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Molecular Dynamics

Molecular Dynamics. Basic Idea. Solve Newton’s equations of motion 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.

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Molecular Dynamics

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  1. Molecular Dynamics

  2. Basic Idea • Solve Newton’s equations of motion • 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

  3. 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, DFT codes, etc

  4. Example of potential used in biomolecular modeling

  5. Ensembles • Micro-canonical Ensemble • Energy is fixed • Canonical ensemble • Need to use “thermostat” to fix temperature • Langevin dynamics • Nosé-Hoover • Generalized Langevin

  6. Langevin Dynamics How to correctly implement the white noise on computer?

  7. Nosé-Hoover Dynamics

  8. Generalized Langevin Σ is known as self-energy

  9. Observables, Statistics • Equilibrium temperature (in micro-canonical ensemble) by equipartition theorem. • Pressure of a fluid (for pair potential) Where d is dimension, Fij is the force acting on particle i from particle j.

  10. Transport Coefficients • The diffusion constant can be computed through velocity correlation function

  11. Transport Coefficients • Thermal conductivity can be computed through energy-current correlation using Green-Kubo formula; or nonequilibrium simulation by directly computing the energy current

  12. 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)

  13. Tutorial Problem Set 12 • Prove the pressure formula (required a great deal of knowledge of statistical mechanics).

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