A Sixth-order Image Approximation to the Ionic Solvent Induced Reaction Field

A recent article by Deng and Cai introduced fourth-order image approximations to the reaction field for a charge inside a dielectric sphere immersed in a solvent of low ionic strength. To represent such a reaction field, the image approximations employ a point charge at the classical Kelvin image point and two line charges that extend from the Kelvin image point along the radial direction to infinity. In this paper, a sixth-order image approximation is developed, using the same point charge with three different line charges. Procedures on how to discretize the line charges by point image charges and how to implement the resulting point image approximation in O(N) complexity for potential and force field calculations are included. Numerical results demonstrate the sixth-order convergence rate of the image approximation and the O(N) complexity of the fast implementation of the point image approximation.