Three-layer dielectric models for generalized Coulomb potential calculation in ellipsoidal geometry

This paper concerns a basic electrostatic problem: how to calculate generalized Coulomb and self-polarization potentials in heterogeneous dielectricmedia. In particular, with simulations of ellipsoidal semiconductor quantum dots and elongated biomacromolecules being its target applications, this paper extends the so-called three-layer dielectric models for generalized Coulomb and self-polarization potential calculation from the spherical and the spheroidal geometries to the triaxial ellipsoidal geometry. Compared to the simple steplike dielectric model, these three-layer dielectric models can overcome the mathematical divergence in the self-polarization energy by employing continuous radial dielectric functions. More specifically, in this paper, the quasiharmonic three-layer dielectric model for the ellipsoidal geometry is discussed, and the explicit analytical series solutions of the corresponding electrostatic problem are obtained in terms of the ellipsoidal harmonics. Then a robust numerical procedure working for general three-layer dielectric models is developed. The key component of the numerical method is to subdivide the transition layer of the underlying three-layer model into multiple sublayers and then in each one of them approximate the select dielectric function of the transition layer by one of the quasiharmonic functional form rather than simply by a constant value as one would normally do. As a result, the numerical method has no numerical divergence.