How to Convert SPME to P3M: Influence Functions and Error Estimates

We demonstrate explicitly how the two seemingly different particle mesh Ewald methods, the smooth particle mesh Ewald (SPME) and the particle-particle particle mesh (P3M), can be mathematically transformed into each other. This allows us in particular to convert the error estimate of the P3M method in the energy-conserving scheme (also known as "P3M with analytic differentiation") into an error estimate for the SPME method, via a simple change of the lattice Green function. Our error estimate is valid for any values of the SPME parameters (mesh size, spline interpolation order, Ewald splitting parameter, real-space cutoff distance), including odd orders of splines. The problem with the self-forces is avoided thanks to an analytical formula that allows to subtract them directly within the particle mesh calculation. Plots of the accuracy of the SPME forces are provided for a wide range of parameter values. The main use of the error estimate is to allow a simulation program to scan quickly the multidimensional parameter space to find the best set of parameters to achieve a target accuracy at the smallest computational cost. As a byproduct, we show how a SPME code can be transformed into a P3M version by changing a few lines of code. We demonstrate also that the P3M lattice Green function can be approximated by a closed form expression, computable on-the-fly, that provides essentially the same accuracy as the full function.