Lecture 3: methods andterminology, part I

Molecular dynamics, internal coordinates, Z-matrices, force fields, periodic boundary conditions, Hartree-Fock theory, basis sets, etc.

Eritis sicut Deus, scientesbonum et malum.

Dr Ilya Kuprov, University of Southampton, 2012

(for all lecture notes and video records see http://spindynamics.org)

Classical molecular dynamics

Classical MD is not a quantum mechanical method. It amounts to solving New-ton’s equations for a collection of balls on springs and bendy surfaces:

Software: AMBER, GROMACS, HyperChem, Chem3D

dv drF F ma a v p mvdt dt

dp F p tdtdr p r tdt m

force

gradient potential acceleration velocity momentum

The record of time dependence of allcoordinates and momenta in the system iscalled the system trajectory.

Advantages Drawbacks

Very fast (millions of atoms) Bond breaking and formation cannot be described.

Can predict statistical properties (density, melting point, vapour pressure, etc.)

Only mechanical, statistical, thermodynamic and some electrostatic properties can be computed.

Easy to understand and interpret. Ignores all quantum mechanical effects.

Classical molecular dynamicsClassical MD is used for systems and properties that do not exhibit stronglyquantum mechanical behaviour: large proteins, cellular membranes, DNA etc.

See AMBER and GROMACS manuals for more information.

Properties that are calculated using mole-cular dynamics simulations are:1. Rotational and internal correlation

functions for large biomolecules.2. Biomolecular geometries from NMR

distance and angle restraints.3. Diffusion constants.4. Binding constants and energies for

simple non-covalent interactions.5. Mechanisms and pathways for simple

conformational transitions.6. Ensemble thermodynamic properties

(enthalpy, entropy, Gibbs energy, etc.)7. Energy minimum geometries, crystal

structures and phase diagrams.Crystal structure of bacterial Photosystem I.

Molecular dynamics force fieldsThe term force field, in the molecular dynamics context, refers to the equationgiving the forces on all atoms in the system as a function of internal coordinates.

2 2b 0 a 0 d

bonds angles

1

dihedral

2 61

0 0,

1 1

s

0

1 1 1( ) ( ) [1 cos( )]2 2 2

2 , 4

N Nij ij i j

i jj i j ij ij ij

k l l k k n

Fr r q qr r r

bonds angles dihedrals

short-rangerepulsion

Van der Waalsinteraction

Coulombinteraction

The parameters of force fields are extracted from accurate quantum mechanicalsimulations of a standard set of small molecules. They are assumed to betransferable to similar structural motifs in larger molecules.A more expensive method is to extract forces from a separate DFT simulation.

See AMBER and GROMACS manuals for more information.

Internal coordinates and Z-matricesIt is often convenient todescribe molecular geometryin terms of bond lengths, bondangles, dihedral angles, etc.,rather than Cartesian coordi-nates of atoms.Internal coordinates are acoordinate system chosen tomatch the structure of themolecule in some way.Z-matrix is a standard set ofinternal coordinates, specify-ing the molecular geometry interms of bond lengths, bondangles and dihedral angles.Coordinate optimizations andtransition state searches oftenconverge considerably faster ininternal coordinates.

User-friendly translation between Cartesians and Z-matrices is done by GaussView.

Cartesian coordinates of methane:C 0.000000 0.000000 0.000000H 0.000000 0.000000 1.089000H 1.026719 0.000000 -0.363000H -0.513360 -0.889165 -0.363000H -0.513360 0.889165 -0.363000

Z-matrix of methane:CH 1 1.089000 H 1 1.089000 2 109.4710 H 1 1.089000 2 109.4710 3 120.0000 H 1 1.089000 2 109.4710 3 -120.0000

bonds angles dihedrals

Redundant internal coordinatesZ-matrix coordinates encoun-ter problems in linear, planarand polycyclic systems, wherethey are inferior to Cartesians.A well-chosen coordinate sys-tem should also minimize notjust the linear and quadratic,but also higher order couplingsbetween coordinates.Redundant internal coordinatesattempt to solve both problemsby starting with a redundantset (e.g. all bonds, all anglesand all dihedrals) and usingSVD to separate out the bestindependent linear combinati-ons. The result is nearly alwayssuperior to Cartesians, andalso allows freezing and scan-ning arbitrary coordinates.

P. Pulay, G. Fogarasi, J. Chem. Phys. 96(4):2856-2860, 1992.

Periodic boundary conditions

Periodic boundary systems are conveniently set up in HyperChem.

Effects of the sample boundary (container walls orvacuum) are often undesirable in bulk propertysimulations. One way of getting around thisproblem is to make atoms and interactions pro-pagate cyclically across the container boundaries –this is known as periodic boundary condition.

The container must be large enough to preventparts of the system from interacting with their ownimages – at least 10 Å of padding is recommended.

A special case is slab boundary conditions, wherethe system is periodic in two dimensions.

The total charge of the system must be zero toavoid summing to an infinite charge. Angularmomentum is not conserved in PBC systems.

The cell geometry need not be rectangular – anytiling of 2D or 3D space (e.g. with truncatedoctahedra) may be used.

Hartree-Fock theoryHartree-Fock theory gives approximate solutions to the following quantummechanical problem:

Software: Gaussian, GAMESS, HyperChem, etc.

electronmomenta

electron-nuclear

attraction

2

21 1 12 2

N N N Nk

k N Nk N N j kN Nk NN jk

Z Z Z r E rm r r r

nuclearmomenta

inter-nuclear

repulsion

inter-electronrepulsion energy

wavefunction

Solving this for a single electron in anaverage electric field of all other electronsis relatively easy, and HF method proce-eds by treating each electron in turn in acycle (see the schematic on the right).

GUESS

Solve forelectron 1

Solve forelectron 2

Solve forelectron 3

Solve forelectron N

The algorithm updates the wavefunctionsfor each electron until the solution no lon-ger changes. This procedure is known asthe self-consistent field method.

Hartree-Fock theoryDrawbacks

Ignores two-electron correlation effects (moving electrons avoid each other in ways that the average

field theory does not properly capture).

Relativistic effects are usually ignored completely.

Nuclei are assumed to be static point charges.

A single configuration ground state is assumed.

Born-Oppenheimer approximation: theelectrons are assumed to react instan-taneously to any rearrangement of thenuclei. As a result, nuclear degrees offreedom can be formally eliminated fromthe equation in the previous slide.

One

of t

he H

artr

ee-F

ock

orbi

tals

in a

fulle

rene

der

ivat

ive.

Slater determinants: a basic constraint on fer-mionic systems requires that wavefunctionschange sign if the labels of any two particles arepermuted. For one-electron orbitals, the simplestway of doing it is to put them into a determinant:

1 1 2 1 1

1 2 2 2 21 2

1 2

( ) ( ) ( )( ) ( ) ( )1( , , , )

!( ) ( ) ( )

N

NN

N N N N

r r rr r r

r r rN

r r r

A. Szabo, N.S. Ostlund, Modern Quantum Chemistry, Chapter 3.

Basis setsA basis set is a way of converting integrodifferential equations into matrixequations, which are much easier to solve on digital computers.

212 k

kU r r E r

k kk

r c r

coefficientswavefunction

basisfunctions

Hamiltonian

21 2k k k k k nk k n

k k k k

c U r r E c r H c Ec

The equation for the wavefunction is replaced by a system of linear equations forthe expansion coefficients:

can be pre-computed simple linearequations

The task of computing multi-dimensional integrals involving basis functions isvery formidable, but...

A. Szabo, N.S. Ostlund, Modern Quantum Chemistry, Chapter 3.

Property Benefits

A product of two Gaussian functions is another Gaussian at their centre of gravity.

All two-electron integrals are analytical. Massive time saving compared to numerical integration.

Gaussians fall off very steeply with distance.

Distance based cut-offs further improve perfor-mance.

Property Drawbacks

Gaussians fall off very steeply with distance.

Incorrect asymptotic behaviour at large distances.

Gaussians have zero derivative at the origin.

Incorrect asymptotic behaviour at small distances.

Gaussians forming the basis are centred at the nuclei.

Basis set superposition error.

Gaussian basis sets

Atom-centred spherical coordinates areusually used in atomic basis sets:

2 2

, , ,, , , nr bn l m l mG r Y e

Linear combinations of Gaussians(called contractions) are often used inthe radial part:

2 2

, , ,, , , nir bn l m l m ni

i

G r Y a e

sphericalharmonic

Gaussian

contractioncoefficients

Gaussian basis sets are used by Gaussian03, GAMESS and similar packages.

Property Explanation

Core functions Basis functions intended to provide the description of the core electrons of an atom. Core electrons (e.g. 1s for carbon) do not participate in chemical bonding.

Valence functions Basis functions intended to provide the description of the valence electrons of an atom that are involved in chemical bonding. Difference to core is often notional.

Zeta number Number of independent sets of radial basis functions in the valence part of the basis (usually 2-4). They are introduced to improve basis flexibility.

Polarization functions Additional functions with a higher angular momentum number, introduced to allow atomic orbitals to be shifted (“polarized”) relative to their nucleus.

Diffuse functions Additional functions with very broad profile, introduced to improve description of the regions that are far away from the nuclei, usually used for anions.

Tight functions Additional functions concentrated tightly around their nuclei to improve the description of the nuclear region for NMR and ESR interaction calculations.

Effective core potentials Effective Coulomb interaction potentials designed to replace inner orbitals of heavy atoms that are not well described by non-relativistic methods.

Rydberg functions Additional extremely diffuse basis functions required for accurate description of Rydberg states.

Properties of atomic basis sets

A. Szabo, N.S. Ostlund, Modern Quantum Chemistry, Section 3.6.

Pople’s basis sets:

What to look for:

Zeta number Polarization functions

Diffusefunctions Tight functions Effective core

potentialsRydberg functions

The more, the better/slower, usually 2-3.

The more, the better/slower,Usually one or two l+1 sets.

The more, the better/slower,

usually a single set for anions.

Only used forcomputing

HFCs and J-couplings.

Used to approximate

inner shells of heavy elements

Only used to describe

Rydberg states

Correlation-consistent basis sets:

6-311G++(3df,3pd)

core Gaussians

valence Gaussians(for each zeta)

aug-cc-pVTZ

diffuseGaussians

polarization functions

“correlation consistent polarized valence triple-zeta basis set augmented with diffuse functions”

Specialized basis sets exist for almost any specificpurpose. Check the literature for any case-specific basissets before using a general purpose one.

Gaussian basis sets

https://bse.pnl.gov/bse/portal (a collection of popular basis sets)

Bigger and more flexible basis sets provide more accurate matrix representations forthe original differential equations and therefore yield more accurate results.

Property Benefits

Plane waves are complex exponentials.

All two-electron integrals can be computed using fast Fourier transforms.

Plane waves are periodic.Plane-wave basis sets are well adapted for periodicboundary conditions.

Plane waves do not decay.Uniform description without the basis set superposition error.

Plane waves are orthogonal.

Self-consistent field convergence is guaranteedin all situations.

Plane waves are parameterized by energy. Simple accuracy criteria.

Property Drawbacks

Plane waves are smooth and non-local.

Wavefunction cusps near nuclei are difficult.

Plane-wave basis sets

Plane wave basis set use crystal latticecoordinates:

iG rG r e

In a given periodic box, only specificfrequencies are allowed in specificspatial directions:

reciprocal lattice vectors

See CASTEP documentation for further information.

Variational principleApplied to quantum chemical calculations, the variational principle states that in agiven class of functions, the one with the lowest expectation value for the energy isthe best approximation to the ground state of a quantum system.

Basis set Energy / Hartree

STO-3G -0.49491

STO-4G -0.49848

STO-5G -0.49951

STO-6G -0.49983

In practice, the variational princi-ple guarantees that a bigger basisset would yield a better approxi-mation, though the law of dimini-shing returns certainly applies.

Because of error compensation effects, thevariational principle does not generallyapply to properties other than energy, andeven to energy differences, as the picture onthe left demonstrates.For property calculations therefore, a biggerbasis set does not necessarily mean a moreaccurate answer, but certainly means alonger processing time.

P.W. Atkins, R.S. Friedman, Molecular Quantum Mechanics, Chapter 6.