Time-propagation algorithms in molecular dynamics

In molecular dynamics simulations, the positions $\mathbf{r}_{i}(t)$ and velocities $\mathbf{v}_{i}(t)$ are monitored as functions of time $t$. 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

$\mathbf{r}_{i}(t+\Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t)\Delta t + \frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2} + \frac{\partial^{3} \mathbf{r}_{i}(t)}{\partial t^{3}}\Delta t^{3} + \mathcal{O}(\Delta t^{4})$

A backward propagation in time by a time step $\Delta t$ can be obtained in a similar way

$\mathbf{r}_{i}(t-\Delta t) = \mathbf{r}_{i}(t) - \mathbf{v}_{i}(t)\Delta t + \frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2} - \frac{\partial^{3} \mathbf{r}_{i}(t)}{\partial t^{3}}\Delta t^{3} + \mathcal{O}(\Delta t^{4})$

Adding these two equation gives and rearrangement gives the Verlet algorithm

$\mathbf{r}_{i}(t+\Delta t) = 2\mathbf{r}_{i}(t)-\mathbf{r}_{i}(t-\Delta t)+\frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2}+\mathcal{O}(\Delta t ^{3})$

The Verlet algorithm can be rearranged to the Velocity-Verlet algorithm by inserting $\mathbf{v}_{i}(t)=\frac{\mathbf{r}_{i}(t)-\mathbf{r}_{i}(t-\Delta t)}{\Delta t}$

$\mathbf{r}_{i}(t+\Delta t) = \mathbf{r}_{i}(t)+ \mathbf{v}_{i}(t)\Delta t+\frac{\mathbf{F}_{i}}{2m}(t)\Delta t^{2}$

Velocity-Verlet Integration scheme

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

1. $\mathbf{v}_{i}(t + \frac{1}{2}\Delta t)=\mathbf{v}_{i}(t)+\frac{\mathbf{F}_{i}(t)}{2m_{i}}\Delta t$
2. $\mathbf{r}_{i}(t + \Delta t) = \mathbf{r}_{i}(t) + \mathbf{v}_{i}(t + \frac{1}{2}\Delta t)\Delta t$
3. compute forces from density functional theory or machine learning
4. $\mathbf{v}_{i}(t + \Delta t)=\mathbf{v}_{i}(t+\frac{1}{2}\Delta t)+\frac{\mathbf{F}_{i}(t+\frac{1}{2}\Delta t)}{2m_{i}}\Delta t$

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