Local molecular field theory for the treatment of electrostatics

TL;DR

This paper explores the theoretical foundations of local molecular field (LMF) theory for electrostatics, showing how it simplifies Coulomb interactions into a single, smoothed potential equation, and demonstrates its accuracy in confined water systems.

Contribution

It provides a detailed derivation of LMF theory from the Yvon-Born-Green hierarchy and extends its application to various confined charged systems with charge smoothing.

Findings

LMF theory accurately approximates electrostatics when interactions vary slowly.

The simplified Poisson equation with smoothed charge density is effective for Coulomb interactions.

Charge smoothing with an optimal sigma improves the accuracy of electrostatic calculations.

Abstract

We examine in detail the theoretical underpinnings of previous successful applications of local molecular field (LMF) theory to charged systems. LMF theory generally accounts for the averaged effects of long-ranged components of the intermolecular interactions by using an effective or restructured external field. The derivation starts from the exact Yvon-Born-Green hierarchy and shows that the approximation can be very accurate when the interactions averaged over are slowly varying at characteristic nearest-neighbor distances. Application of LMF theory to Coulomb interactions alone allows for great simplifications of the governing equations. LMF theory then reduces to a single equation for a restructured electrostatic potential that satisfies Poisson's equation defined with a smoothed charge density. Because of this charge smoothing by a Gaussian of width sigma, this equation may be solved more simply than the detailed simulation geometry might suggest. Proper choice of the smoothing length sigma plays a major role in ensuring the accuracy of this approximation. We examine the results of a basic confinement of water between corrugated wall and justify the simple LMF equation used in a previous publication. We further generalize these results to confinements that include fixed charges in order to demonstrate the broader impact of charge smoothing by sigma. The slowly-varying part of the restructured electrostatic potential will be more symmetric than the local details of confinements.