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Molecular Dynamics Basic Idea • Solve Newton’s equations of motion mj d 2rj dt 2 Fj , V Fj , rj j 1,2, , N • Choose a force field (specified by a potential V) appropriate for the given system under study • Decide a statistical ensemble to use, choice of boundary conditions; collect statistics of observables Commonly Use Force Fields • Lennard-Jones Potential – For noble gas and generic fluids • Tersoff, Brenner, Stillinger-Weber, 3-, 4body potentials – For C, Si, Ge, … • AMBER, CHARMM, GROMOS, MM4, etc – For biomolecules • GULP, DFT codes, etc Example of potential used in biomolecular modeling 2 2 ki k i V (r1,r2 , ,rN ) li li ,0 i i ,0 bonds 2 angles 2 Vn 1 cos(n ) torsions 2 4 ij i j ij rij ij qi q j rij 4 0rij 12 6 Ensembles • Micro-canonical Ensemble – Energy is fixed • Canonical ensemble – Need to use “thermostat” to fix temperature • Langevin dynamics • Nosé-Hoover • Generalized Langevin Langevin Dynamics d 2rj 2 Fj dt m ξ(t ) 0, ξ(t )ξ(t ') T v j ξ, j 1,2, , N 2 kBT I (t t ') m How to correctly implement the white noise on computer? Nosé-Hoover Dynamics rj v j pj , m p j Fj p j , 1K 1 2 2 1 , K mv j K0 j 2 Generalized Langevin t uF r (t t ')u(t ')dt ' (t ) 0, (t ) (t ') i (t t ') † Σ is known as self-energy Observables, Statistics • Equilibrium temperature (in micro-canonical ensemble) by equipartition theorem. 1 1 kBT m j v 2j , , x, y, z 2 2 • Pressure of a fluid (for pair potential) 1 PV NkBT d (r r ) F i j i j ij Where d is dimension, Fij is the force acting on particle i from particle j. Transport Coefficients • The diffusion constant can be computed through velocity correlation function 0 v(t ) v(0) dt lim t r(t ) r(0) 2t 2 3D Transport Coefficients • Thermal conductivity can be computed through energy-current correlation using Green-Kubo formula; or nonequilibrium simulation by directly computing the energy current Textbooks on MD • M P Allen & D J Tildesley, “Computer Simulation of Liquids,” (Oxford, 1987) • D Frenkel & B Smit, “Understanding Molecular Simulation,” 2nd ed (Academic Press, 2002) • A R Leach, “Molecular Modeling, principles and applications” (Addison Wesley Longman, 1996) Tutorial Problem Set 12 • Prove the pressure formula (required a great deal of knowledge of statistical mechanics).