Potential truncation

In molecular dynamics and molecular mechanics calculations, the calculation costs are often dominated by non-bonded long-range interactions. The number of interactions that must be calculated scales with N² where N is the number of atoms. A common approach is to cut off the interaction at a fixed value Rout. However, when this is done there is a discontinuity in the energy and in the force. Since the forces are required to determine the trajectories of the atoms potential truncation can lead to large errors.

For a Lennard-Jones fluid the potential is V(R) = 4[1/R¹² - 1/R⁶]. A plot of this function is shown below

We depict abrupt potential trunction at R = 1.8 below.

To prevent discontinuities in the energy and force a switching function is often used. A switching function is cubic or higher order spline that goes smoothly from 1 to 0 over a range R_in to R_out. The switching function

S(R) = (R_out² - R²)²(R_out² + 2R² - 3R_in²)/( R_out² - R_in²)³

S(R) = 0, R > R_out

is shown below for R_in = 1.8 and R_out = 2.0.

The form of the potential using this type of switching function is shown below

The differences between this functions look important when we scrutinize them, however, we should also look at the big picture shown below. We plot the three potential functions over the important range of interaction between Lennard-Jones atoms and molecules.

These differences would be even smaller if we had made our cut-off distance 2.5 instead of 2.0. As discussed in Allen and Tildesley, 2.5 is a more common cut-off value. For Ar this corresponds to a distance of 2.5 x 3.4 Å = 8.5 Å.

For simple atomic fluids such as argon, neon etc. an analytical long range correction can be calculated.