# Harmonic surface mapping algorithm for electrostatic potentials in an atomistic/continuum hybrid model for electrolyte solutions

## TL;DR

This paper introduces a harmonic surface mapping algorithm that efficiently computes electrostatic potentials in multi-scale electrolyte models, significantly improving accuracy near dielectric interfaces.

## Contribution

The paper presents a novel harmonic surface mapping algorithm combining image charges and charge density discretization for fast electrostatic potential evaluation in hybrid models.

## Key findings

- Improved accuracy by two orders of magnitude over Kirkwood series expansion.
- Efficient evaluation of electrostatic reaction potentials in multi-scale electrolyte models.
- Demonstrated effectiveness using the fast multipole method for Coulomb summation.

## Abstract

Simulating charged many-body systems has been a computational demanding task due to the long-range nature of electrostatic interaction. For the multi-scale model of electrolytes which combines the strengths of atomistic/continuum electrolyte representations, a harmonic surface mapping algorithm is developed for fast and accurate evaluation of the electrostatic reaction potentials. Our method reformulates the reaction potential into a sum of image charges for the near-field, and a charge density on an auxiliary spherical surface for the far-field, which can be further discretized into point charges. Fast multipole method is used to accelerate the pairwise Coulomb summation. The accuracy and efficiency of our algorithm, as well as the choice of relevant numerical parameters are demonstrated in detail. As a concrete example, for charges close to the dielectric interface, our method can improve the accuracy by two orders of magnitudes compared to the Kirkwood series expansion method.