# Particle mesh Ewald: recommended splitting into short and long range

Particle mesh Ewald method for acceleration of solving pairwise interaction by long range forces (electrostatic, gravitational ... ) seem to be very general and easy to implement. The basic principle is well described in many sources.

What is harder to find is how exactly split the potential kernel ( say $V(r)=1/r = V_{SR}(r) + V_{LR}$ ) into short range and long range part. In other words:

• How should I choose short range part $V_{SR}(r)$ so that the long range part $V_{LR} = V(r) - V_{SR}(r)$ is easy to express in Fourier space as some analytic function fast to evaluate numerically.
• It would be nice to use similar splitting in all dimension ( 1D, 2D, 3D ).

e.g. somewhere I read that often is used Yukawa-like potential $V_{SR} = \exp(-\beta r )/r$ or Gaussian like damping $V_{SR} = \exp(-\beta r^2 )/r$.

But how to evaluate Fourier transform of long range $V_{LR}(r)$ efficintly in such case? Or is there any better choice of $V_{SR}$.

## 1 Answer

Check out:

Nijboer, B. R. A., & De Wette, F. W. (1957). On the calculation of lattice sums. Physica, 23(1-5), 309–321. doi:10.1016/S0031-8914(57)92124-9

There they make the case for using a splitting based on the incomplete Gamma function. That splitting generalizes the choice of the error function to arbitrary dimensions.