Electrostatics of charged dielectric spheres with application to biological systems. II. A formalism bypassing Wigner rotation matrices.

Abstract

Given the crucial role of electrostatic forces in biological systems, accurate and rapid calculations of electrostatic forces are imperative in faithfully simulating biological systems. More than a decade ago, we proposed a surface charge method, applied it to a system of an arbitrary number of charged dielectric spheres, and obtained an exact solution for arbitrary configuration of the spheres. The precision depends only on the number of terms kept in a series expansion, and can therefore be controlled at will. However, the numerical implementation can be significantly slowed down by the need to compute a large number of Wigner rotation matrix elements each time the electrostatic energy is calculated. In this paper, we provide the proof of a formula introduced in 1992 and apply it to arrive at a useful closed form for the electrostatic interaction energy without computing any Wigner rotation matrix elements, hence significantly improving the efficiency for numeric implementation of the rigorous surface charge method. This new Wigner-matrix-free formalism may also be used to speed up the computation of the electrostatic energy in the presence of permanent and induced multipoles, which can be very important for atomic modeling with polarization effect included.