Molecular Dynamics Simulation Techniques and Force Fields
Explore the world of molecular dynamics simulation, including the use of force fields and various ensembles like micro-canonical and canonical. Learn about solving Newton's equations of motion, choosing appropriate potentials, and implementing Langevin dynamics for accurate simulations.
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Presentation Transcript
Basic Idea Solve Newton s equations of motion 2 r d dt V r j = = = F F , , 1,2, , m j N j j j 2 j Choose a force field (specified by a potential V) appropriate for the given system under study Decide a statistical ensemble to use, choose boundary conditions; collect statistics of observables
Commonly Use Force Fields Lennard-Jones potential For noble gas and generic fluids Tersoff, Brenner, Stillinger-Weber, 3-, 4- body potentials For C, Si, Ge, AMBER, CHARMM, GROMOS, MM4, etc For biomolecules GULP, LAMMPS, DFT codes (QE), etc
Example of potential used in biomolecular modeling k k ( ) ( ) 2 2 = + ( , , r r r i i , ) V l l 1 2 ,0 ,0 N i i i i 2 2 bonds angles V ( ) + 1 cos( + n ) n 2 torsions 12 6 q q ij ij i j r + + 4 ij 4 r r i j 0 ij ij ij Machine learning based force fields
Ensembles Micro-Canonical Ensemble Total energy is fixed, standard Hamiltonian dynamics Canonical Ensemble Need to use thermostat to fix temperature Langevin dynamics Nos -Hoover Generalized Langevin Stochastic velocity rescaling
Langevin Dynamics 2 r F d dt j j = + = v , 1,2, , j N j j 2 m = ( ) t 0, 2 k T m = T B ( ) ( ') t ( ') t t t I How to correctly implement the white noise on computer?
Solving the Langevin equation F v + = + + v v v B ( ) ( ) t t h h m t + = + + r r ( ) ( ) t ( ) t h h h where h = B ( ) t dt 0 is a Gaussian random vector with each component 2 0, B B m k Th = = = 2 , , , x y z B
Nos-Hoover Dynamics r p d dt d dt d dt j j = = v , j m p j = F p , j j 1 1 2 K K = = 2 j v 1 , K m 2 j 0 = / 2 K N k T 0 f B
Stochastic Velocity Rescaling for Canonical Ensemble 1. Evolve with a time step by a symplectic algorithm (e.g. second order velocity Verlet) 2. Compute kinetic energy K, and evolve K according to ?+ 2 ??0 ????? ? = ? + ?0 ? 3. Rescale velocity ? ??, to enforce the target ?
Generalized Langevin t = + r ( ') ( ') t u t dt ' u F t = ( ) t 0, = ( ) t ( ') t ( ') i t t is known as self-energy
Observables, Statistics Equilibrium temperature (in micro-canonical ensemble) by the equipartition theorem. 1 1 2 2 = = 2 j , , , x y z k T m v , B j Pressure of a fluid (for pair potential) 1 d r = + r F ( ) PV Nk T B i j ij i j Where d is dimension, Fij is the force acting on particle i from particle j.
Transport Coefficients The diffusion constant can be computed through the velocity correlation function 2 r r ( ) t (0) = = v v ( ) t (0) lim t 3 dt D 2 t 0
Transport Coefficients Thermal conductivity can be computed through energy-current correlation using Green-Kubo formula; or nonequilibrium simulation by directly computing the energy current + 1 J V = = ( ) (0) J t J dt T 2 k T V B 0
Conductance of graphene strips Sites 0 to 7 are fixed left lead and sites 28 to 35 are fixed right lead. Heat bath is applied to sites 8 to 15 at temperature TL and site 20 to 27 at TR. Wang, Ni, & Jiang, PRB 2009.
Textbooks on MD M P Allen & D J Tildesley, Computer Simulation of Liquids, (Oxford, 2017) D Frenkel & B Smit, Understanding Molecular Simulation, 3rd (Academic Press, 2023) A R Leach, Molecular Modeling, principles and applications, (Pearson, 2001)
Tutorial Problem Set 12 Prove the pressure formula (required a great deal of knowledge of statistical mechanics). 1 d r = + r F ( ) PV Nk T B i j ij i j assuming pair-wise central force ???
