Materials Modeling at the Molecular Level

Laura Frink

Sandia National Laboratories

Computational Materials Science Dept.

lfrink@cs.sandia.gov

Outline

lecture 1: An introduction to molecular modeling

lab 1: Getting to know MD and MC Codes

lecture 2: What to do with an MD or MC Code ????

lab 2: Computing materials properties

lecture 3: How to write efficient MC and MD codes

lab 3: Implementation of routines for improved efficiency and error estimation

Materials Modeling - Quantum Mechanics

Solve Schrodinger’s equation (eigenvalue problem):

Hartree-Fock (HF)
(Local) Density Functional Theory (LDA)

Assumptions:

Wavefunctions composed of basis functions (Gaussian or plane-wave)

Smoothly varying electron densities (LDA)

Compute:

only "input" is the charge per nuclei & number of associated electrons.

self-consistent iterations

ground-state electronic structure (charge density)

relaxed coordinates of atoms

ab-initio forces for molecular simulations

State of the Art:

LDA: Peter Schultz and Mark Sears (SNL,9225) Peter Feibelman, Ellen Stetchel, Dwight Jennisson (SNL,1100) - up to O(1000) atoms with near linear scaling.

Molecular Scale

Molecular Theory - approximate behavior of correlation functions in known limits

perturbation theory

integral equation theory

density functional theory (perturbation theory for inhomogeneous fluids)

linear response theory (correlation functions)

phonon theory (solids)

Molecular Simulation - attempt to sample configuration space directly and completely

Molecular Dynamics

Monte Carlo

Grand Canonical (Inhomogeneous Fluids)
Gibbs Ensemble (Liquid-Vapor Phase Equilibria)
Thermodynamic Integration MC (Solid Phases)
Gibbs-Duhem Integration (Coexistance lines)

Hybrid MC/MD

Grand Canonical Molecular Dynamics (Diffusion)
hybrid MC
force-bias MC
smart Monte Carlo

Mesoscale Modeling

lattice methods

typically static calculations
a special case of Monte Carlo Methods
sacrifice detail in potential model
computationally efficient
one ‘chunk’ = 100s of atoms

grain growth simulations (Liz Holm, SNL 1800)
self-assembly (complex phase diagrams), K. Dawson
protein folding (Bill Hart and Sorin Istrail, 9200)
lattice polymers
lattice Boltzman dynamics

brownian dynamics

mixtures of particles with very different length/time scales
protein solutions
replace small particle (solvent) with random forces

solvation potentials

calculate the effect of solvent explicitly with density functional theory
potential of mean force
molecular dynamics using the PMF

McMillian-Mayer Theory -- Laura Frink (9200)

A Little Physics

Hamiltonian :

Kinetic Energy:

Potential Energy:

(one-body) + (two-body) + (three-body) + ...

Total Partition Function:

The Kinetic Part (analytical solution):

The Configurational Integral:

Average Properties:

Goal of Molecular Modeling is to Predict <A> !!!

Computing the Configurational Integral

Conventional Quadrature

m = # of equidistant points along coordinate axis = 5

N = 100 particles D = 3-dimensions

O(m^DN = 10²¹⁰) evaluations of integrand !!

Random Sampling

Large number of random samples of f(x) will yield the average <f> and hence the integral

Very inefficient - often samples f(x) = 0.

The Metropolis Algorithm

Ensemble average property:

Probablility density for a certain configuration:

randomly generate points according to

is the number of points generated per unit volume around

Molecular Simulations

Monte Carlo:

Random sampling of configuration space

"ensemble" averaging

NVT is the natural ensemble (constant kinetic energy)

clever methods for overcoming energy barriers - can make large changes in a system quickly (e.g. solid diffusion)

Molecular Dynamics :

Continuous sampling of configuration space with the solution of Newton’s equations of motion

"time" averaging

like an experiment

NVE is the natural ensemble

better for collective motions

may be "trapped" in metastable states

The ergodicity condition: every possible configuration may

be reached by a given algorithm.

The ergodic hypothesis:

time averages = ensemble averages

MD Simulations = MC Simulations

Non-ergodic systems

In principle

glasses

meta-stable phases

harmonic solids

In practice:

Any system with problematical metastable states

Monte Carlo Simulations (the "Metropolis" scheme)

1. Set-up a system in a configuration with a nonzero Boltzmann factor exp[-U/kT].

2. Make a Trial Move

3. Compute the Potential energy of the new configuration

If : ACCEPT MOVE

If [

Compute ratio of probabilities of new and old configurations:

Compare ratio with uniform random number on [0,1]:

if R> random number --> ACCEPT MOVE

otherwise --> REJECT MOVE

]

5. Accumulate Averages (for independent configurations)

Molecular Dynamics

1. Set up initial configuration - positions and momenta

2. Compute forces

O(N²) operation -- calculate force on all particles

3. Displace particles according to Newton’s equations of motion

Verlet algorithm (also velocity Verlet and Leapfrog algorithms)

Higher order: Predictor-Corrector

4. Compute instantaneous properties and averages

5. Repeat 2-4 until averages converge

Evaluating Forces/Energies

non-bonded pairwise interactions +

Lennard-Jones potential

non-bonded Coulombic interactions +

direct calculation with Ewald summations

particle-mesh methods

bonded interactions(harmonic) +

bond-stretch

many-body interactions

bond-bending

torsion

Force / Energy calculation = ~90% of the computational effort

Some Details to Consider

Finite Size Effects

‘Practical’ Simulations 1000-100000 particles
small compared to 10²⁴
use periodic boundary conditions
different ensembles --> different results O(1/N) or O(NlogN)

Cost of an MC trial displacement

too small --> more configurations needed to generate an "independent" sample
too large --> more configurations will be rejected.
rule of thumb: acceptance rate of 20-50% optimizes efficiency

References and Resources

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

2. Understanding Molecular Simulation, D. Frenkel and B. Smit, Academic Press (1996).

3. Computer Simulation in Chemical Physics, ed. by M.P. Allen and D.J. Tildesley, NATO ASI Series, Kluwer Academic Publishers (1993).

4. Applied Statistical Mechanics, T.M. Reed and K.E. Gubbins, McGraw-Hill (1973).

5. Statistical Mechanics, D.A. McQuarrie, Harper Collins Pub., (1976).

Web Pages

Farid Abraham, IBM: www.almaden.ibm.com/vis/fracture/prl.html

www.tc.cornell.edu/~farid/fracture/100million

Steve Plimpton, SNL: www.cs.sandia.gov/~sjplimp/lc.html

Peter Lomdahl, LANL: bigrost.lanl.gov/MD/MD.html

Uzi Landman, GA Tech: www.gtri.gatech.edu/res_news/LUBE.html