

dim

=0,3

dip

ole

_cl

ass

1(i_

dim

,i_cl

ass

)=

dip

ole

_cl

ass

1(i_

dim

,i_cl

ass

)&

+d

ipo

le_

cla

ss(i_

dim

,i_cl

ass

)d

ipo

le_

cla

ss2

(i_

dim

,i_cl

ass

)=

dip

ole

_cl

ass

2(i_

dim

,i_cl

ass

)&

+(d

ipo

le_

cla

ss(i_

dim

,i_cl

ass

))**

2d

ipo

le_

cla

ss4

(i_

dim

,i_cl

ass

)=

dip

ole

_cl

ass

4(i_

dim

,i_cl

ass

)&

+(d

ipo

le_

cla

ss(i_

dim

,i_cl

ass

))**

4e

nd

do

en

dd

o

dip

ole

_co

un

t=dip

ole

_co

un

t+

1

if(i_

time

.eq

.1)

op

en

(33

,file

="p

imd

.pol"

,sta

tus=

"un

kno

wn

")

write

(33

,’(1

5f1

5.7

)’)

r_tim

e,

!1

&a

vg_

dip

ole

(0,2

),!

2&

avg

_d

ipo

le(1

,2),

avg

_d

ipo

le(2

,2),

avg

_d

ipo

le(3

,2),

!3

,4,5

&(d

ipole

(i_

dim

,3

3),

i_d

im=

1,3

),!

6,7

,8&

(dip

ole

(i_

dim

,6

4),

i_d

im=

1,3

)!

9,1

0,1

1

de

allo

cate

(sfo

rce

,r,s

pin

d,n

nlis

t,n

na

t,n

n_

ima

g)

call

forc

e_

on

_sc

al

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!su

bro

utin

ea

pplie

d_

field

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

ne

_a

pp

lied

(n_

dim

)in

teg

er

i_e

_a

pp

lied

_b

ks

e_

ap

plie

d(1

)=

0.

e_

ap

plie

d(2

)=

0.

e_

ap

plie

d(3

)=

0.

!if

(f_e

_a

pp

lied

.gt.0

)th

en

!if

(i_

time

.ge

.n_

rela

x)th

en

!i_

e_

ap

plie

d_

bks

=f_

e_

app

lied

!i_

e_

ap

plie

d_

bks

exi

sts

just

top

lease

the

com

pile

r!

e_

ap

plie

d(1

)=

0.0

1*(

i_tim

e)/

50

00

0!

end

if!

en

dif

do

i_d

im=

1,

n_

dim

!h

art

ree

/a_b

oh

r/e

->kB

K/a

ng

stro

m/e

e_

ap

plie

d(i_

dim

)=5

96

72

4.9

*e_

ap

plie

d(i_

dim

)e

nd

do

do

i_p

art

=1

,n

_p

art

do

i_d

im=

1,

n_

dim

v_in

ter

=v_

inte

r+

cha

rge

(i_

pa

rt)*

r_re

al(i_

dim

,0,i_

pa

rt)

&*

e_

ap

plie

d(i_

dim

)d

oj_

dim

=1

,n

_d

imfo

rce

(i_

dim

,0,i_

pa

rt)

=fo

rce

(i_

dim

,0,i_

pa

rt)

+&

h_

ma

t_in

v(i_

dim

,j_

dim

)*

cha

rge

(i_

pa

rt)

&*

e_

ap

plie

d(j_

dim

)e

nd

do

120


en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

bks

_p

ote

ntia

lim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

call

two

bo

dy_

pote

ntia

l

call

forc

e_

on

_sc

al

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

two

bo

dy_

po

ten

tial

!a

ll2

bo

dy

po

ten

tials

with

sho

rtra

ng

ep

lus

cou

lom

bte

rmim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sio

nv_

inte

r_lo

cd

ou

ble

pre

cisi

on

r_1

,r_

2,p

ot_

loc

do

ub

lep

reci

sio

nr_

2_

ma

xd

ou

ble

pre

cisi

on

mu

_i(n

_p

art

),la

gra

ng

e_

no

m,

lag

ran

ge

_d

en

om

inte

ge

ri_

dim

_m

ax

v_in

ter

=0

.v_

rea

lco

ulo

mb

=0

.v_

sho

rtra

ng

e=

0.

self_

en

erg

y=

0.

do

i_p

art

=1

,n

_p

art

self_

en

erg

y=

self_

en

erg

y+

cha

rge

(i_p

art

)**2

en

dd

ose

lf_e

ne

rgy

=-n

_tr

ot*

ele

c_p

refa

c*a

lph

a*s

elf_

en

erg

y/sq

rt(p

i)

do

i_d

im=

1,

n_

dim

+3

forc

e_

sca

l_in

ter(

i_d

im)

=0

.e

nd

do

if(f

_p

ote

ntia

l.eq

.3)

call

rec_

ew

ald

if((

f_p

ote

ntia

l.ge

.5).

an

d.(

f_p

ote

ntia

l.le

.8))

call

rec_

ew

ald

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

do

i_p

art

=1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

mu

_i(i_

pa

rt)

=q

eq

ele

ctro

ne

g(i_

typ

e)

+q

eq

rec(

i_p

art

)&

+(

qe

qh

ard

ne

ss(i_

typ

e)

&-

2*e

lec_

pre

fac*

alp

ha

/sqrt

(pi)

)*ch

arg

e(i_

pa

rt)

en

dd

oe

nd

if

do

i_p

art

=1

,n_

pa

rti_

typ

e=

type

(i_

pa

rt)

v_in

ter_

loc

=0

.

do

i_n

eig

h=

1,n

eig

hb

or(

0,i_

pa

rt)

j_p

art

=n

eig

hbo

r(i_

ne

igh

,i_p

art

)j_

typ

e=

typ

e(j_

pa

rt)

po

t_lo

c=

0.

do

i_tr

ot

=0,n

_tr

ot-

1

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_

dim

,i_tr

ot,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_d

im)

=d

elta

_ce

ll(i_

dim

)+

1.

en

dif

en

dd

o

r_2

=0

.ca

llca

lc_

dis

t_in

t(r_

2)

if(r_

2.lt

.r_

cuto

ff_

2(i_

typ

e,j_

typ

e))

the

nr_

1=

sqrt

(r_2

)i_

po

ten

tial

=n

int(

n_

po

ten

tial*

r_1

/r_ra

ng

e(i_

typ

e,j_

typ

e))

po

t_lo

c=

po

t_lo

c+

r_p

ot_

1(i_

typ

e,j_

typ

e,i_

po

ten

tial)

&+

cha

rge

(i_

pa

rt)*

cha

rge

(j_

part

)&

*r_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

v_re

alc

ou

lom

b=

v_re

alc

ou

lom

b&

+ch

arg

e(i_

pa

rt)*

cha

rge

(j_

part

)&

*r_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

v_sh

ort

ran

ge

=v_

sho

rtra

ng

e&

+r_

po

t_1

(i_

typ

e,j_

typ

e,i_

pote

ntia

l)

r_d

um

my

=r_

po

t_d

_1

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

&+

cha

rge

(i_

pa

rt)*

cha

rge

(j_

part

)&

*r_

po

t_d

_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

call

calc

_fo

rce

_lo

c(r_

du

mm

y,i_

tro

t,i_

pa

rt,j_

pa

rt)

if(f

_ch

arg

ead

just

.eq

.3)

the

nm

u_

i(i_

pa

rt)

=m

u_

i(i_

pa

rt)

&+

r_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial)

*ch

arg

e(j_

pa

rt)

mu

_i(j_

pa

rt)

=m

u_

i(j_

pa

rt)

&+

r_p

ot_

2(j_

typ

e,i_

typ

e,i_

po

ten

tial)

*ch

arg

e(i_

pa

rt)

en

dif

en

dif

en

dd

ov_

inte

r_lo

c=

v_in

ter_

loc

+p

ot_

loc

en

dd

o

v_in

ter

=v_

inte

r+

v_in

ter_

loc

en

dd

o

v_in

ter

=v_

inte

r+

self_

en

erg

y

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

lag

ran

ge

_n

om

=0

.d

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

lag

ran

ge

_n

om

=la

gra

ng

e_

no

m+

&m

u_

i(i_

pa

rt)

/q

eq

ma

ss(i_

typ

e)

en

dd

ola

gra

ng

e_

de

no

m=

0.

do

i_ty

pe

=1

,n

_ty

pe

lag

ran

ge

_d

en

om

=la

gra

ng

e_

de

no

m+

121

B. Programs&

n_

ato

mty

pe

(i_

typ

e)

/q

eq

mass

(i_

typ

e)

en

dd

oe

nd

if

t_d

yna

mic

_ch

arg

e=

0.

v_ch

arg

e=

0.

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

!---

-ca

lcu

late

forc

eo

nch

arg

es

fQ(i_

pa

rt)

=-m

u_

i(i_

pa

rt)

+la

gra

ng

e_

no

m/

lag

ran

ge

_d

en

om

!---

-ca

lcu

late

en

erg

ies

of

the

ext

en

de

dla

gra

ng

ian

pa

rtt_

dyn

am

ic_

charg

e=

t_d

yna

mic

_ch

arg

e&

+0

.5*

qe

qm

ass

(i_

typ

e)*

(cha

rge

x(1

,i_p

art

)/d

t)**

2e

nd

ifv_

cha

rge

=v_

cha

rge

&+

qeq

ele

ctro

ne

g(i_

typ

e)*

cha

rge

(i_

pa

rt)

&+

0.5

*qe

qh

ard

ne

ss(i_

typ

e)*

charg

e(i_

pa

rt)*

*2e

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

ele

ctro

ne

ga

tivity

_sa

veim

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

ne

n(n

_p

art

),r_

1,

r_2

do

ub

lep

reci

sio

na

vg_

en

(n_

typ

e),

min

_e

n(n

_ty

pe

),m

ax_

en

(n_

type

)in

teg

er

nu

m_p

art

(n_

typ

e)

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

en

(i_

pa

rt)=

0

do

j_p

art

=1

,n

_p

art

if(j_

pa

rt.n

e.i_

pa

rt)

the

nj_

typ

e=

typ

e(j_

pa

rt)

do

i_tr

ot

=0

,n_

tro

t-1

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_d

im,i_

tro

t,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_d

im)

=d

elta

_ce

ll(i_

dim

)+

1.

en

dif

en

dd

o

r_2

=0

.ca

llca

lc_

dis

t_in

t(r_

2)

if(r_

2.lt

.r_cu

toff_

2(i_

typ

e,j_

typ

e))

the

nr_

1=

sqrt

(r_

2)

i_p

ote

ntia

l=

nin

t(n

_p

ote

ntia

l*r_

1/r

_ra

ng

e(i_

typ

e,j_

typ

e))

en

(i_

pa

rt)=

en

(i_

pa

rt)

&+

r_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

tentia

l)*

cha

rge

(j_

pa

rt)

en

dif

en

dd

o

en

dif

en

dd

oe

n(i_

pa

rt)=

en

(i_

pa

rt)+

qe

qe

lect

ron

eg

(i_

typ

e)

&-2

*ele

c_p

refa

c*a

lph

a/s

qrt

(pi)*c

ha

rge

(i_

pa

rt)

&+

cha

rge

(i_

pa

rt)*

qe

qh

ard

ne

ss(i_ty

pe

)e

nd

do

do

i_p

art

=1

,n_

pa

rtw

rite

(*,*

)i_

pa

rt,

en

(i_

pa

rt)

en

dd

o

!h

ier

ste

hts

!w

rite

(62

,’(1

00

0e

15

.7)’)

(en

(i_p

art

),i_

pa

rt=

1,n

_p

art

)

do

i_ty

pe

=1

,n

_ty

pe

avg

_e

n(i_

typ

e)=

0m

ax_

en

(i_

typ

e)=

-1.e

30

min

_e

n(i_

typ

e)=

1.e

30

nu

m_

pa

rt(i_

typ

e)=

0e

nd

do

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

nu

m_

pa

rt(i_

typ

e)=

nu

m_

pa

rt(i_

typ

e)+

1e

nd

do

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

avg

_e

n(i_

typ

e)=

avg

_e

n(i_

typ

e)+

en

(i_

pa

rt)/

nu

m_

pa

rt(i_

typ

e)

if(e

n(i_

pa

rt).

gt.m

ax_

en

(i_

typ

e))

&m

ax_

en

(i_

typ

e)=

en

(i_

pa

rt)

if(e

n(i_

pa

rt).

lt.m

in_

en

(i_

typ

e))

&m

in_

en

(i_

typ

e)=

en

(i_

pa

rt)

en

dd

o

write

(63

,’(2

0e1

5.7

)’)

(avg

_e

n(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&(m

in_e

n(i_

typ

e),

i_ty

pe

=1

,n_

type

),&

(ma

x_e

n(i_

typ

e),

i_ty

pe

=1

,n_

type

)

en

dsu

bro

utin

e

!!!!!!!!!!!!!!!!!!!!!

sub

rou

tine

ext

dla

gra

ng

e_

pre

dic

tor

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

!---

-p

red

ict

cha

rge

sd

oi_

pa

rt=

1,

n_

pa

rtd

oi_

ord

er=

1,

n_

ord

er

cha

rge

(i_

part

)=

cha

rge

(i_

pa

rt)

&+

pre

dic

t_co

ef(

0,i_

ord

er)

*ch

arg

ex(

i_o

rde

r,i_

pa

rt)

en

dd

oe

nd

do

!---

-p

red

ict

hig

he

ro

rde

rch

arg

es

do

i_p

art

=1

,n

_p

art

do

i_o

rde

r=

1,

n_

ord

er-

1d

oj_

ord

er

=i_

ord

er+

1,

n_

ord

er

cha

rge

x(i_

ord

er,

i_p

art

)=

&ch

arg

ex(

i_o

rde

r,i_

pa

rt)

+&

pre

dic

t_co

ef(

i_o

rde

r,j_

ord

er)

*ch

arg

ex(

j_o

rde

r,i_

pa

rt)

122


en

dd

oe

nd

do

en

dd

o

en

dsu

bro

utin

e

!!!!!!!!!!!!!!!!!!!!!

sub

rou

tine

ext

dla

gra

ng

e_

corr

ect

or

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sion

de

ltaQ

2(n

_p

art

)

!---

-su

bra

ctfr

ictio

nd

oi_

pa

rt=

1,n

_p

art

i_ty

pe

=ty

pe(i_

pa

rt)

fQ(i_

pa

rt)

=fQ

(i_

pa

rt)

&-

qe

qm

ass

(i_

typ

e)

*q

eq

fric

tion

*ch

arg

ex(

1,i_

pa

rt)/

dt

en

dd

o

!---

-ca

lcu

late

seco

nd

ord

er

corr

ect

ion

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe(i_

pa

rt)

de

ltaQ

2(i_

pa

rt)

=(d

t_2

*fQ

(i_

pa

rt)/

qeq

ma

ss(i_

typ

e)

/2

&-c

ha

rge

x(2

,i_p

art

))e

nd

do

!---

-co

rre

ctch

arg

es

do

i_p

art

=1

,n

_p

art

cha

rge

(i_

pa

rt)

=ch

arg

e(i_

pa

rt)

&+

corr

ect

_co

ef(

0)*

de

ltaQ

2(i_p

art

)d

oi_

ord

er

=1

,n

_o

rde

rch

arg

ex(

i_ord

er,

i_p

art

)=

cha

rge

x(i_

ord

er,

i_p

art

)&

+co

rre

ct_

coe

f(i_

ord

er)

*de

ltaQ

2(i_

pa

rt)

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!!!!!!!!!!!!!!!!!!!!

sub

rou

tine

ext

en

de

dla

gra

ng

eim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sio

nr_

1,

r_2

do

ub

lep

reci

sio

nq

eq

a(n

_p

art

,n_

pa

rt),

qe

qb

(n_

pa

rt)

do

ub

lep

reci

sio

nm

u_

i(n

_p

art

),su

m_

mu

do

ub

lep

reci

sio

nv_

qa

bh

ae

ng

ig_

ete

stco

mm

on

/ext

lage

test

/v_

qa

bh

ae

ng

ig_

ete

std

ou

ble

pre

cisi

on

qe

qre

c2(n

_p

art

,n_

pa

rt)

com

mo

n/q

eq

rect

est

/q

eq

rec2

inte

ge

ri_

test

do

ub

lep

reci

sio

np

mix

com

mo

n/p

mis

ch/

pm

ixd

ou

ble

pre

cisi

on

avg

_ch

arg

e(n

_ty

pe

),m

ax_

cha

rge

(n_

typ

e),

&m

in_

cha

rge

(n_

typ

e)

do

ub

lep

reci

sio

nv_

ew

ald

2d

ou

ble

pre

cisi

on

qe

qa

12

(n_

pa

rt,n

_p

art

)

!(c

an

be

incl

ud

ed

inp

ote

ntia

lro

utin

ela

ter)

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

j_p

art

=1

,n

_p

art

j_ty

pe

=ty

pe(j_

pa

rt)

i_tr

ot

=0

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_

dim

,i_tr

ot,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_dim

)=

de

lta_

cell(

i_d

im)

+1

.e

nd

ife

nd

do

r_2

=0

.ca

llca

lc_

dis

t_in

t(r_

2)

if(r_

2.le

.r_

cuto

ff_

2(i_

typ

e,j_

typ

e))

the

nr_

1=

sqrt

(r_

2)

i_p

ote

ntia

l=

nin

t(n

_p

ote

ntia

l*r_

1/r

_ra

ng

e(i_

typ

e,j_

typ

e))

qe

qa

(i_

pa

rt,j_

pa

rt)

=&

+r_

po

t_2

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

qe

qa

12

(i_

pa

rt,j_

pa

rt)

=0

.5*(

&+

r_p

ot_

2(i_

typ

e,j_

typ

e,i_

po

ten

tial)*c

ha

rge

(i_

pa

rt)

&*c

ha

rge

(j_

pa

rt)

+r_

po

t_1(i_

typ

e,j_

typ

e,i_

po

ten

tial))

if(i_

po

ten

tial.g

e.n

_p

ote

ntia

l)w

rite

(68

,*)

"gro

sse

Pa

nn

e"

els

e qe

qa

(i_

pa

rt,j_

pa

rt)=

0.0

qe

qa

12

(i_

pa

rt,j_

pa

rt)=

0.0

en

dif

en

dd

oe

nd

do

!ca

lcula

tea

ux

varia

ble

B_

id

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

qe

qb

(i_

pa

rt)

=q

eq

ha

rdn

ess

(i_

typ

e)

&-

2*e

lec_

pre

fac*

alp

ha

/sq

rt(p

i)e

nd

do

!---

-ca

lcu

late

che

mic

al

po

ten

tial

sum

_m

u=

0.

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

mu

_i(i_

pa

rt)

=q

eq

ele

ctro

ne

g(i_

typ

e)

+q

eq

rec(

i_p

art

)&

+q

eqb

(i_

pa

rt)*

cha

rge

(i_

pa

rt)

do

j_p

art

=1,

n_

pa

rtm

u_

i(i_

pa

rt)

=m

u_

i(i_

pa

rt)

+q

eq

a(i_

pa

rt,j_

pa

rt)

&*

cha

rge

(j_

pa

rt)

en

dd

osu

m_

mu

=su

m_

mu

+m

u_

i(i_

pa

rt)

en

dd

o

!---

-ca

lcu

late

resu

ltin

gfo

rce

do

i_p

art

=1

,n

_p

art

fQ(i_

pa

rt)

=-m

u_

i(i_

pa

rt)

+su

m_

mu

/n

_p

art

en

dd

o

!---

-ca

lcula

tee

ne

rgie

s

123

B. Programst_

dyn

am

ic_

cha

rge

=0

.v_

cha

rge

=0.

v_q

ab

ha

en

gig

_e

test

=0

.v_

inte

r=

0.

v_e

wa

ld2

=0.

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

t_d

yna

mic

_ch

arg

e=

t_d

yna

mic

_ch

arg

e&

+0

.5*

qe

qm

ass

(i_

typ

e)*

(ch

arg

ex(

1,i_

pa

rt)/

dt)

**2

v_ch

arg

e=

v_ch

arg

e&

+0

.5*(

mu

_i(i_

pa

rt)+

qe

qe

lect

ron

eg

(i_

typ

e))

*ch

arg

e(i_

part

)&

+q

eqe

lect

ron

eg

(i_

typ

e)*

cha

rge

(i_

pa

rt)

&+

0.5

*qe

qh

ard

ne

ss(i_

typ

e)*

charg

e(i_

pa

rt)*

*2e

nd

do

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

do

j_p

art

=1

,n

_p

art

if(i_

pa

rt.n

e.j_

pa

rt)

the

nj_

typ

e=

type

(j_

pa

rt)

v_q

ab

ha

en

gig

_e

test

=v_

qa

bh

ae

ng

ig_

ete

st+

&0

.5*

qe

qa

(i_

pa

rt,j_

pa

rt)*

cha

rge

(i_

pa

rt)

&*c

ha

rge

(j_

pa

rt)

v_in

ter

=v_

inte

r+

qe

qa

12

(i_

pa

rt,j_

pa

rt)

en

dif

en

dd

ov_

qa

bh

ae

ng

ig_

ete

st=

v_q

ab

ha

en

gig

_e

test

&+

0.5

*q

eq

b(i_

pa

rt)*

cha

rge

(i_p

art

)**2

&+

qeq

ele

ctro

ne

g(i_

typ

e)*

cha

rge

(i_

pa

rt)

v_in

ter

=v_

inte

r-

ele

c_p

refa

c*a

lph

a/s

qrt

(pi)

&*c

harg

e(i_

pa

rt)*

cha

rge

(i_

pa

rt)

&+

0.5

*qe

qre

c(i_

pa

rt)*

cha

rge

(i_

pa

rt)

v_e

wa

ld2

=v_

ew

ald

2+

0.5

*qe

qre

c(i_

pa

rt)*

cha

rge

(i_

pa

rt)

en

dd

ov_

qa

bh

ae

ng

ig_

ete

st=

v_q

ab

ha

en

gig

_e

test

+v_

ew

ald

do

i_ty

pe

=1

,n

_ty

pe

avg

_ch

arg

e(i_

typ

e)=

0m

ax_

cha

rge

(i_

typ

e)=

-1.e

30

min

_ch

arg

e(i_

typ

e)=

1.e

30

en

dd

od

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

avg

_ch

arg

e(i_

typ

e)=

avg

_ch

arg

e(i_

typ

e)+

cha

rge

(i_

pa

rt)

if(c

ha

rge

(i_

pa

rt).

gt.m

ax_

cha

rge

(i_

typ

e))

&m

ax_

cha

rge

(i_

typ

e)=

cha

rge

(i_

pa

rt)

if(c

ha

rge

(i_

pa

rt).

lt.m

in_

cha

rge

(i_

typ

e))

&m

in_ch

arg

e(i_

typ

e)=

cha

rge

(i_

pa

rt)

en

dd

oa

vg_

cha

rge

(1)=

1.d

0*a

vg_

cha

rge

(1)/

n_

silic

avg

_ch

arg

e(2

)=1

.d0

*avg

_ch

arg

e(2

)/n

_oxy

write

(64

,’(i6

,20

e3

0.2

2)’)

i_te

st,

2a

vg_ch

arg

e(1

),3

min

_ch

arg

e(1

),4

ma

x_ch

arg

e(1

),5

avg

_ch

arg

e(2

),6

min

_ch

arg

e(2

),7

ma

x_ch

arg

e(2

),

8p

mix

,9

t_d

yna

mic

_ch

arg

e/n

_p

art

,&

v_ch

arg

e/n

_p

art

,!1

0&

v_q

ab

ha

en

gig

_e

test

/n_

pa

rt,

!11

&v_

ew

ald

/n_

pa

rt,

!12

&v_

inte

r/n

_p

art

,!1

3&

v_e

wa

ld2

/n_

pa

rt!1

4

en

dsu

bro

utin

e

!!!!!!!!!!!!!!!!!!!!!

sub

rou

tine

ma

trix

inve

rsio

nim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

inte

ge

rm

,n,n

p,m

pP

AR

AM

ET

ER

(np

=n

_p

art

,mp

=1

)d

ou

ble

pre

cisi

on

a(n

p,n

p),

b(n

p,m

p)

do

ub

lep

reci

sio

nr_

2,

r_1

inte

ge

rk_

pa

rtd

ou

ble

pre

cisi

on

qe

qre

c2(n

_p

art

,n_

pa

rt)

com

mo

n/q

eq

rect

est

/q

eq

rec2

do

ub

lep

reci

sio

nq

eq

a(n

_p

art

,n_

pa

rt),

qe

qb

(n_

pa

rt)

do

ub

lep

reci

sio

ne

nd

ou

ble

pre

cisi

on

row

_d

um

my

inte

ge

rin

dx(

n_

pa

rt)

n=

np

m=

mp

if(f

_p

ote

ntia

l.ne

.1)

the

nca

llch

eck

_sk

inif

(f_

skin

.eq

.1)

the

nca

llb

inn

ing

en

dif

en

dif

call

ma

trix

_re

c(q

eq

rec2

)

b(1

,1)=

0.

!Q

_to

tal

a(1

,1)=

1.

j_ty

pe

=ty

pe

(1)

do

i_p

art

=2

,n

_p

art

!fir

stro

w1

i_ty

pe

=ty

pe

(i_

pa

rt)

a(1

,i_p

art

)=1

.b

(i_

pa

rt,1

)=-(

qe

qe

lect

ron

eg

(i_

typ

e)-

qe

qe

lect

ron

eg

(j_

typ

e))

en

dd

o

i_ty

pe

=ty

pe

(1)

!fir

stco

lum

n-B

1d

oi_

pa

rt=

2,

n_

pa

rta

(i_

pa

rt,1

)=-(

qe

qh

ard

ne

ss(i_

typ

e)-

&2

*ele

c_p

refa

c*a

lph

a/s

qrt

(pi)

+q

eq

rec2

(1,1

))q

eq

b(1

)=-a

(i_

pa

rt,1

)e

nd

do

i_p

art

=1

!i=2

..n

xj=

2..n

-A1

ji_

typ

e=

typ

e(i_

pa

rt)

do

j_p

art

=2

,n

_p

art

124


j_ty

pe

=ty

pe

(j_

pa

rt)

i_tr

ot

=0

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_

dim

,i_tr

ot,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_d

im)

=d

elta

_ce

ll(i_

dim

)+

1.

en

dif

en

dd

o

r_2

=0

.ca

llca

lc_

dis

t_in

t(r_

2)

if(r_

2.le

.r_

cuto

ff_

2(i_

typ

e,j_

typ

e))

the

nr_

1=

sqrt

(r_

2)

i_p

ote

ntia

l=

nin

t(n

_p

ote

ntia

l*r_

1/r

_ra

ng

e(i_

typ

e,j_

typ

e))

do

k_p

art

=2

,n_

pa

rta

(k_

pa

rt,j_

pa

rt)=

-r_

po

t_2

q(i_

typ

e,j_

typ

e,i_

po

ten

tial)

&-q

eq

rec2

(i_

pa

rt,j_

part

)e

nd

do

els

e do

k_p

art

=2

,n_

pa

rta

(k_

pa

rt,j_

pa

rt)=

-qe

qre

c2(i_

pa

rt,j_

part

)e

nd

do

en

dif

qe

qa

(1,j_

part

)=-a

(2,j_

pa

rt)

en

dd

o

do

i_p

art

=2

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

j_p

art

=1,

n_

pa

rtif

(i_

pa

rt.n

e.j_

pa

rt)

the

n!

with

ou

td

iag

on

al:

Aij

j_ty

pe

=ty

pe

(j_

pa

rt)

i_tr

ot

=0

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

part

)-

r0(i_

dim

,i_tr

ot,j_

pa

rt)

if(d

elta

_ce

ll(i_

dim

).g

t.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_d

im)

=d

elta

_ce

ll(i_

dim

)+

1.

en

dif

en

dd

o

r_2

=0

.ca

llca

lc_

dis

t_in

t(r_

2)

!w

rite

(*,*

)i_

pa

rt,

j_p

art

,sq

rt(r

_2

)

if(r_

2.le

.r_

cuto

ff_

2(i_

typ

e,j_

typ

e))

the

nr_

1=

sqrt

(r_

2)

i_p

ote

ntia

l=

nin

t(n

_p

ote

ntia

l*r_

1/r

_ra

ng

e(i_

typ

e,j_

typ

e))

a(i_

pa

rt,j_

pa

rt)=

a(i_

pa

rt,j_

pa

rt)

&+

r_p

ot_

2q

(i_

typ

e,j_

type

,i_p

ote

ntia

l)&

+q

eq

rec2

(i_

pa

rt,j_

pa

rt)

qe

qa

(i_

pa

rt,j_

pa

rt)=

r_p

ot_

2q

(i_

typ

e,j_

typ

e,i_

po

ten

tial)

&+

qe

qre

c2(i_

pa

rt,j_

pa

rt)

els

ea

(i_

pa

rt,j_

pa

rt)=

a(i_

pa

rt,j_

pa

rt)

&+

qe

qre

c2(i_

pa

rt,j_

pa

rt)

qe

qa

(i_

pa

rt,j_

pa

rt)=

qe

qre

c2(i_

pa

rt,j_

pa

rt)

en

dif

els

e!

dia

go

na

l:B

ia

(i_

pa

rt,i_

pa

rt)=

a(i_

pa

rt,i_

pa

rt)

+q

eq

ha

rdn

ess

(i_

typ

e)

&-

2*e

lec_

pre

fac*

alp

ha

/sq

rt(p

i)+

qe

qre

c2(i_

pa

rt,i_

pa

rt)

qe

qb

(i_

pa

rt)

=q

eq

ha

rdn

ess

(i_

typ

e)

&-

2*e

lec_

pre

fac*

alp

ha

/sq

rt(p

i)+

qe

qre

c2(i_

pa

rt,i_

pa

rt)

en

dif

en

dd

oe

nd

do

call

lud

cmp

(a,n

,np

,ind

x,ro

w_

du

mm

y)ca

lllu

bks

b(a

,n,n

p,in

dx,

b)

v_q

eq

self=

0.

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

cha

rge

(i_

pa

rt)=

b(i_

pa

rt,1

)v_

qe

qse

lf=v_

qe

qse

lf&

+ch

arg

e(i_

pa

rt)*

qe

qe

lect

ron

eg

(i_

typ

e)

&+

0.5

*ch

arg

e(i_

pa

rt)*

cha

rge

(i_

pa

rt)*

qe

qh

ard

ne

ss(i_

typ

e)

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

sub

rou

tine

ma

trix

_re

c(q

eq

rec2

)im

plic

itn

one

incl

ud

e"p

imd

.co

m.f"

inte

ge

ri_

q(n

_d

im),

tro

t_m

ax

do

ub

lep

reci

sio

nq

_ve

c(n

_d

im),

q_

2,lo

c_p

refa

c,re

c_p

refa

cd

ou

ble

pre

cisi

on

two

_re

c_p

refa

cco

mp

lex*

16

plu

s_iq

r(n

_p

art

),m

inu

s_iq

r(n

_p

art

),&

diff

_iq

r(n

_p

art

,n_

pa

rt)

do

ub

lep

reci

sio

nq

eq

rec2

(n_

pa

rt,n

_p

art

)

do

i_p

art

=1

,n

_p

art

do

j_p

art

=1,

n_

pa

rtq

eq

rec2

(i_

pa

rt,j_

pa

rt)=

0.

en

dd

oe

nd

do

loc_

pre

fac

=e

lec_

pre

fac*

two

pi/v

olu

me

j_tr

ot

=m

od(in

t(n

_tr

ot*

ran

do

m(1

)),n

_tr

ot)

do

i_re

c_ve

c=

1,n

_re

c_ve

cq

_2

=0

.d

oi_

dim

=1

,n_d

imi_

q(i_

dim

)=

rec_

vec_

list(

i_re

c_ve

c,i_

dim

)q

_ve

c(i_

dim

)=

i_q

(i_

dim

)*d

q(i_

dim

,i_d

im)

q_

2=

q_

2+

q_

vec(

i_d

im)*

*2e

nd

do

rec_

pre

fac

=lo

c_p

refa

c*e

xp(-

q_

2/(

2*a

lph

a)*

*2)/

q_

2if(

rec_

vec_

list(

i_re

c_ve

c,3

).n

e.0

)re

c_p

refa

c=

2*

rec_

pre

fac

two

_re

c_p

refa

c=

2*r

ec_

pre

fac

tro

t_m

ax

=n

_tr

ot-

1if

(f_

op

t_re

c.eq

.1)

tro

t_m

ax

=0

125

B. Programsd

oi_

tro

t=

0,

tro

t_m

ax

if(f

_o

pt_

rec.

eq

.0)

j_tr

ot

=i_

tro

t

do

i_p

art

=1

,n

_p

art

r_d

um

my

=0

.if

(f_

op

t_re

c.eq

.0)

the

nd

oi_

dim

=1

,n_

dim

r_d

um

my

=r_

du

mm

y+

i_q

(i_

dim

)*r0

(i_

dim

,j_tr

ot,i_

pa

rt)

en

dd

oe

lse

do

i_d

im=

1,

n_

dim

r_d

um

my

=r_

du

mm

y+

i_q

(i_

dim

)*

&(

r_re

f(i_

dim

,i_p

art

)+

u0

(i_

dim

,1,i_

pa

rt)/

sqrt

p)

en

dd

oe

nd

ifr_

du

mm

y=

dm

od

(r_

du

mm

y,1

.d0

)i_

du

mm

y=

nin

t(r_

du

mm

y*n

_p

ote

ntia

l)p

lus_

iqr(

i_p

art

)=!

&ta

b_

exp

_im

a(i_

du

mm

y)&

exp

(2.*

pi*

(0.,1

.)*r

_d

um

my)

i_d

um

my

=n

int(

-r_

du

mm

y*n

_p

ote

ntia

l)m

inu

s_iq

r(i_

pa

rt)=

!&

tab

_e

xp_

ima

(i_

du

mm

y)&

exp

(-2

.*p

i*(0

.,1

.)*r

_d

um

my)

en

dd

o

go

to2

34

do

i_p

art

=1

,n

_p

art

do

j_p

art

=1

,n

_p

art

do

i_d

im=

1,n

_d

imd

elta

_ce

ll(i_

dim

)=

&r0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r0

(i_

dim

,i_tr

ot,j_

part

)if(

de

lta_

cell(

i_d

im).

gt.0

.5)

the

nd

elta

_ce

ll(i_

dim

)=

de

lta_

cell(

i_d

im)

-1

.e

lse

if(d

elta

_ce

ll(i_

dim

).lt.

-0.5

)th

en

de

lta_

cell(

i_dim

)=

de

lta_

cell(

i_d

im)

+1

.e

nd

ife

nd

do

r_d

um

my

=0.

do

i_d

im=

1,n

_d

imr_

du

mm

y=

r_d

um

my

+i_

q(i_

dim

)&

*de

lta_

cell(

i_d

im)

en

dd

o!

r_d

um

my

=d

mo

d(r

_d

um

my,

1.d

0)

!i_

du

mm

y=

nin

t(r_

du

mm

y*n

_p

ote

ntia

l)!

diff

_iq

r(i_

pa

rt,j_

pa

rt)=

!&

tab

_e

xp_

ima

(i_

dum

my)

!&

exp

(2.*

pi*

(0.,1

.)*r

_d

um

my)

diff

_iq

r(i_

pa

rt,j_

pa

rt)=

!&

tab

_e

xp_

ima

(i_

dum

my)

&e

xp(2

.*p

i*(0

.,1

.)*r

_d

um

my)

en

dd

oe

nd

do

23

4co

ntin

ue

do

i_p

art

=1

,n

_p

art

do

j_p

art

=1,

n_

pa

rtq

eq

rec2

(i_

pa

rt,j_

pa

rt)=

qe

qre

c2(i_

pa

rt,j_

pa

rt)

&+

two

_re

c_p

refa

c*d

ble

(plu

s_iq

r(i_

pa

rt)*

&m

inu

s_iq

r(j_

pa

rt))

!q

eq

rec2

(i_

pa

rt,j_

pa

rt)=

qe

qre

c2(i_

pa

rt,j_

pa

rt)

!&

+tw

o_

rec_

pre

fac*

db

le(d

iff_

iqr(

i_p

art

,j_pa

rt))

en

dd

oe

nd

do

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

therm

ost

at

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

!---

-o

nly

n_tr

ot*

n_

pa

rtin

de

pe

nd

en

t(r

an

do

m)

forc

es

!---

-re

al

part

of

i_tr

ot

=0

,i_

tro

t=

n_

tro

t/2

!---

-b

oth

part

so

f1

.le.

i_tr

ot

.le.

n_

tro

t/2

do

j_tr

ot

=0

,n

_tr

ot-

1i_

tro

t=

j_tr

ot

+1

!---

cho

ose

larg

er

da

mp

ing

for

inte

rna

lm

od

es

if(i_

tro

t.eq

.1)

the

nr_

du

mm

y=

1.

/n

_tim

e_

sca

le

lse

r_d

um

my

=sq

rt(1

.*n

_tim

e_

sca

l)if(

i_tr

ot.e

q.2

)i_

tro

t=

n_

tro

t+

1e

nd

if

i_d

um

my

=(i_

tro

t-1

)/2

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

call

r25

0(m

z,ra

nd

om

,n_

dim

,n_

dim

,kp

tr)

do

i_d

im=

1,n

_d

imf0

(i_

dim

,i_tr

ot,i_

pa

rt)

=f0

(i_

dim

,i_tr

ot,i_

pa

rt)

&-

ma

ss(i_

du

mm

y,i_

typ

e)

*fr

ictio

n(i_

typ

e)

*r_

du

mm

y**2

&*

(ux(

1,i_

dim

,i_tr

ot,i_

pa

rt)/

dt)

&-

k_e

igen

(i_

du

mm

y,i_

typ

e)

*u

0(i_

dim

,i_tr

ot,i_

pa

rt)

fr(i_

dim

,i_tr

ot,i_

pa

rt)

=r_

du

mm

y&

*(2

*ra

nd

om

(i_

dim

)-1

.)/

ma

ss(i_d

um

my,

i_ty

pe

)!-

---

wid

tho

fra

nd

om

forc

eis

know

nin

pre

dic

tor

en

dd

oe

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

+e

_d

imfo

rce

_sc

al(i_

dim

)=

forc

e_

sca

l(i_

dim

)&

-m

ass

_b

ox

*fr

ic_

bo

x*

(sca

l_x(

1,i_

dim

)/d

t)e

nd

do

if(f

ric_

bo

x.gt.0

.)ca

llr2

50

(mz,

ran

dom

,n_

dim

,n_

dim

,kp

tr)

do

i_d

im=

1,

n_

dim

fr_

bo

x(i_

dim

)=

(2*r

an

do

m(i_

dim

)-1

.)/

ma

ss_

bo

xe

nd

do

if(e

_d

im.e

q.3

)th

en

call

r25

0(m

z,ra

nd

om

,n_

dim

,n_

dim

,kp

tr)

do

i_d

im=

1,

3fr

_b

ox(

i_d

im+

n_

dim

)=

(2*r

an

do

m(i_

dim

)-1

.)/

ma

ss_

bo

xe

nd

do

en

dif

en

dsu

bro

utin

e

126


!!

!!

!!

!!

!!

!

sub

rou

tine

ob

se_

init

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

inte

ge

rjs

ee

dco

mm

on

/cjs

ee

d/

jse

ed

t_d

yna

mic

_1

=0.

t_d

yna

mic

_2

=0.

t_d

iag

_1

=0

.t_

nd

iag

_1

=0

.

v_in

ter_

1=

0.

v_in

ter_

2=

0.

viri_

est

_1

=0

.d0

viri_

est

_2

=0

.d0

prim

_e

st_

1=

0.d

0p

rim

_e

st_

2=

0.d

0

do

i_d

im=

1,

n_

dim

+3

sca

l_1

(i_

dim

)=

0.

do

j_d

im=

1,n

_d

im+

3sc

al_

2(j_

dim

,i_d

im)

=0

.e

nd

do

en

dd

ovo

l_1

=0

.vo

l_2

=0

.

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

stra

in_

1(i_

dim

,j_d

im)

=0

.e

la_

kin

(i_

dim

,j_d

im)

=0

.d

ok_

dim

=1

,n

_d

imd

ol_

dim

=1

,n

_d

ime

la_

bo

rn_

1(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

0.

stra

in_

2(i_d

im,

j_d

im,

k_d

im,

l_d

im)

=0

.e

nd

do

en

dd

oe

nd

do

en

dd

o

di_

time

=1

ieff_

time

=0

do

i_tr

ot

=0

,n_

tro

t-1

cor_

ifun

c(i_

tro

t)=

0.d

0e

nd

do

do

i_co

r=

0,n

_co

rco

r_o

bs(

i_co

r)=

0.d

0q

do

t_co

r(i_

cor)

=0

.d0

do

i_ty

pe

=1,

n_

typ

eco

r_fu

nc(

i_co

r,i_

typ

e)

=0

.d0

en

dd

oco

r_fu

nc_

A(i_co

r)=

0.d

0co

r_fu

nc_

B(i_co

r)=

0.d

0e

nd

do

do

i_ty

pe

=1

,n_

typ

ed

oj_

typ

e=

1,n

_ty

pe

do

i_g

rid

=0

,n_

grid

+1

g_

r(i_

typ

e,j_

typ

e,i_

grid

)=

0.

en

dd

oe

nd

do

en

dd

o

time

_o

bs

=0.d

0co

un

t_ifu

nc

=0

i_m

ea

sure

=n

_co

r

time

_b

orn

=0

.

ord

_p

ar_

1=

0.d

0o

rd_

pa

r_2

=0

.d0

ord

_p

ar_

4=

0.d

0o

rd_

pa

rB_

1=

0.d

0o

rd_

pa

rB_

2=

0.d

0o

rd_

pa

rB_

4=

0.d

0o

rd_

pa

rC_

1=

0.d

0o

rd_

pa

rC_

2=

0.d

0o

rd_

pa

rC_

4=

0.d

0

!m

ovi

ein

itif

(f_

mo

vie

.eq

.1)

the

no

pe

n(4

6,file

="m

ovi

e.x

yz",

sta

tus=

"new

")o

pe

n(4

7,file

="m

ovi

e.p

ol"

,sta

tus=

"new

")w

rite

(46

,*)

n_

pa

rtw

rite

(46

,*)

en

dif

if(f

_co

nfm

ovi

e.e

q.1

)th

en

op

en

(48

,file

="m

ovi

e.c

on

f",s

tatu

s="n

ew

")e

nd

if

if(i_

run

.eq

.1)

the

no

pe

n(2

0,file

="p

imd

.ou

t",s

tatu

s="u

nkn

ow

n")

write

(20

,21

0)

f_p

ote

ntia

l,"f_

po

ten

tial,

info

rma

tion

ava

ilab

le"

write

(20

,21

0)

f_ch

arg

ea

dju

st,"

f_ch

arg

ea

dju

st"

write

(20

,21

0)

n_

rela

x,"

#o

fre

laxa

tion

ste

ps"

write

(20

,21

0)

n_

ob

ser,

"#

of

ob

serv

atio

nst

ep

s"w

rite

(20

,21

0)

n_

safe

,"#

of

ste

ps

be

twe

en

con

figu

ratio

no

utp

ut"

write

(20

,21

0)

jse

ed

+2

,"ra

nd

om

#in

it"w

rite

(20

,*)

write

(20

,20

1)

dt*

n_

time

_sc

al,"

time

ste

p"

write

(20

,20

1)

tk,"

tem

pe

ratu

re"

write

(20

,*)

write

(20

,20

2)

(ma

ss(0

,i_ty

pe

),i_

typ

e=

1,n

_ty

pe

),"

ma

sse

s"w

rite

(20

,20

2)

(frict

ion

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

"fr

ictio

n"

write

(20

,20

2)

ma

ss_

bo

x/n

_tr

ot,

fric

_b

ox,

"m

ass

+fr

.o

fb

ox"

if(f_

po

ten

tial.g

t.1

)th

en

write

(20

,20

1)

write

(20

,20

1)

pre

ss,"

pre

ssu

re"

write

(20

,21

0)

f_p

ress

ure

,"0

=N

VT

;1

=N

pT

"if

(f_

po

ten

tial.e

q.3

.or.

(&

(f_

po

ten

tial.g

e.5

).a

nd

.(f_

po

tentia

l.le

.8))

)&

write

(20

,210

)f_

op

t_re

c,"

1=

op

timiz

ed

Ew

ald

"w

rite

(20

,*)

if(f

_ch

arg

ead

just

.ne

.1)

the

n

127

B. Programsw

rite

(20

,20

2)

(qe

qe

lect

ron

eg

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

ele

ctro

ne

ga

tivity

"w

rite

(20

,20

2)

(qe

qh

ard

ne

ss(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

ha

rdne

sso

fE

N"

en

dif

write

(20

,*)

if(f

_ch

arg

ea

dju

st.e

q.3

)th

en

write

(20

,20

2)

(qe

qm

ass

(i_

typ

e),

i_ty

pe=

1,n

_ty

pe

),&

"fic

titio

us

cha

rge

ma

ss"

write

(20

,20

1)

qe

qfr

ictio

n,

"ch

arg

efr

ictio

n"

en

dif

write

(20

,*)

if(f_

po

ten

tial.e

q.2

)th

en

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(ep

sil(i_

typ

e,j_

typ

e)/

4,j_

typ

e=

1,n

_ty

pe

),&

"e

ne

rgy

pa

ram

ete

rs"

en

dd

ow

rite

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(sig

ma

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

len

gth

sca

lep

ara

me

ters

"e

nd

do

els

eif(

f_p

ote

ntia

l.eq

.3)

the

nd

oi_

typ

e=

1,n

_ty

pe

write

(20

,20

2)

(ep

sil(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

exp

.e

ne

rgy

pa

ram

ete

rs"

en

dd

ow

rite

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(sig

ma

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

inve

rse

exp

.le

ng

ht

sca

les"

en

dd

ow

rite

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(c_

van

dw

aa

ls(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

van

de

Wa

als

stre

ng

th"

en

dd

oe

lse

if((f

_po

ten

tial.g

e.5

).a

nd

.(f_

po

ten

tial.l

e.7

))th

en

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(d0

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

MS

bo

nd

stre

ng

thD

0"

en

dd

ow

rite

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(msr

0(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

MS

bo

nd

len

gth

R0

"e

nd

do

write

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(ga

mm

a(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

MS

forc

eco

nst

an

tg

am

ma

"e

nd

do

write

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(msz

eta

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

slate

ro

rbita

le

xpo

ne

nt

zeta

"e

nd

do

write

(20

,*)

write

(20

,22

0)

(msn

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

prin

cip

al

qu

an

tum

nu

mb

er

n"

els

eif(

f_p

ote

ntia

l.eq

.8)

the

nd

oi_

typ

e=

1,n

_ty

pe

write

(20

,20

2)

(va

sh_

h(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

rep

uls

ive

stre

ng

thH

"e

nd

do

write

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(va

sh_

eta

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

rep

uls

ive

exp

on

en

tse

ta"

en

dd

ow

rite

(20

,*)

write

(20

,20

2)

(va

sh_

alp

ha

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

po

lariza

bili

tya

lph

a"

write

(20

,20

2)

(va

sh_

B(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

thre

eb

od

yB

"w

rite

(20

,20

2)

(va

sh_

l(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

thre

eb

od

yl"

write

(20

,20

2)

(va

sh_

cth

eta

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

&"

thre

eb

od

yco

sth

eta

"w

rite

(20

,20

2)

(va

sh_

r0(i_

typ

e),

i_ty

pe=

1,n

_ty

pe

),&

"th

ree

bo

dy

r0"

en

dif

write

(20

,*)

if(f_

po

ten

tial.e

q.3

.or.

&((

f_p

ote

ntia

l.ge

.5).

an

d.(

f_po

ten

tial.l

e.9

)))

the

nw

rite

(20

,20

2)

alp

ha

,v_

cut,"

alp

ha

,v_

cut"

en

dif

if(f

_p

ote

ntia

l.le

.2.o

r.f_

po

ten

tial.e

q.4

)th

en

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(r_

cuto

ff(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

cuto

ffra

diu

s"e

nd

do

en

dif

write

(20

,*)

do

i_ty

pe

=1

,n_

typ

ew

rite

(20

,20

2)

(r_

skin

(i_

typ

e,j_

typ

e),

j_ty

pe

=1

,n_

typ

e),

&"

skin

rad

ius"

en

dd

oe

nd

ifw

rite

(20

,*)

write

(20

,21

0)

n_

silic

,"#

of

silic

ium

ato

ms"

write

(20

,21

0)

n_

oxy

,"#

of

oxy

ge

nato

ms"

write

(20

,21

0)

n_

tro

t,"

Tro

tte

r#

"w

rite

(20

,21

0)

n_

ord

er,

"o

rde

ro

fin

teg

rato

r"w

rite

(20

,21

0)

n_

time

_sc

al,"

time

sca

lese

pa

ratio

nin

t/e

xt"

if(f

_p

ote

ntia

l.eq

.3.o

r.&

((f_

po

ten

tial.g

e.5

).a

nd

.(f_

pote

ntia

l.le

.8))

)th

en

write

(20

,*)

write

(20

,20

2)

alp

ha

,sq

rt(q

_cu

t_2

),"

alp

ha

,q_

cut"

write

(20

,20

2)

erf

cc(a

lph

a*r

_cu

toff(1

,1))

&,e

xp(-

q_

cut_

2/(

2*a

lph

a)*

*2),

"m

ax.

ind

.e

rro

rs"

write

(20

,20

1)

r_cu

t_sh

ort

,"r_

cut_

short

"w

rite

(20

,21

0)

f_sy

mm

etr

y,"

sym

me

try"

en

dif

en

dif

if(f

_p

ote

ntia

l.eq

.3.o

r.&

((f_

po

ten

tial.g

e.5

).a

nd

.(f_

pote

ntia

l.le

.8))

)&

call

his

t_a

ng

_in

it

dip

ole

_cl

ass

1=

0.d

0d

ipo

le_

cla

ss2

=0

.d0

dip

ole

_cl

ass

4=

0.d

0

128


dip

ole

_co

un

t=

0

op

en

(86

,file

=’p

imd

.sio

si’,s

tatu

s=’u

nkn

ow

n’)

op

en

(87

,file

=’p

imd

.osi

o’,s

tatu

s=’u

nkn

ow

n’)

!o

pe

n(6

5,file

=’p

imd

.dip

’,sta

tus=

’un

kno

wn

’)

20

1fo

rma

t(1

e14

.6,a

)2

02

form

at(

2e

14

.6,a

)2

20

form

at(

2i1

4,a

)2

03

form

at(

3e

14

.6,a

)2

10

form

at(

1i1

4,a

)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

his

t_a

ng

_in

itim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

i_g

rid

=1

,n

_g

rid

do

i_d

um

my

=1

,3

his

t_a

ng

(i_d

um

my,

i_g

rid

)=

0.

en

dd

oe

nd

do

ne

igh

_tim

e=

0.

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

ob

serv

atio

nim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

inte

ge

rj_

cor,

k_co

r,j_

me

asu

re,

i_cl

ass

do

ub

lep

reci

sion

r_2

,r_

1,

vel(n

_d

im),

pb

c_n

orm

inte

ge

rp

bc_

x,pb

c_y,

pb

c_z

t_d

yna

mic

_1

=t_

dyn

am

ic_

1+

t_d

yna

mic

*dt

t_d

yna

mic

_2

=t_

dyn

am

ic_

2+

t_d

yna

mic

**2

*dt

t_d

iag

_1

=t_

dia

g_

1+

t_d

iag

*dt

t_n

dia

g_

1=

t_nd

iag

_1

+t_

nd

iag

*dt

v_in

ter_

1=

v_in

ter_

1+

(v_

inte

r-v_

ga

uge

)*d

tv_

inte

r_2

=v_

inte

r_2

+(v

_in

ter-

v_g

au

ge

)**2

*dt

if(f

_p

ote

ntia

l.gt.1

)ca

llvi

ri_

est

ima

tor

viri_

est

_1

=vi

ri_

est

_1

+vi

ri_

est

*dt

viri_

est

_2

=vi

ri_

est

_2

+vi

ri_

est

**2

*dt

call

prim

_e

stim

ato

r!

r_d

um

my

=n

_d

im*n

_p

art

*n_

tro

t*tk

p/2

-v_

cha

inr_

du

mm

y=

t_d

yna

mic

-v_

cha

inp

rim

_e

st_

1=

prim

_e

st_

1+

r_d

um

my*

dt

prim

_e

st_

2=

prim

_e

st_

2+

r_d

um

my*

*2*d

t

do

i_d

im=

1,

n_

dim

+3

sca

l_1

(i_

dim

)=

sca

l_1

(i_

dim

)+

sca

l_0

(i_

dim

)*d

td

oj_

dim

=1

,n_

dim

+3

sca

l_2

(j_

dim

,i_d

im)

=sc

al_

2(j_

dim

,i_d

im)

&+

sca

l_0

(j_

dim

)*sc

al_

0(i_

dim

)*d

te

nd

do

en

dd

ovo

l_1

=vo

l_1

+vo

lum

e*d

tvo

l_2

=vo

l_2

+vo

lum

e**

2*d

t

time

_o

bs

=tim

e_

ob

s+

dt

if(m

od

(ie

ff_

time

,ni_

time

).e

q.0

)th

en

!---

-im

agin

ary

time

corr

ela

tion

fun

ctio

n

call

r25

0(m

z,ra

nd

om

,n_

dim

,1,k

ptr

)i_

tro

t=

int(

n_tr

ot*

ran

do

m(1

))if

(i_

tro

t.e

q.n

_tr

ot)

i_tr

ot

=0

cou

nt_

ifun

c=

cou

nt_

ifun

c+

1

!---

-T

ER

Mm

ea

sco

r_ifu

nc

do

i_d

um

my

=0

,n

_tr

ot-

1j_

tro

t=

mo

d(i_tr

ot+

i_d

um

my,

n_

tro

t)d

oi_

pa

rt=

1,n

_p

art

i_ty

pe

=ty

pe(i_

pa

rt)

do

i_d

im=

1,n

_d

imco

r_ifu

nc(

i_d

um

my)

=co

r_ifu

nc(

i_d

um

my)

+m

ass

(0,i_

typ

e)

*(

&r_

rea

l(i_

dim

,i_tr

ot,i_

pa

rt)

-r_

real(i_

dim

,j_tr

ot,i_

pa

rt)

)**

2e

nd

do

en

dd

oe

nd

do

!---

-ce

ntr

oid

time

corr

ela

tion

funct

ion

!---

--T

ER

Mm

ea

sco

nf_

cen

trif(

i_m

ea

sure

.eq.n

_co

r)th

en

do

i_p

art

=1,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

do

i_d

im=

1,n

_d

imco

nf_

cen

tr(i_d

im,i_

pa

rt)

=0

.q

do

t0(i_

dim

,i_p

art

)=

0.

!Q

’(t=

0)

do

j_d

im=

1,

n_

dim

if(

(e_

dim

.eq

.3)

.or.

(i_

dim

.eq

.j_dim

))

the

nco

nf_

cen

tr(i_

dim

,i_p

art

)=

con

f_ce

ntr

(i_

dim

,i_p

art

)&

+h

_m

at(

i_d

im,j_

dim

)*

ux(

1,j_

dim

,1,i_

pa

rt)

qd

ot0

(i_

dim

,i_p

art

)=

qd

ot0

(i_

dim

,i_p

art

)&

+ch

arg

ex(

1,i_

pa

rt)

en

dif

en

dd

oco

nf_

cen

tr(i_d

im,i_

pa

rt)

=co

nf_

cen

tr(i_

dim

,i_p

art

)/

dt

qd

ot0

(i_

dim

,i_p

art

)=

qd

ot0

(i_

dim

,i_pa

rt)

/d

te

nd

do

en

dd

oi_

me

asu

re=

0co

r_fu

nc(

n_

cor,

1)

=co

r_fu

nc(

n_

cor,

1)

+1

.q

do

t_co

r(n

_co

r)=

qd

ot_

cor(

n_

cor)

+1.

cor_

fun

c_A

(n_co

r)=

cor_

fun

c_A

(n_

cor)

+1

.e

nd

if

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

i_d

im=

1,n

_d

imve

l(i_

dim

)=

0.

do

j_d

im=

1,

n_

dim

if(

(e_

dim

.eq

.3)

.or.

(i_

dim

.eq

.j_d

im)

)&

vel(i_

dim

)=

vel(i_

dim

)&

+h

_m

at(

i_d

im,j_

dim

)*

ux(

1,j_

dim

,1,i_

pa

rt)

129

B. Programse

nd

do

vel(i_

dim

)=

vel(i_

dim

)/

dt

cor_

fun

c(i_

me

asu

re,i_

typ

e)

&=

cor_

fun

c(i_

me

asu

re,i_

typ

e)

+m

ass

(0,i_

typ

e)

*&

con

f_ce

ntr

(i_

dim

,i_p

art

)*

vel(i_

dim

)/

n_

tro

tq

do

t_co

r(i_

me

asu

re)

=q

do

t_co

r(i_

me

asu

re)

+q

eq

ma

ss(i_

typ

e)

*&

qd

ot0

(i_

dim

,i_p

art

)*

cha

rge

x(1

,i_

pa

rt)

/d

tco

r_fu

nc_

A(i_

me

asu

re)

&=

cor_

fun

c_A

(i_

me

asu

re)

+m

ass

(0,i_

typ

e)

*&

con

f_ce

ntr

(i_

dim

,i_p

art

)*

vel(i_

dim

)/

n_

tro

t!-

--m

eth

od

B:

be

tte

rst

atis

tics,

less

ind

ep

en

de

nce

:st

ore

d_

vel(i_

dim

,i_m

ea

sure

,i_p

art

)=ve

l(i_

dim

)if

(i_

time

.gt.(n

_re

lax+

n_

cor)

)th

en

cor_

fun

c_B

(n_

cor)

=co

r_fu

nc_

B(n

_co

r)+

1.

do

j_m

ea

sure

=0

,i_

me

asu

reco

r_fu

nc_

B(j_

me

asu

re)=

cor_

fun

c_B

(j_

me

asu

re)

+&

ma

ss(0

,i_ty

pe

)*ve

l(i_

dim

)&

*sto

red

_ve

l(i_

dim

,i_m

ea

sure

-j_

me

asu

re,i_

pa

rt)

&/

n_

tro

te

nd

do

do

j_m

ea

sure

=i_

me

asu

re+

1,

n_

cor-

1co

r_fu

nc_

B(j_

me

asu

re)=

cor_

fun

c_B

(j_

me

asu

re)

+&

ma

ss(0

,i_ty

pe

)*ve

l(i_

dim

)&

*sto

red

_ve

l(i_

dim

,n_

cor+

i_m

ea

sure

-j_

me

asu

re,i_

part

)&

/n

_tr

ot

en

dd

oe

nd

if

en

dd

oe

nd

do

i_m

ea

sure

=i_

me

asu

re+

1

en

dif

!---

-m

ea

sure

rad

ial

corr

ela

tion

fun

ctio

nif

((f

_p

ote

ntia

l.ge

.2)

.an

d.

(mo

d(i_

time

,5).

eq

.0)

)th

en

!---

-ch

ose

ran

do

mtr

otte

rtim

eca

llr2

50

(mz,

ran

do

m,n

_d

im,1

,kp

tr)

i_tr

ot

=in

t(ra

nd

om

(1)*

n_

tro

t)if(

i_tr

ot.e

q.n

_tr

ot)

i_tr

ot

=0

g_

r(1

,1,n

_g

rid

+1

)=

g_

r(1

,1,n

_g

rid

+1

)+

dt

if(f

_p

ote

ntia

l.eq

.9)

the

n!

Sca

nd

olo

:sm

all

syst

em

size

,co

un

tp

erio

dic

ima

ge

sa

sw

ell

pb

c_n

orm

=2

.0!*

(2.0

*n_

pb

c+1

.0)*

*3d

oi_

pa

rt=

1,n

_p

art

do

j_p

art

=1

,n_

pa

rtd

op

bc_

x=

-n_

pb

c,n

_p

bc

do

pb

c_y

=-n

_p

bc,

n_

pb

cd

op

bc_

z=

-n_

pb

c,n

_p

bc

i_ty

pe

=ty

pe

(i_

pa

rt)

j_ty

pe

=ty

pe

(j_

pa

rt)

call

calc

_d

ist_

pb

c(r_

2,i_

tro

t,i_

pa

rt,j_

pa

rt,

&p

bc_

x,p

bc_

y,p

bc_

z)r_

1=

sqrt

(r_

2)

i_g

rid

=n

int(

n_

grid

*r_

1/r

_cu

toff(i_

type

,j_ty

pe

))if(

i_g

rid

.le.n

_g

rid

)&

g_

r(i_

typ

e,j_

typ

e,i_

grid

)=

&g

_r(

i_ty

pe

,j_ty

pe

,i_g

rid

)&

+d

t/p

bc_

no

rme

nd

do

en

dd

oe

nd

do

en

dd

oe

nd

do

els

e do

i_p

art

=1,n

_p

art

i_ty

pe

=ty

pe

(i_

pa

rt)

do

i_n

eig

h=

1,n

eig

hb

or(

0,i_

pa

rt)

j_p

art

=n

eig

hb

or(

i_n

eig

h,i_

pa

rt)

j_ty

pe

=ty

pe

(j_

pa

rt)

call

calc

_d

ista

nce

(r_

2,i_

tro

t,i_

pa

rt,j_

pa

rt)

r_1

=sq

rt(r

_2)

i_g

rid

=n

int(

n_

grid

*r_

1/r

_cu

toff(i_

typ

e,j_

typ

e))

if(i_

grid

.le.n

_g

rid

)&

g_

r(i_

typ

e,j_

typ

e,i_

grid

)=

&g

_r(

i_ty

pe

,j_ty

pe

,i_g

rid

)+

dt

en

dd

oe

nd

do

en

dif

en

dif

if((

f_p

ote

ntia

l.eq

.3).

or.

(&

(f_

po

ten

tial.g

e.5

).a

nd

.(f_

po

ten

tial.l

e.9

)))

the

nif

(f_

po

ten

tial.n

e.8

)ca

llb

on

d_

an

aly

sis

if((

f_sy

mm

etr

y.e

q.6

).o

r.(f

_sy

mm

etr

y).e

q.7

)th

en

ord

_c

=o

rd_c

+1

.d0

ord

_p

ar

=o

rd_

pa

r-

45

.o

rd_

pa

r_1

=o

rd_

pa

r_1

+o

rd_

pa

ro

rd_

pa

r_2

=o

rd_

pa

r_2

+o

rd_

pa

r**2

ord

_p

ar_

4=

ord

_p

ar_

4+

ord

_p

ar*

*4

ord

_p

arC

_1

=o

rd_

pa

rC_

1+

ord

_p

arC

ord

_p

arC

_2

=o

rd_

pa

rC_

2+

ord

_p

arC

**2

ord

_p

arC

_4

=o

rd_

pa

rC_

4+

ord

_p

arC

**4

en

dif

en

dif

if(f

_p

ote

ntia

l.eq

.9)

the

nif

((f_

sym

metr

y.e

q.6

).o

r.(f

_sy

mm

etr

y).e

q.7

)th

en

ord

_p

arB

=0.d

0d

oi_

pa

rt=

1,

n_

pa

rtif

((ty

pe

(7)=

=2

).a

nd

.(ty

pe

(i_

pa

rt)=

=2

))th

en

i_cl

ass

=m

od

(i_

pa

rt-1

,18

)-5

if((

i_cl

ass

.eq

.2).

or.

(i_

cla

ss.e

q.4

).o

r.(i_

cla

ss.e

q.6

).o

r.&

(i_

cla

ss.e

q.8

).o

r.(i_cl

ass

.eq

.9).

or.

(i_

cla

ss.e

q.1

2))

&th

en

ord

_p

arB

=ord

_p

arB

+d

ipo

le(3

,i_p

art

)!

write

(81

,*)

i_p

art

,i_

cla

ss,

!&

dip

ole

(3,i_

part

)e

lse

if&

((i_

cla

ss.e

q.1

).o

r.(i_

class

.eq

.3).

or.

(i_

cla

ss.e

q.5

).o

r.&

(i_

cla

ss.e

q.7

).o

r.(i_

cla

ss.e

q.1

0).

or.

(i_

cla

ss.e

q.1

1))

&th

en

ord

_p

arB

=ord

_p

arB

-d

ipo

le(3

,i_p

art

)!

write

(82

,*)

i_p

art

,i_

cla

ss,

!&

dip

ole

(3,i_

part

)e

lse st

op

"co

uld

no

ta

ssig

ncl

ass

top

art

icle

"e

nd

ife

nd

if

130


!if

(typ

e(i_

pa

rt)=

=2

)th

en

!o

rd_

pa

rB=

ord

_p

arB

+a

bs(

dip

ole

(3,i_

pa

rt))

!e

nd

ife

nd

do

ord

_p

arB

=o

rd_

pa

rB/

n_

oxy

ord

B_

c=

ord

B_

c+

1.d

0o

rd_

pa

rB_

1=

ord

_p

arB

_1

+o

rd_

pa

rBo

rd_

pa

rB_

2=

ord

_p

arB

_2

+o

rd_

pa

rB**

2o

rd_

pa

rB_

4=

ord

_p

arB

_4

+o

rd_

pa

rB**

4e

nd

ife

nd

if!

write

mo

vie

if(f

_m

ovi

e.e

q.1

)th

en

do

i_p

art

=1

,n

_p

art

if(ty

pe

(i_

pa

rt).

eq

.1)

the

nw

rite

(46

,20

4)

"Si

",(r

_re

al(i_

dim

,0,i_

pa

rt),

i_d

im=

1,n

_d

im)

els

eif(

typ

e(i_

pa

rt).

eq

.2)

the

nw

rite

(46

,20

4)

"O",

(r_

rea

l(i_

dim

,0,i_

pa

rt),

i_d

im=

1,n

_d

im)

en

dif

write

(47

,10

3)

(dip

ole

(i_

dim

,i_p

art

),i_

dim

=1

,n_

dim

)e

nd

do

write

(46

,*)

"EN

D"

write

(47

,*)

"EN

D"

en

dif

if(f

_co

nfm

ovi

e.e

q.1

)th

en

if(m

od

(i_

time

,10

).e

q.9

)th

en

write

(48

,103

)(s

cal_

0(i_

dim

),i_

dim

=1,n

_d

im+

3)

write

(48

,103

)(s

cal_

x(1

,i_d

im)/

dt,i_

dim

=1

,n_

dim

+3

)d

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

pa

rt=

1,n

_p

art

write

(48

,14

1)

typ

e(i_

pa

rt),

&(r

0(i_

dim

,i_tr

ot,i_

pa

rt),

i_d

im=

1,n

_d

im)

en

dd

oe

nd

do

en

dif

en

dif

10

3fo

rma

t(3

e1

3.4

)2

04

form

at(

a,3

e1

3.4

)1

41

form

at(

i5,3

f14

.6)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

kiha

ra_

bo

nd

sim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

inte

ge

rj_

du

mm

y,k_

pa

rtd

ou

ble

pre

cisi

on

r_2

,s_

2,

de

lta_

s(n

_d

im)

do

ub

lep

reci

sio

nr_

avg

2(n

_d

im),

r_a

vg6

(n_

dim

),r_

avg

(n_

pa

rt,n

_d

im),

&r_

avg

_re

al(n

_p

art

,n_

dim

)

inte

ge

ri,j

,k,

cou

nt

!e

qu

iva

len

tnu

mb

er

pim

da

nd

nu

mb

er

kiha

ra:

!si

1o

6o

1o

2o

5!

31

01

31

18

!5

15

71

71

4

!ki

ha

ralis

ted

bo

nd

s!

o1

-si-o

2,

o1

-si-o

6,

o1

-si-o

5,

o2

-si-o

5

!si

1-o

1-s

i3=

3-

13

-6

;5

-7

-2

cou

nt

=0

do

i=

1,

3r_

avg

2(i)=

0.

r_a

vg6

(i)=

0.

en

dd

o

do

i=0

,n_

bin

_x-

1d

oj=

0,n

_b

in_

y-1

do

k=0

,n_

bin

_z-

1r_

avg

2(1

)=r_

avg

2(1

)+(r

0(1

,0,1

8*c

ou

nt+

2)

-i*

1./n

_b

in_

x)r_

avg

2(2

)=r_

avg

2(2

)+(r

0(2

,0,1

8*c

ou

nt+

2)

-j*

1./n

_b

in_

y)r_

avg

2(3

)=r_

avg

2(3

)+(r

0(3

,0,1

8*c

ou

nt+

2)

-k*

1./n

_b

in_

z)

r_a

vg6

(1)=

r_a

vg6

(1)+

(r0

(1,0

,18

*co

un

t+6

)-

i*1

./n

_b

in_

x)r_

avg

6(2

)=r_

avg

6(2

)+(r

0(2

,0,1

8*c

ou

nt+

6)

-j*

1./n

_b

in_

y)r_

avg

6(3

)=r_

avg

6(3

)+(r

0(3

,0,1

8*c

ou

nt+

6)

-k*

1./n

_b

in_

z)

cou

nt

=co

un

t+

1e

nd

do

en

dd

oe

nd

do

r_a

vg2

(1)=

r_avg

2(1

)/

cou

nt

r_a

vg2

(2)=

r_avg

2(2

)/

cou

nt

r_a

vg2

(3)=

r_avg

2(3

)/

cou

nt

r_a

vg6

(1)=

r_avg

6(1

)/

cou

nt

r_a

vg6

(2)=

r_avg

6(2

)/

cou

nt

r_a

vg6

(3)=

r_avg

6(3

)/

cou

nt

cou

nt

=0

do

i=0

,n_

bin

_x-

1d

oj=

0,n

_b

in_y-

1d

ok=

0,n

_b

in_z-

1r_

avg

(18

*cou

nt+

2,1

)=r_

avg

2(1

)+

i*1

./n

_b

in_

xr_

avg

(18

*cou

nt+

2,2

)=r_

avg

2(2

)+

j*1

./n

_b

in_

yr_

avg

(18

*cou

nt+

2,3

)=r_

avg

2(3

)+

k*1

./n

_b

in_

z

r_a

vg(1

8*c

ou

nt+

6,1

)=r_

avg

6(1

)+

i*1

./n

_b

in_

xr_

avg

(18

*cou

nt+

6,2

)=r_

avg

6(2

)+

j*1

./n

_b

in_

yr_

avg

(18

*cou

nt+

6,3

)=r_

avg

6(3

)+

k*1

./n

_b

in_

z

cou

nt

=co

un

t+

1e

nd

do

en

dd

oe

nd

do

do

i_p

art

=1,n

_p

art

do

i_d

im=

1,

n_

dim

r_a

vg_

rea

l(i_

pa

rt,i_

dim

)=

sca

l_0

(i_

dim

)*

&(r

_a

vg(i_

part

,i_d

im)

+1

.d0

*pb

c0(i_

dim

,0,i_

pa

rt))

en

dd

oif

(e_

dim

.eq

.3)

the

nr_

avg

_re

al(i_

pa

rt,1

)=

r_a

vg_

rea

l(i_

pa

rt,1

)+

&sc

al_

0(4

)*

(r_

avg

(i_

pa

rt,2

)+

1.d

0*p

bc0

(2,0

,i_p

art

))+

&sc

al_

0(6

)*

(r_

avg

(i_

pa

rt,3

)+

1.d

0*p

bc0

(3,0

,i_p

art

))

131

B. Programsr_

avg

_re

al(i_

pa

rt,2

)=

r_a

vg_

rea

l(i_

pa

rt,2

)+

&sc

al_

0(4

)*

(r_

avg

(i_

pa

rt,1

)+

1.d

0*p

bc0

(1,0

,i_p

art

))+

&sc

al_

0(5

)*

(r_

avg

(i_

pa

rt,3

)+

1.d

0*p

bc0

(3,0

,i_p

art

))r_

avg

_re

al(i_

pa

rt,3

)=

r_a

vg_

rea

l(i_

pa

rt,3

)+

&sc

al_

0(6

)*

(r_

avg

(i_

pa

rt,1

)+

1.d

0*p

bc0

(1,0

,i_p

art

))+

&sc

al_

0(5

)*

(r_

avg

(i_

pa

rt,2

)+

1.d

0*p

bc0

(2,0

,i_p

art

))e

nd

ife

nd

do

do

i_p

art

=1,

n_

pa

rtif

((m

od

(i_

pa

rt,1

8).

eq

.3).

or.

(mo

d(i_

pa

rt,1

8).

eq

.5))

the

nd

oi_

du

mm

y=

1,

4d

oj_

du

mm

y=

1,

4j_

pa

rt=

i_ne

igh

_o

ri(i_

pa

rt,i_

du

mm

y)k_

pa

rt=

i_ne

igh

_o

ri(i_

pa

rt,j_

du

mm

y)i=

mo

d(i_

pa

rt,1

8)

j=m

od

(j_

pa

rt,1

8)

k=m

od

(k_

pa

rt,1

8)

if(

((i.e

q.3

).a

nd

.(j.e

q.1

3).

an

d.(

k.eq

.11

)).o

r.&

((i.e

q.3

).a

nd

.(j.e

q.1

3).

an

d.(

k.e

q.1

0))

.or.

&((

i.eq

.3).

an

d.(

j.eq

.13

).a

nd

.(k.

eq

.8))

.or.

&((

i.eq

.3).

an

d.(

j.eq

.11

).a

nd

.(k.

eq

.8))

.or.

&((

i.eq

.5).

an

d.(

j.eq

.7).

an

d.(

k.e

q.1

7))

.or.

&((

i.eq

.5).

an

d.(

j.eq

.7).

an

d.(

k.e

q.1

5))

.or.

&((

i.eq

.5).

an

d.(

j.eq

.7).

an

d.(

k.e

q.1

4))

.or.

&((

i.eq

.5).

an

d.(

j.eq

.17

).a

nd

.(k.

eq

.14

)))

the

n

i_ty

pe

=ty

pe

(i_

pa

rt)

call

calc

_d

ista

nce

(s_

2,0

,i_p

art

,j_p

art

)d

oi_

dim

=1

,n

_d

imd

elta

_s(

i_d

im)

=d

elta

_ce

ll(i_

dim

)e

nd

do

call

calc

_d

ista

nce

(r_

2,0

,i_p

art

,k_

pa

rt)

r_d

um

my

=0

.d

oi_

dim

=1

,n

_d

imr_

du

mm

y=

r_du

mm

y+

me

t_te

n_

voi(i_

dim

)*

&d

elta

_s(

i_d

im)

*d

elta

_ce

ll(i_

dim

)e

nd

do

if(e

_d

im.e

q.3

)th

en

r_d

um

my

=r_

du

mm

y&

+m

et_

ten

_vo

i(4

)*

(&

de

lta_

s(1

)*d

elta

_ce

ll(2

)&

+d

elta

_s(

2)

*d

elta

_ce

ll(1

))

/2

&+

me

t_te

n_

voi(5

)*

(&

de

lta_

s(2

)*d

elta

_ce

ll(3

)&

+d

elta

_s(

3)

*d

elta

_ce

ll(2

))

/2

&+

me

t_te

n_

voi(6

)*

(&

de

lta_

s(3

)*d

elta

_ce

ll(1

)&

+d

elta

_s(

1)

*d

elta

_ce

ll(3

))

/2

en

dif

r_d

um

my

=r_

dum

my

/sq

rt(

r_2

*s_

2)

r_d

um

my

=a

cos(

r_d

um

my)

write

(87

,’(3

i,f1

4.5

)’)

i,j,k

,r_

du

mm

y*18

0./p

ie

nd

ife

nd

do

en

dd

oe

nd

ifif

(mo

d(i_

time

,15

).e

q.0

)th

en

if((

mo

d(i_

pa

rt,1

8).

eq

.13

).o

r.(m

od

(i_p

art

,18

).e

q.7

))th

en

do

i_d

um

my

=1

,2

do

j_d

um

my

=1

,2

j_p

art

=i_

ne

igh

_o

ri(i_

pa

rt,i_

du

mm

y)k_

pa

rt=

i_n

eig

h_

ori(i_

pa

rt,j_

du

mm

y)i=

mo

d(i_

pa

rt,1

8)

j=m

od

(j_

pa

rt,1

8)

k=m

od

(k_

pa

rt,1

8)

if(

((i.e

q.1

3).

an

d.(

j.eq

.3).

an

d.(

k.e

q.6

)).o

r.&

((i.e

q.7

).a

nd

.(j.e

q.5

).a

nd

.(k.

eq

.2))

)th

en

write

(88

,’(3e

16

.7)’)

&r_

rea

l(1

,0,i_

part

)-r_

rea

l(1

,0,k

_p

art

),&

r_re

al(2

,0,i_

part

)-r_

rea

l(2

,0,k

_p

art

),&

r_re

al(3

,0,i_

part

)-r_

rea

l(3

,0,k

_p

art

)w

rite

(89

,’(3e

16

.7)’)

&r_

rea

l(1

,0,i_

part

)-r_

avg

_re

al(k_

pa

rt,1

),&

r_re

al(2

,0,i_

part

)-r_

avg

_re

al(k_

pa

rt,2

),&

r_re

al(3

,0,i_

part

)-r_

avg

_re

al(k_

pa

rt,3

)e

nd

ife

nd

do

en

dd

oe

nd

ife

nd

ife

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

bo

nd

_a

na

lysi

sim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

inte

ge

rj_

du

mm

y,k_

pa

rt,

i_a

ng

,iii

,jjj

do

ub

lep

reci

sio

nd

r_n

eig

h,

r_2

,s_

1,

s_2

,d

elta

_s(

n_

dim

)d

ou

ble

pre

cisi

on

rord

_d

um

my

do

ub

lep

reci

sio

nrij(3

),rik(

3)

call

kih

ara

_b

on

ds

ord

_p

ar

=0

.d0

ord

_p

arC

=0

.d0

dr_

ne

igh

=(r

_n

eig

h_

ma

x-r_

ne

igh

_m

in)

/n

_g

rid

ne

igh

_tim

e=

ne

igh

_tim

e+

1.

do

i_p

art

=1

,n

_p

art

i_a

ng

=0

i_ty

pe

=ty

pe

(i_

pa

rt)

if(i_

typ

e.e

q.1

)th

en

!--

-S

i do

i_d

um

my

=1,

4j_

pa

rt=

i_ne

igh

_o

ri(i_

pa

rt,i_

du

mm

y)ca

llca

lc_

dis

tan

ce(s

_2

,0,i_

pa

rt,j_

pa

rt)

do

i_d

im=

1,

n_

dim

de

lta_

s(i_

dim

)=

de

lta_

cell(

i_d

im)

en

dd

os_

1=

sqrt

(s_

2)

r_d

um

my

=s_

1i_

grid

=n

int(

(r_

du

mm

y-r_

ne

igh

_m

in)

/d

r_n

eig

h)

if((i_

grid

.gt.1

).a

nd

.(i_

grid

.lt.n

_g

rid

))

&h

ist_

an

g(1

,i_g

rid

)=

his

t_a

ng

(1,i_

grid

)+

1.d

0d

oj_

du

mm

y=

i_d

um

my

+1

,4

i_a

ng

=i_

an

g+

1

132


if(i_

an

g.g

t.6

)st

op

"mo

reth

an

6a

ng

les

aro

un

dS

ia

tom

"k_

pa

rt=

i_n

eig

h_

ori(i_

pa

rt,j_

du

mm

y)ca

llca

lc_

dis

tan

ce(r

_2

,0,i_

pa

rt,k

_p

art

)r_

du

mm

y=

0.

do

i_d

im=

1,

n_

dim

r_d

um

my

=r_

dum

my

+m

et_

ten

_vo

i(i_

dim

)*

&d

elta

_s(

i_d

im)

*d

elta

_ce

ll(i_

dim

)e

nd

do

if(e

_d

im.e

q.3

)th

en

r_d

um

my

=r_

dum

my

&+

me

t_te

n_

voi(4

)*

(&

de

lta_s(

1)

*de

lta_

cell(

2)

+d

elta

_s(

2)

*d

elta

_ce

ll(1

))

/2

&+

me

t_te

n_

voi(5

)*

(&

de

lta_s(

2)

*de

lta_

cell(

3)

+d

elta

_s(

3)

*d

elta

_ce

ll(2

))

/2

&+

me

t_te

n_

voi(6

)*

(&

de

lta_s(

3)

*de

lta_

cell(

1)

+d

elta

_s(

1)

*d

elta

_ce

ll(3

))

/2

en

dif

r_d

um

my

=r_

du

mm

y/

sqrt

(r_

2*

s_2

)

!st

ore

variab

les

for

3b

od

yp

ote

ntia

lif

(f_

po

tentia

l.eq

.8)

the

ntr

iple

_co

s(i_

pa

rt,

i_a

ng

)=

r_d

um

my

trip

le_

pa

rtj(i_

pa

rt,

i_a

ng

)=

j_p

art

trip

le_

rij(i_

pa

rt,

i_a

ng

)=

s_1

trip

le_

pa

rtk(

i_p

art

,i_

an

g)

=k_

pa

rttr

iple

_rik(

i_pa

rt,

i_a

ng

)=

sqrt

(r_

2)

do

i_d

im=

1,

n_d

imtr

iple

_d

elta

ij(i_

pa

rt,

i_a

ng

,i_

dim

)=

de

lta_

s(i_

dim

)tr

iple

_d

elta

ik(i_

pa

rt,

i_a

ng

,i_

dim

)=

de

lta_

cell(

i_d

im)

en

dd

oe

nd

if

r_d

um

my

=aco

s(r_

du

mm

y)

i_g

rid

=n

int(

n_

grid

*r_

du

mm

y/p

i)

if((i_

grid.g

e.1

).a

nd

.(i_

grid

.le.n

_grid

))

&h

ist_

ang

(3,i_

grid

)=

his

t_a

ng

(3,i_

grid

)+

1.d

0e

nd

do

en

dd

oe

lse

if(i_

typ

e.e

q.2

)th

en

!--

-O

i_a

ng

=1

j_p

art

=i_

ne

igh

_o

ri(i_

pa

rt,1

)ca

llca

lc_

dis

tan

ce(s

_2

,0,i_

pa

rt,j_

part

)d

oi_

dim

=1

,n

_d

imd

elta

_s(

i_d

im)

=d

elta

_ce

ll(i_

dim

)e

nd

do

!--

--q

ua

rtz

eva

lua

teo

rd_

pa

ram

if((

f_sy

mm

etr

y.e

q.6

).o

r.(f

_sy

mm

etr

y.e

q.7

))th

en

if(

((ty

pe

(7).

eq

.1).

an

d.

!1

/3n

Si,

2/3

nO

Co

nfig

&((

mo

d(j_

pa

rt,6

).e

q.3

).o

r.(m

od

(j_

pa

rt,6

).e

q.4

))).

or.

&((

typ

e(7

).e

q.2

).a

nd

.!

6S

i,1

2O

,6

Si,

12

Oco

nfig

&((

mo

d(j_

pa

rt,1

8).

eq

.1).

or.

(mo

d(j_

pa

rt,1

8).

eq

.4))

))

&th

en

rord

_d

um

my

=d

elta

_ce

ll(2

)/

&sq

rt(

de

lta_

cell(

2)*

*2+

de

lta_

cell(

3)*

*2)

if(d

elta

_ce

ll(2

).g

t.0

.)th

en

if(d

elta

_ce

ll(3

).g

t.0

.)th

en

rord

_d

um

my

=a

cos(

rord

_d

um

my)

els

ero

rd_

du

mm

y=

asi

n(r

ord

_d

um

my)

en

dif

ord

_p

ar

=ord

_p

ar

+ro

rd_

du

mm

yo

rd_

pa

rC=

ord

_p

arC

+a

bs(

rord

_d

um

my*

18

0./p

i-

45

.)!

write

(44

,*)

i_p

art

,ty

pe

(i_

pa

rt),

j_p

art

,ty

pe

(j_

pa

rt),

!&

mo

d(j_

pa

rt,1

8),

18

0*r

ord

_d

um

my/

pi-4

5e

nd

ife

nd

ife

nd

ifk_

pa

rt=

i_n

eig

h_

ori(i_

pa

rt,2

)ca

llca

lc_

dis

tan

ce(r

_2

,0,i_

pa

rt,k

_p

art

)r_

du

mm

y=

0.

do

i_d

im=

1,

n_

dim

r_d

um

my

=r_

du

mm

y+

me

t_te

n_

voi(i_

dim

)*

&d

elta

_s(

i_d

im)

*d

elta

_ce

ll(i_

dim

)e

nd

do

!--

--q

uart

ze

valu

ate

ord

_p

ara

mif

((f_

sym

me

try.

eq

.6).

or.

(f_

sym

me

try.

eq

.7))

the

nif

(((

typ

e(7

).e

q.1

).a

nd

.!

1/3

nS

i,2

/3n

OC

on

fig&

((m

od

(k_

pa

rt,6

).e

q.3

).o

r.(m

od

(k_

pa

rt,6

).e

q.4

))).

or.

&((

type

(7).

eq

.2).

an

d.

!6

Si,

12

O,

6S

i,1

2O

con

fig&

((m

od

(k_

pa

rt,1

8).

eq

.1).

or.

(mo

d(k

_p

art

,18

).e

q.4

)))

)&

then

rord

_d

um

my

=d

elta

_ce

ll(2

)/

&sq

rt(

de

lta_

cell(

2)*

*2+

de

lta_

cell(

3)*

*2)

if(d

elta

_ce

ll(2

).g

t.0

.)th

en

if(d

elta

_ce

ll(3

).g

t.0

.)th

en

rord

_d

um

my

=a

cos(

rord

_d

um

my)

els

ero

rd_

du

mm

y=

asi

n(r

ord

_d

um

my)

en

dif

ord

_p

ar

=ord

_p

ar

+ro

rd_

du

mm

yo

rd_

pa

rC=

ord

_p

arC

+a

bs(

rord

_d

um

my*

18

0./p

i-

45

.)!

write

(44

,*)

i_p

art

,ty

pe(i_

pa

rt),

k_p

art

,ty

pe

(k_p

art

),!

&m

od

(k_

pa

rt,1

8),

18

0*r

ord

_d

um

my/

pi-4

5e

nd

ife

nd

ife

nd

if

if(e

_d

im.e

q.3

)th

en

r_d

um

my

=r_

du

mm

y&

+m

et_

ten

_vo

i(4

)*

(&

de

lta_

s(1

)*d

elta

_ce

ll(2

)+

de

lta_

s(2

)*

de

lta_

cell(

1)

)/

2&

+m

et_

ten

_vo

i(5

)*

(&

de

lta_

s(2

)*d

elta

_ce

ll(3

)+

de

lta_

s(3

)*

de

lta_

cell(

2)

)/

2&

+m

et_

ten

_vo

i(6

)*

(&

de

lta_

s(3

)*d

elta

_ce

ll(1

)+

de

lta_

s(1

)*

de

lta_

cell(

3)

)/

2e

nd

ifr_

du

mm

y=

r_du

mm

y/

sqrt

(r_

2*

s_2

)

!st

ore

varia

ble

sfo

r3

bo

dy

po

ten

tial

if(f

_p

ote

ntia

l.eq

.8)

the

ntr

iple

_co

s(i_

pa

rt,

i_a

ng

)=

r_d

um

my

trip

le_

pa

rtj(i_

pa

rt,

i_a

ng

)=

j_p

art

trip

le_

rij(i_

pa

rt,

i_a

ng

)=

sqrt

(s_

2)

trip

le_

pa

rtk(

i_p

art

,i_

an

g)

=k_

pa

rttr

iple

_rik(

i_p

art

,i_

an

g)

=sq

rt(r

_2)

do

i_d

im=

1,

n_

dim

trip

le_

de

ltaij(

i_p

art

,i_

an

g,

i_d

im)

=d

elta

_s(

i_d

im)

trip

le_

de

ltaik

(i_

pa

rt,

i_a

ng

,i_

dim

)=

de

lta_

cell(

i_d

im)

en

dd

oe

nd

if

r_d

um

my

=a

cos(

r_d

um

my)

133

B. Programs!

write

vect

or

fro

mm

idd

leo

fsi

-si

con

ne

ctio

nlin

eto

o

do

iii=

1,3

rij(iii

)=0

.rik(

iii)=

0.

do

jjj=

1,3

rij(iii

)=rij(iii

)+h

_m

at(

iii,jj

j)*d

elta

_s(

jjj)

rik(

iii)=

rik(

iii)+

h_

ma

t(iii

,jjj)*d

elta

_ce

ll(jjj

)e

nd

do

en

dd

o

if(i_

pa

rt.e

q.1

5)

&w

rite

(86

,’(3

i,3f1

4.5

)’)

i_p

art

,j_p

art

,k_

pa

rt,

&(0

.5*(

rij(iii

)+rik(

iii))

,iii=

1,3

)

i_g

rid

=n

int(

n_

grid

*r_

du

mm

y/p

i)

if((i_

grid

.gt.0

).a

nd

.(i_

grid

.lt.n

_g

rid

))

&h

ist_

an

g(2

,i_g

rid

)=

his

t_a

ng

(2,i_

grid

)+

1.d

0e

nd

ife

nd

do

if((

f_sy

mm

etr

y.e

q.6

).o

r.(f

_sy

mm

etr

y.e

q.7

))th

en

ord

_p

ar

=1

80

*o

rd_

pa

r/

pi

ord

_p

ar

=1

.5*

ord

_p

ar

/n

_si

lico

rd_

pa

rC=

1.5

*o

rd_

pa

rC/

n_

silic

en

dif

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

viri_

est

ima

tor

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nr_

loca

l(n

_d

im),

r_h

elp

(n_

dim

),f_

loca

l(n

_d

im)

!---

-ch

an

ge

tra

nsf

orm

viri_

est

=0.

!---

-te

rmm

ea

svi

ri_

est

do

i_p

art

=1

,n_

pa

rtd

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

dim

=1

,n_d

imr_

loca

l(i_

dim

)=

0.

f_lo

cal(i_

dim

)=

0.

r_h

elp

(i_

dim

)=

r0(i_

dim

,i_tr

ot,i_

pa

rt)

&+

1.

*p

bc0

(i_

dim

,i_tr

ot,i_

pa

rt)

-r_

ref(

i_d

im,i_

pa

rt)

en

dd

od

oi_

dim

=1

,n_

dim

do

j_d

im=

1,n

_d

imr_

loca

l(i_

dim

)=

r_lo

cal(i_

dim

)+

&h

_m

at(

i_d

im,j_

dim

)*

r_h

elp

(j_

dim

)f_

loca

l(i_

dim

)=

f_lo

cal(i_

dim

)+

&h

_m

at(

i_d

im,j_

dim

)*

(fo

rce

(j_

dim

,i_tr

ot,i_

pa

rt)

)!

&0

*m

ass

(i_

du

mm

y,i_

typ

e)

*fr

ictio

n(i_

typ

e)

*r_

du

mm

y**2

!&

*(u

x(1

,j_d

im,i_

tro

t,i_

pa

rt)/

dt)

)e

nd

do

!6

66

line

sa

bo

vee

nd

do

do

i_d

im=

1,

n_

dim

viri_

est

=vi

ri_

est

+r_

loca

l(i_

dim

)*

f_lo

cal(i_

dim

)e

nd

do

en

dd

o

en

dd

ovi

ri_

est

=-v

iri_

est

/2

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

prim

_e

stim

ato

rim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

v_ch

ain

=0

.d

oi_

tro

t=

0,

n_

tro

t-1

j_tr

ot

=m

od(i_

tro

t+1

,n_

tro

t)d

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

!---

-T

ER

Mm

ea

sp

rim

_e

std

oi_

dim

=1

,n

_d

imv_

cha

in=

v_ch

ain

+k_

qu

an

t(i_

typ

e)

*(

&r_

rea

l(i_

dim

,i_tr

ot,i_

pa

rt)

-r_

rea

l(i_

dim

,j_tr

ot,i_

pa

rt)

)**2

en

dd

oe

nd

do

en

dd

ov_

cha

in=

v_ch

ain

/2

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

ob

ser_

ou

tim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

inte

ge

rn

um

be

r_sp

ec(

n_

typ

e),

i_cl

ass

do

ub

lep

reci

sio

nn

orm

_sp

ec(

n_

typ

e,n

_ty

pe

)

do

i_ty

pe

=1

,n_

typ

en

um

be

r_sp

ec(

i_ty

pe

)=

0e

nd

do

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

nu

mb

er_

spe

c(i_

typ

e)

=n

um

be

r_sp

ec(

i_ty

pe

)+

1e

nd

do

write

(20

,*)

write

(20

,20

1)

time

_o

bs,

"tim

eo

bse

rve

d"

t_d

yna

mic

_1

=t_

dyn

am

ic_

1/tim

e_

ob

st_

dyn

am

ic_

2=

t_d

yna

mic

_2

/tim

e_

ob

s-

t_d

yna

mic

_1

**2

t_d

yna

mic

_1

=t_

dyn

am

ic_

1/

(n_

pa

rt*n

_tr

ot)

t_d

yna

mic

_2

=t_

dyn

am

ic_

2/

(n_

pa

rt*n

_tr

ot*

tkp

**2

)w

rite

(20

,*)

write

(20

,20

2)

t_d

yna

mic

_1

,t_

dyn

am

ic_

2,"

dyn

am

icki

n.

en

erg

y"

t_d

iag

_1

=t_

dia

g_

1/tim

e_

ob

st_

nd

iag

_1

=t_

nd

iag

_1

/tim

e_

ob

sw

rite

(20

,20

2)

t_d

iag

_1

,t_

nd

iag

_1

,"d

iag

an

dn

on

_d

iag

T_

kin

"

v_in

ter_

1=

v_in

ter_

1/

time

_o

bs

v_in

ter_

2=

v_in

ter_

2/

time

_o

bs

-v_

inte

r_1

**2

v_in

ter_

1=

v_in

ter_

1/

(n_

pa

rt*n

_tr

ot)

v_in

ter_

2=

v_in

ter_

2/

(n_

pa

rt*n

_tr

ot*

tkp

**2

)w

rite

(20

,*)

134


write

(20

,20

2)

v_in

ter_

1,

v_in

ter_

2,"

inte

ract

ion

en

erg

y"

viri_

est

_1

=vi

ri_

est

_1

/tim

e_

ob

svi

ri_

est

_2

=vi

ri_

est

_2

/tim

e_

ob

s-

viri_

est

_1

**2

viri_

est

_1

=vi

ri_

est

_1

/(n

_p

art

*n_

trot)

viri_

est

_2

=vi

ri_

est

_2

/(n

_p

art

*n_

trot*

tkp

**2

)w

rite

(20

,*)

write

(20

,20

2)

viri_

est

_1

,vi

ri_

est

_2

,"vi

ria

le

stim

ato

r"

prim

_e

st_

1=

prim

_e

st_

1/

time

_o

bs

prim

_e

st_

2=

prim

_e

st_

2/

time

_o

bs

-p

rim

_e

st_

1**

2p

rim

_e

st_

1=

prim

_e

st_

1/

(n_

pa

rt*n

_tr

ot)

prim

_e

st_

2=

prim

_e

st_

2/

(n_

pa

rt*n

_tr

ot*

tkp

**2

)w

rite

(20

,20

2)

prim

_e

st_

1,

prim

_e

st_

2,"

Prim

itive

en

erg

y"

if((

f_sy

mm

etr

y.e

q.6

).o

r.(f

_sy

mm

etr

y.e

q.7

))th

en

ord

_p

ar_

1=

ord

_p

ar_

1/

ord

_c

ord

_p

ar_

2=

ord

_p

ar_

2/

ord

_c

ord

_p

ar_

4=

ord

_p

ar_

4/

ord

_c

write

(20

,*)

"ord

_p

ar

de

lta_

ph

i",

ord

_p

ar_

1,

ord

_p

ar_

2,

ord

_p

ar_

4o

rd_

pa

r_2

=ord

_p

ar_

2-

ord

_p

ar_

1**

2o

rd_

pa

r_2

=ord

_p

ar_

2/

tkw

rite

(20

,*)

"su

ssz.

",o

rd_

pa

r_2

write

(20

,*)

ord

_p

arB

_1

=o

rd_

pa

rB_

1/

ord

B_

co

rd_

pa

rB_

2=

ord

_p

arB

_2

/o

rdB

_c

ord

_p

arB

_4

=o

rd_

pa

rB_

4/

ord

B_

cw

rite

(20

,*)

"ord

_p

ar

mu

_z"

,o

rd_

pa

rB_1

,o

rd_

pa

rB_

2,

ord

_p

arB

_4

ord

_p

arB

_2

=o

rd_

pa

rB_

2-

ord

_p

arB

_1

**2

ord

_p

arB

_2

=o

rd_

pa

rB_

2/

tkw

rite

(20

,*)

"su

ssz.

",o

rd_

pa

rB_

2w

rite

(20

,*)

ord

_p

arC

_1

=o

rd_

pa

rC_

1/

ord

_c

ord

_p

arC

_2

=o

rd_

pa

rC_

2/

ord

_c

ord

_p

arC

_4

=o

rd_

pa

rC_

4/

ord

_c

write

(20

,*)

"ord

_p

ar

mu

_z"

,o

rd_

pa

rC_1

,o

rd_

pa

rC_

2,

ord

_p

arC

_4

ord

_p

arC

_2

=o

rd_

pa

rC_

2-

ord

_p

arC

_1

**2

ord

_p

arC

_2

=o

rd_

pa

rC_

2/

tkw

rite

(20

,*)

"su

ssz.

",o

rd_

pa

rC_

2e

nd

if

if(f

_p

ote

ntia

l.eq

.9)

the

nd

oi_

dim

=0

,3d

oi_

cla

ss=

1,1

2d

ipo

le_

cla

ss1

(i_

dim

,i_cl

ass

)=&

dip

ole

_cl

ass

1(i_

dim

,i_cl

ass

)/d

ipo

le_

cou

nt

en

dd

oe

nd

do

write

(20

,*)

do

i_cl

ass

=1

,12

write

(20

,209

)(d

ipo

le_

cla

ss1

(i_

dim

,i_cl

ass

),i_

dim

=0

,3),

&"

oxy

ge

nty

pe

",i_

cla

sse

nd

do

en

dif

if(f

_p

ote

ntia

l.gt.1

)th

en

vol_

1=

vol_

1/

time

_o

bs

vol_

2=

vol_

2/

time

_o

bs

vol_

2=

vol_

2-

vol_

1**

2vo

l_2

=(v

ol_

2/v

ol_

1)

/tk

write

(20

,*)

write

(20

,20

2)

vol_

1/n

_p

art

,vo

l_2

,"vo

lum

e"

do

i_d

im=

1,

n_

dim

+3

sca

l_1

(i_

dim

)=

sca

l_1

(i_

dim

)/

time

_o

bs

en

dd

od

oi_

dim

=1

,n

_d

im+

3d

oj_

dim

=1

,n

_d

im+

3sc

al_

2(j_

dim

,i_d

im)

=sc

al_

2(j_

dim

,i_d

im)

/tim

e_

ob

ssc

al_

2(j_

dim

,i_d

im)

=(

sca

l_2

(j_

dim

,i_d

im)

-&

sca

l_1

(j_

dim

)*

sca

l_1

(i_

dim

))

/tk

en

dd

oe

nd

do

r_d

um

my

=sc

al_

1(1

)*sc

al_

1(2

)*sc

al_

1(3

)+

&2

*sca

l_1

(4)*

sca

l_1

(5)*

sca

l_1

(6)

-sc

al_

1(4

)**2

*sca

l_1

(3)

&-

sca

l_1

(5)*

*2*s

cal_

1(1

)-

sca

l_1

(6)*

*2*s

cal_

1(2

)w

rite

(20

,20

2)

r_d

um

my/

n_

pa

rt

write

(20

,*)

if(f

_p

ote

ntia

l.gt.1

)th

en

write

(20

,*)

"sca

l.va

ria

ble

"w

rite

(20

,20

6)

(sca

l_1

(i_

dim

),i_

dim

=1,

n_

dim

+3

)if

(f_

pre

ssu

re.e

q.1

)th

en

write

(20

,*)

do

i_d

im=

1,

n_

dim

+3

write

(20

,20

6)

(sca

l_2

(i_

dim

,j_d

im),

j_d

im=

1,

n_

dim

+3

)e

nd

do

en

dif

en

dif

en

dif

if(

(f_

po

ten

tial.g

t.1

).a

nd

.(f

_p

ress

ure

.eq

.0)

)th

en

write

(20

,*)

write

(20

,*)

"ela

stic

con

sta

nts

"w

rite

(20

,*)

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

stra

in_

1(i_

dim

,j_d

im)

=st

rain

_1

(i_

dim

,j_d

im)

/tim

e_

bo

rne

nd

do

en

dd

od

oi_

dim

=1

,n

_d

imd

oj_

dim

=i_

dim

,n

_d

imd

ok_

dim

=i_

dim

,n

_d

imd

ol_

dim

=k_

dim

,n

_d

imif

(i_

dim

+j_

dim

.le.k

_d

im+

l_d

im)

the

ne

la_

bo

rn_

1(i_d

im,

j_d

im,

k_d

im,

l_d

im)

=&

ela

_b

orn

_1

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)/

time

_b

orn

stra

in_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

&st

rain

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)/

time

_b

orn

stra

in_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

&st

rain

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)-

&st

rain

_1

(i_

dim

,j_

dim

)*

stra

in_

1(k

_d

im,

l_d

im)

stra

in_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

-be

ta*

&st

rain

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)/

vol_

1w

rite

(20

,’(4

i2,3

e1

4.4

)’)

&i_

dim

,j_

dim

,k_

dim

,l_

dim

,&

ela

_b

orn

_1

(i_

dim

,j_

dim

,k_

dim

,l_

dim

),&

stra

in_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

),&

ela

_b

orn

_1

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)+

&st

rain

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)e

nd

ife

nd

do

en

dd

o

135

B. Programse

nd

do

en

dd

ow

rite

(20

,*)

write

(20

,*)

"str

ain

"w

rite

(20

,*)

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

write

(20

,’(2i2

,1f1

0.4

)’)

i_d

im,

j_d

im,

&st

rain

_1

(i_d

im,

j_d

im)/

(n_

pa

rt*n

_tr

ot)

en

dd

oe

nd

do

en

dif

if(i_

run

.eq.n

_ru

n)

clo

se(2

0)

if(n

_ru

n.g

e.2

)w

rite

(71

,*)

tk,

v_in

ter_

1,

viri_

est

_1

,&

prim

_e

st_

1,

vol_

1/n

_p

art

,v_

inte

r_2

,vo

l_2

if(i_

run

.eq

.1)

op

en

(21

,file

=’p

imd

.ima’,s

tatu

s=’u

nkn

ow

n’)

do

i_tr

ot

=0

,n_

tro

t-1

write

(21

,’(2

e15

.5)’)

i_tr

ot*

be

tap

,&

cor_

ifun

c(i_

tro

t)/c

ou

nt_

ifun

c/n

_p

art

en

dd

ow

rite

(21

,’(2e

15

.5)’)

be

ta,c

or_

ifun

c(0

)/co

un

t_ifu

nc/

n_

pa

rtif(

i_ru

n.e

q.n

_ru

n)

clo

se(2

1)

op

en

(22

,file

=’p

imd

.co

r.S

iO’,s

tatu

s=’u

nkn

ow

n’)

op

en

(23

,file

=’p

imd

.co

r.q

’,sta

tus=

’un

kno

wn

’)o

pe

n(2

5,file

=’p

imd

.co

r_B

’,sta

tus=

’un

kno

wn

’)o

pe

n(2

6,file

=’p

imd

.co

r_A

’,sta

tus=

’un

kno

wn

’)

do

i_co

r=

0,i_

me

asu

re-1

do

i_ty

pe

=1

,n_

typ

eco

r_fu

nc(

i_co

r,i_

typ

e)

=2

.*co

r_fu

nc(

i_co

r,i_

typ

e)/

&co

r_fu

nc(

n_

cor,

1)/

nu

mb

er_

spec(

i_ty

pe

)e

nd

do

qd

ot_

cor(

i_co

r)=

2.*

qd

ot_

cor(

i_co

r)/q

do

t_co

r(n

_co

r)/n

_p

art

cor_

fun

c_A

(i_

cor)

=2

.*co

r_fu

nc_

A(i_

cor)

&/c

or_

fun

c_A

(n_

cor)

/n_

pa

rtco

r_fu

nc_

B(i_

cor)

=2

.*co

r_fu

nc_

B(i_

cor)

/n_

pa

rt/c

or_

fun

c_B

(n_

cor)

write

(22

,’(3

e15

.5)’)

ni_

time

*i_

cor*

dt/di_

time

,&

(co

r_fu

nc(

i_co

r,i_

typ

e),

i_ty

pe=

1,n

_ty

pe

)w

rite

(23

,’(2

e15

.5)’)

ni_

time

*i_

cor*

dt/di_

time

,qd

ot_

cor(

i_co

r)w

rite

(25

,’(2

e15

.5)’)

ni_

time

*i_

cor*

dt/di_

time

,co

r_fu

nc_

B(i_

cor)

write

(26

,’(2

e15

.5)’)

ni_

time

*i_

cor*

dt/di_

time

,co

r_fu

nc_

A(i_

cor)

en

dd

oif(

i_m

ea

sure

+1

.lt.n

_co

r)th

en

do

i_co

r=

i_m

ea

sure

,n_

cor-

1d

oi_

typ

e=

1,

n_

typ

eco

r_fu

nc(

i_co

r,i_

typ

e)

=2

.*co

r_fu

nc(

i_co

r,i_

typ

e)/

&(c

or_

fun

c(n

_co

r,1

)-1

.)/n

um

be

r_sp

ec(

i_ty

pe

)e

nd

do

qd

ot_

cor(

i_co

r)=

2.*

qd

ot_

cor(

i_co

r)/(

qd

ot_

cor(

n_

cor)

-1.)

/n_

pa

rtco

r_fu

nc_

A(i_

cor)

=2

.*co

r_fu

nc_

A(i_

cor)

/(co

r_fu

nc_

A(n

_co

r)-1

.)&

/n_

part

cor_

fun

c_B

(i_

cor)

=2

.*co

r_fu

nc_

B(i_

cor)

/n_

pa

rt/c

or_

fun

c_B

(n_

cor)

write

(22

,’(3e

15

.5)’)

ni_

time

*i_

cor*

dt/d

i_tim

e,

&(c

or_

fun

c(i_

cor,

i_ty

pe

),i_

typ

e=

1,n

_ty

pe

)w

rite

(23

,’(2e

15

.5)’)

ni_

time

*i_

cor*

dt/d

i_tim

e,q

do

t_co

r(i_

cor)

write

(25

,’(2e

15

.5)’)

ni_

time

*i_

cor*

dt/d

i_tim

e,c

or_

fun

c_B

(i_

cor)

write

(26

,’(2e

15

.5)’)

ni_

time

*i_

cor*

dt/d

i_tim

e,c

or_

fun

c_A

(i_

cor)

en

dd

o

en

dif

if(i_

run

.eq

.n_ru

n)

clo

se(2

2)

if(i_

run

.eq

.n_ru

n)

clo

se(2

3)

if(f_

po

ten

tial.g

t.1

)th

en

if(i_

run

.eq

.1)

op

en

(24

,file

=’p

imd

.g_r’,s

tatu

s=’u

nkn

ow

n’)

!---

-co

un

tn

um

be

ro

fp

art

icle

sfo

rg

ive

nty

pe

sd

oi_

typ

e=

1,n

_ty

pe

do

j_ty

pe

=1

,n_

typ

e!-

---

de

fine

no

rma

liza

tion

fact

or

if(n

um

be

r_sp

ec(

i_ty

pe

)*n

um

be

r_sp

ec(

j_ty

pe

).n

e.0

)th

en

no

rm_

spe

c(i_

typ

e,j_

typ

e)

=re

al(n

_p

art

)/

&(n

um

be

r_sp

ec(

i_ty

pe

)*n

um

be

r_sp

ec(

j_ty

pe

))e

lse

no

rm_

spe

c(i_

typ

e,j_

typ

e)

=0

.e

nd

ifn

orm

_sp

ec(

i_ty

pe

,j_ty

pe

)=

&2

*(n

orm

_sp

ec(

i_ty

pe

,j_ty

pe

)/(n

_p

art

/vo

l_1

))*

&(r

ea

l(n

_grid

)/r_

ran

ge

(i_

typ

e,j_

typ

e))

/g_

r(1

,1,n

_g

rid

+1

)!-

---

sym

metr

ize

toyi

eld

be

tte

rst

atis

tics

if(i_

typ

e.lt

.j_ty

pe

)th

en

do

i_g

rid

=0

,n_

grid

g_

r(i_

typ

e,j_

typ

e,i_

grid

)=

&(

g_

r(i_

typ

e,j_

typ

e,i_

grid

)+

g_

r(j_

typ

e,i_

typ

e,i_

grid

))

/2

g_

r(j_

typ

e,i_

typ

e,i_

grid

)=

g_

r(i_

typ

e,j_

typ

e,i_

grid

)e

nd

do

en

dif

!---

-a

pp

lyn

orm

aliz

atio

nto

g(r

)d

oi_

grid

=0

,n

_g

rid

g_

r(i_

typ

e,j_

typ

e,i_

grid

)=

g_

r(i_

type

,j_ty

pe

,i_g

rid

)&

*n

orm

_sp

ec(

i_ty

pe

,j_ty

pe

)e

nd

do

en

dd

oe

nd

do

!---

-fo

rma

tr,

g(r

)/4

{\p

i}r*

*2d

oi_

grid

=1

,n

_g

rid

write

(24

,’(6

e12

.4)’)

&i_

grid*r

_cu

toff(1

,1)/

n_

grid

,g_

r(1

,1,i_

grid

)&

/(4

*pi*

(i_

grid

*r_

cuto

ff(1

,1)/

n_

grid

)**2

),

&i_

grid*r

_cu

toff(1

,2)/

n_

grid

,g_

r(1

,2,i_

grid

)&

/(4

*pi*

(i_

grid

*r_

cuto

ff(1

,2)/

n_

grid

)**2

),

&i_

grid*r

_cu

toff(2

,2)/

n_

grid

,g_

r(2

,2,i_

grid

)&

/(4

*pi*

(i_

grid

*r_

cuto

ff(2

,2)/

n_

grid

)**2

)e

nd

do

if(i_

run

.eq

.n_

run

)cl

ose

(24

)e

nd

if

if(f

_p

ote

ntia

l.eq

.3)

call

bo

nd

_o

ut

if(f

_p

ote

ntia

l.eq

.5)

call

bo

nd

_o

ut

if(f

_p

ote

ntia

l.eq

.6)

call

bo

nd

_o

ut

if(f

_p

ote

ntia

l.eq

.7)

call

bo

nd

_o

ut

if(f

_p

ote

ntia

l.eq

.8)

call

bo

nd

_o

ut

if(f

_p

ote

ntia

l.eq

.9)

call

bo

nd

_o

ut

20

1fo

rma

t(1

e1

5.7

,a)

20

2fo

rma

t(2

e1

5.7

,a)

20

3fo

rma

t(3

e1

5.7

,a)

20

4fo

rma

t(4

e1

5.7

,a)

20

6fo

rma

t(6

e1

3.5

,a)

20

9fo

rma

t(4

e1

3.5

,a,i3

)

136


en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

bo

nd_

ou

tim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sion

dr_

ne

igh

dr_

ne

igh

=(r

_ne

igh

_m

ax-

r_n

eig

h_

min

)/

n_

grid

op

en

(41

,file

=’p

imd

.bo

n’,s

tatu

s=’u

nkn

ow

n’)

do

i_g

rid

=1

,n

_g

rid

r_d

um

my

=r_

ne

igh

_m

in+

i_g

rid

*d

r_ne

igh

write

(41

,’(5

e1

5.6

)’)

r_d

um

my,

&h

ist_

an

g(1

,i_g

rid

)/

(dr_

ne

igh

*n_

silic

*ne

igh

_tim

e)

/r_

du

mm

y**2

,&

i_g

rid

*18

0.

/n

_g

rid

,&

n_

grid

*his

t_a

ng

(2,i_

grid

)/

(18

0.

*n

_o

xy*

ne

igh

_tim

e),

!S

i-O

-Si

&n

_g

rid

*his

t_a

ng

(3,i_

grid

)/

(6*

18

0.

*n

_si

lic*

ne

igh

_tim

e)

!O

-Si-O

en

dd

ocl

ose

(41

)e

nd

sub

rou

tine

!!

!!

!!

!!

!!

!

sub

rou

tine

che

ck_

skin

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

inte

ge

rf_

up

da

te(0

:n_

tro

t-1

,n_

pa

rt)

com

mo

n/c

f_u

pd

ate

/f_

up

da

te

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

do

i_tr

ot

=0,n

_tr

ot-

1r_

du

mm

y=

0.

f_u

pd

ate

(i_tr

ot,i_

pa

rt)

=0

do

i_d

im=

1,n

_d

imd

elta

_r(

i_d

im)

=r_

rea

l(i_

dim

,i_tr

ot,i_

pa

rt)

&-

r0_

old

(i_

dim

,i_tr

ot,i_

pa

rt)

r_d

um

my

=r_

du

mm

y+

de

lta_

r(i_

dim

)**2

en

dd

oif(

r_d

um

my.

gt.(0

.5*r

_sk

in(i_

typ

e,i_

typ

e))

**2

)th

en

f_sk

in=

1f_

up

da

te(i_

tro

t,i_

pa

rt)

=1

en

dif

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

pre

dic

tim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

ub

lep

reci

sion

loc_

forc

e(n

_d

im)

!---

-p

rop

ag

ate

on

lyin

de

pe

nd

en

tp

art

s

do

j_tr

ot

=0

,n_

tro

t-1

i_tr

ot

=j_

tro

t+

1if(

i_tr

ot.e

q.2

)i_

tro

t=

n_

tro

t+

1

!---

-fir

stp

osi

tion

sd

oj_

ord

er

=1

,n_

ord

er

do

i_d

im=

1,n

_d

imd

oi_

pa

rt=

1,n

_p

art

u0

(i_

dim

,i_tr

ot,i_

pa

rt)

=u

0(i_

dim

,i_tr

ot,i_

pa

rt)

+&

pre

dic

t_co

ef(

0,j_

ord

er)

*u

x(j_

ord

er,

i_d

im,i_

tro

t,i_

pa

rt)

en

dd

oe

nd

do

en

dd

o

!---

-a

dd

ran

do

mfo

rce

top

osi

tion

sa

nd

velo

citie

si_

du

mm

y=

(i_

tro

t-1

)/2

do

i_p

art

=1

,n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

!---

-te

rmst

och

forc

ed

oi_

dim

=1

,n

_d

imlo

c_fo

rce

(i_

dim

)=

0.

do

j_d

im=

1,

n_

dim

loc_

forc

e(i_

dim

)=

loc_

forc

e(i_

dim

)+

&h

_m

at_

inv(

i_d

im,j_

dim

)*

fr(j_

dim

,i_tr

ot,i_

pa

rt)

en

dd

oe

nd

do

do

i_d

im=

1,n

_d

imu

0(i_

dim

,i_tr

ot,i_

pa

rt)

=u

0(i_

dim

,i_tr

ot,i_

pa

rt)

+&

loc_

forc

e(i_

dim

)*

dt_

2/

2&

*rf

_w

idth

(i_

du

mm

y,i_

typ

e)

ux(

1,i_

dim

,i_tr

ot,i_

pa

rt)

=u

x(1

,i_d

im,i_

tro

t,i_

pa

rt)

+&

loc_

forc

e(i_

dim

)*

dt_

2*

rf_

wid

th(i_

du

mm

y,i_

typ

e)

en

dd

oe

nd

do

!---

-th

en

de

riva

tive

sd

oi_

ord

er

=1,

n_

ord

er-

1d

oj_

ord

er

=i_

ord

er

+1

,n

_o

rde

rd

oi_

dim

=1

,n_

dim

do

i_p

art

=1

,n_

pa

rtu

x(i_

ord

er,

i_d

im,i_

tro

t,i_

pa

rt)

=&

ux(

i_o

rde

r,i_

dim

,i_tr

ot,i_

pa

rt)

+&

pre

dic

t_co

ef(

i_o

rde

r,j_

ord

er)

*u

x(j_

ord

er,

i_d

im,i_

tro

t,i_

part

)e

nd

do

en

dd

oe

nd

do

en

dd

o

en

dd

o

call

sym

me

triz

e

!6

67

con

tinu

e

!---

pre

dic

tb

ox

sha

pe

if(f_

pre

ssu

re.e

q.1

)th

en

do

j_o

rde

r=

1,

n_

ord

er

do

i_d

im=

1,n

_d

im+

e_

dim

sca

l_0

(i_

dim

)=

sca

l_0

(i_

dim

)+

&p

red

ict_

coef(

0,j_

ord

er)

*sc

al_

x(j_

ord

er,

i_d

im)

en

dd

oe

nd

do

do

i_d

im=

1,

n_

dim

+e

_d

imif

(i_

dim

.le.n

_d

im)

the

n

137

B. Programssc

al_

0(i_

dim

)=

sca

l_0

(i_

dim

)&

+rf

b_

wid

th*

fr_

bo

x(i_

dim

)*

dt_

2/

2sc

al_

x(1

,i_d

im)

=sc

al_

x(1

,i_d

im)

&+

rfb

_w

idth

*fr

_b

ox(

i_d

im)

*d

t_2

els

esc

al_

0(i_

dim

)=

sca

l_0

(i_

dim

)&

+rf

b_

wid

th*

fr_

bo

x(i_

dim

)*

dt_

2/

2/

sqrt

2sc

al_

x(1

,i_d

im)

=sc

al_

x(1

,i_d

im)

&+

rfb

_w

idth

*fr

_b

ox(

i_d

im)

*d

t_2

/sq

rt2

en

dif

en

dd

o

do

i_o

rde

r=

1,

n_

ord

er-

1d

oj_

ord

er

=i_

ord

er

+1

,n

_o

rde

rd

oi_

dim

=1

,n

_d

im+

e_

dim

sca

l_x(

i_o

rde

r,i_

dim

)=

sca

l_x(

i_o

rder,

i_d

im)

&+

pre

dic

t_co

ef(

i_o

rde

r,j_

ord

er)

*sc

al_

x(j_

ord

er,

i_d

im)

en

dd

oe

nd

do

en

dd

o

call

sca

l_0

_u

pd

ate

en

dif

if(i_

time

.lt.4

00

0)

the

ni_

typ

e=

typ

e(1

)j_

typ

e=

typ

e(n

_p

art

)w

rite

(51

,’(1

0e1

6.7

)’)

r_tim

e,

&u

x(1

,1,1

,1)

/d

t/

v_th

erm

(0,i_

typ

e),

&u

x(1

,1,2

*n_

tro

t-1

,1)

/d

t/

v_th

erm

(n_

tro

t-1

,i_ty

pe

),&

ux(

1,1

,1,n

_pa

rt)

/d

t/

v_th

erm

(0,j_

typ

e),

&u

x(1

,1,2

*n_

tro

t-1

,n_

pa

rt)

/d

t/

v_th

erm

(n_

tro

t-1

,j_ty

pe

),&

volu

me

/n

_pa

rt,

1.*

i_ty

pe

,1

.*j_

type

els

eif

(i_

time

.eq

.40

00

)th

en

clo

se(5

1)

en

dif

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

corr

ect

imp

licit

no

ne

incl

ud

e"p

imd

.co

m.f"

do

ub

lep

reci

sio

nco

r_m

ax,

cor_

ma

x_m

in,c

or_

ma

x_m

ax

inte

ge

rn

um

_p

art

(n_

typ

e)

do

ub

lep

reci

sio

na

vg_

cha

rge

(n_

typ

e),

ma

x_ch

arg

e(n

_ty

pe

),&

min

_ch

arg

e(n

_ty

pe

)d

ou

ble

pre

cisi

on

fstr

ess

(3,3

)in

teg

er

ki_

dim

!---

cha

nge

!p

ara

me

ter

(co

r_m

ax_

min

=2

.e-0

9,c

or_

ma

x_m

ax=

1.e

-06

)p

ara

me

ter

(co

r_m

ax_

min

=1

.e-0

9,c

or_

ma

x_m

ax=

0.5

e-0

6)

inte

ge

rco

r_m

ax_

cou

nt

com

mo

n/c

cor_

ma

x_co

un

t/co

r_m

ax_

cou

nt

inte

ge

rfa

ilure

(3)

t_d

yna

mic

=0

.d0

if(i_

time

.le.1

)co

r_m

ax_

cou

nt

=0

!---

he

re,

f0h

as

me

an

ing

of

diff

ere

nce

be

twe

en

pre

dic

ted

!a

cce

lera

tion

an

dm

ea

sure

da

cce

lera

tion

cor_

ma

x=

0.

do

i_p

art

=1

,n_

pa

rti_

typ

e=

type

(i_

pa

rt)

do

j_tr

ot

=0

,n_

tro

t-1

i_tr

ot

=j_

trot

+1

if(i_

tro

t.e

q.2

)i_

tro

t=

n_

tro

t+

1i_

du

mm

y=

(i_

tro

t-1

)/2

do

i_d

im=

1,n

_d

imf0

(i_

dim

,i_tr

ot,i_

pa

rt)

=-u

x(2

,i_d

im,i_

tro

t,i_

pa

rt)

&+

f0(i_d

im,i_

tro

t,i_

pa

rt)/

ma

ss(i_

du

mm

y,i_

typ

e)*

dt_

2/

2r_

du

mm

y=

(f0

(i_

dim

,i_tr

ot,i_

pa

rt)*

corr

ect

_co

ef(

1)*

dt/

&v_

the

rm(i_

du

mm

y,i_

typ

e))

**2

!---

cha

ng

eif(

r_d

um

my.

gt.co

r_m

ax)

the

nco

r_m

ax

=r_

du

mm

yfa

ilure

(1)

=i_

pa

rtfa

ilure

(2)

=i_

tro

tfa

ilure

(3)

=i_

dim

en

dif

cor_

ma

x=

max(

cor_

ma

x,r_

du

mm

y)e

nd

do

en

dd

oe

nd

do

if(c

or_

ma

x.g

t.co

r_m

ax_

ma

x)th

en

writE

(*,’(

1e1

2.4

,3i5

)’)

cor_

ma

x,fa

ilure

sto

P"m

on

om

er

mo

ve"

en

dif

!---

-ch

an

ge

!g

oto

667

do

j_tr

ot

=0

,n_

tro

t-1

i_tr

ot

=j_

tro

t+

1if(

i_tr

ot.e

q.2

)i_

tro

t=

n_

tro

t+

1d

oi_

pa

rt=

1,n

_p

art

do

i_d

im=

1,n

_d

imu

0(i_

dim

,i_tr

ot,i_

pa

rt)

=u

0(i_

dim

,i_tr

ot,i_

pa

rt)

&+

corr

ect

_co

ef(

0)

*f0

(i_

dim

,i_tr

ot,i_

pa

rt)

en

dd

oe

nd

do

do

i_o

rde

r=

1,n

_o

rde

rd

oi_

pa

rt=

1,n

_p

art

do

i_d

im=

1,n

_d

imu

x(i_

ord

er,

i_dim

,i_tr

ot,i_

pa

rt)

=u

x(i_

ord

er,

i_d

im,i_

tro

t,i_

pa

rt)

&+

corr

ect

_co

ef(

i_o

rde

r)*

f0(i_

dim

,i_tr

ot,i_

pa

rt)

en

dd

oe

nd

do

en

dd

oe

nd

do

call

sym

me

triz

e

!6

67

con

tinu

e

t_d

yna

mic

=0

.d0

!---

-te

rmm

ea

sd

yna

mic

do

i_p

art

=1

,n_

pa

rt

138


i_ty

pe

=ty

pe(i_

pa

rt)

do

i_tr

ot

=0,n

_tr

ot-

1d

oi_

dim

=1

,n_

dim

forc

e(i_

dim

,i_tr

ot,i_

pa

rt)

=0

.d0

t_d

yna

mic

=t_

dyn

am

ic+

&m

ass

(i_

tro

t,i_

typ

e)

*m

et_

ten

_vo

i(i_

dim

)*

&(u

x(1

,i_d

im,2

*i_

tro

t+1

,i_p

art

)**2

+&

ux(

1,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)*

*2)

en

dd

oif

(e_

dim

.eq

.3)

the

nt_

dyn

am

ic=

t_d

yna

mic

&+

ma

ss(i_

tro

t,i_

typ

e)

*m

et_

ten

_vo

i(4

)*

&(

ux(

1,1

,2*i

_tr

ot+

1,i_

pa

rt)

*u

x(1

,2,2

*i_

tro

t+1

,i_p

art

)&

+u

x(1

,1,2

*i_

tro

t+2

,i_p

art

)*

ux(

1,2

,2*i

_tr

ot+

2,i_

pa

rt)

)&

+m

ass

(i_

tro

t,i_

typ

e)

*m

et_

ten

_vo

i(5

)*

&(

ux(

1,2

,2*i

_tr

ot+

1,i_

pa

rt)

*u

x(1

,3,2

*i_

tro

t+1

,i_p

art

)&

+u

x(1

,2,2

*i_

tro

t+2

,i_p

art

)*

ux(

1,3

,2*i

_tr

ot+

2,i_

pa

rt)

)&

+m

ass

(i_

tro

t,i_

typ

e)

*m

et_

ten

_vo

i(6

)*

&(

ux(

1,3

,2*i

_tr

ot+

1,i_

pa

rt)

*u

x(1

,1,2

*i_

tro

t+1

,i_p

art

)&

+u

x(1

,3,2

*i_

tro

t+2

,i_p

art

)*

ux(

1,1

,2*i

_tr

ot+

2,i_

pa

rt)

)e

nd

ife

nd

do

en

dd

ot_

dyn

am

ic=

t_d

yna

mic

/(2

*dt_

2)

if(f

_p

ress

ure

.eq

.1)

the

n!-

---

corr

ect

bo

xsh

ap

ed

oi_

dim

=1

,n

_d

im+

e_

dim

forc

e_

sca

l(i_

dim

)=

-sc

al_

x(2

,i_d

im)

&+

forc

e_

sca

l(i_

dim

)/

ma

ss_

bo

x*

dt_

2/

2r_

du

mm

y=

(fo

rce

_sc

al(i_

dim

)*co

rre

ct_

coe

f(1

)*d

t/

w_

the

rm)*

*2if

(r_

du

mm

y.g

t.1

00

*co

r_m

ax_

ma

x)th

en

writE

(*,*

)r_

du

mm

yst

oP

"vo

lum

em

ove

"e

nd

ife

nd

do

do

i_d

im=

1,

n_

dim

+e

_d

imsc

al_

0(i_

dim

)=

sca

l_0

(i_

dim

)+

&co

rre

ct_

coe

f(0

)*

forc

e_

sca

l(i_

dim

)e

nd

do

do

i_o

rde

r=

1,n

_o

rde

rd

oi_

dim

=1

,n_

dim

+e

_d

imsc

al_

x(i_

ord

er,

i_d

im)

=sc

al_

x(i_

ord

er,

i_d

im)

&+

corr

ect

_co

ef(

i_o

rde

r)*

forc

e_

sca

l(i_

dim

)e

nd

do

en

dd

o

t_d

iag

=0

.t_

nd

iag

=0

.!-

---

term

mea

sb

ox

dyn

do

i_d

im=

1,

n_

dim

+e

_d

imif

(i_

dim

.le.n

_d

im)

t_d

iag

=t_

dia

g&

+m

ass

_b

ox

*sc

al_

x(1

,i_d

im)*

*2if

(i_

dim

.gt.n

_d

im)

t_n

dia

g=

t_n

dia

g&

+2

*m

ass

_b

ox

*sc

al_

x(1

,i_d

im)*

*2e

nd

do

t_d

iag

=t_

dia

g/

(2*d

t_2

)t_

nd

iag

=t_

nd

iag

/(2

*dt_

2)

t_d

yna

mic

=t_

dyn

am

ic+

t_d

iag

+t_

ndia

ge

nd

if

do

i_ty

pe

=1

,n

_ty

pe

avg

_ch

arg

e(i_

typ

e)=

0n

um

_p

art

(i_ty

pe

)=0

en

dd

od

oi_

pa

rt=

1,

n_

pa

rti_

typ

e=

typ

e(i_

pa

rt)

avg

_ch

arg

e(i_

typ

e)=

avg

_ch

arg

e(i_

type

)+ch

arg

e(i_

pa

rt)

nu

m_

pa

rt(i_ty

pe

)=n

um

_p

art

(i_

typ

e)+

1e

nd

do

do

i_ty

pe

=1

,n

_ty

pe

avg

_ch

arg

e(i_

typ

e)=

avg

_ch

arg

e(i_

typ

e)/

nu

m_

pa

rt(i_

typ

e)

en

dd

o

po

l_to

tal

=0

.d0

po

l_d

ipo

les

=0

.d0

do

i_p

art

=1

,n

_p

art

do

i_d

im=

1,

n_

dim

po

l_to

tal(i_

dim

)=

po

l_to

tal(i_

dim

)+

r_re

al(i_

dim

,0,i_

pa

rt)

&*

charg

e(i_

pa

rt)

po

l_d

ipo

les(

i_d

im)

=p

ol_

dip

ole

s(i_

dim

)+

dip

ole

(i_

dim

,i_p

art

)e

nd

do

en

dd

op

ol_

tota

l=

po

l_to

tal

+p

ol_

dip

ole

s

do

i_d

im=

1,

3d

oj_

dim

=1

,3fs

tre

ss(i_

dim

,j_d

im)=

0.d

0d

ok_

dim

=1

,3if

(i_

dim

.eq.k

_d

im)

the

nki

_d

im=

i_d

ime

lse

if(i_

dim

*k_

dim

.eq

.2)

the

nki

_d

im=

4e

lse

if(i_

dim

*k_

dim

.eq

.6)

the

nki

_d

im=

5e

lse

if(i_

dim

*k_

dim

.eq

.3)

the

nki

_d

im=

6e

nd

iffs

tre

ss(i_

dim

,j_d

im)=

fstr

ess

(i_

dim

,j_d

im)

+&

h_

ma

t(j_

dim

,k_

dim

)*fo

rce

_sc

al_

inte

r(ki

_d

im)

&/v

olu

me

en

dd

oe

nd

do

en

dd

o

!g

oto

10

1if

(mo

d(i_

time

,1).

eq

.0)

the

nca

llp

rim

_est

ima

tor

write

(60

,’(3

6e

15

.7)’)

r_tim

e,

!if

(mo

d(i_

time

,10

).e

q.9

)w

rite

(60

,’(3

0e

15

.7)’)

r_tim

e,

2(t

_d

yna

mic

+v_

cha

in+

v_in

ter+

t_d

yna

mic

_ch

arg

e+

v_ch

arg

e+

&v_

pre

ss-v

_g

au

ge

)/n

_p

art

/n_

tro

t,3

t_d

yna

mic

/n_

pa

rt/n

_tr

ot,

4v_

cha

in/n

_p

art

/n_

tro

t,5

(v_

inte

r-v_

gau

ge

)/n

_p

art

/n_

tro

t,6

v_p

ress

/n_

pa

rt/n

_tr

ot,

7v_

ew

ald

/n_

pa

rt/n

_tr

ot,

8vo

lum

e/n

_p

art

,9

(sca

l_0

(i_

dim

),i_

dim

=1

,n_

dim

+e

_d

im),

!co

l9

-14

&v_

rea

lco

ulo

mb/n

_p

art

,v_

sho

rtra

ng

e/n

_p

art

,!

col

15

+1

6&

v_th

ree

bo

dy/

n_

pa

rt,

0.,

!co

l1

7+

18

&se

lf_e

ne

rgy/

n_

pa

rt,

v_q

eq

self/

n_

pa

rt,

!co

l1

9+

20

&(a

vg_

cha

rge

(i_

typ

e),

i_ty

pe

=1

,n_

typ

e),

!co

l2

1+

22

139

B. Programs&

t_d

yna

mic

_ch

arg

e/n

_p

art

/n_

tro

t,!

col

23

&v_

cha

rge

/n_

pa

rt/n

_tr

ot,

!co

l2

4&

(po

l_to

tal(i_

dim

),i_

dim

=1

,3),

!co

l2

5-2

7&

(po

l_d

ipo

les(

i_d

im),

i_d

im=

1,3

),!

col

28

-30

&fs

tre

ss(1

,1),

fstr

ess

(2,2

),fs

tre

ss(3

,3),

!co

l3

1-3

3&

0.5

*(fs

tre

ss(1

,2)+

fstr

ess

(2,1

)),

!co

l3

4&

0.5

*(fs

tre

ss(2

,3)+

fstr

ess

(3,2

)),

!co

l3

5&

0.5

*(fs

tre

ss(1

,3)+

fstr

ess

(3,1

))!

col

36

els

eif(

i_tim

e.e

q.n

_tim

e_

sca

l*4

00

0)

the

ncl

ose

(60

)e

nd

if

if((m

od

(i_

time

,10

).e

q.9

).a

nd

.&

((f_

sym

me

try.

eq

.6).

or.

(f_

sym

me

try.

eq

.7))

)&

write

(61

,’(4

f12

.6)’)

r_tim

e,

ord

_p

ar,

ord

_p

arB

,o

rd_

pa

rC

if(m

od

(i_

time

,10

).e

q.9

)th

en

do

i_ty

pe

=1

,n

_ty

pe

avg

_ch

arg

e(i_ty

pe

)=0

ma

x_ch

arg

e(i_ty

pe

)=-1

.e3

0m

in_

cha

rge

(i_ty

pe

)=1

.e3

0e

nd

do

do

i_p

art

=1

,n

_p

art

i_ty

pe

=ty

pe(i_

pa

rt)

avg

_ch

arg

e(i_ty

pe

)=a

vg_

cha

rge

(i_

typ

e)+

cha

rge

(i_

pa

rt)

if(c

ha

rge

(i_p

art

).g

t.m

ax_

cha

rge

(i_

typ

e))

&m

ax_

cha

rge

(i_

typ

e)=

cha

rge

(i_

pa

rt)

if(c

ha

rge

(i_p

art

).lt.

min

_ch

arg

e(i_

typ

e))

&m

in_ch

arg

e(i_

typ

e)=

cha

rge

(i_

pa

rt)

en

dd

oa

vg_

cha

rge

(1)=

avg

_ch

arg

e(1

)/n

_si

lica

vg_

cha

rge

(2)=

avg

_ch

arg

e(2

)/n

_o

xy

write

(62

,’(2

0e1

5.7

)’)

r_tim

e,

2a

vg_ch

arg

e(1

),3

min

_ch

arg

e(1

),4

ma

x_ch

arg

e(1

),5

avg

_ch

arg

e(2

),6

min

_ch

arg

e(2

),7

ma

x_ch

arg

e(2

)e

nd

if

!1

01

con

tinu

e

ieff_

time

=ie

ff_

time

+d

i_tim

e

!---

-in

cre

ase

time

ste

po

nly

ifo

nm

ulti

ple

inte

ge

rso

fn

i_tim

eif(

cor_

ma

x.lt.

cor_

ma

x_m

in)

the

nif(

mo

d(ie

ff_

time

,ni_

time

).n

e.0

)co

r_m

ax

=2

*co

r_m

ax_

min

en

dif

!---

-ch

an

ge

go

to1

00

!---

-m

ake

sure

tha

ttim

est

ep

do

esn

’tb

eco

me

too

larg

e/s

ma

llif(

(co

r_m

ax.

lt.co

r_m

ax_

min

).a

nd

.(d

i_tim

e.e

q.n

i_tim

e))

the

nco

r_m

ax

=2

*cor_

ma

x_m

ine

lse

if((c

or_

ma

x.g

t.co

r_m

ax_

ma

x).a

nd.(

di_

time

.eq

.1))

the

nst

op

"Tim

eco

rre

lca

nn

ot

be

me

asu

red

an

ylo

ng

er.

"e

nd

if

!---

-m

ake

sure

ieff_

time

isn

ot

too

big

if(ie

ff_

time

.le.0

)st

op

"ie

ff_

time

isin

fty"

!---

cha

ng

e!

write

(61

,*)

r_tim

e,d

t,co

r_m

ax,

ieff_

time

,fa

ilure

!---

-ch

an

ge

!in

clu

de

wid

tho

fb

ox

de

fom

ove

sif(

cor_

ma

x.g

t.co

r_m

ax_

ma

x)th

en

!---

-d

ecr

ease

time

ste

pco

r_m

ax_

cou

nt

=1

!---

-ch

an

ge

!if(

i_tim

e.lt

.10

00

0)

write

(61

,*)

r_tim

e,d

t,co

r_m

ax,

ieff_tim

ed

t=

dt/2

di_

time

=d

i_tim

e/2

!---

-ch

an

ge

!if(

i_tim

e.lt

.10

00

0)

write

(61

,*)

r_tim

e,d

t,co

r_m

ax,

ieff_tim

ed

t_2

=d

t**2

do

i_ty

pe

=1

,n_

typ

ed

oi_

tro

t=

0,n

_tr

ot-

1rf

_w

idth

(i_

trot,i_

typ

e)

=rf

_w

idth

(i_

tro

t,i_

typ

e)

*2

.d0

**0

.5d

0e

nd

do

en

dd

od

oi_

ord

er

=1

,n_

ord

er

r_d

um

my

=1

./2**

i_o

rde

r!

if(i_

ord

er.

gt.2

)r_

du

mm

y=

0.

do

i_p

art

=1

,n_

pa

rtd

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

dim

=1

,n_

dim

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

=&

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

1,i_

part

)*

r_d

um

my

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

=&

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

2,i_

part

)*

r_d

um

my

en

dd

oe

nd

do

en

dd

od

oi_

dim

=1

,n_

dim

sca

l_x(

i_o

rde

r,i_

dim

)=

sca

l_x(

i_o

rder,

i_d

im)

*r_

du

mm

ye

nd

do

en

dd

oe

lse

if(co

r_m

ax.

lt.co

r_m

ax_

min

)th

en

!---

-in

cre

ase

time

ste

pco

r_m

ax_

cou

nt

=1

!---

-ch

an

ge

!if(

i_tim

e.lt

.10

00

0)

write

(61

,*)

r_tim

e,d

t,co

r_m

ax,

ieff_tim

ed

t=

dt*

2d

i_tim

e=

di_

time

*2!-

---

cha

ng

e!

if(i_

time

.lt.1

00

00

)w

rite

(61

,*)

r_tim

e,d

t,co

r_m

ax,

ieff_tim

ed

t_2

=d

t**2

do

i_ty

pe

=1

,n_

typ

ed

oi_

tro

t=

0,n

_tr

ot-

1rf

_w

idth

(i_

trot,i_

typ

e)

=rf

_w

idth

(i_

tro

t,i_

typ

e)

/2

.d0

**0

.5d

0e

nd

do

en

dd

od

oi_

ord

er

=1

,n_

ord

er

r_d

um

my

=2

**i_

ord

er

!if(

i_o

rde

r.g

t.2

)r_

du

mm

y=

0.

do

i_p

art

=1

,n_

pa

rtd

oi_

tro

t=

0,n

_tr

ot-

1d

oi_

dim

=1

,n_

dim

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

=

140


&u

x(i_

ord

er,

i_d

im,2

*i_

tro

t+1

,i_p

art

)*

r_d

um

my

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

=&

ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

*r_

du

mm

ye

nd

do

en

dd

oe

nd

do

do

i_d

im=

1,n

_d

imsc

al_

x(i_

ord

er,

i_d

im)

=sc

al_

x(i_

ord

er,

i_d

im)

*r_

du

mm

ye

nd

do

en

dd

oe

nd

if!

if(i_

time

.eq

.10

00

)cl

ose

(61

)

10

0co

ntin

ue

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

sym

me

triz

eim

plic

itn

on

ein

clu

de

"pim

d.c

om

.f"

do

i_tr

ot

=1

,n_

tro

t/2

-1d

oi_

dim

=1

,n_d

im!-

---

first

po

sitio

ns

do

i_p

art

=1,n

_p

art

u0

(i_

dim

,2*(

n_

tro

t-i_

tro

t)+

1,i_

pa

rt)

=&

u0

(i_

dim

,2*i

_tr

ot+

1,i_

pa

rt)

u0

(i_

dim

,2*(

n_

tro

t-i_

tro

t)+

2,i_

pa

rt)

=&

-u0

(i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

en

dd

o!-

---

the

nd

eriva

tive

sd

oi_

ord

er

=1

,n_

ord

er

do

i_p

art

=1,n

_p

art

ux(

i_o

rde

r,i_

dim

,2*(

n_

tro

t-i_

tro

t)+

1,i_

pa

rt)

=&

ux(

i_o

rder,

i_d

im,2

*i_

tro

t+1

,i_p

art

)u

x(i_

ord

er,

i_d

im,2

*(n

_tr

ot-

i_tr

ot)

+2,i_

pa

rt)

=&

-ux(

i_o

rde

r,i_

dim

,2*i

_tr

ot+

2,i_

pa

rt)

en

dd

oe

nd

do

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

SU

BR

OU

TIN

EIN

R25

0(M

Z,IS

EE

D,K

PT

R)

C CIN

ITIA

LIZ

AT

ION

RO

UT

INE

FO

RR

25

0R

AN

DO

MN

UM

BE

RG

EN

ER

AT

OR

CF

OR

32

BIT

CO

MP

UT

ER

SO

NL

Y!

CU

SE

SS

UN

FO

RS

TA

RT

UP

CIN

TE

GE

RM

Z(2

50

)IN

TE

GE

RIS

EE

D,

KP

TR

CIN

TE

GE

RIU

F(3

1),

IGF

(31

)IN

TE

GE

RIA

DD

R1

(10

00

),IA

DD

R2

(10

00

)D

OU

BL

EP

RE

CIS

ION

RR

AN

F(2

00

0),

rco

mm

on

/cin

r25

0/

r

C CK

PT

R=

1C

DO

10

0I

=1,3

0IG

F(I

)=

2**

(I-

1)

IUF

(I)

=2

**I

-1

10

0C

ON

TIN

UE

CIG

F(3

1)

=2

**3

0IU

F(3

1)

=2

**3

0+

(2**

30

-1

)C

CA

LL

SU

N(I

SE

ED

,RR

AN

F,2

50

)C C

MA

PR

EA

LS

TO

PO

SIT

IVE

INT

EG

ER

SC

DO

20

0I

=1

,25

0M

Z(I

)=

IDN

INT

(DB

LE

(RR

AN

F(I

))*D

BL

E(F

LO

AT

(IU

F(3

1))

))2

00

CO

NT

INU

EC C

SP

EC

IAL

TR

EA

TM

EN

TO

FF

IRS

T3

1N

UM

BE

RS

CD

O3

00

I=

1,3

1M

Z(I

)=

IOR

(MZ

(I),

IGF

(I))

30

0C

ON

TIN

UE

CD

O4

00

I=

1,3

1M

Z(I

)=

IAN

D(M

Z(I

),IU

F(I

))4

00

CO

NT

INU

EC C

MIX

ING

CC

AL

LS

UN

(IS

EE

D,R

RA

NF

,20

00

)C

DO

50

0I

=1

,10

00

IAD

DR

1(I

)=

1.

+2

50

.*

RR

AN

F(I

)IA

DD

R2

(I)

=1.

+2

50

.*

RR

AN

F(I

+1

000

)5

00

CO

NT

INU

EC

DO

60

0I

=1

,10

00

NT

EM

P=

MZ

(IA

DD

R1

(I))

MZ

(IA

DD

R1

(I))

=M

Z(I

AD

DR

2(I

))M

Z(I

AD

DR

2(I

))=

NT

EM

P6

00

CO

NT

INU

EC C

WA

RM

ING

UP

CD

O8

00

I=

1,8

CD

O7

10

K=

1,1

47

MZ

(K)

=IE

OR

(MZ

(K),

MZ

(K+

10

3))

71

0C

ON

TIN

UE

CD

O7

20

K=

148

,25

0M

Z(K

)=

IEO

R(M

Z(K

),M

Z(K

-1

47

))7

20

CO

NT

INU

EC

80

0C

ON

TIN

UE

CE

ND

C CC

CC

C

141

B. ProgramsC

SU

BR

OU

TIN

ES

UN

(IS

EE

D,R

AN

F,N

)C C

ST

OR

ES

NR

EA

LR

AN

DO

MN

UM

BE

RS

INR

AN

FC

ISE

ED

ISA

ST

AR

TU

PV

AL

UE

CIM

PL

ICIT

DO

UB

LE

PR

EC

ISIO

N(A

-H,

O-Z

)C

INT

EG

ER

ISE

ED

DO

UB

LE

PR

EC

ISIO

NR

AN

F(N

),r

com

mo

n/c

inr2

50

/r

C C CD

AT

AF

AC

TO

R/4

14

75

55

7.0

D0

/,T

WO

28

/268

43

54

56

.0D

0/

C C CIF

(IS

EE

D.G

E.

0)

TH

EN

R=

DB

LE

(IS

EE

D)/

TW

O2

8R

=D

MO

D(R

*FA

CT

OR

,1.0

D0

)IS

EE

D=

-1E

ND

IFC

DO

10

0I

=1

,NR

=D

MO

D(R

*FA

CT

OR

,1.0

D0

)R

AN

F(I

)=

SN

GL

(R)

10

0C

ON

TIN

UE

CE

ND

C C CC

CC

CC

SU

BR

OU

TIN

ER

25

0(M

Z,R

AN

,n_

dim

,N,K

PT

R)

C CS

TO

RE

SN

RE

AL

RA

ND

OM

NU

MB

ER

SIN

RA

NU

SIN

GIN

TE

GE

RS

INM

ZC

RA

ND

OM

NU

MB

ER

SH

AV

EU

NIF

OR

MD

IST

RIB

UT

ION

ININ

TE

RV

AL

(0,1

)C

PA

RA

ME

TE

R(R

INV

=1

./2

14

74

83

64

7.)

INT

EG

ER

MZ

(250

)IN

TE

GE

RN

,KP

TR

DO

UB

LE

PR

EC

ISIO

NR

AN

(n_

dim

)IN

TE

GE

RI,L

C CI

=0

L=

N+

KP

TR

-1

C1

0C

ON

TIN

UE

CK

MIN

=K

PT

RK

MA

X=

MIN

(L,1

47

)D

O1

00

K=

KM

IN,K

MA

XM

Z(K

)=

IEO

R(M

Z(K

),M

Z(K

+1

03

))I

=I

+1

RA

N(I

)=

MZ

(K)

*R

INV

10

0C

ON

TIN

UE

CK

MIN

=M

AX

(KM

IN,K

MA

X+

1)

KM

AX

=M

IN(L

,25

0)

*VO

CL

LO

OP

,NO

VR

EC

DO

20

0K

=K

MIN

,KM

AX

MZ

(K)

=IE

OR

(MZ

(K),

MZ

(K-

14

7))

I=

I+

1R

AN

(I)

=M

Z(K

)*

RIN

V2

00

CO

NT

INU

EC

IF(K

MA

X.E

Q.

25

0)

TH

EN

KP

TR

=1

L=

L-

25

0G

OT

O1

0E

ND

IFC

KP

TR

=K

MA

X+

1C

EN

DC !

!!

!!

!!

!!

sub

rou

tine

jaco

bi(d

im,

ma

trix

,ro

t_m

at,

eig

en

,ite

r_m

ax)

!d

iag

ona

lize

ssy

mm

etr

icm

atr

ixm

atr

ixw

ithg

ive

nd

ime

nsi

on

!u

sin

gite

ratio

nsc

he

me

of

jaco

bi

! !ro

t_m

at

ism

atr

ixco

inta

inin

geig

en

vect

ors

of

ma

trix

!m

atr

ixth

us

tra

nsf

orm

sfr

om

"ort

ho

go

na

l"sp

ace

tog

ive

nsp

ace

! !e

ige

nis

ma

trix

inn

ew

coo

rdin

ate

syst

em

inte

ge

rd

im,

dim

_m

ax,

i_d

im,

j_d

im,

k_d

im,

l_d

imp

ara

me

ter

(dim

_m

ax=

3)

do

ub

lep

reci

sio

nm

atr

ix(d

im_

ma

x,d

im_

max)

do

ub

lep

reci

sio

nro

t_m

at(

dim

_m

ax,

dim

_m

ax)

,e

ige

n(d

im_

ma

x,d

im_

max)

do

ub

lep

reci

sio

nh

_m

at(

dim

_m

ax,

dim

_m

ax)

,lo

c_ro

(dim

_m

ax,

dim

_m

ax)

inte

ge

rite

rat,

iter_

ma

xin

teg

er

i_d

_m

ax,

i_o

_m

ax,

j_o

_m

ax

do

ub

lep

reci

sio

nm

ax_

dia

g,

ma

x_o

ff,

tole

ran

cep

ara

me

ter

(to

lera

nce

=1

.d-1

3)

do

ub

lep

reci

sio

np

hi

if(d

im.n

e.d

im_

ma

x)st

oP

"Allo

catio

npro

ble

min

jaco

bi"

do

i_d

im=

1,

dim

_m

ax

do

j_d

im=

1,

dim

_m

ax

h_

ma

t(i_

dim

,j_d

im)

=0

.ro

t_m

at(

i_d

im,j_

dim

)=

0.

eig

en

(i_

dim

,j_d

im)

=m

atr

ix(i_

dim

,j_d

im)

en

dd

oro

t_m

at(

i_d

im,i_

dim

)=

1.

en

dd

o

if(ite

r_m

ax.

eq

.0)

iter_

ma

x=

4*d

im_

max*

dim

_m

ax

do

10

0ite

rat

=1

,ite

r_m

ax

c---

-g

et

big

ge

std

iag

on

al

an

do

ff-d

iag

on

al

ele

me

nt

ma

x_d

iag

=ab

s(e

ige

n(1

,1))

i_d

_m

ax

=1

ma

x_o

ff=

abs(

eig

en

(1,2

))i_

o_

ma

x=

1j_

o_

ma

x=

2d

oi_

dim

=1,

dim

_m

ax-

1if

(ab

s(e

ige

n(i_

dim

,i_d

im))

.gt.m

ax_

dia

g)

the

nm

ax_

dia

g=

ab

s(e

ige

n(i_

dim

,i_d

im))

142


i_d

_m

ax

=i_

dim

en

dif

do

j_d

im=

i_d

im+

1,

dim

_m

ax

if(a

bs(

eig

en(i_

dim

,j_d

im))

.gt.m

ax_

off)

the

nm

ax_

off

=ab

s(e

ige

n(i_

dim

,j_d

im))

i_o

_m

ax

=i_

dim

j_o

_m

ax

=j_

dim

en

dif

en

dd

oe

nd

do

if(a

bs(

eig

en(d

im_

ma

x,d

im_

ma

x)).

gt.m

ax_

dia

g)

the

nm

ax_

dia

g=

eig

en

(dim

_m

ax,

dim

_m

ax)

i_d

_m

ax

=dim

_m

ax

en

dif

if(a

bs(

ma

x_off/m

ax_

dia

g).

lt.to

lera

nce

)re

turn

!d

efin

elo

cal

rota

tion

ma

trix

,!

set

h_m

at

=0

-ma

trix

,lo

c_ro

=1

-ma

trix

do

j_d

im=

1,

dim

_m

ax

do

i_d

im=

1,

dim

_m

ax

loc_

ro(i_

dim

,j_d

im)

=0

.d0

h_

ma

t(i_

dim

,j_d

im)

=0

.d0

en

dd

olo

c_ro

(j_

dim

,j_d

im)

=1

.d0

en

dd

op

hi

=0

.5*

ata

n(

2*e

ige

n(i_

o_

ma

x,j_

o_

ma

x)/

&(e

ige

n(i_

o_

ma

x,i_

o_

ma

x)-e

ige

n(j_

o_

ma

x,j_

o_

ma

x))

)lo

c_ro

(i_

o_

max,

i_o

_m

ax)

=d

cos(

ph

i)lo

c_ro

(j_

o_

max,

j_o

_m

ax)

=lo

c_ro

(i_

o_m

ax,

i_o

_m

ax)

loc_

ro(i_

o_

max,

j_o

_m

ax)

=-s

in(p

hi)

loc_

ro(j_

o_

max,

i_o

_m

ax)

=-lo

c_ro

(i_

o_

ma

x,j_

o_

ma

x)!

rota

tem

atr

ixe

ige

nd

oj_

dim

=1

,d

im_

ma

xd

ol_

dim

=1

,d

im_

ma

xif

(lo

c_ro

(l_d

im,j_

dim

).n

e.0

.d0

)th

en

do

k_d

im=

1,

dim

_m

ax

if(e

ige

n(l_

dim

,k_

dim

).n

e.0

.d0

)th

en

do

i_d

im=

1,

dim

_m

ax

h_

ma

t(i_

dim

,j_d

im)

=h

_m

at(

i_d

im,j_

dim

)&

+lo

c_ro

(k_

dim

,i_d

im)*

eig

en

(k_

dim

,l_d

im)*

loc_

ro(l_

dim

,j_d

im)

en

dd

oe

nd

ife

nd

do

en

dif

en

dd

oe

nd

do

do

j_d

im=

1,

dim

_m

ax

do

i_d

im=

1,

dim

_m

ax

eig

en

(i_

dim

,j_d

im)

=h

_m

at(

i_d

im,j_

dim

)h

_m

at(

i_d

im,j_

dim

)=

0.d

0e

nd

do

en

dd

o

do

j_d

im=

1,

dim

_m

ax

do

k_d

im=

1,

dim

_m

ax

if(lo

c_ro

(k_d

im,j_

dim

).n

e.0

.d0

)th

en

do

i_d

im=

1,

dim

_m

ax

h_

ma

t(i_

dim

,j_d

im)

=h

_m

at(

i_d

im,j_

dim

)&

+ro

t_m

at(

i_d

im,k

_d

im)

*lo

c_ro

(k_

dim

,j_d

im)

en

dd

oe

nd

ife

nd

do

en

dd

o

do

i_d

im=

1,

dim

_m

ax

do

j_d

im=

1,

dim

_m

ax

rot_

ma

t(i_

dim

,j_d

im)

=h

_m

at(

i_d

im,j_

dim

)e

nd

do

en

dd

o

10

0co

ntin

ue

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!su

bro

utin

ead

dn

um

tost

rin

g(s

trin

g,n

um

be

r)im

plic

itn

one

inte

ge

r*4

i,str

len

,nu

mb

er,

no

dig

,nu

m,s

nu

mch

ara

cte

r(*)

strin

g

snu

m=

nu

mb

er

do

i=le

n(s

trin

g),

1,-

1if(

strin

g(i:i)

/=’

’)g

oto

10

en

dd

o1

0st

rle

n=

i

no

dig

=in

t(lo

g1

0(1

.0*s

nu

m+

0.1

))+

1d

oi=

no

dig

,1,-

1n

um

=sn

um

/10

**(i-1

)st

rin

g(s

trle

n+

1:s

trle

n+

1)=

cha

r(4

8+

nu

m)

strle

n=

strle

n+

1sn

um

=sn

um

-nu

m*1

0**

(i-1

)e

nd

do

retu

rne

nd

!!!!!!!!!!!!

pim

d.co

m.f

!---

-w

he

reto

pa

ya

tte

ntio

nw

he

nn

ew

po

ten

tial

ge

tsb

uild

in!-

---

!---

-lo

ok

for

pla

ces

wh

ere

f_p

ote

ntia

l.eq

.2o

ccu

rs! !-

---

initi

aliz

ed

efa

ult:

ma

ss,in

tera

ctio

np

ara

me

ters

,cu

t_o

ff,s

kin

!te

mp

era

ture

,tim

est

ep

,p

ress

ure

inp

ara

_in

it!

--->

cho

sere

aso

na

ble

dyn

am

icm

od

em

ass

es

for

each

pa

rtic

lety

pe

!--

->ca

lcu

late

shift

sin

en

erg

y!-

---

initi

aliz

ed

efa

ult

stru

ctu

rean

dd

efa

ult

che

mic

al

com

po

sitio

n!-

---

ma

keo

utp

ut

file

an

din

pu

tfil

eco

mp

atib

le!-

---

write

ne

wfo

rce

/en

erg

yro

utin

e

rea

lp

i,tw

opi,s

qrt

2,s

qrt

3p

ara

me

ter

(pi=

3.1

41

59

26

5,tw

op

i=2

*pi,s

qrt

2=

1.4

14

21

35

6,

&sq

rt3

=1

.73

20

50

81

)

do

ub

lep

reci

sio

nm

ass

_a

mu

,jou

le,e

v,e

lec_

pre

fac

pa

ram

ete

r(m

ass

_a

mu

=0

.02

06

1,jo

ule

=7

.24

32

e2

2,e

V=

1.1

60

5e

4,

&e

lec_

pre

fac=

1.6

70

9e

5)

143

B. Programsin

teg

er

i_d

um

my

do

ub

lep

reci

sio

nr_

du

mm

yco

mp

lex

c_d

um

my

cha

ract

er(

64)

cha

r_d

um

my

inte

ge

rf_

sym

me

try

inte

ge

rn

_b

in_

x,n

_b

in_

y,n

_b

in_

zin

teg

er

i_si

lic,n

_si

lic,i_

oxy

,n_

oxy

!---

-f_

sym

me

try

=1

:fc

csy

mm

etr

y!

pa

ram

ete

r(f

_sy

mm

etr

y=1

)!

pa

ram

ete

r(n

_b

in_

x=5

,n_

bin

_y=

5,n

_b

in_

z=5

)!

pa

ram

ete

r(n

_si

lic=

n_

bin

_x*

n_b

in_

y*n

_b

in_

z,n

_o

xy=

3*n

_si

lic)

!---

-tr

idym

it!

pa

ram

ete

r(f

_sy

mm

etr

y=2

)!

pa

ram

ete

r(n

_b

in_

x=4

,n_

bin

_y=

3,n

_b

in_

z=3

)!

pa

ram

ete

r(n

_si

lic=

8*n

_b

in_

x*n

_b

in_

y*n

_b

in_

z,n

_o

xy=

2*n

_si

lic)

!---

-cr

isto

ba

lita

lon

gc-

axi

s:so

metim

es

con

vert

sto

kre

atin

e!

pa

ram

ete

r(f

_sy

mm

etr

y=3

)!

pa

ram

ete

r(n

_b

in_

x=4

,n_

bin

_y=

3,n

_b

in_

z=2

)!

pa

ram

ete

r(n

_si

lic=

12

*n_

bin

_x*

n_

bin

_y*

n_

bin

_z,

n_

oxy

=2*n

_si

lic)

!---

-cr

isto

ba

litin

cub

icfo

rm!

pa

ram

ete

r(f

_sy

mm

etr

y=4

)!

pa

ram

ete

r(n

_b

in_

x=3

,n_

bin

_y=

n_

bin

_x,

n_

bin

_z=

n_

bin

_x)

!p

ara

me

ter

(n_

silic

=8

*n_

bin

_x*

n_

bin

_y*

n_

bin

_z,

n_

oxy

=2

*n_

silic

)!-

---

be

taq

ua

rz!

pa

ram

ete

r(f

_sy

mm

etr

y=6

)!

pa

ram

ete

r(n

_b

in_

x=5

,n_

bin

_y=

3,n

_b

in_

z=4

)!

pa

ram

ete

r(n

_b

in_

x=6

,n_

bin

_y=

4,n

_b

in_

z=5

)!

pa

ram

ete

r(n

_si

lic=

6*n

_b

in_

x*n

_b

in_

y*n

_b

in_

z,n

_o

xy=

2*n

_si

lic)

!---

-a

lph

aq

ua

rzp

ara

me

ter

(f_

sym

me

try=

7)

pa

ram

ete

r(n

_b

in_

x=4

,n_

bin

_y=

3,n

_b

in_z=

4)

!p

ara

me

ter

(n_

bin

_x=

5,n

_b

in_

y=3

,n_

bin

_z=

4)

!p

ara

me

ter

(n_

bin

_x=

6,n

_b

in_

y=4

,n_

bin

_z=

5)

pa

ram

ete

r(n

_si

lic=

6*n

_b

in_

x*n

_b

in_

y*n

_b

in_

z,n

_o

xy=

2*n

_si

lic)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

inte

ge

ri_

part

,j_p

art

,n_

pa

rt!

pa

ram

ete

r(n

_si

lic=

1,n

_o

xy=

2)

pa

ram

ete

r(n

_p

art

=n

_si

lic+

n_

oxy

)

inte

ge

ri_

neig

h,n

_n

eig

hp

ara

me

ter

(n_

ne

igh

=1

00

0)

inte

ge

rn

eig

hb

or(

0:n

_n

eig

h,n

_p

art

)co

mm

on

/cn

eig

hb

or/

ne

igh

bo

rin

teg

er

n_

bin

_m

ax,

n_

ne

igh

_m

ax

com

mo

n/c

n_

bin

ne

igh

/n

_b

in_

ma

x,n

_n

eig

h_

ma

x

inte

ge

ri_

tro

t,j_

tro

t,n

_tr

ot,n

_tim

e_

sca

lp

ara

me

ter

(n_

tro

t=1

,n_

time

_sc

al=

1)

do

ub

lep

reci

sio

nsq

rtp

com

mo

n/c

sqrt

p/

sqrt

p

inte

ge

ri_

dim

,j_d

im,k

_d

im,l_

dim

,n_

dim

,e_

dim

pa

ram

ete

r(n

_d

im=

3,e

_d

im=

3)

do

ub

lep

reci

sio

nd

elta

_r(

n_

dim

),d

elta

_ce

ll(n

_d

im)

com

mo

n/c

de

lta_

rc/

de

lta_

r,d

elta

_ce

ll

do

ub

lep

reci

sio

nr0

(n_

dim

,0:n

_tr

ot-

1,n

_p

art

),&

r0_

old

(n_

dim

,0:n

_tr

ot-

1,n

_p

art

),r_

rea

l(n

_d

im,0

:n_

tro

t-1

,n_

part

),&

forc

e(n

_d

im,0

:n_

tro

t-1

,n_

pa

rt)

do

ub

lep

reci

sio

nu

0(n

_d

im,2

*n_

tro

t,n

_pa

rt),

forc

e_

loc(

n_

dim

)co

mm

on

/cr0

u0

forc

e/

r0,r

0_

old

,r_

rea

l,u0

,fo

rce

,fo

rce

_lo

cin

teg

er

pb

c0(n

_d

im,0

:n_

tro

t-1

,n_

pa

rt),

f_sk

inco

mm

on

/cp

bc0

/p

bc0

,f_

skin

do

ub

lep

reci

sio

nv_

rea

l(n

_d

im,2

*n_

tro

t,n

_p

art

)co

mm

on

/cvr

ea

l/v_

rea

ld

ou

ble

pre

cisi

on

r_re

al_

avg

(n_

dim

,0:n

_tr

ot-

1,n

_p

art

)co

mm

on

/cre

ala

vg/

r_re

al_

avg

inte

ge

ri_

ord

er,

j_o

rde

r,n

_o

rde

rp

ara

me

ter

(n_

ord

er=

2)

do

ub

lep

reci

sio

nu

x(n

_o

rde

r,n

_d

im,2

*n_tr

ot,n

_p

art

),&

fr_

bo

x(n

_d

im+

3),

&f0

(n_

dim

,2*n

_tr

ot,n

_p

art

),fr

(n_

dim

,2*n

_tr

ot,n

_p

art

)co

mm

on

/cu

f0/

ux,

f0,fr,

fr_

bo

x

inte

ge

ri_

time,n

_re

lax,

n_

ob

ser,

n_

safe

com

mo

n/c

itim

e/

i_tim

e,n

_re

lax,

n_

ob

ser,

n_

safe

do

ub

lep

reci

sio

nd

t,d

t_2

,r_

time

com

mo

n/c

dtim

e/

dt,d

t_2

,r_

time

do

ub

lep

reci

sio

ntk

,tkp

,be

ta,b

eta

pco

mm

on

/cth

erm

/tk

,tkp

,be

ta,b

eta

p

inte

ge

ri_

typ

e,j_

typ

e,n

_ty

pe

,typ

e(n

_p

art

)p

ara

me

ter

(n_

typ

e=

2)

do

ub

lep

reci

sio

nrf

_w

idth

(0:n

_tr

ot-

1,n

_ty

pe

),rf

b_

wid

thco

mm

on

/cty

pe

/ty

pe

,rf_

wid

th,r

fb_

wid

th

do

ub

lep

reci

sio

nm

ass

(0:n

_tr

ot-

1,n

_ty

pe

),fr

ictio

n(n

_ty

pe

),&

ep

sil(n

_ty

pe

,n_

typ

e),

sig

ma

(n_

typ

e,n

_ty

pe

),k_

qu

an

t(n

_ty

pe

),&

k_e

ige

n(0

:n_

tro

t-1

,n_

typ

e),

sig

ma

_2

(n_

typ

e,n

_ty

pe

),&

e_

shift

(n_

typ

e,n

_ty

pe

),v_

the

rm(0

:n_tr

ot-

1,n

_ty

pe

),&

c_va

nd

wa

als

(n_

typ

e,n

_ty

pe

),w

_th

erm

,&

cha

rge

np

(n_p

art

),ch

arg

en

m(n

_p

art

)d

ou

ble

pre

cisi

on

cha

rge

(n_

pa

rt),

cha

rge

x(n

_o

rde

r,n

_p

art

),&

fQ(n

_p

art

)co

mm

on

/cp

rop

ert

ies/

ma

ss,frict

ion

,ep

sil,s

igm

a,k

_q

ua

nt,

&k_

eig

en

,sig

ma

_2

,e_

shift

,v_

the

rm,c

_va

nd

wa

als

,w_

the

rm,

&ch

arg

e,c

ha

rge

x,&

cha

rge

np

,cha

rge

nm

,&

fQ do

ub

lep

reci

sio

nva

sh_

h(n

_ty

pe

,n_

typ

e),

vash

_e

ta(n

_ty

pe

,n_

typ

e),

&va

sh_

alp

ha

(n_

typ

e),

vash

_B

(n_

typ

e),

vash

_l(n

_ty

pe

),&

vash

_ct

he

ta(n

_ty

pe

),va

sh_

r0(n

_ty

pe

)co

mm

on

/va

shis

hta

/va

sh_

h,v

ash

_e

ta,v

ash

_a

lph

a,v

ash

_B

,&

vash

_l,

vash

_ct

he

ta,

vash

_r0

inte

ge

rn

_a

ng

ula

r,n

_a

ng

(n_

typ

e)

pa

ram

ete

r(n

_an

gu

lar=

6)

do

ub

lep

reci

sio

ntr

iple

_co

s(n

_p

art

,n_

an

gu

lar)

,&

trip

le_

rij(n

_p

art

,n_

an

gu

lar)

,&

trip

le_

rik(

n_

pa

rt,n

_a

ng

ula

r),

&tr

iple

_d

elta

ij(n

_p

art

,n_

an

gu

lar,

n_

dim

),&

trip

le_

de

ltaik

(n_p

art

,n_

an

gu

lar,

n_

dim

)in

teg

er

trip

le_

pa

rtj(n

_p

art

,n_

an

gu

lar)

,&

trip

le_

pa

rtk(

n_

part

,n_

an

gu

lar)

com

mo

n/th

ree

bo

dy/

n_

an

g,

trip

le_

cos,

trip

le_

rij,

trip

le_

rik,

144


&tr

iple

_d

elta

ij,tr

iple

_d

elta

ik,

trip

le_

pa

rtj,

trip

le_

part

kd

ou

ble

pre

cisi

on

r_cu

toff(n

_ty

pe

,n_

typ

e),

r_sk

in(n

_ty

pe

,n_

typ

e),

&r_

cuto

ff_

2(n

_ty

pe

,n_

typ

e),

r_ra

ng

e_

2(n

_ty

pe

,n_

typ

e),

&r_

ran

ge

(n_

typ

e,n

_ty

pe

),r_

cuto

ff_

ma

xco

mm

on

/ccu

toffsk

in/

r_cu

toff,r

_sk

in,r

_cu

toff_

2,r

_ra

ng

e_

2,

&r_

ran

ge

,r_

cuto

ff_

ma

xd

ou

ble

pre

cisi

on

tra

ns_

un

its,d

0(n

_ty

pe

,n_

typ

e),

&m

sze

ta(n

_ty

pe

,n_

typ

e),

&m

sr0

(n_

typ

e,n

_ty

pe

),g

am

ma

(n_

typ

e,n

_ty

pe

)in

teg

er

msn

(n_

typ

e)

com

mo

n/m

ors

est

retc

h/

d0

,msr

0,g

am

ma

,msn

,msz

eta

pa

ram

ete

r(t

rans_

un

its=

50

3.2

28

62

)d

ou

ble

pre

cisi

on

dip

ole

(n_

dim

,n_

pa

rt),

avg

_d

ipo

le(0

:n_

dim

,n_

typ

e),

&a

vg_

dip

ole

_p

art

(0:n

_d

im,n

_p

art

),p

ol_

tota

l(n

_d

im),

&p

ol_

dip

ole

s(n

_d

im)

com

mo

n/c

po

larize

/d

ipo

le,a

vg_

dip

ole

,avg

_d

ipo

le_

pa

rt,

&p

ol_

tota

l,p

ol_

dip

ole

s

inte

ge

ris

ee

d,m

z(2

50

),kp

trd

ou

ble

pre

cisi

on

ran

do

m(n

_d

im)

com

mo

n/c

ran

do

m/

mz,

ise

ed

,kp

tr

do

ub

lep

reci

sion

pre

dic

t_co

ef(

0:n

_o

rde

r-1

,0:n

_o

rde

r),

&co

rre

ct_

coe

f(0

:n_

ord

er)

com

mo

n/c

pre

dic

t/p

red

ict_

coe

f,co

rre

ct_co

ef

do

ub

lep

reci

sion

t_d

yna

mic

,t_

dyn

am

ic_

1,t_

dyn

am

ic_

2,t_

dia

g,t_

nd

iag

&,t_

dia

g_

1,t_

nd

iag

_1,t_

dyn

am

ic_

cha

rge

com

mo

n/c

tdym

ic/

t_d

yna

mic

,t_

dyn

am

ic_

1,t_

dyn

am

ic_

2,t_

dia

g,t_

nd

iag

&,t_

dia

g_

1,t_

nd

iag

_1,t_

dyn

am

ic_

cha

rge

do

ub

lep

reci

sion

v_in

ter,

v_in

ter_

1,v

_in

ter_

2,

v_g

au

ge

com

mo

n/c

inte

r/v_

inte

r,v_

inte

r_1

,v_

inte

r_2

,v_

ga

ug

ed

ou

ble

pre

cisi

on

v_ch

ain

,prim

_e

st,p

rim

_e

st_

1,p

rim

_e

st_

2co

mm

on

/cp

rim

_est

/v_

cha

in,p

rim

_e

st,p

rim

_e

st_

1,p

rim

_e

st_

2d

ou

ble

pre

cisi

on

viri_

est

,viri_

est

_1

,viri_

est

_2

com

mo

n/c

viri_

est

/vi

ri_

est

,viri_

est

_1

,viri_

est

_2

do

ub

lep

reci

sion

v_q

eq

self,

v_re

alc

ou

lom

b,

v_sh

ort

ran

ge

,v_

cha

rge

com

mo

n/c

vqe

q/

v_q

eq

self,

v_re

alc

ou

lom

b,

v_sh

ort

ran

ge

,v_

cha

rge

do

ub

lep

reci

sion

v_th

ree

bo

dy

com

mo

n/v

thre

e/

v_th

ree

bo

dy

do

ub

lep

reci

sion

dip

ole

_cl

ass

(0:n

_d

im,1

2),

&dip

ole

_cl

ass

1(0

:n_

dim

,12

),&

dip

ole

_cl

ass

2(0

:n_

dim

,12

),d

ipo

le_

cla

ss4

(0:n

_d

im,1

2),

&d

ipo

le_

cou

nt

com

mo

n/c

dip

ole

cla

ss/

dip

ole

_cl

ass

,d

ipo

le_

cla

ss1

,&

dip

ole

_cl

ass

2,

dip

ole

_cl

ass

4,

dip

ole

_co

un

t

inte

ge

ri_

cor,

n_

cor,

i_m

ea

sure

pa

ram

ete

r(n

_co

r=2

04

8)

do

ub

lep

reci

sion

con

f_ce

ntr

(n_

dim

,n_

pa

rt)

do

ub

lep

reci

sion

cor_

ob

s(0

:n_

cor)

,co

r_fu

nc(

0:n

_co

r,n

_ty

pe

)d

ou

ble

pre

cisi

on

qd

ot_

cor(

0:n

_co

r),

qd

ot0

(n_

dim

,n_

pa

rt)

do

ub

lep

reci

sion

cor_

fun

c_A

(0:n

_co

r)co

mm

on

/cco

r/co

nf_

cen

tr,c

or_

ob

s,co

r_fu

nc,

i_m

ea

sure

com

mo

n/c

cor2

/co

r_fu

nc_

A,q

do

t_co

r,q

do

t0d

ou

ble

pre

cisi

on

cor_

fun

c_B

(0:n

_co

r),

&st

ore

d_

vel(n

_d

im,0

:n_

cor,

n_p

art

)co

mm

on

/cco

rfu

ncb

/co

r_fu

nc_

B,

sto

red

_ve

l

inte

ge

rd

i_tim

e,n

i_tim

e,ie

ff_

time

com

mo

n/c

rea

l_tim

e/

di_

time

,ieff_

time

pa

ram

ete

r(n

i_tim

e=

n_

time

_sc

al)

do

ub

lep

reci

sio

nco

r_ifu

nc(

0:n

_tr

ot-

1)

com

mo

n/c

icor/

cor_

ifun

c

do

ub

lep

reci

sio

ntim

e_

ob

sco

mm

on

/ctim

e_

ob

s/tim

e_

ob

s

inte

ge

rco

unt_

ifun

cco

mm

on

/cco

un

t_ifu

nc/

cou

nt_

ifun

c

inte

ge

rn

_p

bc

com

mo

n/c

n_

pb

c/n

_p

bc

inte

ge

rf_

movi

e!

f_m

ovi

e=

1;

write

mu

ltip

lefr

am

es

xyz-

file

!f_

mo

vie

=0

;d

on

ot

pa

ram

ete

r(f_

mo

vie

=0

)

inte

ge

rf_

con

fmo

vie

!f_

confm

ovi

e=

1;

write

mu

ltip

lefr

am

es

con

f-fil

e!

f_co

nfm

ovi

e=

0;

do

no

tp

ara

me

ter(

f_co

nfm

ovi

e=

0)

inte

ge

rf_

e_a

pp

lied

!f_

e_

ap

plie

d=

0;

no

ext

ern

al

ele

ctric

field

!f_

e_

ap

plie

d=

1;

ele

ctric

field

inx

dire

ctio

n!

f_e

_ap

plie

d=

2;

ele

ctric

field

iny

dire

ctio

n!

f_e

_ap

plie

d=

3;

ele

ctric

field

inz

dire

ctio

np

ara

me

ter(

f_e_

ap

plie

d=

1)

!C

AU

TIO

N:

for

Sca

nd

olo

po

ten

tial,

ple

ase

set

!=

==

==

==

tele

ctric

inm

etr

ic.f

/m

od

ule

!e

lect

ric_

field

acc

ord

ing

ly!!!

inte

ge

rf_

po

ten

tial

!---

-f_

po

ten

tial

=1

:H

arm

on

ico

scill

ato

r!

f_p

ote

ntia

l=

2;

Le

nn

ard

Jon

es

!f_

po

ten

tial

=3

;B

KS

po

ten

tial

!f_

po

ten

tial

=4

;A

ziz

!f_

po

ten

tial

=5

;M

SQ

Eq

po

tentia

l!

f_p

ote

ntia

l=

6;

MS

fluc-

Qp

ote

ntia

l!

f_p

ote

ntia

l=

7;

MS

DQ

Eq

po

ten

tial

!f_

po

ten

tial

=8

;V

ash

ish

tap

ote

ntia

l!

f_p

ote

ntia

l=

9;

Sca

nd

olo

po

ten

tial

pa

ram

ete

r(f_

po

ten

tial=

9)

inte

ge

rf_

charg

ea

dju

st!-

---

f_ch

arg

ea

dju

st=

1:

con

sta

nt

cha

rge

s!

f_ch

arg

ea

dju

st=

2;

self

con

sist

en

cy!

f_ch

arg

ea

dju

st=

3;

ext

en

de

dla

gra

ng

ian

!f_

charg

ea

dju

st=

4;

ma

trix

inve

rsio

nm

eth

od

pa

ram

ete

r(f_

cha

rge

ad

just

=1

)

do

ub

lep

reci

sio

nq

eq

ele

ctro

ne

g(n

_ty

pe),

qe

qh

ard

ne

ss(n

_ty

pe

),&

qe

qla

mb

da

i(n_

pa

rt),

qe

qla

mb

da

,q

eq

rec(

n_

pa

rt),

&q

eq

ma

ss(n

_ty

pe

),q

eq

old

cha

rge

(n_

pa

rt,0

:3),

qe

qfr

ictio

nin

teg

er

n_

ato

mty

pe

(n_

typ

e)

com

mo

n/q

eq

/q

eq

ele

ctro

ne

g,

qe

qh

ard

ness

,&

qe

qla

mb

da

i,q

eq

lam

bd

a,

qe

qre

c,q

eq

old

cha

rge

,&

qe

qm

ass

,q

eqfr

ictio

n,

n_

ato

mty

pe

do

ub

lep

reci

sio

nco

nst

an

t_ch

arg

e(n

_ty

pe

)co

mm

on

/flu

cqe

q/

con

sta

nt_

cha

rge

145

B. Programsd

ou

ble

pre

cisi

on

forc

e_

sca

l(n

_d

im+

3),

forc

e_

sca

l_in

ter(

n_

dim

+3)

&,s

cal_

0(n

_d

im+

3),

sca

l_x(

n_

ord

er,

n_

dim

+3

)d

ou

ble

pre

cisi

on

pre

ssu

re(n

_d

im+

3),

mass

_b

ox,

pre

ss,fric_

bo

xin

teg

er

f_p

ress

ure

com

mo

n/c

pre

ss/

forc

e_

sca

l_in

ter,

sca

l_0

,sca

l_x,

pre

ssu

re,

&fo

rce

_sc

al,p

ress

,f_

pre

ssu

reco

mm

on

/cp

ress

2/

ma

ss_

bo

x,fr

ic_

bo

x

do

ub

lep

reci

sio

nh

_m

at(

3,3

),h

_m

at_

2(3

,3),

h_

ma

t_d

ot(

3,3

),&

h_

ma

t_2

d(3

,3),

h_

ma

t_2

_in

v(3

,3),

g_

ma

t_do

t(3

,3),

h_

ma

t_in

v(3

,3)

com

mo

n/c

tenso

r/h

_m

at,h

_m

at_

2,h

_m

at_

do

t,h

_m

at_

2d

,h_

ma

t_2

_in

v,&

g_

ma

t_d

ot,h

_m

at_

inv

do

ub

lep

reci

sio

nm

et_

ten

_vo

i(n

_d

im+

3),

g_

voi_

do

t(n

_d

im+

3)

com

mo

n/c

voig

t/m

et_

ten

_vo

i,g_

voi_

do

t

do

ub

lep

reci

sio

nsc

al_

1(n

_d

im+

3),

volu

me

,vo

l_1

,vo

l_2

,&

sca

l_2(n

_d

im+

3,n

_d

im+

3),

v_p

ress

com

mo

n/c

pre

ss_

ob

s/sc

al_

1,s

cal_

2,v

_pre

ss,v

olu

me

,vo

l_1

,vo

l_2

do

ub

lep

reci

sio

nr_

ref(

n_

dim

,n_

pa

rt)

com

mo

n/c

r_re

f/r_

ref

inte

ge

ri_

grid

,n_

grid

pa

ram

ete

r(n

_g

rid

=2

56

)d

ou

ble

pre

cisi

on

g_

r(n

_ty

pe

,n_

typ

e,0

:n_

grid

+1

)co

mm

on

/cg

_r/

g_

r

inte

ge

ri_

pote

ntia

l,n_

po

ten

tial

pa

ram

ete

r(n

_p

ote

ntia

l=1

00

00

)d

ou

ble

pre

cisi

on

r_p

ot(

n_

typ

e,n

_ty

pe

,0:n

_p

ote

ntia

l),

&r_

po

t_d

(n_

typ

e,n

_ty

pe

,0:n

_p

ote

ntia

l)d

ou

ble

pre

cisi

on

r_p

ot_

1(n

_ty

pe

,n_

type

,0:n

_p

ote

ntia

l),

&r_

po

t_2

(n_

typ

e,n

_ty

pe

,0:n

_p

ote

ntia

l),

&r_

po

t_2

q(n

_ty

pe

,n_

typ

e,0

:n_

po

tentia

l),

&r_

po

t_d

_1

(n_

typ

e,n

_ty

pe

,0:n

_p

ote

ntia

l),

&r_

po

t_d

_2

(n_

typ

e,n

_ty

pe

,0:n

_p

ote

ntia

l),

&r_

po

t_d

_2

q(n

_ty

pe

,n_

typ

e,0

:n_

po

ten

tial)

com

mo

n/c

r_po

td/

r_p

ot_

1,r

_p

ot_

2,r

_p

ot_

2q

,r_

po

t_d

_1

,&

r_p

ot_

d_2

,r_

po

t_d

_2

q,r

_p

ot,r_

pot_

dco

mp

lex

tab

_e

xp_

ima

(-n

_p

ote

ntia

l:n_

pote

ntia

l)co

mm

on

/cta

b_

exp

_im

/ta

b_

exp

_im

a

inte

ge

ri_

run

,n_

run

com

mo

n/in

_ru

n/

i_ru

n,n

_ru

n

do

ub

lep

reci

sio

ne

rfcc

do

ub

lep

reci

sio

na

lph

a,q

_cu

t_2

,se

lf_en

erg

y,v_

ew

ald

,&

dq

(n_

dim

,n_

dim

),v_

cut,a

lph

a_

pre

fac

com

mo

n/c

ew

ald

/a

lph

a,q

_cu

t_2

,se

lf_e

ne

rgy,

v_e

wa

ld,d

q,v

_cu

t&

,alp

ha

_p

refa

c

inte

ge

rn

q_

ma

x(n

_d

im),

n_

rec_

vec,

n_

rec_

vec_

ma

x,i_

rec_

vec

com

mo

n/c

rec_

ew

ald

/n

q_

ma

x,n

_re

c_ve

cp

ara

me

ter

(n_

rec_

vec_

ma

x=1

2*n

_p

art

)in

teg

er

rec_

vec_

list(

n_

rec_

vec_

ma

x,n

_d

im)

com

mo

n/c

rec_

vec_

list/

rec_

vec_

list

do

ub

lep

reci

sio

nre

c_fo

rce

_o

ffse

t(n

_dim

+3

),re

c_p

ot_

offse

tco

mm

on

/re

c_o

ffse

t/re

c_fo

rce

_o

ffse

t,re

c_p

ot_

offse

t

do

ub

lep

reci

sio

nr_

cut_

sho

rt,e

_sh

ift_sh

ort

(n_

typ

e,n

_ty

pe

)

pa

ram

ete

r(r

_cu

t_sh

ort

=9

.0)

do

ub

lep

reci

sio

ne

la_

bo

rn(n

_d

im,n

_d

im,n

_d

im,n

_d

im),

&e

la_

bo

rn_

1(n

_d

im,n

_d

im,n

_d

im,n

_d

im),

time

_b

orn

com

mo

n/c

ela

stic

/e

la_

bo

rn,e

la_

bo

rn_

1,tim

e_

bo

rnd

ou

ble

pre

cisi

on

stra

in(n

_d

im,n

_d

im),

stra

in_

1(n

_d

im,n

_d

im),

&st

rain

_2

(n_

dim

,n_

dim

,n_

dim

,n_

dim

)co

mm

on

/cst

rain

/st

rain

,str

ain

_1

,str

ain

_2

inte

ge

rf_

op

t_re

cco

mm

on

/cf_

op

t/f_

op

t_re

c

do

ub

lep

reci

sio

nro

t_m

at(

n_

dim

,n_

dim

),e

ige

n(n

_d

im,n

_d

im)

com

mo

n/c

jaco

bi/

rot_

ma

t,e

ige

n

do

ub

lep

reci

sio

ne

la_

kin

(n_

dim

,n_

dim

)co

mm

on

/ce

la_

kin

/e

la_

kin

do

ub

lep

reci

sio

nfo

r_sc

a_

mfa

_co

rco

mm

on

/cfo

r_sc

a_

mfa

_co

r/fo

r_sc

a_

mfa

_co

r

inte

ge

ri_

ne

igh

_o

ri(n

_p

art

,4)

do

ub

lep

reci

sio

nh

ist_

an

g(3

,n_

grid

)co

mm

on

/ci_

ne

igh

_o

ri/

his

t_a

ng

,i_

ne

igh

_o

rid

ou

ble

pre

cisi

on

r_n

eig

h_

min

,r_

ne

igh

_m

ax,

ne

igh

_tim

eco

mm

on

/cr_

ne

igh

/r_

ne

igh

_m

in,

r_n

eig

h_

ma

x,n

eig

h_

time

do

ub

lep

reci

sio

no

rd_

pa

r,o

rd_

pa

r_1

,ord

_p

ar_

2,o

rd_

pa

r_4

,ord

_c

com

mo

n/c

ord

_pa

r/o

rd_

pa

r,o

rd_

pa

r_1

,ord

_p

ar_

2,o

rd_

pa

r_4

,ord

_c

do

ub

lep

reci

sio

no

rd_

pa

rB,o

rd_

pa

rB_

1,o

rd_

pa

rB_

2,

&o

rd_

pa

rB_

4,o

rdB

_c

com

mo

n/c

ord

_pa

rB/

ord

_p

arB

,ord

_p

arB

_1,o

rd_

pa

rB_

2,

&o

rd_

pa

rB_

4,o

rdB

_c

do

ub

lep

reci

sio

no

rd_

pa

rC,o

rd_

pa

rC_

1,o

rd_

pa

rC_

2,o

rd_

pa

rC_

4,o

rdC

_c

com

mo

n/c

ord

parC

/o

rd_

pa

rC,o

rd_

pa

rC_

1,o

rd_

pa

rC_

2,o

rd_

pa

rC_

4,o

rdC

_c

146


B.1.1. Inputfiles

Inputfile pimd.inp for BKS potential

3 f_potential, information available1 f_chargeadjust

10000 # of relaxation steps200000 # of observation steps

10000 # of steps between configuration output48674 random # init

0.130910E-03 time step3.000000e+02 temperature

0.579141E+00 0.329760E+00 masses0.763884E+02 0.763884E+02 friction0.464498E+01 0.763884E+02 mass + fr. of box

0.000000e+00 pressure1 0 = NVT; 1 = NpT0 1 = optimized Ewald

0.000000E+00 0.208934E+09 exp. energy parameters0.208934E+09 0.161167E+08 exp. energy parameters

0.000000E+00 0.487318E+01 inverse exp. lenght scales0.487318E+01 0.276000E+01 inverse exp. lenght scales

0.000000E+00 0.154971E+07 van de Waals strength0.154971E+07 0.203088E+07 van de Waals strength

0.300000E+00 0.200000E+02 alpha, v_cut

0.450000E+00 0.450000E+00 skin radius0.450000E+00 0.450000E+00 skin radius

Inputfile pimd.inp for fluctuating charge potential


50000 # of relaxation steps200000 # of observation steps

10000 # of steps between configuration output48748 random # init

0.130910E-03 time step3.000000e+02 temperature


0.000000e+00 pressure1 0 = NVT; 1 = NpT0 1 = optimized Ewald

0.483676E+05 0.101435E+06 electronegativity0.809298E+05 0.155083E+06 hardness of EN

0.400000E-01 0.400000E-01 fictitious charge mass0.763884E+03 charge friction

0.295600E+00 0.459970E+02 MS bond strength D00.459970E+02 0.536300E+00 MS bond strength D0

0.341030E+01 0.161480E+01 MS bond length R00.161480E+01 0.378350E+01 MS bond length R0

0.117139E+02 0.880220E+01 MS force constant gamma0.880220E+01 0.104112E+02 MS force constant gamma

0.148810E+01 0.148810E+01 slater orbital exponent zeta0.186846E+01 0.186846E+01 slater orbital exponent zeta

3 2 principal quantum number n

0.300000E+00 0.000000E+00 alpha, v_cut


147

B. Programs

Inputfile pimd.inp for fluctuating dipole potential


1000 # of relaxation steps6000 # of observation steps1000 # of steps between configuration output6811 random # init

0.130910E-03 time step0.300000E+03 temperature


0.217290E+04 pressure1 0 = NVT; 1 = NpT

0.300000E+00 0.200000E+02 alpha, v_cut


B.2. Analysis Programs

examf60.f

Reads the output filefort.60 and writes analysis output to standard out.The output contains averaged box dimensions, lattice constants, stress, volume perunit cell, elastic constants, polarization, dielectric constant and piezoelectric stress andstrain coefficients.The gaussj routine from “Numerical Recipes in Fortran” [74] needs to be included inthe code.

calcstrain.f

Reads the output filefort.60 and writes analysis output tofort.59 . Needs areference box matrix from standard in.The output is the strain with time, relative to a reference boxh0. The reference box isprompted at the start of the program.

148


exam

f60.

fp

rog

ram

exa

m_

f60

imp

licit

no

ne

inte

ge

rn

_b

in_

x,n

_b

in_

y,n

_b

in_

zp

ara

me

ter

(n_

bin

_x=

4,

n_

bin

_y=

3,

n_

bin

_z=

4)

inte

ge

ri_

rela

x,n

_re

lax,

n_

ob

ser

inte

ge

ri_

dim

,j_

dim

,n

_d

im,

e_

dim

,iv

_d

im,

voi_

ind

pa

ram

ete

r(n

_d

im=

3,

e_

dim

=3

)d

ou

ble

pre

cisi

on

vol_

1co

mm

on

/cvo

l_1

/vo

l_1

do

ub

lep

reci

sion

x(8

),sc

al(n

_d

im+

e_

dim

),sc

al_

1(n

_d

im+

e_

dim

),&

sca

l_2

(n_

dim

+e

_d

im,

n_

dim

+e

_d

im),

tem

p,

y(1

0),

&p

ol_

tota

l(n

_d

im),

po

l_d

ipo

les(

n_

dim

),p

ol_

fluc(

n_

dim

,n_

dim

),&

po

l_a

vg(n

_d

im),

po

lstr

ain

(n_

dim

+e

_dim

,n_

dim

)d

ou

ble

pre

cisi

on

a,b

,c,d

a,d

b,d

c,re

dp

ol_

avg

(n_

dim

),re

dp

ol(n

_d

im)

do

ub

lep

reci

sion

h_

ma

t(3

,3),

h_

i(3

,3),

h1

(3,3

),h

1_

i(3

,3)

&,

h1

_i_

avg

_i(3

,3),

h1

_i_

avg

(3,3

)d

ou

ble

pre

cisi

on

h(3

,3),

h0

(3,3

),h

t(3

,3),

h0

i(3

,3),

h0

ti(3

,3),

&h

0t(

3,3

)d

ou

ble

pre

cisi

on

s1(3

,3),

s2(3

,3),

stra

inm

(3,3

),&

stra

in(6

),st

rain

_1

(6),

stre

ss(6

),p

ols

tre

ss(6

,3),

stre

ss_

1(6

)

write

(*,*

)"H

ow

ma

ny

ste

ps

tod

isca

rd?

"re

ad

(*,*

)n

_re

lax

write

(*,*

)"T

em

p."

rea

d(*

,*)

tem

p

!fir

stru

n do

i_d

im=

1,

n_

dim

red

po

l_a

vg(i_d

im)=

0.d

0d

oj_

dim

=1

,n

_d

imh

1_

i_a

vg(i_

dim

,j_d

im)

=0

.d0

en

dd

oe

nd

do

do

i_re

lax

=1

,n

_re

lax

rea

d(6

0,’(

36

e1

5.7

)’)

(x(i_

dim

),i_

dim

=1

,8

),&

(sca

l(j_

dim

),j_

dim

=1

,n

_d

im+

e_

dim

),(y

(i_

dim

),i_

dim

=1

,10),

&(p

ol_

tota

l(i_

dim

),i_

dim

=1

,3),

(pol_

dip

ole

s(i_

dim

),i_

dim

=1

,3),

&(s

tre

ss(i_

dim

),i_

dim

=1

,6)

en

dd

o

n_

ob

ser

=0

do

i_d

im=

1,

n_

dim

+e

_d

imsc

al_

1(i_

dim

)=

0.

en

dd

o

90

1co

ntin

ue

rea

d(6

0,’(

36

e1

5.7

)’,e

nd

=9

02

)(x

(i_

dim

),i_

dim

=1

,8

),&

(sca

l(j_

dim

),j_

dim

=1

,n

_d

im+

e_

dim

),(y

(i_

dim

),i_

dim

=1

,10),

&(p

ol_

tota

l(i_

dim

),i_

dim

=1

,3),

(pol_

dip

ole

s(i_

dim

),i_

dim

=1

,3),

&(s

tre

ss(i_

dim

),i_

dim

=1

,6)

n_

ob

ser

=n

_ob

ser

+1

call

tra

ns_

to_

ten

s(sc

al,

h_

ma

t)ca

llin

v_m

at(

h_

ma

t,h

_i)

do

i_d

im=

1,

n_

dim

+e

_d

im

sca

l_1

(i_

dim

)=

sca

l_1

(i_

dim

)+

sca

l(i_

dim

)e

nd

do

call

tra

ns_

to_

ten

s(sc

al,

h1

)ca

llin

v_m

at(

h1

,h

1_

i)

do

i_d

im=

1,

n_

dim

po

l_to

tal(i_

dim

)=

po

l_to

tal(i_

dim

)-

po

l_d

ipo

les(

i_d

im)

!u

nco

mm

en

tfo

llow

ing

line

toco

nsi

de

ro

nly

po

lariza

tion

du

eto

dip

ole

s:!

po

l_to

tal(i_

dim

)=

po

l_dip

ole

s(i_

dim

)e

nd

do

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

h1

_i_

avg

(i_d

im,j_

dim

)=

&h1

_i_

avg

(i_

dim

,j_d

im)

+h1

_i(i_

dim

,j_d

im)

red

po

l_a

vg(i_

dim

)=re

dp

ol_

avg

(i_

dim

)+

h_

i(i_

dim

,j_d

im)

&*

po

l_to

tal(j_

dim

)e

nd

do

en

dd

o

go

to9

01

90

2co

ntin

ue

do

i_d

im=

1,

n_

dim

red

po

l_a

vg(i_

dim

)=re

dp

ol_

avg

(i_

dim

)/

n_

ob

ser

do

j_d

im=

1,

n_

dim

h1

_i_

avg

(i_

dim

,j_d

im)

=h

1_

i_a

vg(i_

dim

,j_d

im)

/n

_o

bse

re

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

+e

_d

imsc

al_

1(i_

dim

)=

sca

l_1

(i_

dim

)/

n_

ob

ser

en

dd

o

call

inv_

ma

t(h

1_

i_a

vg,

h1

_i_

avg

_i)

!ca

lcu

late

h0

,h

0ˆ-

1,

h0

’,h

0’ˆ-1

for

stra

inca

lltr

an

s_to

_te

ns(

sca

l_1

,h0

)w

rite

(*,*

)"h

0",

h0

(1,1

),h

0(2

,2),

h0(3

,3),

h0

(1,2

),h

0(2

,3),

h0(1

,3)

call

inv_

ma

t(h

0,h

0i)

call

tra

ns_

ma

t(h

0,h

0t)

call

inv_

ma

t(h

0t,h

0ti)

write

(*,*

)"h

0ti"

,h

0ti(

1,1

),h

0ti(

2,2

),h

0ti(

3,3

)

clo

se(6

0)

!se

con

dru

n

do

i_re

lax

=1

,n

_re

lax

rea

d(6

0,’(

36

e15

.7)’)

(x(i_

dim

),i_

dim

=1,

8),

&(s

cal(j_

dim

),j_

dim

=1

,n

_d

im+

e_

dim

),(y

(i_

dim

),i_

dim

=1

,10

),&

(po

l_to

tal(i_

dim

),i_

dim

=1

,3),

(po

l_d

ipo

les(

i_d

im),

i_d

im=

1,3

),&

(str

ess

(i_

dim

),i_

dim

=1

,6)

en

dd

o

n_

ob

ser

=0

do

i_d

im=

1,

n_

dim

+e

_d

imsc

al_

1(i_

dim

)=

0.

stra

in_

1(i_

dim

)=

0.

stre

ss_

1(i_

dim

)=

0.

149

B. Programsd

oj_

dim

=1,

n_

dim

+e

_d

imsc

al_

2(i_

dim

,j_d

im)

=0

.e

nd

do

en

dd

od

oj_

dim

=1

,n

_d

imp

ol_

avg

(j_

dim

)=0

.d

oi_

dim

=1

,6p

ols

tra

in(i_

dim

,j_d

im)=

0.

po

lstr

ess

(i_

dim

,j_d

im)=

0.

en

dd

od

oi_

dim

=1

,3p

ol_

fluc(

i_d

im,j_

dim

)=0

.e

nd

do

en

dd

o

10

0co

ntin

ue

rea

d(6

0,’(

36

e15

.7)’,e

nd

=2

00

)(x

(i_

dim

),i_

dim

=1

,8

),&

(sca

l(j_

dim

),j_

dim

=1

,n

_d

im+

e_

dim

),(y

(i_

dim

),i_

dim

=1

,10

),&

(po

l_to

tal(i_

dim

),i_

dim

=1

,3),

(po

l_d

ipo

les(

i_d

im),

i_d

im=

1,3

),&

(str

ess

(i_

dim

),i_

dim

=1

,6

)n

_o

bse

r=

n_

obse

r+

1d

oi_

dim

=1

,n

_d

im+

e_

dim

sca

l_1

(i_

dim

)=

sca

l_1

(i_

dim

)+

sca

l(i_

dim

)d

oj_

dim

=1,

n_

dim

+e

_d

imsc

al_

2(i_

dim

,j_d

im)

=sc

al_

2(i_

dim

,j_d

im)

+&

sca

l(i_

dim

)*

sca

l(j_

dim

)e

nd

do

en

dd

o

call

tra

ns_

to_te

ns(

sca

l,h

_m

at)

call

inv_

ma

t(h_

ma

t,h

_i)

do

i_d

im=

1,

n_

dim

po

l_to

tal(i_

dim

)=

po

l_to

tal(i_

dim

)-

po

l_d

ipo

les(

i_d

im)

!u

nco

mm

en

tfo

llow

ing

line

toco

nsi

de

ro

nly

po

lariza

tion

du

eto

dip

ole

s:!

po

l_to

tal(i_

dim

)=

po

l_dip

ole

s(i_

dim

)e

nd

do

do

i_d

im=

1,

n_

dim

red

po

l(i_

dim

)=

0.

do

j_d

im=

1,

n_

dim

iv_

dim

=vo

i_in

d(i_

dim

,j_d

im)

red

po

l(i_

dim

)=

red

po

l(i_

dim

)+

h_

i(i_

dim

,j_d

im)

*&

po

l_to

tal(j_

dim

)e

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

po

l_to

tal(i_

dim

)=

0.

do

j_d

im=

1,

n_

dim

po

l_to

tal(i_

dim

)=

po

l_to

tal(i_

dim

)&

+h

1_

i_a

vg_

i(i_

dim

,j_d

im)

*&

(re

dp

ol(j_

dim

)-

red

po

l_avg

(j_

dim

))e

nd

do

!co

mm

en

tou

tfo

llow

ing

line

toco

nsi

de

ro

nly

po

lariza

tion

du

eto

cha

rge

s:p

ol_

tota

l(i_

dim

)=

po

l_to

tal(i_

dim

)+

po

l_d

ipo

les(

i_d

im)

en

dd

o

do

i_d

im=

1,

n_

dim

po

l_a

vg(i_

dim

)=

po

l_a

vg(i_

dim

)+

po

l_to

tal(i_

dim

)e

nd

do

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_d

imp

ol_

fluc(

i_d

im,j_

dim

)=

po

l_flu

c(i_

dim

,j_d

im)

+&

po

l_to

tal(i_

dim

)*p

ol_

tota

l(j_

dim

)e

nd

do

en

dd

o

call

tra

ns_

to_

ten

s(sc

al,h

)ca

lltr

an

s_m

at(

h,h

t)ca

llm

ult_

mat(

h,h

0i,s

1)

call

mu

lt_m

at(

ht,s1

,s2

)ca

llm

ult_

mat(

h0

ti,s2

,str

ain

m)

do

i_d

im=

1,3

do

j_d

im=

1,3

if(i_

dim

.eq

.j_d

im)

&st

rain

m(i_

dim

,j_d

im)

=st

rain

m(i_

dim

,j_d

im)

-1.d

0st

rain

m(i_

dim

,j_d

im)

=0

.5d

0*

stra

inm

(i_

dim

,j_d

im)

en

dd

oe

nd

do

call

tra

ns_

to_

voig

t(st

rain

m,s

tra

in)

do

i_d

im=

1,

6st

rain

_1

(i_

dim

)=

stra

in_

1(i_

dim

)+

stra

in(i_

dim

)st

ress

_1

(i_

dim

)=

stre

ss_

1(i_

dim

)+

stre

ss(i_

dim

)e

nd

do

write

(44

,’(7f1

5.5

)’)

x(1

),(s

tra

in(i_

dim

),i_

dim

=1

,6)

do

i_d

im=

1,

6d

oj_

dim

=1

,3

po

lstr

ain

(i_

dim

,j_d

im)

=p

ols

tra

in(i_d

im,j_

dim

)+

&st

rain

(i_

dim

)*

po

l_to

tal(j_

dim

)p

ols

tre

ss(i_

dim

,j_d

im)

=p

ols

tre

ss(i_d

im,j_

dim

)+

&st

ress

(i_

dim

)*

po

l_to

tal(j_

dim

)e

nd

do

en

dd

og

oto

10

02

00

con

tinu

e

write

(*,*

)"O

bs.

Ste

ps"

,n

_o

bse

rd

oi_

dim

=1

,n

_d

im+

e_

dim

sca

l_1

(i_

dim

)=

sca

l_1

(i_

dim

)/

n_

obse

rst

rain

_1

(i_

dim

)=

stra

in_

1(i_

dim

)/

n_

ob

ser

stre

ss_

1(i_

dim

)=

stre

ss_

1(i_

dim

)/

n_

ob

ser

en

dd

ow

rite

(*,*

)"s

tra

in_

1",

(str

ain

_1

(i_

dim

),i_

dim

=1

,n_

dim

)w

rite

(*,*

)"s

tre

ss_

1",

(str

ess

_1

(i_

dim

),i_

dim

=1

,n_

dim

)

do

i_d

im=

1,

n_

dim

po

l_a

vg(i_

dim

)=

po

l_a

vg(i_

dim

)/

n_

ob

ser

en

dd

o

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

po

l_flu

c(i_

dim

,j_d

im)

=p

ol_

fluc(

i_d

im,j_

dim

)/

n_

ob

ser

po

l_flu

c(i_

dim

,j_d

im)

=(p

ol_

fluc(

i_dim

,j_d

im)

-&

pol_

avg

(i_

dim

)*p

ol_

avg

(j_

dim

))e

nd

do

en

dd

o

150


do

i_d

im=

1,

6d

oj_

dim

=1

,3p

ols

tra

in(i_d

im,j_

dim

)=p

ols

tra

in(i_

dim

,j_d

im)

/n

_o

bse

rp

ols

tra

in(i_d

im,j_

dim

)=(p

ols

tra

in(i_d

im,j_

dim

)-

&st

rain

_1

(i_

dim

)*p

ol_

avg

(j_

dim

))p

ols

tre

ss(i_d

im,j_

dim

)=p

ols

tre

ss(i_

dim

,j_d

im)

/n

_o

bse

rp

ols

tre

ss(i_d

im,j_

dim

)=(p

ols

tre

ss(i_d

im,j_

dim

)-

&st

ress

_1

(i_

dim

)*p

ol_

avg

(j_

dim

))e

nd

do

en

dd

o

do

i_d

im=

1,

n_

Dim

+e

_d

imd

oj_

dim

=1

,n

_d

im+

e_

dim

sca

l_2

(i_

dim

,j_d

im)

=sc

al_

2(i_

dim

,j_d

im)

/n

_o

bse

rsc

al_

2(i_

dim

,j_d

im)

=(

sca

l_2

(i_

dim

,j_d

im)

-&

sca

l_1

(i_d

im)

*sc

al_

1(j_

dim

))

en

dd

oe

nd

do

call

calc

_vo

l(sc

al_

1,v

ol_

1)

call

pie

zo_

coe

ff(p

ol_

avg

,sca

l_2

,sca

l_1

,te

mp

,po

lstr

ain

,po

lstr

ess

)ca

lld

iele

ctric_

con

st(p

ol_

fluc,

po

l_a

vg,s

cal_

2,s

cal_

1,te

mp

)

write

(*,*

)"A

vg.

Po

lariza

tion

"d

oi_

dim

=1

,3

write

(*,*

)p

ol_

avg

(i_

dim

)/vo

l_1

,"+

-",

&sq

rt(p

ol_

fluc(

i_d

im,i_

dim

))/v

ol_

1e

nd

do

write

(*,*

)"A

vg.

Sca

l.V

aria

b."

do

i_d

im=

1,

n_

dim

+e

_d

imw

rite

(*,’(

f15

.9a

f15

.9)’)

sca

l_1

(i_

dim

),"

+-"

,&

sqrt

(sca

l_2

(i_

dim

,i_d

im))

en

dd

oa

=sc

al_

1(1

)/n

_bin

_x

b=

sca

l_1

(2)/

sqrt

(3.)

/n_

bin

_y

c=sc

al_

1(3

)/n

_bin

_z

da

=sq

rt(s

cal_

2(1

,1))

/n_

bin

_x

db

=sq

rt(s

cal_

2(2

,2))

/sq

rt(3

.)/n

_b

in_

yd

c=sq

rt(s

cal_

2(3

,3))

/n_

bin

_z

write

(*,*

)"a

",a

,"+

-",

da

write

(*,*

)"b

",b

,"+

-",

db

write

(*,*

)"c

",c,

"+

-",

dc

write

(*,*

)"c

/a",

1.

/(s

qrt

(a*b

)/

c),"

+-

",sq

rt(

&(.

25

*((d

a/a

)**2

+(d

b/b

)**2

)+

dc*

*2)/

(a*b

))w

rite

(*,*

)"v

ol

",sq

rt(3

.)/2

*a*b

*c,"

+-

",sq

rt(.

75

*&

((d

a*b

*c)*

*2+

(a*d

b*c

)**2

+(a

*b*d

c)**

2))

write

(*,’(

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

a)’)

&"h

={"

,sca

l_1

(1),

",",

sca

l_1

(2),

",",

&sc

al_

1(3

),",

",sc

al_

1(4

),",

",sc

al_

1(5

),",

",sc

al_

1(6

),"}

"w

rite

(*,’(

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

a)’)

&"d

h=

{",s

qrt

(sca

l_2

(1,1

)),"

,",s

qrt

(sca

l_2

(2,2

)),"

,"&

,sq

rt(s

cal_

2(3

,3))

,","

,sq

rt(s

cal_

2(4

,4))

,","

,&

sqrt

(sca

l_2

(5,5

)),"

,",s

qrt

(sca

l_2

(6,6

)),"

}"w

rite

(*,’(

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

a)’)

&"P

={"

,po

l_a

vg(1

)/vo

l_1

,","

,pol_

avg

(2)/

vol_

1,"

,",

&p

ol_

avg

(3)/

vol_

1,"

}"w

rite

(*,’(

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

af1

5.9

a)’)

&"d

P=

{",s

qrt

(po

l_flu

c(1

,1))

/vol_

1,"

,",

&sq

rt(p

ol_

fluc(

2,2

)/vo

l_1

),",

",sq

rt(p

ol_

fluc(

3,3

)/vo

l_1

),"}

"

write

(*,*

)"A

vg.

Flu

c.T

en

s."

do

i_d

im=

1,

n_

Dim

+e

_d

imd

oj_

dim

=1

,n

_d

im+

e_

dim

sca

l_2

(i_

dim

,j_d

im)

=sc

al_

2(i_

dim

,j_d

im)

/te

mp

en

dd

ow

rite

(*,’(

6e

13

.5)’)

&(

sca

l_2

(i_

dim

,j_d

im),

j_d

im=

1,

n_

dim

+e

_d

im)

en

dd

o

call

calc

_st

rain

(sca

l_2

,sc

al_

1)

call

calc

_e

lco

(sca

l_2

,sc

al_

1)

en

dp

rog

ram

!!

!!

!!

!!

!

sub

rou

tine

calc

_vo

l(sc

al_

1,v

ol_

1)

do

ub

lep

reci

sio

nsc

al_

1,

vol_

1,

h_

ma

t(3

,3)

call

tra

ns_

to_

ten

s(sc

al_

1,

h_

ma

t)vo

l_1

=d

ete

rm(h

_m

at)

en

dsu

bro

utin

e

!!

!!

!!

!!

!

sub

rou

tine

die

lect

ric_

con

st(p

ol_

fluc,

po

l_a

vg,

sca

l_2

,sca

l_1

,te

mp

)in

teg

er

i_d

im,

j_d

im,

n_

dim

,e

_d

im,

iv_

dim

,vo

i_in

din

teg

er

k_d

im,

l_d

im,

ij_d

im,

kl_

dim

pa

ram

ete

r(n

_d

im=

3,

e_

dim

=3

)d

ou

ble

pre

cisi

on

po

l_flu

c(n

_d

im,n

_d

im),

dk(

n_

dim

+e

_d

im)

do

ub

lep

reci

sio

nvo

l_1

,r_

du

mm

y,p

ol_

avg

(n_

dim

),te

mp

do

ub

lep

reci

sio

np

olc

orr

ect

(n_

dim

,n_

dim

),sm

at(

6,6

),&

sca

l_1

(n_

dim

+e

_d

im),

sca

l_2

(n_d

im+

e_

dim

,n

_d

im+

e_

dim

)co

mm

on

/cvo

l_1

/vo

l_1

call

calc

_sm

at(

sca

l_2

,sc

al_

1,

sma

t)

do

i_d

im=

1,

n_

dim

do

k_d

im=

1,

n_

dim

po

lco

rre

ct(i_

dim

,k_

dim

)=

0.

do

j_d

im=

1,

n_

dim

do

l_d

im=

1,

n_

dim

ij_d

im=

voi_

ind

(i_

dim

,j_d

im)

kl_

dim

=vo

i_in

d(k

_d

im,l_

dim

)p

olc

orr

ect

(i_

dim

,k_

dim

)=

po

lco

rre

ct(i_

dim

,k_

dim

)+

&sm

at(

ij_d

im,

kl_

dim

)*

po

l_a

vg(j_

dim

)*

po

l_a

vg(l_

dim

)e

nd

do

en

dd

oe

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

write

(*,*

)(p

olc

orr

ect

(i_

dim

,k_

dim

),k_

dim

=1

,3)

en

dd

o

write

(*,*

)"d

k_ij"

151

B. Programsd

oi_

dim

=1,

n_

dim

do

j_d

im=

1,

n_

dim

iv_

dim

=vo

i_in

d(i_

dim

,j_d

im)

!(a

ng

stro

m*

e)*

*2/(

ep

silo

n0

an

gst

rom

**3

kBK

)=

20

99

84

1d

k(iv

_d

im)

=(p

ol_

fluc(

i_d

im,

j_d

im))

&*2

09

98

41

/vo

l_1

/te

mp

write

(*,*

)i_

dim

,j_

dim

,(p

ol_

fluc(

i_d

im,

j_d

im)

&-p

olc

orr

ect

(i_

dim

,j_

dim

))&

*20

998

41

/vo

l_1

/te

mp

if(iv_

dim

.le.3

)d

k(iv

_d

im)

=d

k(iv

_d

im)

+1

en

dd

oe

nd

do

write

(*,*

)"A

vg.

die

lect

ric

con

sta

nt"

write

(*,’(

6f1

3.5

)’)

(dk(

iv_

dim

),iv

_d

im=

1,

6)

en

dsu

bro

utin

e

!!

!!

!!

!!

!

sub

rou

tine

pie

zo_

coe

ff(t

em

p,p

ols

tra

in,

po

lstr

ess

)in

teg

er

j_d

im,

k_d

imp

ara

me

ter

(n_

dim

=3

,e

_d

im=

3)

do

ub

lep

reci

sio

np

ols

tra

in(n

_d

im+

e_

dim

,n_

dim

),&

tem

p,

pols

tre

ss(6

,3)

write

(*,*

)"A

vg.

pie

zoe

lect

ric

stre

ssC

oe

ffic

ien

ts[C

b/m

ˆ2]"

do

k_d

im=

1,3

write

(*,’(

6f1

3.5

)’)

&(p

ols

tre

ss(j_

dim

,k_

dim

)*1

6.0

21

77

/te

mp

,j_d

im=

1,6

)e

nd

do

write

(*,*

)"A

vg.

pie

zoe

lect

ric

stra

inC

oe

ffic

ien

ts[1

0ˆ-

12

m/V

]"d

ok_

dim

=1

,3w

rite

(*,’(

6f1

3.5

)’)

&(p

ols

tra

in(j_

dim

,k_

dim

)*1

.16

04

5d

-6*1

.d1

2/

tem

p,j_

dim

=1,6

)e

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_st

rain

(sca

l_2

,sc

al_

1)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,k_

dim

,l_

dim

,n

_d

im,

jv_

dim

,iv

_d

im,

&a

_d

im,

voi_

ind

pa

ram

ete

r(n

_d

im=

3)

do

ub

lep

reci

sio

nsc

al_

2(6

,6),

h_

ma

t(3

,3),

sca

l_1

(6),

&h

_i(3

,3),

stra

in(6

),s_

2(3

,3,3

,3)

call

tra

ns_

to_

ten

s(sc

al_

1,

h_

ma

t)ca

llin

v_m

at(

h_

ma

t,h

_i)

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

iv_

dim

=vo

i_in

d(i_

dim

,j_d

im)

do

k_d

im=

1,

n_

dim

do

l_d

im=

1,

n_

dim

jv_

dim

=vo

i_in

d(k

_d

im,l_

dim

)s_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

scal_

2(iv_

dim

,jv_

dim

)e

nd

do

en

dd

oe

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

iv_

dim

=vo

i_in

d(i_

dim

,j_d

im)

stra

in(iv_

dim

)=

0.

do

a_

dim

=1,

n_

dim

stra

in(iv_

dim

)=

stra

in(iv_

dim

)+

(&

sqrt

(s_

2(i_

dim

,a_

dim

,i_d

im,a

_d

im))

*h_

i(a

_d

im,j_

dim

)+

&sq

rt(s

_2(a

_d

im,i_

dim

,a_

dim

,i_d

im))

*h_

i(j_

dim

,a_

dim

))/2

.e

nd

do

en

dd

oe

nd

do

write

(*,*

)"s

tra

in"

write

(*,’(

6e

19.1

1)’)

(str

ain

(iv_

dim

),iv

_d

im=

1,6

)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

tran

s_to

_vo

igt(

ma

trix

,ve

ctor)

imp

licit

no

ne

do

ub

lep

reci

sio

nm

atr

ix(3

,3),

vect

or(

6)

vect

or(

1)

=m

atr

ix(1

,1)

vect

or(

2)

=m

atr

ix(2

,2)

vect

or(

3)

=m

atr

ix(3

,3)

vect

or(

4)

=2

*m

atr

ix(1

,2)

vect

or(

5)

=2

*m

atr

ix(2

,3)

vect

or(

6)

=2

*m

atr

ix(3

,1)

en

dsu

bro

utin

e

!!

!!

!!

!!

!

sub

rou

tine

tran

s_to

_te

ns(

vect

or,

ma

trix

)im

plic

itn

on

ed

ou

ble

pre

cisi

on

ma

trix

(3,3

),ve

cto

r(6)

ma

trix

(1,1

)=

vect

or(

1)

ma

trix

(2,2

)=

vect

or(

2)

ma

trix

(3,3

)=

vect

or(

3)

ma

trix

(1,2

)=

vect

or(

4)

ma

trix

(2,1

)=

vect

or(

4)

ma

trix

(2,3

)=

vect

or(

5)

ma

trix

(3,2

)=

vect

or(

5)

ma

trix

(3,1

)=

vect

or(

6)

ma

trix

(1,3

)=

vect

or(

6)

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

inv_

ma

t(m

at,m

at_

inv)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,si

gn

do

ub

lep

reci

sio

nm

at(

3,3

),m

at_

inv(

3,3

),d

et_

ma

tre

al

de

term

do

ub

lep

reci

sio

nvo

l_1

com

mo

n/c

vol_

1/

vol_

1

152


de

t_m

at

=d

ete

rm(m

at)

vol_

1=

de

t_m

at

do

i_d

im=

1,

3d

oj_

dim

=1

,3

ma

t_in

v(i_

dim

,j_d

im)

=0

.d0

en

dd

oe

nd

do

ma

t_in

v(1

,1)

=m

at(

2,2

)*m

at(

3,3

)-

ma

t(2

,3)*

*2m

at_

inv(

2,2

)=

ma

t(3

,3)*

ma

t(1

,1)

-m

at(

3,1

)**2

ma

t_in

v(3

,3)

=m

at(

1,1

)*m

at(

2,2

)-

ma

t(1

,2)*

*2m

at_

inv(

1,2

)=

-(m

at(

2,1

)*m

at(

3,3

)-

ma

t(2

,3)*

ma

t(3

,1))

ma

t_in

v(2

,3)

=-

(ma

t(3

,2)*

ma

t(1

,1)

-m

at(

3,1

)*m

at(

1,2

))m

at_

inv(

1,3

)=

+(m

at(

1,2

)*m

at(

2,3

)-

ma

t(1

,3)*

ma

t(2

,2))

do

i_d

im=

1,

3d

oj_

dim

=i_

dim

,3

ma

t_in

v(i_

dim

,j_d

im)

=m

at_

inv(

i_d

im,j_

dim

)/

de

t_m

at

ma

t_in

v(j_

dim

,i_d

im)

=m

at_

inv(

i_d

im,j_

dim

)e

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

ma

t(m

at,m

at_

tra

ns)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

do

ub

lep

reci

sion

ma

t(3

,3),

ma

t_tr

an

s(3

,3)

do

i_d

im=

1,

3d

oj_

dim

=1

,3m

at_

tra

ns(

i_d

im,j_

dim

)=m

at(

j_d

im,i_

dim

)e

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

rea

lfu

nct

ion

de

term

(ma

t)im

plic

itn

on

ed

ou

ble

pre

cisi

on

ma

t(3

,3)

de

term

=m

at(

1,1

)*m

at(

2,2

)*m

at(

3,3

)+

2*m

at(

1,2

)*m

at(

2,3

)*m

at(

3,1

)&

-ma

t(1

,1)*

ma

t(2

,3)*

*2-m

at(

2,2

)*m

at(

3,1

)**2

-ma

t(3

,3)*

ma

t(1,2

)**2

en

dfu

nct

ion

!!

!!

!!

!!

!!

!

sub

rou

tine

mu

lt_m

at(

ma

t1,m

at2

,mu

lt)im

plic

itn

on

ein

teg

er

i_d

im,

j_d

im,

k_d

imd

ou

ble

pre

cisi

on

ma

t1(3

,3),

ma

t2(3

,3),

mu

lt(3

,3)

do

i_d

im=

1,

3d

oj_

dim

=1

,3m

ult(

i_d

im,j_

dim

)=0

.d0

do

k_d

im=

1,3

mu

lt(i_

dim

,j_dim

)=

mu

lt(i_

dim

,j_d

im)

&+

ma

t1(i_

dim

,k_

dim

)*

ma

t2(k

_d

im,j_

dim

)

en

dd

oe

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

sub

rou

tine

calc

_e

lco

(sca

l_2

,sc

al_

1)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,k_

dim

,l_

dim

,m

_d

im,

n_

dim

pa

ram

ete

r(n

_d

im=

3)

do

ub

lep

reci

sio

nsc

al_

2(6

,6),

h_

1(6

,6),

sca

l_1

(6),

s_2

(6,6

,6,6

)d

ou

ble

pre

cisi

on

vol_

1,

rot_

6(6

,6),

eig

_6

(6,6

),sm

at(

6,6

)co

mm

on

/cvo

l_1

/vo

l_1

rea

lc_

11

,c_

12

,c_

13

,c_

14

,c_

33

,c_

44

,c_

66

call

calc

_sm

at(

sca

l_2

,sc

al_

1,

sma

t)w

rite

(*,*

)"s

_m

atr

ix"

do

i_d

im=

1,

6d

oj_

dim

=1

,6

sca

l_2

(i_

dim

,j_d

im)

=vo

l_1

*sm

at(

i_d

im,j_

dim

)e

nd

do

write

(*,’(

6e

13.5

)’)

(sca

l_2

(i_

dim

,j_d

im),

j_d

im=

1,6

)e

nd

do

call

ga

uss

j(sc

al_

2)

write

(*,*

)"E

l.C

on

st.

inG

Pa

"d

oi_

dim

=1,

6d

oj_

dim

=1

,6

if(i_

dim

.gt.3

)sc

al_

2(i_

dim

,j_d

im)

=sc

al_

2(i_

dim

,j_d

im)

/2

if(j_

dim

.gt.3

)sc

al_

2(i_

dim

,j_d

im)

=sc

al_

2(i_

dim

,j_d

im)

/2

en

dd

ow

rite

(*,’(

6e

13.5

)’)

(sca

l_2

(i_

dim

,j_d

im)/

72

.43

,j_d

im=

1,6

)e

nd

do

write

(*,*

)"#

c_1

1,

c_1

2,

c_1

3,

c_1

4,

c_3

3,

c_4

4,

c_6

6&

(Q

ua

rtz

)"

c_1

1=

(sc

al_

2(1

,1)

+sc

al_

2(2

,2)

)/

72

.43

/2

c_1

2=

sca

l_2

(1,2

)/

72

.43

c_1

3=

(sc

al_

2(1

,3)

+sc

al_

2(2

,3)

)/

72

.43

/2

c_1

4=

(sc

al_

2(1

,5)

-sc

al_

2(2

,5)

+sc

al_

2(4

,6)

)/

72

.43

/3

c_3

3=

sca

l_2

(3,3

)/

72

.43

c_4

4=

(sc

al_

2(5

,5)

+sc

al_

2(6

,6)

)/

72

.43

/2

c_6

6=

sca

l_2

(4,4

)/

72

.43

write

(*,"

(7f7

.1)"

)c_

11

,c_

12

,c_

13

,c_

14

,c_

33

,c_

44

,c_

66

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

calc

_sm

at(

sca

l_2

,sc

al_

1,

sma

t)im

plic

itn

one

inte

ge

ri_

dim

,j_

dim

,k_

dim

,l_

dim

,n_

dim

,iv

_d

im,

jv_

dim

,&

voi_

ind

,a

_d

im,

b_

dim

pa

ram

ete

r(n

_d

im=

3)

do

ub

lep

reci

sio

ntk

,sc

al_

2(6

,6),

h_

ma

t(3

,3),

sca

l_1

(6),

&s_

2(3

,3,3

,3),

h_

i(3

,3),

e_

2(3

,3,3

,3),

sma

t(6

,6)

call

tra

ns_

to_

ten

s(sc

al_

1,

h_

ma

t)ca

llin

v_m

at(

h_

ma

t,h

_i)

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

153

B. Programsiv

_d

im=

voi_

ind

(i_

dim

,j_d

im)

do

k_d

im=

1,

n_

dim

do

l_d

im=

1,

n_

dim

jv_

dim

=vo

i_in

d(k

_d

im,l_

dim

)s_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

scal_

2(iv_

dim

,jv_

dim

)e

nd

do

en

dd

oe

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

do

k_d

im=

1,

n_

dim

do

l_d

im=

1,

n_

dim

e_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

0.

do

a_

dim

=1

,n

_d

imd

ob

_d

im=

1,

n_

dim

e_

2(i_

dim

,j_d

im,k

_d

im,l_

dim

)=

&e

_2

(i_

dim

,j_d

im,k

_d

im,l_

dim

)+

&s_

2(i_

dim

,a_

dim

,b_

dim

,l_d

im)*

h_

i(a

_d

im,j_

dim

)*h

_i(b

_d

im,k

_d

im)

+&

s_2

(i_

dim

,a_

dim

,k_

dim

,b_

dim

)*h

_i(a

_d

im,j_

dim

)*h

_i(b

_d

im,l_

dim

)+

&s_

2(a

_d

im,j_

dim

,b_

dim

,l_d

im)*

h_

i(a

_d

im,i_

dim

)*h

_i(b

_d

im,k

_d

im)

+&

s_2

(a_

dim

,j_d

im,k

_d

im,b

_d

im)*

h_

i(a

_d

im,i_

dim

)*h

_i(b

_d

im,l_

dim

)e

nd

do

en

dd

oe

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)=

&e

_2

(i_

dim

,j_

dim

,k_

dim

,l_

dim

)/

4e

nd

do

en

dd

oe

nd

do

en

dd

o

do

i_d

im=

1,

n_

dim

do

j_d

im=

1,

n_

dim

iv_

dim

=vo

i_in

d(i_

dim

,j_d

im)

do

k_d

im=

1,

n_

dim

do

l_d

im=

1,

n_

dim

jv_

dim

=vo

i_in

d(k

_d

im,l_

dim

)sm

at(

iv_

dim

,jv_

dim

)=

e_

2(i_

dim

,j_

dim

,k_

dim

,l_

dim

)e

nd

do

en

dd

oe

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

inte

ge

rfu

nct

ion

voi_

ind

(i_

dim

,j_d

im)

inte

ge

ri_

dim

,j_

dim

if(i_

dim

.eq.j_

dim

)th

en

voi_

ind

=i_

dim

els

eif

(i_

dim

*j_

dim

.eq

.2)

the

nvo

i_in

d=

4e

lse

if(i_

dim

*j_

dim

.eq

.6)

the

nvo

i_in

d=

5e

lse

if(i_

dim

*j_

dim

.eq

.3)

the

nvo

i_in

d=

6e

lse

sto

P"I

nd

.in

voi_

ind

w.o

.se

nse

"e

nd

ife

nd

fun

ctio

n

calc

stra

in.f

pro

gra

mca

lcst

rain

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,n

_d

im,

e_

dim

pa

ram

ete

r(n

_dim

=3

,e

_d

im=

3)

do

ub

lep

reci

sio

nh

(3,3

),h

0(3

,3),

ht(

3,3

),h

0i(3

,3),

h0

ti(3

,3),

&h

0t(

3,3

)d

ou

ble

pre

cisi

on

s1(3

,3),

s2(3

,3),

stra

in(3

,3)

do

ub

lep

reci

sio

nx(

8),

sca

l(n

_d

im+

e_d

im),

y(1

0),

&p

ol_

tota

l(n

_d

im),

po

l_d

ipo

les(

n_

dim

)

write

(*,*

)"h

0(1

,1)

?"

rea

d(*

,*)

h0

(1,1

)w

rite

(*,*

)"h

0(2

,2)

?"

rea

d(*

,*)

h0

(2,2

)w

rite

(*,*

)"h

0(3

,3)

?"

rea

d(*

,*)

h0

(3,3

)w

rite

(*,*

)"h

0(1

,2)

?"

rea

d(*

,*)

h0

(1,2

)w

rite

(*,*

)"h

0(2

,3)

?"

rea

d(*

,*)

h0

(2,3

)w

rite

(*,*

)"h

0(1

,3)

?"

rea

d(*

,*)

h0

(1,3

)

h0

(2,1

)=

h0

(1,2

)h

0(3

,2)

=h

0(2

,3)

h0

(3,1

)=

h0

(1,3

)

call

inv_

ma

t(h0

,h0

i)ca

lltr

an

s_m

at(

h0

,h0

t)ca

llin

v_m

at(

h0

t,h

0ti)

10

0co

ntin

ue

rea

d(6

0,’(

30e

15

.7)’,e

nd

=2

00

)(x

(i_

dim

),i_

dim

=1

,8

),&

(sca

l(j_

dim

),j_

dim

=1

,n

_d

im+

e_

dim

),(y

(i_

dim

),i_

dim

=1

,10

),&

(po

l_to

tal(i_

dim

),i_

dim

=1

,3),

(po

l_d

ipo

les(

i_d

im),

i_d

im=

1,3

)

call

tra

ns_

to_

ten

s(sc

al,h

)ca

lltr

an

s_m

at(

h,h

t)ca

llm

ult_

mat(

h,h

0i,s

1)

call

mu

lt_m

at(

ht,s1

,s2

)ca

llm

ult_

mat(

h0

ti,s2

,str

ain

)

do

i_d

im=

1,3

do

j_d

im=

1,3

if(i_

dim

.eq

.j_d

im)

&st

rain

(i_

dim

,j_d

im)

=st

rain

(i_

dim

,j_d

im)

-1

.d0

stra

in(i_

dim

,j_d

im)

=0

.5d

0*

stra

in(i_

dim

,j_d

im)

en

dd

oe

nd

do

write

(59

,’(30

e1

5.7

)’)

x(1

),st

rain

(1,1

),st

rain

(2,2

),&

stra

in(3

,3),

stra

in(1

,2),

stra

in(2

,3),

stra

in(1

,3)

154


go

to1

00

20

0co

ntin

ue

en

dp

rog

ram

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

to_

ten

s(ve

cto

r,m

atr

ix)

imp

licit

no

ne

do

ub

lep

reci

sion

ma

trix

(3,3

),ve

cto

r(6

)m

atr

ix(1

,1)

=ve

cto

r(1

)m

atr

ix(2

,2)

=ve

cto

r(2

)m

atr

ix(3

,3)

=ve

cto

r(3

)m

atr

ix(1

,2)

=ve

cto

r(4

)m

atr

ix(2

,1)

=ve

cto

r(4

)m

atr

ix(2

,3)

=ve

cto

r(5

)m

atr

ix(3

,2)

=ve

cto

r(5

)m

atr

ix(3

,1)

=ve

cto

r(6

)m

atr

ix(1

,3)

=ve

cto

r(6

)e

nd

sub

rou

tine

!!

!!

!!

!!

!!

!

sub

rou

tine

tra

ns_

ma

t(m

at,m

at_

tra

ns)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

do

ub

lep

reci

sion

ma

t(3

,3),

ma

t_tr

an

s(3

,3)

do

i_d

im=

1,

3d

oj_

dim

=1

,3m

at_

tra

ns(

i_d

im,j_

dim

)=m

at(

j_d

im,i_

dim

)e

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

rea

lfu

nct

ion

de

term

(ma

t)im

plic

itn

on

ed

ou

ble

pre

cisi

on

ma

t(3

,3)

de

term

=m

at(

1,1

)*m

at(

2,2

)*m

at(

3,3

)+

2*m

at(

1,2

)*m

at(

2,3

)*m

at(

3,1

)&

-ma

t(1

,1)*

ma

t(2

,3)*

*2-m

at(

2,2

)*m

at(

3,1

)**2

-ma

t(3

,3)*

ma

t(1,2

)**2

en

dfu

nct

ion

!!

!!

!!

!!

!!

!

sub

rou

tine

mu

lt_m

at(

ma

t1,m

at2

,mu

lt)im

plic

itn

on

ein

teg

er

i_d

im,

j_d

im,

k_d

imd

ou

ble

pre

cisi

on

ma

t1(3

,3),

ma

t2(3

,3),

mu

lt(3

,3)

do

i_d

im=

1,

3d

oj_

dim

=1

,3m

ult(

i_d

im,j_

dim

)=0

.d0

do

k_d

im=

1,3

mu

lt(i_

dim

,j_dim

)=

mu

lt(i_

dim

,j_d

im)

&+

ma

t1(i_

dim

,k_

dim

)*

ma

t2(k

_d

im,j_

dim

)e

nd

do

en

dd

oe

nd

do

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

sub

rou

tine

inv_

ma

t(m

at,m

at_

inv)

imp

licit

no

ne

inte

ge

ri_

dim

,j_

dim

,si

gn

do

ub

lep

reci

sio

nm

at(

3,3

),m

at_

inv(

3,3

),d

et_

ma

tre

al

de

term

do

ub

lep

reci

sio

nvo

l_1

com

mo

n/c

vol_

1/

vol_

1

de

t_m

at

=d

ete

rm(m

at)

vol_

1=

de

t_m

at

do

i_d

im=

1,

3d

oj_

dim

=1,

3m

at_

inv(

i_d

im,j_

dim

)=

0.d

0e

nd

do

en

dd

om

at_

inv(

1,1

)=

ma

t(2

,2)*

ma

t(3

,3)

-m

at(

2,3

)**2

ma

t_in

v(2

,2)

=m

at(

3,3

)*m

at(

1,1

)-

mat(

3,1

)**2

ma

t_in

v(3

,3)

=m

at(

1,1

)*m

at(

2,2

)-

mat(

1,2

)**2

ma

t_in

v(1

,2)

=-

(ma

t(2

,1)*

ma

t(3

,3)

-m

at(

2,3

)*m

at(

3,1

))m

at_

inv(

2,3

)=

-(m

at(

3,2

)*m

at(

1,1

)-

ma

t(3

,1)*

ma

t(1

,2))

ma

t_in

v(1

,3)

=+

(ma

t(1

,2)*

ma

t(2

,3)

-m

at(

1,3

)*m

at(

2,2

))d

oi_

dim

=1,

3d

oj_

dim

=i_

dim

,3

ma

t_in

v(i_

dim

,j_d

im)

=m

at_

inv(

i_d

im,j_

dim

)/

de

t_m

at

ma

t_in

v(j_

dim

,i_d

im)

=m

at_

inv(

i_d

im,j_

dim

)e

nd

do

en

dd

o

en

dsu

bro

utin

e

!!

!!

!!

!!

!!

!

inte

ge

rfu

nct

ion

voi_

ind

(i_

dim

,j_d

im)

inte

ge

ri_

dim

,j_

dim

if(i_

dim

.eq.j_

dim

)th

en

voi_

ind

=i_

dim

els

eif

(i_

dim

*j_

dim

.eq

.2)

the

nvo

i_in

d=

4e

lse

if(i_

dim

*j_

dim

.eq

.6)

the

nvo

i_in

d=

5e

lse

if(i_

dim

*j_

dim

.eq

.3)

the

nvo

i_in

d=

6e

lse

sto

P"I

nd

.in

voi_

ind

w.o

.se

nse

"e

nd

ife

nd

fun

ctio

n

155

B. Programs

156

List of Figures

0.1. Phase diagram for the silica system.. . . . . . . . . . . . . . . . . . 3

1.1. Pair Correlation Function ofα-Quartz . . . . . . . . . . . . . . . . . 19

2.1. Averageβ-Quartz Structure. . . . . . . . . . . . . . . . . . . . . . . 242.2. Averageα-Quartz Structure. . . . . . . . . . . . . . . . . . . . . . . 242.3. Fourth-order cumulant for the fluctuating dipole potential. . . . . . . 262.4. Order parameter〈|Φ|〉 against Temperature for fluc-dipole. . . . . . 272.5. Fourth-order cumulant for the fluctuating charge potential. . . . . . . 282.6. Order parameter〈|Φ|〉 against Temperature for fluc-Q. . . . . . . . . 292.7. Radial distribution function at the Quartz Transition. . . . . . . . . . 302.8. Volume against Temperature at theα-β Transition . . . . . . . . . . . 312.9. c/a Ratio against Temperature at theα-β Transition . . . . . . . . . . 312.10.c/a Ratio against Temperature for fluc-Q. . . . . . . . . . . . . . . . 322.11. Order parameter against Temperature for fluc-dipole. . . . . . . . . . 332.12. Volume against Temperature for fluc-dipole. . . . . . . . . . . . . . 332.13.c/a Ratio in Quartz as a Function of Temperature. . . . . . . . . . . 342.14. O-Si-O bond angle against Temperature. . . . . . . . . . . . . . . . 352.15. Fourth Order Cumulant against Temperature. . . . . . . . . . . . . . 352.16. Definition of the atom numbering. . . . . . . . . . . . . . . . . . . . 362.17.c/a ratio in quartz as a function of the volume. . . . . . . . . . . . . 372.18. Models showing the formation ofπ orbitals in SiO4. . . . . . . . . . . 382.19. Correlation between bond distance and oxygen angle. . . . . . . . . 382.20. Average dipoles in theα- and theβ-quartz phase.. . . . . . . . . . . 392.21. Elastic constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.22. Elastic constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.23.α quartz: Phonon Density of States BKS. . . . . . . . . . . . . . . . 452.24.α quartz: Phonon Density of States fluc-charge. . . . . . . . . . . . 452.25.α quartz: Phonon Density of States fluc-dipole. . . . . . . . . . . . . 46

3.1. Snapshot of the Quartz II configuration. . . . . . . . . . . . . . . . . 483.2. Silicon Coordination in Quartz II. . . . . . . . . . . . . . . . . . . . 493.3. Pressure Driven Transition Path Ways in Quartz with fluc-Q. . . . . 503.4. Pressure Driven Transition Path Ways in Quartz with fluc-dipole. . . 51

157

List of Figures

3.5. Si-O Pair correlation around the Quartz II - Quartz II b transition. . . 523.6. Energy dispersive x-ray diffraction patterns of Quartz II. . . . . . . . 533.7. Comparison of X-Ray spectra at various pressures. . . . . . . . . . . 543.8. Interplanar spacing for the compression ofα-quartz . . . . . . . . . . 55

4.1. Time evolution of the volume in cristobalite. . . . . . . . . . . . . . 574.2. Time evolution of the volume in tridymite. . . . . . . . . . . . . . . 584.3. Snapshots of Tridymite. . . . . . . . . . . . . . . . . . . . . . . . . 584.4. Stishovite Equation of State. . . . . . . . . . . . . . . . . . . . . . . 594.5. c/a Ratio in Stishovite against Pressure. . . . . . . . . . . . . . . . . 604.6. Birch Coefficients of Stishovite against Pressure. . . . . . . . . . . . 614.7. Lattice Constantsa andb of Stishovite . . . . . . . . . . . . . . . . . 62

5.1. Illustration of the definition of the dipole. . . . . . . . . . . . . . . . 685.2. Comparison of piezo induced strain for different approaches. . . . . 705.3. Piezoelectrical strain coefficient against temperature. . . . . . . . . . 725.4. Distribution of the tilt angle in quartz. . . . . . . . . . . . . . . . . . 735.5. Pressure dependence ofα-quartz.. . . . . . . . . . . . . . . . . . . . 745.6. Relation between dipole and Si-O-Si angle. . . . . . . . . . . . . . . 755.7. Sketch of bond angle and dipole orientation. . . . . . . . . . . . . . 765.8. Relation between dipole orientation and absolute value. . . . . . . . 76

A.1. Sketch for Two Center Integrals. . . . . . . . . . . . . . . . . . . . 83

158

Bibliography

[1] M. P. Allen and D. J. Tildesley.Computer Simulation of Liquids. Oxford Sci-ence Publications, Oxford, 1987.

[2] D. Andrault, G. Fiquet, F. Guyot, and M. Hanfland.Science, 282:720, 1998.

[3] J. C. Axe and G. Shirane.Phys. Rev. B, 1:342, 1970.

[4] J. P. Bachheimer and G. Dolino.Phys. Rev. B, 11:3195, 1975.

[5] E. J. Banda, R. A. Craven, R. D. Parks, P. M. Horn, and M. Blume.Solid StateComm., 17:11, 1975.

[6] A. G. Beda.Soviet Physics - Solid State, 9:5, 1967.

[7] M. Benoit and W. Kob.Europhys. Lett., 60:269, 2002.

[8] K. Binder. Phys. Rev. Lett., 47:693, 1981.

[9] W. Bragg and R. E. Gibbs.Proc. Roy. Soc. London, A109:414, 1925.

[10] W. G. Cady.Piezoelectricity. Dover Publications, New York, 1964.

[11] R. Car and M. Parrinello.Phys. Rev. Lett., 55:2471, 1985.

[12] M. A. Carpenter, R. J. Hemley, and H. kwang Mao.J. Geophys. Res.,105:10807, 2000.

[13] M. A. Carpenter, E. K. H. Salje, A. Graeme-Barber, B. Wrucki, M. T. Dove,and K. S. Knight.Am. Mineral., 83:2, 1998.

[14] E. C. T. Chao, J. J. Fahey, J. Littler, and D. J. Milton.J. Geophys. Res., 67:419,1962.

[15] R. E. Cohen.Geophys. Res. Lett., 14:37, 1987.

[16] B. Coluzzi and P. Verrocchio.J. Chem. Phys., 116:3789, 2002.

[17] R. K. Cook and P. G. Weissler.Phys. Rev., 80:4, 1950.

159

Bibliography

[18] D. W. J. Cruickshank.J. chem. Soc., 1077:5486, 1961.

[19] E. Demiralp, 2001. private communication.

[20] E. Demiralp, T. Cagin, and W. A. Goddard III.Phys. Rev. Lett., 82:1708, 1999.

[21] T. Demuth, Y. Jeanvoine, J. Hafner, and J. G. Angyan.J. Phys.: Condens.Matter, 11:3833, 1999.

[22] G. Dolino, B. Berge, M. Vallade, and F. Moussa.J. de Physique I, 2:1461, 1992.

[23] W. A. Dollase.Acta Cryst., 23:617, 1967.

[24] M. T. Dove, D. A. Kreen, A. C. Hannon, and I. P. Swainson.Phys. Chem.Miner., 24:311, 1997.

[25] F. Ercolessi and J. B. Adams.Europhys. Lett., 26:583, 1994.

[26] P. P. Ewald.Ann. Phys., 64:253, 1921.

[27] P. W. Fowler and P. A. Madden.Phys. Rev. B, 31:5443, 1985.

[28] D. Frenkel and B. Smit.Understanding Molecular Simulation. Academic Press,London, 2nd edition, 2002.

[29] J. D. Gale.Phil. Mag. B, 73:3, 1996.

[30] J. D. Gale.J. Chem. Soc. Faraday Trans., 93:629, 1997.

[31] J. D. Gale.Mol. Simul., 29:291, 2003.

[32] F. Galeener.Phys. Rev. B, 19:4292, 1979.

[33] C. W. Gear. Numerical Initial Value Problems in ordinary Differential Equa-tions. Prentice Hall, Englewood Cliffs, NJ, 1971. Chap. 9.

[34] G. V. Gibbs, M. M. Hamil, S. J. Louisnathan, L. S. Bartell, and H. Yow.Amer.Mineral., 57:1578, 1972.

[35] H. Grimm and B. Dorner.J. Phys. Chem. Solids, 36:407, 1975.

[36] J. M. Haile.Molecular Dynamics Simulation: Elementary Methods. John Wileyand Sons, Inc., New York, wiley professional paperback series edition, 1997.

[37] J. Haines, O. Cambon, D. A. Keen, M. G. Tucker, and M. T. Dove.Appl. Phys.Lett., 81:2968, 2002.

160

Bibliography

[38] J. Haines, J. M. Leger, F. Gorelli, and M. Hanfland.Phys. Rev. Lett., 87:155503,2001.

[39] P. J. Heaney. Structure and chemistry of the low-pressure silica polymorphs. InP. J. Heaney, C. T. Prewitt, and G. V. Gibbs, editors,Silica - Physical Behaviour,Geochemistry, and Materials Applications, volume 29 ofRev. Mineral., page 1.Mineral. Soc. of Am., Washington D. C., 1994.

[40] R. H. Hemley, A. P. Jephcoat, H. K. Mao, L. C. Ming, and M. H. Manghnani.Nature, 334:52, 1988.

[41] R. J. Hemley. Pressure dependence of raman spectra in sio2 polymorphs:α-quartz, coesite, and stishovite. In M. H. Manghnani and Y. Syono, editors,HighPressure Research in Mineral Physics, volume 39 ofGeophys. Monogr. Ser.,page 347. AGU, Washington D. C., 1987.

[42] R. J. Hemley, M. D. Jackson, and R. G. Gordon.Eos Trans. AGU, 66:357, 1985.

[43] R. J. Hemley, C. T. Prewitt, and K. J. Kingma. High-pressure behaviour ofsilica. In P. J. Heaney, C. T. Prewitt, and G. V. Gibbs, editors,Silica - Phys-ical Behaviour, Geochemistry, and Materials Applications, volume 29 ofRev.Mineral., page 41. Mineral. Soc. of Am., Washington D. C., 1994.

[44] R. J. Hill and G. V. Gibbs.Acta Cryst., B35:25, 1979.

[45] U. T. Hochli and J. F. Scott.Phys. Rev. Lett., 26:1627, 1971.

[46] J. Horbach and W. Kob.Phys. Rev. B, 60:3169, 1999.

[47] J. B. Jones and E. R. Segnit.J. Geol. Soc. Australia, 18:419, 1972.

[48] P. Jund and R. Jullien.Phys. Rev. Lett., 83:2210, 1999.

[49] K. Kihara. Eur. J. Miner., 2:63, 1990.

[50] K. Kihara. Phys. Chem. Min., 19:492, 1993.

[51] K. J. Kingma, R. E. Cohen, R. J. Hemley, and H. K. Mao.Nature, 374:243,1995.

[52] K. J. Kingma, R. J. Hemley, H. K. Mao, and D. R. Veblen.Phys. Rev. Lett.,70:3927, 1993.

[53] K. J. Kingma, H. K. Mao, and R. J. Hemley.High Press. Res., 14:363, 1996.

[54] Klein and Hurlbut. Manual of Mineralogy, p. 527. John Wiley & Sons, 21stedition, 1962.

161

Bibliography

[55] M. B. Kruger and R. Jeanloz.Science, 249:647, 1990.

[56] D. J. Lacks and R. G. Gordon.J. Geophys. Res., 98:22147, 1993.

[57] A. Laio, S. Bernard, G. L. Chiarotti, S. Scandolo, and E. Tosatti.Science,287:1027, 2000.

[58] D. P. Landau and K. Binder, editors.A Guide to Monte Carlo Simulations inStatistical Physics. Cambridge University Press, Cambridge, 2000.

[59] S. J. Louisnathan and G. V. Gibbs.Amer. Mineral., 57:1614, 1972.

[60] M. H. M. Wilson, P. A. Madden and C. A. Angell.Phys. Rev. Lett., 77:4023,1996.

[61] C. H. Macgillavry, G. D. Rieck, and K. Lonsdale, editors.International Tablesof X-Ray Crystallography. Kynoch Press, Birmingham, 1962.

[62] H. K. Mao, J. Shu, J. Hu, and R. J. Hemley.Eos Trans. AGO, Fall Meet. Suppl.,75(44):662, 1994.

[63] R. M. Martin. Phys. Rev. B, 5:1607, 1972.

[64] L. McNeil and M. Grimsditch.Phys. Rev. Lett., 68:83, 1992.

[65] M. H. Muser. Average structure vs. real structure: Molecular dynamics studiesof silica. In K. A. Gernoth and M. L. Ristig, editors,Particle Scattering, X-RayDiffraction, and Microstructure of Solids and Liquids, volume 610 ofLectureNotes in Physics, pages 75–100. Springer, Berlin, 2003.

[66] M. H. Muser and K. Binder.Phys. Chem. Miner., 28:746, 2001.

[67] L. Nagel and M. O’Keeffe.Mater. Res. Bull., 6:1317, 1971.

[68] E. L. Nave, H. E. Stanley, and F. Sciortino.Phys. Rev. Lett., 88:035501, 2002.

[69] S. Nose.J. Chem. Phys., 81:511, 1984.

[70] R. G. Parr and W. Yang.Density-Functional Theory of Atoms and Molecules.Oxford Science, Oxford, 1989.

[71] M. Parrinello and A. Rahman.Phys. Rev. Lett., 45:1196, 1980.

[72] M. Parrinello and A. Rahman.J. Chem. Phys., 76:2662, 1982.

[73] A. Pasquarello and R. Car.Phys. Rev. Lett., 80:5145, 1998.

162

Bibliography

[74] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling.NumericalRecipes in Fortran. Cambridge University Press, Cambridge, 2nd edition, 1992.

[75] D. L. Price and J. M. Carpenter.J. Non-Cryst. Solids, 92:153, 1987.

[76] A. K. Rappe, 2001. private communication.

[77] A. K. Rappe and W. A. Goddard III.J. Phys. Chem., 95:3358, 1991.

[78] S. W. Rick, S. J. Stuart, and B. J. Berne.J. Chem. Phys., 101:6141, 1994.

[79] G. Roma, Y. Limoge, and S. Baroni.Phys. Rev. Lett., 86:4564, 2001.

[80] N. L. Ross, J. F. Shu, and R. M. Hazen.Am. Min., 75:739, 1990.

[81] E. K. H. Salje, A. Ridgwell, B. Guttler, B. Wruck, M. T. Dove, and G. Dolino.J. Phys. Cond. Mat., 4:571, 1992.

[82] R. T. Sanderson.Science, 114:670, 1951.

[83] P. Schoffel. Diplomarbeit, Johannes Gutenberg-Universitat Mainz, 2000.

[84] D. Sebastiani. PhD thesis, Universitat Stuttgart, 2001. http://elib.uni-stuttgart.de/opus/volltexte/2001/792.

[85] J. Sefcik, E. Demiralp, T. Cagin, and W. A. Goddard III.J. Comput. Chem.,23:1507, 2002.

[86] G. S. Smith.Acta Cryst., 16:542, 1963.

[87] D. R. Spearing, I. Farnan, and J. F. Stebbins.Phys. Chem. Miner., 19:307, 1992.

[88] M. Sprik and M. L. Klein.J. Chem. Phys., 89:7556, 1988.

[89] S. M. Stishov and S. V. Popova.Geochem. (USSR), 10:923, 1961.

[90] P. Tangney and S. Scandolo.J. Chem. Phys., 117:8898, 2002.

[91] S. N. Taraskin and S. R. Elliot.Phys. Rev. B, 56:8605, 1997.

[92] Y. Tezuka, S. Shin, and M. Ishigame.Phys. Rev. Lett., 66:2356, 1991.

[93] M. P. Tosi.Solid St. Phys., 16:1, 1964.

[94] A. Toukmaji, C. Sagui, J. Board, and T. Darden.J. Chem. Phys., 113:10913,2000.

[95] J. S. Tse and D. D. Klug.Phys. Rev. Lett., 67:3559, 1991.

163

Bibliography

[96] J. S. Tse and D. D. Klug.J. Chem. Phys., 95:9176, 1991.

[97] J. S. Tse, D. D. Klug, and Y. L. Page.Phys. Rev. Lett., 69:3647, 1992.

[98] Y. Tsuchida and T. Yagi.Nature, 340:217, 1989.

[99] Y. Tsuchida and T. Yagi.Nature, 347:267, 1990.

[100] S. Tsuneyuki, H. Aoki, M. Tsukada, and Matsui.Phys. Rev. Lett., 64:776, 1990.

[101] B. van Beest, G. Kramer, and R. van Santen.Phys. Rev. Lett., 64:1955, 1990.

[102] W. Voigt. Lehrbuch der Kristallphysik. Teubner, Berlin, 1910.

[103] K. Vollmayr, W. Kob, and K. Binder.Phys. Rev. B, 54:15808, 1996.

[104] K. Vollmayr, J. D. Reger, M. Scheucher, and K. Binder.Z. Phys. B, 91:113,1993.

[105] R. M. Wentzcovitch, C. da Silva, J. R. Chelikowsky, and N. Binggeli.Phys.Rev. Lett., 80:2149, 1998.

[106] M. Wilson and P. A. Madden.J. Phys.: Condens. Matt., 5:2687, 1993.

[107] A. F. Wright and M. S. Lehmann.J. Solid. State. Chem., 36:372, 1981.

164

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