A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums.

Christian J Burnham,Niall J English

doi:10.3390/ijms21010277

Christian J Burnham, Niall J English

Open Access

https://doi.org/10.3390/ijms21010277

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Abstract

We present a novel derivation of the multipole interaction (energies, forces and fields) in spherical harmonics, which results in an expression that is able to exactly reproduce the results of earlier Cartesian formulations. Our method follows the derivations of Smith (W. Smith, CCP5 Newsletter 1998, 46, 18.) and Lin (D. Lin, J. Chem. Phys. 2015, 143, 114115), who evaluate the Ewald sum for multipoles in Cartesian form, and then shows how the resulting expressions can be converted into spherical harmonics, where the conversion is performed by establishing a relation between an inner product on the space of symmetric traceless Cartesian tensors, and an inner product on the space of harmonic polynomials on the unit sphere. We also introduce a diagrammatic method for keeping track of the terms in the multipole interaction expression, such that the total electrostatic energy can be viewed as a ‘sum over diagrams’, and where the conversion to spherical harmonics is represented by ‘braiding’ subsets of Cartesian components together. For multipoles of maximum rank n, our algorithm is found to have scaling of vs. for our most optimised Cartesian implementation.

Highlights

Point-charge electrostatic models have been a mainstay of molecular simulation for years
Increasing in sophistication, many authors have implemented dipole interactions, such that the electrostatics on each nuclear site is modelled by the combination of a dipole and a charge, which requires that the simulation code be capable of calculating both charge–dipole and dipole–dipole interactions in addition to the usual charge–charge interactions required for point charge models
We have presented a non-technical—almost trivial—derivation of the multipole interaction in spherical harmonics, in a form suitable for use with Ewald-sum methods

Summary

Introduction

Point-charge electrostatic models have been a mainstay of molecular simulation for years. Increasing in sophistication, many authors have implemented dipole interactions, such that the electrostatics on each nuclear site is modelled by the combination of a dipole and a charge, which requires that the simulation code be capable of calculating both charge–dipole and dipole–dipole interactions in addition to the usual charge–charge interactions required for point charge models. This presented particular problems for treatment of long-range interactions, as typically handled by an Ewald sum, or related schemes, as such algorithms were initially developed for the case of point charge models, and needed to be modified in order to account for dipole interactions. The problem is faced—how to implement these terms in a simulation code, and how to modify an Ewald sum, or similar algorithm, such that it can properly handle the higher order multipoles interactions

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