Time-propagation algorithms in molecular dynamics

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In molecular dynamics simulations, the positions and velocities are monitored as functions of time . This time dependence is obtained by integrating Newton's equations of motion. To solve the equations of motion, various integration algorithms have been developed. The time dependence of a particle can be expressed in a Taylor expansion

A backward propagation in time by a time step can be obtained in a similar way

Adding these two equation gives and rearrangement gives the Verlet algorithm

The Verlet algorithm can be rearranged to the Velocity-Verlet algorithm by inserting

Velocity-Verlet Integration scheme

The Velocity-Verlet algorithm can be decomposed into the following steps:

  1. compute forces from density functional theory or machine learning

From these equations it can be seen that the velocity and the position vectors are synchronous in time.

Leap-Frog Integration scheme

Another form of the Verlet algorithm can be written in the form of the Leap-Frog algorithm. The Leap-Frog algorithm consists of the following steps:


MDALGO thermostat integration algorithm
0 Nose-Hoover Velocity-Verlet
1 Andersen Leap-Frog
2 Nose-Hoover Leap-Frog
3 Langevin Velocity-Verlet
4 NHC Leap-Frog
5 CSVR Leap-Frog