Analytical Hessians for Ewald and particle mesh Ewald electrostatics

Abstract

The particle mesh Ewald (PME) method has become ubiquitous in the molecular simulation community due to its ability to deliver long range electrostatics accurately with ON ⁡log(N) complexity. Despite this widespread use, spanning more than two decades, second derivatives (Hessians) have not been available. In this work, we describe the theory and implementation of PME Hessians, which have applications in normal mode analysis, characterization of stationary points, phonon dispersion curve calculation, crystal structure prediction, and efficient geometry optimization. We outline an exact strategy that requires O(1) effort for each Hessian element; after discussing the excessive memory requirements of such an approach, we develop an accurate, efficient approximation that is far more tractable on commodity hardware.

FIG. 1.

The error in the minimax denominator decomposition for different numbers of decomposition terms, k.

FIG. 2.

The error in the harmonic vibrational frequencies for crambin, as a function of the number of terms used in the denominator decomposition, k.

FIG. 3.

The error in the minimax denominator decomposition using double- and single-precision arithmetic, for k = 10.

FIG. 4.

The sparsity structure of the Hessian elements for crambin, demonstrated by plotting ${\log }_{10}\left(|H/\left({\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}{\AA }^{-2}\right)|\right)$ for a range of systems (a) in the gas phase, (b) with periodic boundary conditions and a 16 Å cutoff, and (c) with PME.