Time-propagation algorithms in molecular dynamics: Difference between revisions

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   <li><math>\mathbf{v}_{i}(t + \frac{1}{2}\Delta t)=\mathbf{v}_{i}(t)+\frac{\mathbf{F}_{i}(t)}{2m_{i}}\Delta t</math></li>
   <li><math>\mathbf{v}_{i}(t + \frac{1}{2}\Delta t)=\mathbf{v}_{i}(t)+\frac{\mathbf{F}_{i}(t)}{2m_{i}}\Delta t</math></li>
   <li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
   <li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
   <li>compute forces from density functional theory or machine learning</li>
   <li>compute forces <math> \mathbf{r}_{i}(t)</math> from density functional theory or machine learning</li>
   <li><math>\mathbf{v}_{i}(t + \Delta t)=\mathbf{v}_{i}(t+\frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t+\frac{1}{2}\Delta t)}{2m_{i}}\Delta t</math></li>
   <li><math>\mathbf{v}_{i}(t + \Delta t)=\mathbf{v}_{i}(t+\frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t+\frac{1}{2}\Delta t)}{2m_{i}}\Delta t</math></li>
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=== Leap-Frog Integration scheme ===
=== 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:
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:
 
<ol>
  <li> compute forces <math> \mathbf{r}_{i}(t)</math> from density functional theory or machine learning </li>
  <li><math>\mathbf{v}_{i}(t + \frac{1}{2}\Delta t)=\mathbf{v}_{i}(t-+ \frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t)}{m_{i}}\Delta t</math></li>
  <li><math>\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t</math></li>
</ol>


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Revision as of 19:06, 16 October 2024

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:

  1. compute forces from density functional theory or machine learning
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