# Introduction to Atomistic Modeling and Computation

Introduction to Atomistic Modeling and Computation Weiqing Ren

1. Introduction to molecular dynamics (MD):

• Equation of motion - Newton’s equation in Cartesian coordinate (an example of Hamiltonian dynamics);

• Force field: empirical force field for modeling bio-molecules (bonded and nonbonded interactions), empirical forces for coarse-grained particles, hydrodynamics interaction;

• Computing averages from trajectories, canonical and microcanonical ensembles.

2. Geometric properties of Hamiltonian dynamics:

• Conserved quantities or first integrals; • Flow map of Hamiltonian dynamics - symplectic map.

3. Numerical method - Geometric integrators:

• Construction of geometric integrators by Hamiltonian splitting; Verlet algorithm;

• Modified equation of geometric integrators and backward error analysis ( to understand for example why geometric integrators “nearly” conserve the total energy).

4. Numerical method - Implementation issues:

• Boundary conditions;

• Efficient computation of short-ranged forces (e.g. Lennard-Jones interaction) using cell or Verlet list;

• Ewald sum for long-ranged forces (e.g. Coulombic force, hydrodynamics interactions).

5. Simulation methods in canonical ensemble:

• Iso-kinetic MD, Anderson thermostat;

• Langevin dynamics (stochastic method by adding friction and noise to Newton’s equation);

• Noss-Hoover dynamics (deterministic dynamics of an extended system).

6. Constrained molecular dynamics: • Disparity of time scales in molecular dynamics;

• Imposing holonomic constraints (e.g. rigid bond or bond angle); SHAKE and RATTLE algorithms;

• Computing constrained averages.

References: 1. Understanding molecular simulation, by Frenkel and Smit 2. Computer simulation of liquids, by Allen and Tildesley 3. Simulating Hamiltonian dynamics, by Leimkuhler and Reich