MSc in High Performance ComputingMSc in High Performance ComputingComputational Chemistry ModuleComputational Chemistry Module

Introduction to Molecular DynamicsIntroduction to Molecular Dynamics

Bill SmithComputational Science and Engineering

STFC Daresbury LaboratoryWarrington WA4 4AD

● MD is the solution of the classical equations of motion for atoms and molecules to obtain the time evolution of the system.

● Applied to many-particle systems - a general analytical solution not possible. Must resort to numerical methods and computers

● Classical mechanics only - fully fledged many-particle time-dependent quantum method not yet available

● Maxwell-Boltzmann averaging process for thermodynamic properties (time averaging).

What is Molecular Dynamics?What is Molecular Dynamics?

Example: Simulation of ArgonExample: Simulation of Argon

rrcutcut

612

4)(rr

rV

Pair Potential:Pair Potential:

Lagrangian:Lagrangian:

L r v m v V ri i i ii

N

ijj ii

N

( , ) ( )

1

22

1

Lennard -Jones PotentialLennard -Jones Potential

612

4)(rr

rV

V(r)V(r)

rr

rrcutcut

Equations of MotionEquations of Motion

d

dt

L

v

L

ri i

N

ijiji

iii

fF

Fam

Lagrange

ijijijij

ijiij rrrr

rVf

612

2224)(

Newton

Lennard-Jones

Periodic Boundary ConditionsPeriodic Boundary Conditions

Minimum Image ConventionMinimum Image Convention

iijjj’j’

rrcutcut

LL

rrcutcut < L/2 < L/2

Use rUse rij’ij’ not r not rijij

xxij ij = x= xij ij - L* Nint(x- L* Nint(xijij/L)/L)

Nint(a)=nearest integer to aNint(a)=nearest integer to a

Integration Algorithms: Essential IdeaIntegration Algorithms: Essential Idea

r (t)

r (t+t)v (t)t

f(t)t2/m

Net displacement

r’ (t+t)

[r (t), v(t), f(t)] [r (t+t), v(t+t), f(t+t)]

Time step t chosen to balance efficiencyand accuracy of energy conservation

Integration Algorithms (i)Integration Algorithms (i)

r r rtm

F t

vtr r t

in

in

in

iin

in

in

in

1 12

4

1 1 2

2

12

( )

( ) ( )

Verlet algorithm

Integration Algorithms (ii)Integration Algorithms (ii)

)(

)(

42/11

32/12/1

tvtrr

tFm

tvv

ni

ni

ni

ni

i

ni

ni

Leapfrog Verlet Algorithm

Integration AlgorithmsIntegration Algorithms

)()(2

)(2

211

42

1

tFFm

tvv

tFm

tvtrr

ni

ni

i

ni

ni

ni

i

ni

ni

ni

Velocity VerletAlgorithm

12/11

2/11

2/1

2

2

ni

i

ni

ni

ni

ni

ni

ni

i

ni

ni

Fm

tvv

vtrr

Fm

tvv

As Applied

Verlet Algorithm: DerivationVerlet Algorithm: Derivation

)4()(2/)()(

:or

)()(2)()(

:(1) from (2)Subtract

)3()(/)()()(2)(

:or

)()()(2)()(

:(2) and (1) Add

2)()()()()()(

1)()()()()()(

:expansions sTaylor'

2

3

42

42

43612

21

43612

21

tOtttrttrv(t)

tOttrttrttr

tOmttfttrtrttr

tOttrtrttrttr

tOttrttrttrtrttr

tOttrttrttrtrttr

Key Stages in MD SimulationKey Stages in MD Simulation

●Set up initial systemSet up initial system●Calculate atomic forcesCalculate atomic forces●Calculate atomic motionCalculate atomic motion●Calculate physical propertiesCalculate physical properties●Repeat !Repeat !●Produce final summaryProduce final summary

InitialiseInitialise

ForcesForces

MotionMotion

PropertiesProperties

SummariseSummarise

MD – Further CommentsMD – Further Comments

Constraints and ShakeIf certain motions are considered unimportant, constrained MD can be more efficient e.g. SHAKE algorithm - bond length constraintsRigid bodies can be used e.g. Eulers methods and quaternion algorithms

Statistical MechanicsThe prime purpose of MD is to sample the phase space of the statistical mechanics ensemble.Most physical properties are obtained as averages of some sort.Structural properties obtained from spatial correlation functions e.g. radial distribution function.Time dependent properties (transport coefficients) obtained via temporal correlation functions e.g. velocity autocorrelation function.

System Properties: Static (1)System Properties: Static (1)

● Thermodynamic Properties– Kinetic Energy:

– Temperature:

K E m vi ii

N

. . 1

22

TNk

K EB

2

3. .

System Properties: Static (2)System Properties: Static (2)

– Configuration Energy:

– Pressure:

– Specific Heat

U V rc ijj i

N

i

( )

1

13

1 N

i

N

ijijijB frTNkPV

( ) ( )U Nk TNk

Cc NVE BB

v

2 2 23

21

3

2

System Properties: Static (3)System Properties: Static (3)

● Structural Properties– Pair correlation (Radial Distribution Function):

– Structure factor:

Note: S(k) available from x-ray diffraction

g rn r

r r

V

Nr rij

j i

N

i

( )( )

( )

4 2 2

drrrgkr

krkS 2

01)(

)sin(41)(

Radial Distribution FunctionRadial Distribution Function

RR

RR

g(r)g(r)

separation (r)separation (r)

1.01.0

Typical RDFTypical RDF

Free Energies?Free Energies?

● All above calculable by molecular dynamics or Monte Carlo simulation. But NOT Free Energy:

where

is the Partition Function.But can calculate a free energy difference!

A V T k T Q V TB e N( , ) log ( , )

Q V TN h

H r p dr dpN NN N N N( , )

!exp ( , )

13

● The bulk of these are in the form of Correlation Functions :

2

0

)0()()(

or

)()(1

)(

av

T

avav

fftftC

dffftfT

tC

System Properties: Dynamic (1)System Properties: Dynamic (1)

System Properties: Dynamic (2)System Properties: Dynamic (2)

● Mean squared displacement (Einstein relation)

● Velocity Autocorrelation (Green-Kubo relation)

2|)0()(|3

12 ii tDt rr

dttD ii

0

)0()(3

1vv

time (ps)time (ps)

<|r<|r

ii(t)-

r(t

)-r ii(0

)|(0

)|22 >

(A

>

(A22 ))

SolidSolid

LiquidLiquid

Typical MSDsTypical MSDs

1.01.0

<v<v ii(t

).v

(t).

v ii(0)>

(0)>

0.00.0t (ps)t (ps)

Typical VAFTypical VAF

Recommended TextbooksRecommended Textbooks

● The Art of Molecular Dynamics Simulation, D.C. Rapaport, Camb. Univ. Press (2004)

● Understanding Molecular Simulation, D. Frenkel and B. Smit, Academic Press (2002).

● Computer Simulation of Liquids, M.P. Allen and D.J. Tildesley, Oxford (1989).

● Theory of Simple Liquids, J.-P. Hansen and I.R. McDonald, Academic Press (1986).

● Classical Mechanics, H. Goldstein, Addison Wesley (1980)

The DL_POLY PackageThe DL_POLY Package

A General Purpose Molecular Dynamics Simulation Package

DL_POLY BackgroundDL_POLY Background

● General purpose parallel MD code to meet needs of CCP5 (academic collaboration)

● Authors W. Smith, T.R. Forester & I. Todorov● Over 3000 licences taken out since 1995● Available free of charge (under licence) to University

researchers.

DL_POLY VersionsDL_POLY Versions

● DL_POLY_2– Replicated Data, up to 30,000 atoms– Full force field and molecular description

● DL_POLY_3– Domain Decomposition, up to 10,000,000

atoms– Full force field but no rigid body description.

● I/O files cross-compatible (mostly)● DL_POLY_4

– New code under development– Dynamic load balancing

Supported Molecular EntitiesSupported Molecular Entities

Point ionsand atoms

Polarisableions (core+shell)

Flexiblemolecules

Rigidbonds

Rigidmolecules

Flexiblylinked rigidmolecules

Rigid bondlinked rigidmolecules

DL_POLY is for Distributed DL_POLY is for Distributed Parallel MachinesParallel Machines

M1 P1

M2 P2

M3 P3

M0 P0 M4P4

M5P5

M6P6

M7P7

DL_POLY: Target SimulationsDL_POLY: Target Simulations

● Atomic systems● Ionic systems● Polarisable ionics● Molecular liquids● Molecular ionics● Metals

● Biopolymers and macromolecules

● Membranes● Aqueous solutions● Synthetic polymers● Polymer electrolytes

DL_POLY Force FieldDL_POLY Force Field● Intermolecular forces

– All common van der Waals potentials– Finnis_Sinclair and EAM metal (many-body) potential

(Cu3Au)– Tersoff potential (2&3-body, local density sensitive, SiC)– 3-body angle forces (SiO2)– 4-body inversion forces (BO3)

● Intramolecular forces– bonds, angle, dihedrals, improper dihedrals, inversions– tethers, frozen particles

● Coulombic forces– Ewald* & SPME (3D), HK Ewald* (2D), Adiabatic shell model,

Neutral groups*, Bare Coulombic, Shifted Coulombic, Reaction field

● Externally applied field– Electric, magnetic and gravitational fields, continuous and

oscillating shear fields, containing sphere field, repulsive wall field

* Not in DL_POLY_3

Algorithms and EnsemblesAlgorithms and Ensembles

Algorithms● Verlet leapfrog● Velocity Verlet● RD-SHAKE● Euler-Quaternion*● No_Squish*● QSHAKE*● [Plus combinations]

* Not in DL_POLY_3

Ensembles● NVE● Berendsen NVT● Hoover NVT● Evans NVT● Berendsen NPT● Hoover NPT● Berendsen NT● Hoover NT● PMF

The DL_POLY Java GUIThe DL_POLY Java GUI

The DL_POLY WebsiteThe DL_POLY Website

http://www.ccp5.ac.uk/DL_POLY/

The End