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Simulation of structural phase transitions by metadynamics

R. Martonak,1, ∗ A. Laio,1 M. Bernasconi,2 C. Ceriani,3 P. Raiteri,1 and M. Parrinello1

1Computational Science, Department of Chemistry and Applied Biosciences,

ETH Zurich, USI Campus, Via Giuseppe Buffi 13, CH-6900 Lugano, Switzerland2Dipartimento di Scienza dei Materiali and Istituto Nazionale per la Fisica della Materia,

Universita degli Studi di Milano-Bicocca, Via Cozzi 53, I-20125 Milano, Italy3Department of Chemical, Physical and Mathematical Science,

University of Insubria at Como, Via Lucini 3, 22100 Como, Italy

(Dated: February 2, 2008)

We describe here in detail the recently introduced methodology for simulation of structural tran-sitions in crystals. The applications of the new scheme are illustrated on various kinds of crystalsand the advantages with respect to previous schemes are emphasized. The relevance of the newmethod for the problem of crystal structure prediction is also discussed.

I. INTRODUCTION

The constant-pressure molecular dynamics (MD)method of Parrinello and Rahman[1] enabled for the firsttime the study of structural phase transitions in bulksolid crystals by computer simulation. Starting from aknown initial structure it allowed the identification ofpossible candidates for the new structure without anyprevious knowledge. It thus achieved predictive power,in particular in combination with ab-initio methods[2],and has been successfully applied many times to a vari-ety of crystalline systems (for few selected applications,see Refs.[3, 4, 5, 6, 7, 8, 9]). In the practical use of themethod, however, several problems arise related mainlyto the fact that structural transitions are often first order.This is by necessity the case when the symmetries of thetwo crystal structures are not in a group-subgroup rela-tion. Experimentally, such transitions proceed via nucle-ation of the new phase, which often starts on the surfaceor on structural defects. For simulations of crystals, how-ever, periodic boundary conditions that eliminate surfaceare commonly used. The systems used in simulationsare typically relatively small and therefore contain nostructural defects. The simulation setup therefore sup-presses the possibilities for a heterogeneous nucleation ofthe new phase. As a consequence, the transformationof the system proceeds in a collective way, involving allatoms, which results in a high barrier. This might causea metastability of the initial phase far beyond the ther-modynamic transition point and large hysteretic effectsare frequently observed. Should there be a substantialdifference in volume between the phases, increasing pres-sure favors the new phase with a smaller volume, due tothe contribution of the PV term in the Gibbs potential.In order to observe the transition within the accessiblesimulation time one often has to overpressurize the sys-tem close to the point of mechanical instability[10]. Un-

∗Permanent address: Department of Physics (FEI), Slovak Univer-sity of Technology, Ilkovicova 3, 812 19 Bratislava, Slovakia

der such conditions one or more intermediate phases maybe skipped[2, 11], which reduces the predictive power ofthe method. In other cases, where the volume differencebetween the phases is not so pronounced, even overpres-surization might not help to force the transition to occur.For the above reasons, there is still a need for developingspecific methods aimed at simulating structural transi-tions in crystals, given the great theoretical and prac-tical relevance of the closely related problem of crystalstructure prediction.

In this paper we review recent progress in this field re-lated to the application of the new approach of Laio andParrinello, called metadynamics[12]. Rather than givinga ready-to-use recipe we would like to highlight the possi-bilities offered by the new methodology which allows thedesign of suitable algorithms for different kinds of sys-tems. In section II we review the generic algorithm de-veloped in Ref.[13] in which metadynamics is performedusing the simulation box as order parameter. The use ofthe method will be illustrated with examples of zeoliteand benzene crystals. In section III A we discuss a differ-ent variant of the method (Ref.[14]) suitable for systemswhere an internal order parameter has to be used insteadof the simulation box. This case is illustrated by the ex-ample of graphite-to-diamond conversion. Finally, in thelast part we draw some conclusions and suggest possibledirections for further development.

II. METADYNAMICS USING A SIMULATION

BOX AS ORDER PARAMETER

Since the Parrinello-Rahman method represents ageneric constant-pressure MD simulation method, theproblems related to its application to structural transi-tions originate in the lack of efficiency of standard MDin crossing high barriers. During a standard equilibriumsimulation the system explores only a small part of itsfree energy surface, corresponding to thermal fluctua-tions around a locally stable minimum. Consequently,a spontaneous passage to another minimum separatedby a barrier substantially larger than the thermal energykBT is extremely unlikely on the time scale of a typical

2

MD simulation, which is of the order of ns in the case ofclassical force fields and ps in the case of ab-initio meth-ods.

A new general approach for escaping free-energy min-ima and systematic exploration of free-energy surfaceshas recently been developed by Laio and Parrinello[12].It has been called metadynamics and is capable of dra-matically speeding up the simulation of activated pro-cesses involving barrier crossing[15, 16, 17, 18], includingfirst-order phase transitions. The general algorithm hasbeen adapted by Martonak, Laio and Parrinello [13] forsimulating structural transitions in crystals. We notethat this method represents a conceptual extension ofthe idea of constant-pressure simulation. Instead of thelatter, we perform a search for new minima by exploringthe surface of the Gibbs potential. In order to reduce thecomplexity of the problem, an order parameter or col-lective variable is needed that allows us to discriminatebetween different crystalline structures. A natural onefor a crystal is the structure factor S(k) which providesa unique fingerprint of the spatial arrangement of theatoms in a periodic lattice. The structure factor, how-ever, is not a convenient order parameter to use for ourpurpose, because it has high dimensionality (in principleinfinite). Instead, we follow in this point the approach ofParrinello and Rahman and use as order parameter di-rectly the three supercell edges a,b, c, arranged as a 3×3matrix h = (a,b, c). For relatively small systems, wherethe creation of defects is too costly, the box matrix h islikely to be simply related to the unit cell u via relationh = um, where m is an integer matrix. The matrix h

can therefore distinguish between different unit cells andcrystal structures. Out of the 9 independent degrees offreedom of the matrix h only 6 determine the shape ofthe box, while the remaining 3 are related to the globalrotation of the box. The latter is irrelevant and only com-plicates the analysis of the results. Therefore it is conve-nient to freeze it, thus reducing the number of degrees offreedom to 6. In Ref.[13] we followed the idea of Nose andKlein [19] and used a symmetric matrix h. An equallygood choice is to constrain h to an upper triangular formand here we follow this way. Since our aim is to simulatea phase transition at a pressure P and a temperature Twe will explore the Gibbs potential G(h) = F(h)+PV asa function of h where F(h) is the Helmholtz free energyof the system at fixed box and V = det(h) is the volumeof the box. The six non-zero components of h can beconveniently arranged as h = (h11, h22, h33, h12, h13, h23)and represent a 6-dimensional order parameter. This dis-tinguishes the different local minima of G correspondingat pressure P to stable or metastable crystal structures.

The metadynamics requires calculation of derivativesof the free energy with respect to the order parameter.In our case such a derivative has a simple form

−∂G

∂hij= V

[

h−1(p − P )]

ji, (1)

where p is the internal pressure tensor, which can be

easily evaluated in MD or Monte Carlo simulations atconstant h from the averaged microscopic virial tensor[20].

The order parameter is now evolved according to asteepest-descent-like discrete evolution with a steppingparameter δh (metadynamics)

ht+1 = ht + δhφt

|φt|. (2)

Here, the driving force φt = −∂Gt

∂his derived from a

history-dependent Gibbs potential Gt where a Gaussianhas been added to G(h) at every point ht′ already visitedin order to discourage it from being visited again. Hencewe have

Gt(h) = G(h) +∑

t′<t

We−|h−h

t′ |2

2δh2 (3)

and the force φt is thus a sum of a thermodynamicaldriving force F = −∂G

∂hand the term Fg coming from a

potential constructed as a superposition of Gaussians. Astime proceeds the history-dependent term in Eq.(3) fillsthe initial well of the free-energy surface and the systemis driven out of the local minimum.

The metadynamics algorithm described above can beimplemented as follows (Fig.1). We start from an equi-librated box h containing the initial structure at a givenpressure P and temperature T and evaluate the pressuretensor p in a constant h MD run long enough to allow re-laxation to equilibrium and sufficient averaging of p. Thebox h is then updated using the forces (1) and metady-

namics equations (2,3) to a new value h′

. After the boxis modified the particle positions are rescaled in orderto fit into the new box using the relation ~r

′

= h′

h−1~r.In the case of molecular crystals we only scale the cen-ters of mass of the molecules but not the intramolecu-lar degrees of freedom. As the initial free energy well isgradually filled the box undergoes a sequence of progres-sively larger deformations until a transition occurs andthe system enters into the basin of attraction of a newstate. This can be detected by monitoring the structurefactor S(k) and is often apparent also on a visual inspec-tion of the atomic configuration. At this stage one canswitch off the Gaussian term, so that the metadynamicsbecomes purely steepest-descent-like and drives the sys-tem towards the equilibrium state for the new structure.In this equilibrium state the pressure will be equal to P .Once the new structure is characterized one can switchthe Gaussians on again, thus filling the new minimum,and move to other minima, if available.

The important point for the success of the method isthe judicious choice of the parameters W and δh. Inprinciple, these depend on the G(h) landscape. The pa-rameter δh determines the resolution in h and should besmaller than the typical size of the well. A simple wayto estimate the order of magnitude of δh is to performa short Parrinello-Rahman simulation and compute the

3

FIG. 1: Flowchart diagram of the metadynamics algorithmfor seeking new crystal structures.

fluctuations of the box h. However, we note that a verysmall value of δh is not likely to be useful. Since thevolume of the 6-dimensional Gaussian is proportional tothe sixth power of its size, a small δh means that manyGaussians would be needed to fill the well, resulting in along run. In order to achieve the necessary energy reso-lution, W should be chosen as a fraction of the relevantenergy barriers. These are, however, not known at thebeginning, since one does not know the exact mechanismof the transition to the new phase. A practical guidelinefor the choice of W and δh can be based on the require-ment that the Gaussians should not be too “sharp”, orin other words, for an optimal filling the curvature of theGaussians should be smaller than that of the well (seeRef.[12] for a more general discussion of the choice of Wand δh). This leads to the condition W

δh2 ≤ K where K isthe smallest eigenvalue of the G(h) Hessian at the min-imum h0. For a cubic system we can estimate K fromthe approximate expansion of G(h) around h0

G(∆h) ≈ G(h0) +1

2V c(

∆h

L)2 (4)

where L is the cell edge and c is of the order of magnitude

of the elastic constants. This provides an estimate K ≈Lc and a condition

W

δh2 . Lc ; (5)

we stress here that the right-hand side of the inequalitycontains the box size L and therefore the choice of theparameters W and δh is dependent on the size of thesimulation box. In the examples that follow the criterion(5) was found to provide a good working procedure.

We now discuss several aspects of the algorithm. Firstwe note an important difference with respect to the gen-eral method [12]. In principle, metadynamics may al-low the free energy profile to be recovered. As shown inRef.[12], the sum of the Gaussians in Eq.(3) convergesto −G(h) up to an additive constant, provided W andδh are properly chosen, the system is confined to a finiteregion of the order parameter space and several crossingsbetween the minima occur. Such information would, ofcourse, be very valuable, since it would allow an accu-rate determination of the transition pressure. The lat-ter is, in general, difficult to calculate once the anhar-monic effects start to play a role and one cannot ap-ply the quasiharmonic or self-consistent quasiharmonicapproximation. The recovering of the free energy, how-ever, implicitly assumes (similarly to thermodynamic in-tegration) that the system always evolves in a reversibleway, and the free energy never decreases discontinuously.This is generally not guaranteed in the case of solid-solidtransitions when box h is used as an order parameter.In this case the metadynamics should be effectively re-garded as an “engine” which by inducing a progressivelylarger deformation of the initial structure guarantees atsome point the transition to a new one. Whether such atransition proceeds in a reversible or irreversible way, aswell as the particular mechanism involved, is clearly verymuch system dependent. It can proceed via the creationof defects, such as grain boundaries or stacking faults,or in a collective way, as when a mechanical instabilityis reached by increasing the pressure in the Parrinello-Rahman method. There is, however, a substantial differ-ence related to the symmetry. In the constant-pressureParrinello-Rahman simulation the internal stress tensoris on average equal to a prescribed value, which is usu-ally chosen as the hydrostatic pressure P . Such compres-sion preserves the initial symmetry of the system and un-less the system is close to an instability point, the sym-metry is only slightly broken by instantaneous thermalfluctuations. Metadynamics, however, is not a constant-pressure simulation but rather a method aimed at ex-ploring the free-energy surface. Starting from an initialminimum, the exploration consists of applying varioussymmetry-breaking deformations of the initial structure,which clearly facilitates reaching a structure with a dif-ferent symmetry. The internal stress tensor during theexploration may become substantially anisotropic and itsfluctuations around the prescribed (hydrostatic) value Pare much stronger than the thermal ones. For this rea-son, the role of P does not appear to be critical. As

4

long as a given structure is at least metastable at pres-sure P , it can in principle be found by our algorithm;as discussed at the beginning of the previous section, thephases are usually metastable in a broad pressure intervalaround the equilibrium transition pressure Pc. However,the value of P affects the number of steps necessary toreach the transition. If its value is too low compared toPc, a lot of Gaussians may be needed to fill the initialwell before a transition is observed.

A second remark should be made concerning the pos-sibility that the system might change box h without re-ally changing the structure. This usually proceeds viaplane sliding and is a manifestation of the fact that onestructure can be described by many equivalent choicesof box h, known as modular invariance[21]. This phe-nomenon cannot be avoided completely as it is related tothe very nature of the order parameter h. Its frequentoccurrence, however, together with a failure of the algo-rithm to find a new structure, might be a manifestationof either too wrong a pressure or the fact that box h isnot a good order parameter for a given system (see sec-tion III). According to our experience the choice of a nottoo small value of δh, inducing substantial volume fluc-tuations, helps to suppress the transitions to the samestructure, as these obviously conserve the volume. Toolarge a value might, however, cause a transition to anamorphous structure; a suitable compromise has to befound by trial and error.

Our original application to an atomic system, siliconcrystal, is described in the paper [13]; the method workedvery well in that case. To assess its applicability to awider class of systems we also performed a case studyof two other crystals of a rather different kind. Theseare presented in the following subsections. Zeolite is anexample of a crystal having an extended network whilebenzene represents an organic molecular crystal.

A. Application to zeolite

As our first example we provide here a summary ofthe main results obtained by the application of the abovemethod to reconstructive transitions on a complex frame-work structure like a zeolite[22].

Zeolites are tectosilicates, with a structure formed bycorner sharing SiO4 or AlO4 tetrahedra and characterizedby having cages or channels able to accommodate alkalineor earth-alkaline cations and small molecules (generallywater) [23, 24]. We have focused on the Li-ABW ze-olite (Li[AlSiO4]·H2O), first synthesized by Barrer andWhite in 1951 [25]. The framework structure has anorthorhombic symmetry [26] and is formed by directlyconnected hexagonal rings sheets in the bc plane. Wa-ter molecules and Li cations are located inside monodi-mensional 8 membered channels developing along the caxis. If the temperature is raised in dry environment, theLi-ABW zeolite undergoes two phase transitions. A firstdisplacive transition occurs to the framework when dehy-

dration starts, leading to a structure known as anhydrousLi-ABW. At a temperature around 650C a second recon-structive transition drives the system to a new structure,the γ-eucryptite. This transition is accompanied by asymmetry change, leading to a monoclinic crystal. Theframework, formed by 8- and 4-membered rings of tetra-hedral units in the anhydrous Li-ABW, transforms in toa 6-membered rings structure. Because of the high sta-bility of the Al-O and Si-O bonds a Parrinello-Rahmanapproach is not able to reproduce the reconstructive tran-sition; in order to provide enough thermal energy to ob-serve a spontaneous breaking of some bonds, the temper-ature has to be increased so much that only a collapse ofthe structure is observed.

The simulation of the anhydrous Li-ABW → γ-eucryptite transition has been carried out within themetadynamics approach employing a classical potential,using the form and the parameters proposed in literatureby Zirl and Garofalini [27]. Details about the structureof the potential and its adaptation for our purposes canbe found in Ref.[22].

The metadynamics run was performed at externalpressure P = 0 for a system consisting of 56 atoms, us-ing the parameters for the Gaussian term equal to W= 9.94kcal/mol and δh = 0.4 A. The total simulationtime for each NVT run was 10 ps with an equilibrationof 2.5 ps. During the first 67 steps of metadynamicsthe deformation of the simulation cell does not lead tobond breaking, but only to displacive rearrangement ofthe framework. After step 67 a reconstructive event isrecognized by a sudden drop of the strongest diffractionpeaks of the original structure (see Fig.2). The transitionfrom anhydrous Li-ABW to γ-eucryptite is completed inthree metadynamics steps and it involves only processesof bond switching between some Al-O bonds (Fig.3). Inparticular, half of the Al atoms of the simulation cellmove in a concerted way, leading to four bond switch,such that the Li-ABW tetrahedral units are broken. Af-ter few picoseconds, a second bond switch is observedfor each of the Al atoms already involved in the firststep, leading to the formation of new tetrahedral unitswith the topology of the eucryptite. While the Al atomsinvolved in the bond breaking move significantly, explor-ing three main positions, the other atoms at the centerof the unbroken tetrahedral units move very little fromtheir starting crystallographic positions, in spite of thefact that each tetrahedron undergoes a rigid rotation inorder to complete the transition. The Li atoms also par-ticipate in the transformation with a displacement fromtheir starting position near to the Al involved in the bondswitch. The transition pathway observed with metady-namics is in good agreement with the experimental obser-vations reported in Ref. [26]. A more detailed descriptionof the calculations and the transition path are given inRef. [22].

The above results demonstrate the ability of metady-namics to uncover the detailed microscopic mechanismof a phase transition in a complex crystal structure. An-

5

8

10

12ce

ll le

ngth

s (Å

)

abc

80

90

100

cell

angl

es (

deg)

αβγ

50 55 60 65 70 75 80metasteps

0

500

1000

1500

|S(h

,k,l)

|2

(620)(620)(400)(004)

FIG. 2: Zeolite: time evolution of cell lengths, angles andselected strong peaks of the structure factor during the meta-dynamics simulation.

other attractive feature of the procedure we outlined inthe previous section is that it allows a systematic ex-ploration of the different polymorphs in which a givensystem can exist. Starting from an initial structure, sev-eral metadynamics runs can be performed at differentconditions of pressure and temperature until a transitionis observed. Any classical potential providing a quali-tative description for the system can be employed, andperforming the metadynamics with an accurate model(e.g. DFT) might be a waste of computer time. In fact,the Gibbs free energy difference between the candidatestructures found by metadynamics can be computed withthe high level method once the structures are known ata relatively small computational price.

For the system under investigation, we performed sev-eral metadynamics runs starting from the Li-ABW struc-ture in the pressure range between 10 and 100 kbar, ob-taining seven new structures characterized by a differ-ent connectivity of the atoms. To illustrate the excel-lent capability of the technique to find new polymorphswe report here a brief description of two of the struc-tures we find (these results will be described in detail ina separate publication [28]). The phase pictured in Fig.4(a) is formed by ordered sheets, balanced by the pres-ence of Li cations in the intra-layers space. Each layer isformed by alternating Si-centered and Al-centered tetra-hedra that form four-membered rings. The structurereported in Fig.4 (b) has the same connectivity of γ-eucryptite, but in this case there is no perfect alterna-tion of Si-centered and Al-centered tetrahedra, whereasthere are some Al-O-Al and Si-O-Si bridges. The Al-O-Al bridge is normally not found in zeolitic frameworks(Lowenstein’s rule[29]) because it shows larger stabilityfor angles around 180 which are generally not present inzeolites, but in conditions of high external pressure suchconfiguration can be stabilized. For these two phases andthe other five new structures found by the metadynamicsprocedure we determined the ab-initio equation of state

FIG. 3: Zeolite: three main steps involved in the transitionpath. The blue tetrahedra are the Al-centered ones, while thegrey tetrahedra are those that are Si-centered. The tetrahe-dra directly involved in the bond breaking are pictured in ball-and-stick representation, where the red spheres represent theoxygen atoms. The Li cations inside the cavities are omittedfor clarity. (a) The anhydrous Li-ABW structure (8- and 4-membered rings), (b) an intermediate structure, formed dur-ing the first bond switching, (c) the final γ-eucryptite struc-ture (6-membered rings).

and the phonon spectrum in order to estimate the en-tropic contribution to the free energy, thus obtaining aphase diagram for the system at the DFT level and with-out any a priori knowledge of the possible phases of thesystem[28].

6

FIG. 4: Zeolite: two phases obtained by performing metady-namics at high external pressure. More details will be pro-vided in a forthcoming publication[28].

B. Application to benzene

The possibility of predicting all of the stable crys-talline structures and thus the physical and chemicalproperties for a given compound is extremely impor-tant for pharmaceuticals[30, 31] and benzene is a goodcase study to prove the applicability of our techniqueto organic molecules. Benzene, though being a sim-ple molecule has a quite complex phase diagram. Sincethe late 60’s many authors studied solid benzene and inthe literature two different notations for its crystallinephases can be found. In this paper we follow the no-tation used by Thiery and Leger in Ref. [32]. At roomtemperature and ambient pressure benzene crystallizesin an orthorhombic structure, Pbca, which is stable upto 1.4 GPa [32]. Beyond this pressure a sluggish transi-tion to benzene II is observed [32, 33, 34]. Determiningthe crystalline structure of this phase was a big challengeand only 10 years after its first observation theoreticalcalculations could determine that it is orthorhombic andbelongs to the P43212 space group [35]. Benzene II isstable up to 4 GPa and then it transforms into benzeneIII, which is monoclinic and belongs to the P21/c spacegroup [32, 33, 34]. Upon increasing pressure two morephases are observed: benzene III’, which is stable be-tween 11 and 24 GPa, and benzene IV, which is stableat even higher pressure[32, 34]. These latter two struc-tures are not well characterized and there is still somedebate on whether they are real phases or not. In partic-ular, benzene III’ is supposed to be only a modification ofbenzene III[32] and benzene IV to be polymorphic [33].Beyond these five, another phase has been hypothesized:benzene I’, which should be stable at low temperatures(∼100 K) [32] and normal pressure.

In this work we focus our attention only on benzeneI, II and III. These are indeed the only three phases forwhich the structure is well understood both by experi-ment and by theory. In a different publication we tacklethe problem of determining all stable structures of ben-zene [36] and show that the method can indeed identifythe structure of all experimentally proposed phases.

All our MD simulations were done using the GRO-MOS96 force field[37]. The molecule is fully flexible andthe long range electrostatic interaction is calculated withthe Particle Mesh Ewald (PME) summation. We notethat the intermolecular potential can reproduce the sta-bility of all the three phases, though not their correcttransition pressures. At 0 K, benzene II is less stablethan benzene III and, at variance with the experiments,the common tangent construction suggested a transitionpressure of about 1.5 GPa for both the I to II and I toIII transformation[32, 33].

We apply metadynamics to study the transitionsamong those phases. During the metadynamics both theequilibration and the averaging of the internal stress ten-sor are performed by NVT runs of 1 ps, the externalpressure P is always fixed at 2 GPa and the temperatureat 300 K. For what concerns the parameters entering thetime dependent potential, we tried a wide range of val-ues, however, good results are obtained only when theGaussian width, δh, is chosen between 2 and 3 A andthe Gaussian height, W, between 120 and 600 kcal/mol.Given the elastic constants of benzene of about 6 GPaand the box vectors varying between about 10 and 50 Awe notice that the working parameters fulfill the empiri-cal guideline given in section II (Eq. 5). We experiencedthat the correct choice of δh is extremely important. Infact, if δh < 2 A during metadynamics run we only ob-serve a useless plane sliding, which is indeed a much lessexpensive deformation than a genuine solid–solid phasetransition. For molecular solid this is even more easythan in covalent crystals because of the weak nature ofthe intermolecular interactions. Moreover, if δh > 3 Athe deformation of the cell was too large to be accom-modated by the molecules and amorphization is alwaysobserved. On the other hand, choosing a too small valuefor W causes a slow escaping from the energy basin re-sulting in a waste of time.

The structure factor provides a unique fingerprint ofeach crystalline structure and can be compared with theexperimental x-ray powder diffraction pattern. Charac-teristic reflection peaks of a given structure often vanishfor other crystalline structures and by monitoring theintensity of the strongest reflections during the metady-namics loop we can easily detect when a phase transitionoccurs. For a typical metadynamics run, where the phaseI transforms into phase III, we show in Figure 5 the be-havior of the box lengths and angles (top panels) as wellas the intensity of a selected peak of the structure factor,relative to the strongest reflection.

We tried tens of starting configurations with either ofthe three phases and with different dimensions of thesimulation cell, containing from 4 to 256 molecules. Weobserved that by increasing the cell dimension the num-ber of metasteps necessary to escape from the energybasin decreases. At the same time the formation of de-fects becomes easier and in the cells larger than abouthundred molecules we sometimes observe stacking faults.We stress, however, that we never observe “point” de-

7

2.5

3.0

3.5

4.0La

ttice

Vec

tor

(nm

)I III

abc

80

90

100

110

Ang

le (

deg)

αβγ

0 10 20 30

Metastep

0.0

0.2

0.4

0.6

0.8

1.0

I[4 3

4] /

I 0

FIG. 5: Benzene: behavior of the box lengths and angles ina typical metadynamics run with 192 molecules in the sim-ulation cell (top and middle panels). In the bottom panelwe report the intensity of the (434) reflection relative to thestrongest peak observed in that metastep. The sudden dropof the intensity clearly indicates the occurrence of a phasetransition at the metastep 22.

fects, such as vacancies or misoriented molecules. In par-ticular, we could never observe a clean transition betweenbenzene I and II if the simulation cell was larger than 32molecules. With larger cells benzene I always transformseither to benzene III, which is lower in energy than ben-zene II, or to a defective structure. In Figure 6 we showthe plane stacking along the c axis for benzene II and IIIand also for a typical defective configuration obtained bymetadynamics.

For comparison, we performed also a limited amountof Parrinello-Rahman simulations, starting from phaseI. We observed in this case only a transition to phaseIII, but not to phase II. Moreover, no phase transition isobserved at pressures lower than 10 GPa. The results ofmetadynamics simulations clearly demonstrate that thenew method works very well also for molecular crystals.

III. METADYNAMICS USING AN INTERNAL

ORDER PARAMETER

The new method described in section II still suf-fers from some limitations in common with the originalParrinello-Rahman method. For instance it is less effec-tive for the study of phase transitions for which the pri-mary order parameter is an internal coordinate instead ofthe cell edges. This is the case for phase transformationsunder pressure described in terms of solid state chemicalreactions such as the 2D [38] and 3D [39] polymerizationsof C60 or the topochemical solid-state polymerizations ofalkenes, alkynes and aromatic hydrocarbons [8, 40]. Forinstance, in the 2D polymerization of C60 the activationbarrier for the [2+2] cycloaddition reaction is overcomeby a suitable deformation of the fullerenic cage which isnot induced by simply decreasing the intercage distancesdown to the density of the 2D polymer [41]. In the per-spective to address the study of phase transformations inthis class of materials, we have extended the metadynam-ics scheme recently devised by Iannuzzi, Laio and Par-rinello (ILP) [42] to constant-pressure MD simulations[14]. The ILP scheme can be dubbed reactive moleculardynamics since suitably defined reaction coordinates areintroduced as dynamical variables.

By combining the ideas behind the Parrinello-Rahmanand ILP methods we have introduced a constant-pressurereactive MD described by a Lagrangian of the form

L =1

2

N∑

i=1

mi(stih

thsi) − E(si,h) +1

2WcTrhth

− pΩ +∑

α

1

2Mαη2

α −∑

α

1

2kα

(

ηα(si,h) − ηα

)2

− V (t, ηα), (6)

where the first line is the Parrinello-Rahman Lagrangian[1] and the second line is the ILP Lagrangian [42]. Here,si are scaled ionic coordinates, Ω is the cell volume, pthe external pressure and ηα are collective variables as inthe ILP scheme [42] with a fictitious kinetic energy andmass (Mα). A harmonic potential restrains the values ofthe collective coordinates ηα(si,h) close to the corre-sponding dynamical collective variables ηα. The valuesof Mα and kα control the time scale of the evolution ofthe collective variables and are chosen according to theprescription given in Ref. [42]. The collective coordi-nates ηα(si,h) are functions of the scaled ionic coordi-nates and of the cell edges and should be able to discrim-inate between the initial and final phases. E(si,h) isthe total internal energy while V (ηα, t) is the history-dependent potential acting on the collective variables andgiven by

V (t, ηα) =∑

t′<t

W∏

α

e−

|ηα−ηt′α |2

2σ2α , (7)

where W and σα are suitably chosen parameters as de-

8

a b a ba b

Benzene IIIBenzene II

c a b cd da b

Metadynamics

a a*b a’b d bd a’a*

FIG. 6: Benzene: comparison of the molecular arrangement of phases II and III with that observed in a typical defectivestructure obtained during metadynamics. Note that benzene II has a typical a b c d plane stacking while benzene III has an a

b stacking. The defective structure can be viewed either as a crystal of benzene II with two stacking faults (along the planesmarked by a star) or as benzene III with two stacking faults (in the planes marked by a prime).

scribed in Ref. [12]. The equations of motion correspond-ing to the Lagrangian (6) are reported in Ref. [14].

A. Application to graphite

To demonstrate the validity of the new scheme de-scribed above we have simulated the conversion of car-bon from graphite to diamond under pressure [14]. Thistransformation can be seen as driven by an internal orderparameter such as the corrugation of the graphitic planeswhich leads to the change of hybridization of carbon fromsp2 to sp3. As a measure of the hybridization type of thecarbon atoms, we have defined as collective variable thecoordination number of the atoms of a single graphiticplane in the simulation cell with respect to the atoms ofthe two neighboring planes, i.e.

η =∑

i∈plane

∑

j /∈plane

cij , (8)

with

cij =1 − (

rij

d )6

1 − (rij

d )12, (9)

where rij is the distance between atoms i and j and

d = 2.2 A. The graphite to diamond conversion has beenalready reproduced in the ab-initio Parrinello-Rahmanmolecular dynamics simulations of Ref. [9], although ata pressure of 90 GPa, six times larger than the experi-mental estimate (15 GPa, [9]), due to the aforementionedlimitations of the Parrinello-Rahman method. In thepresent work graphite is described by the tight-binding(TB) potential of Ref. [43] supplemented by an empiri-cal two-body van der Waals (vdW) interaction, necessaryto describe the interplanar distance in graphite [14] [44].

This model Hamiltonian describes with good accuracythe lattice parameters and compressibility of graphiteand diamond at equilibrium, but drastically overesti-mates the compressibility of graphite at high pressure.As a consequence the theoretical transition pressure todiamond is as high as 129 GPa and the volume jump atthe transition pressure is very small, the volume being4.70 A3/atom for graphite and 4.60 A3/atom for dia-mond (cfr. the ab-initio EOS of graphite and diamondof Ref. [9]). Nevertheless this model graphite representsa good testing case for the new simulation scheme. Infact, the small volume jump at the theoretical transitionpressure prevents the reduction of the activation barrierby overpressurization and consequently the transition todiamond does not take place in a Parrinello-Rahman sim-ulation (70 ps long) even by increasing the pressure upto 700 GPa and temperature up to 1000 K. Conversely,within the new simulation scheme the transformation todiamond occurs very close to the theoretical transitiondensity. We have performed a metadynamics simulationof graphite at 15 GPa and 300 K with a supercell contain-ing 128 atoms initially arranged in the graphite structurewith four graphitic planes per cell in the ABAB (hexago-nal) stacking [9]. During the simulation run, 28 ps long,we have observed several (of the order of 15) forward andbackward transitions between graphite and diamond. Aclear monitoring of the phase transition is given by thetime evolution of the indicator χ which provides a sharpdistinction between the hybridization states of carbonatoms as

χ =1

N

∑

i

1

ni

∑

j>k

nijnikcos3θjik (10)

where

ni =∑

j 6=i

nij , (11)

9

nij =1

1 + e(rij−d)/∆, (12)

with d = 1.8 A and ∆ = 0.05 A. The index i runs overall the N atoms of the simulation cell and ni is the co-ordination number of atom i-th. The index k and j runover atoms neighboring to atom i-th and θjik is the an-gle subtended by the tern jik whose contribution to χis weighted by the product of the partial coordinationnumber nik and nij . The cosine function in (10) is ableto discriminate between the hybridization sp2 and sp3.In fact the indicator χ has a value of -0.125 for graphiteand -0.06 for diamond at 15 GPa and 300 K. For the sakeof clarity the time evolution of χ is reported in Fig. 7only for the first part of the run. Two transitions fromgraphite to diamond are clearly identified at 10.5 ps andat 14.0 ps. Similarly, other twelve transformations can beidentified in the other 12 ps of simulation. The Gaussianpotential fills first the free-energy basin corresponding tographite and, once the system is driven to the new phase,the basin corresponding to the final structure is then pro-gressively filled. Once both basins are filled, the systemis able to oscillate from one structure to the other. Inagreement with the ab-initio Parrinello-Rahman simula-tion of Ref.[9], some of the structures observed duringthe oscillations from graphite to diamond are a mixtureof cubic and hexagonal diamond. The transformation at10.5 ps (14.0 ps) starts at a volume of 4.45 A3/atom (4.70A3/atom), very close to the theoretical transition volumeof 4.70 A3/atom obtained from the theoretical equationof state of graphite and diamond. These results demon-strate the effectiveness of the metadynamics scheme pre-sented above. The phase transition occurs spontaneouslyat the theoretical transition density whereas it does nottake place in a Parrinello-Rahman simulation (within thetight-binding model) even by overpressurizing the systemup to five times the theoretical transition pressure. Thismetadynamics scheme would be particularly suitable andprobably superior to the one based on the h matrix (aspresented in section II) for the simulation of phase transi-tions described in terms of solid state chemical reactions.

IV. CONCLUSIONS AND OUTLOOK

The methodology based on exploration of the free en-ergy surface results in a substantial improvement of simu-lation of structural transitions in crystals. Using severalinorganic and organic crystals as examples we demon-strated its general applicability as well as heuristic valuein various cases where the plain constant-pressure sim-ulations fail. The main advantages that emerge can be

summarized as follows. Starting from an initial structure,the new approach is able to find a number of realisticpolymorphs, as demonstrated by the examples of zeoliteand benzene crystals. For the search for new structures,a classical force-field can be used resulting in a simulationcost comparable to that of a standard classical MD sim-ulation. The mechanism of transitions between different

7 8 9 10 11 12 13 14 15t [ps]

-0.12

-0.1

-0.08

-0.06

χFIG. 7: Graphite: time evolution of the indicator χ (seeEq. (10) in the text) which discriminates between the struc-tures of graphite and diamond. Only the first part of thesimulation is reported for sake of clarity. Along the wholerun 28 ps long, we have seen 15 jumps of χ similar to thosereported in the figure which equally correspond to oscillationsbetween graphite and diamond.

phases can be understood at conditions close to those ofexperiment, as shown by the example of graphite.

Clearly, there still remains a lot of room for improve-ment, including the algorithm itself. As we see it, themajor difficulty in the discrete version of the algorithmdescribed in section II is the choice of parameters W andδh. One of possible alternatives is to use also in this casea continuous version of metadynamics with an adaptivemechanism for the choice of Gaussian parameters (includ-ing anisotropic Gaussians), based on monitoring the cellfluctuations and estimating the shape of the well. Thework in this direction is in progress. Also, an accuratecalculation of a transition pressure at finite temperaturestill remains a challenge. The ultimate goal in this fieldshould be a crystal structure prediction from “scratch”,based only on the knowledge of the chemical compositionof the system. We believe that methods described in thispaper represent a substantial step forward towards thisgoal.

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