Relationships between spherical and bispherical harmonics, and an electrostatic T-matrix for dimers

Abstract

Bispherical harmonics are the solutions to Laplace’s equation in bispherical coordinates. We investigate the relationships between spherical harmonics and bispherical harmonics in terms of radial inversion and derive new series expansions between the harmonics. The series coefficients are the Delannoy numbers encountered in combinatorics, and we exploit the series to derive for them a new second order homogeneous recurrence relation. We use the T-matrix/null field method to solve for the potential of two different spheres in an arbitrary static external electric field, as a series of bispherical harmonics where the coefficients are related via matrix transformations. The matrix elements are expressed as surface integrals of bispherical harmonics, for which analytic expressions are derived in terms of Legendre functions. The resonant values of the dielectric function are computed via an eigenvalue problem which is found to be more stable than the solution using difference equations. The rate of convergence of the matrix formulation is compared to the re-expansion method using spherical harmonics for a uniform field an a dipole in the gap; each series converges faster in a different region of space, which we discuss in terms of the boundaries of convergence of each solution and the image singularities of the scattered field.