Stochastic Parameter Optimization for Empirical Molecular Potentials

function optimization

simulated annealing

tight binding parameters

Motivation

simulate dynamics of atomic structuresderive total energy and forces acting on atoms

empirical potentials + fit parameters to experimentsoft spheres: only distance dependent

quantum mechanics: electrons dominate bondingmillions of atoms: approximate electronic degree of freedom

semi-empirical: capture QM origin of bonding

tight binding: provides directional bonding

fit simulated properties to experimental onesmore approximations: more parameters to adjust

BOP4 potential : 11 parameters [material/compound]

automatic fit procedure providing one or more good parameter sets

Optimization

find optimal solution to given problem such as:

economy: shortest itinerary between number of cities (traveling salesman)

engineering:

drug design/ circuit design

quantify the problem‘goodness’ of solution depends on parameters objective function

set of parameters state in vector space

goal: find best local minimum on Potential Energy Surface (PES)

cost function :recover exp. properties, some better than others

find point in 11-D continous space

Deterministic Methods (downhill only)

1D Golden Section Search

higher dimensions:Steepest DescentConjugate GradientVariable Metricdownhill simplex (no derivative)

Monte Carlo

statistical physics: access ensemble averagesmagnetization of Ising model

higher energy states less probable

trick: don’t weigh all possible states , but only representative subset

simple sampling: waste time on states, that don’t contribute

importance sampling: arithmetic mean

?how to judge importance without

prior knowledge of energy reference?

Metropolis Algorithm

judge upon relative energy-difference to previous state guarantee detailed balance of hopping between states

Metropolis-function: transition probability

Metropolis et al. (1953) : find optimal wiring (min. length) on chip

allow for uphill climbing: move to neighboring local minima

Simulated Annealing

in analogy to anneal process of metals:slower cooling: better crystalization (energetically lower state)

faster cooling: freezing small crystals (higher, local minimum)

Kirkpatrick et al. (1983) added T-schedule to Metropolis search

search parameter space at successively lower temperature (higher ) :T controls:

scale on which parameters are randomly changed:

prob. at which costly uphill moves are accepted:

find global minimum on PES for

logarithmic annealing (single crystal)

in practice: simulated quenching

with exponential cooling scheme

propose new state

accept reject

update TopList

lower Tin intervals

Traveling Salesman

visit all cities: combinatorial problemminimize salesman’s way

different cost for crossing the river:

minimize salesman’s cost

equal weight:

smuggler:

river penalty:

Variations of the Theme: Statistic Tunneling (ST)

simulated quenching is prone to freezing

process is trapped in a deep local (but not global) minimum, that is surrounded by higher intermediate states

-or-

very good (perhaps global) minimum is surrounded by higher states (on mountain top) and might never be found

transform PES:

‘tunnel’ through forbidden, higher regions

preserve/amplify lower lying regions

effectively raising T in higher regions

Tight Binding (TB) Parameters

molecular wavefunction is linear

combination of atomic wf.

replace hopping integral with parameter

angular dependence was given by Slater and Koster (1954) and is fitted to band structures of periodic systems

dynamic modeling needs continuous distance dependence

heuristic shape guided by radial solutions such as:

choice of dist. dep. is the integral part of TB

total energy:

Radial Dependence

repulsive potential and bond integral scale with same functional form

separate scaling parameter for -bonds and repulsive potential following common cut-off parameter #of parameters for s-p-bonded system:3x2(scaling)+1(cutoff)+3(screening)+1(promotion)=11strong repulsion at andstrong attraction at equilibrium at

Fitting BOP4

cost-function: equilibrium values ofbulk modulus

rem. elastic constants

lattice parameter

cohesive energy

lattice parameter for graphitic and -tin phase

for diamond phase

T-dependent criterion: „distance in vector space“

distinguish btw truly different sets and slight variation from same local minimum

Summary

Simulated annealing invaluable to handle our multi-variable optimization

drawback: may run to forbidden areas in parameters space many times, since only TopList and two current states are stored (blind search)

genetic algorithm: interchange subset of parameters btw good parameterization, once annealing process is finished/frozen

general strategy: locate various minima with SA at high T

refine once with SA at lower T

use variable metric method to find „bottom“ of local minima