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Bill Smith Computational Science and Engineering STFC Daresbury Laboratory Warrington WA4 4AD

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MSc in High Performance ComputingComputational Chemistry ModuleIntroduction to Molecular Dynamics

Bill Smith

Computational Science and Engineering

STFC Daresbury Laboratory

Warrington 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).

rcut

Pair Potential:

Lagrangian:

V(r)

s

r

e

rcut

Lagrange

Newton

Lennard-

Jones

Use rij’ not rij

L

xij = xij - L* Nint(xij/L)

rcut

j’

j

i

Nint(a)=nearest integer to a

rcut < L/2

r’ (t+Dt)

r (t+Dt)

v (t)Dt

Net displacement

r (t)

f(t)Dt2/m

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

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

Time step Dt chosento balance efficiency

and accuracy of energy conservation

Verlet algorithm

Leapfrog Verlet Algorithm

Velocity Verlet

Algorithm

As Applied

Initialise

Forces

Motion

Properties

Summarise

- Set up initial system
- Calculate atomic forces
- Calculate atomic motion
- Calculate physical properties
- Repeat !
- Produce final summary

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

- Statistical Mechanics
- The 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.

Thermodynamic Properties

Kinetic Energy:

Temperature:

Configuration Energy:

Pressure:

Specific Heat

Structural Properties

Pair correlation (Radial Distribution Function):

Structure factor:

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

R

R

g(r)

1.0

separation (r)

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!

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

Mean squared displacement (Einstein relation)

Velocity Autocorrelation (Green-Kubo relation)

Liquid

<|ri(t)-ri(0)|2> (A2)

Solid

time (ps)

<vi(t).vi(0)>

0.0

t (ps)

1.0

- 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 Package

A General Purpose Molecular Dynamics Simulation Package

- 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_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

Rigid

molecules

Point ions

and atoms

Flexibly

linked rigid

molecules

Polarisable

ions (core+

shell)

Rigid bond

linked rigid

molecules

Flexible

molecules

Rigid

bonds

M4

P4

M0

P0

M5

P5

M1

P1

M6

P6

M2

P2

M7

P7

M3

P3

Atomic systems

Ionic systems

Polarisable ionics

Molecular liquids

Molecular ionics

Metals

Biopolymers and macromolecules

Membranes

Aqueous solutions

Synthetic polymers

Polymer electrolytes

- 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

- Electric, magnetic and gravitational fields, continuous and oscillating shear fields, containing sphere field, repulsive wall field

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

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

The End