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. When integrating the equations of motions it is important to use symplectic algorithms which conserve phase space volume. To solve the equations of motion under symplectic conditions, 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 $\mathbf{F}_{i}(t)$ 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:

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

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