Precomputed grid potential (PGP) method: Exact reciprocal-space acceleration of particle mesh Ewald for Monte Carlo simulations

When conducting Monte Carlo (MC) simulations using a fixed or infrequently updated charge environment, each insertion or deletion of a particle requires an O(N log N) particle mesh Ewald (PME) long-range electrostatics calculation. To address this, we have developed Precomputed Grid Potential (PGP). This approach is a mathematically exact redesign of the smoothed PME (SPME) stationary-mobile reciprocal cross-term, which has an O(1) reciprocal-space cost in the stationary region size per insertion and achieves no loss of accuracy when compared to the discrete SPME calculation using the same mesh. After precomputing the static contribution of the stationary charges on a 3D grid employing a B-spline interpolation operator, particle insertions only require O(1) interpolations of the stationary potential and not a Fourier transform. The PGP and PME cross-terms are algebraically the same, and the only source of numerical variation between them is the position-dependent mesh self-artifact included in discrete PME. This is eliminated by construction in PGP's cross-term formulation. We have compared PGP to PME for water sampling in protein systems of up to 56K atoms, showing through grand canonical Monte Carlo (GCMC) simulations that, after independent calibration of the chemical potential, all distribution measurements, including mean occupancy and interaction energy, were statistically indistinguishable, while accept/reject ratios differed by no more than 0.23%. For these fixed-stationary region GCMC workloads, PGP reduces the per-trial reciprocal-space cost (4678× to 9857×) at an identical SPME Hamiltonian, mesh size, and order of B-spline.