Verlet method

NAMD uses the velocity form of the Verlet (leapfrog) method for integration. Beginning at a timestep n and given the position, velocity, and force acting on each atom, X_n, V_n, and F_n, the following equations are used to obtain values for the next step.
$$\begin{align*}V_{n+\frac{1}{2}} &= V_n + {\frac{\Delta t}{2}} {M^{-1}}{F_n} \\ ... ...{n+1} &= V_{n+\frac{1}{2}} + {\frac{\Delta t}{2}} {M^{-1}}{F_{n+1}} \end{align*}$$
There are slightly different formulations of the Verlet algorithm that rely on positions X_n and X_n-1, but it is felt that such formulations suffer from a higher degree of round-off error than the velocity formulation.

Multiple time stepping can be used along with FMA (DPMTA) to make full electrostatics cheaper to compute. (Computing full electrostatics directly using the FullDirect option is meant only for very small systems or testing purposes.) Full electrostatic simulations are enabled by setting the FMA parameter on. The long range electrostatic forces are evaluated only at the cycle boundaries, that is to say, every stepspercycle steps. The parameter MTSAlgorithm determines the multiple time stepping method to be used:

• naive -- The long range force for each step is determined using constant extrapolation from the most recent cycle boundary step.
• verletI -- The long range force scaled by stepspercycle is added as an impulse only on cycle boundary steps.
• verletX -- The long range force for each step is determined using linear extrapolation from the previous two cycle boundary steps.

See [3] for more details about the Verlet I and Verlet X methods.

The long and short range electrostatic forces may be smoothly combined using a splitting function defined by the longSplitting parameter:

• sharp -- No smoothing is done. The forces are simply added together.
• xplor -- The X-PLOR switching function for the van der Waals potential is applied to the electrostatic potential. See section 5.1.4 for details.
• c1 -- Each electrostatic force interaction is split into
  $$\begin{gather*}\vec F_{\text{elec}}^{\text{short}} = S(\vert\vec r_{ij}\vert)\ve... ...}^{\text{long}} = [1-S(\vert\vec r_{ij}\vert)]\vec F_{\text{elec}} \end{gather*}$$
  by the continuously differentiable scalar function
  
  $$S(\vert\vec r_{ij}\vert) = \begin{cases} 1 & \text{if $\vert... ...\text{if $R_{\text{off}} < \vert\vec r_{ij}\vert$.} \end{cases}$$
  
  

We recommend the use of the c1 splitting [7].