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

Abstract

This paper concerns a basic electrostatic problem: How to calculate generalized Coulomb and self-polarization potentials in heterogeneous dielectric media. In particular, with simulations of ellipsoidal semi-conductor quantum dots and elongated bio-macromolecules 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 tri-axial ellipsoidal geometry. Compared to the simple step-like dielectric model, these threelayer dielectric models can overcome the mathematical divergence in the self-polarization energy by employing continuous radial dielectric functions. More specifically, in this paper, the novel quasi-harmonic three-layer dielectric model for the ellipsoidal geometry is first 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 quasi-harmonic functional form rather than simply by a constant value as one would normally do. As the result, the numerical method has no numerical divergence.

Computational science and re-discovery: open-source implementation of ellipsoidal harmonics for problems in potential theory

“…or a similar form if ||r|| < ||r ′ || (see, e.g., 8 ). The normal derivative at the ellipsoid surface defined by λ = a is computed as 21…”

“…Our implementations, written in Python and MATLAB 6 , employ a variety of insights developed over more than a century of research 1,7,8 ; many important results were published during the mid-19th century and early 20th by greats like Heine 9 , and Charles Darwin's son Sir George Howard Darwin 10 . From the authors' perspective, these surprisingly direct ties to mathematical history offer an unusual and inspiring aspect to their study.…”

“…Unsurprisingly, the more general shape allows ellipsoidal-harmonic expansions to be more accurate than ones based on spherical harmonics 7,12,13 . Successful applications cover most of potential theory: gravity 7,10 , electrostatics 8,[14][15][16][17][18] , electromagnetics 19-22 , hydrodynamics 23-26 , and elasticity 27 . Specific uses in molecular and biological sciences include solving the Schrödinger equation 28 , modeling van der Waals (close packing) interactions between molecules 12 , design of magnetic resonance imaging (MRI) devices 29 and analysis of clinical electroencephalography (EEG) and magnetoencephalography (MEG) data 21,22 .…”

Generalized image charge solvation model for electrostatic interactions in molecular dynamics simulations of aqueous solutions

“…Semi-analytical series solutions in terms of ellipsoidal harmonics can be easily found [57], but unfortunately, they are too complicated to be used in actual biomolecular dynamics simulations. Therefore, in the same spirit as what we have done for the prolate spheroidal case, we propose the following single image approximation for the reaction field Φ RF inside the ellipsoid $${\mathrm{boldR}}_{\mathrm{ind}}=\frac{F_0^1\left({\xi }_{\mathrm{normalb}}\right)}{F_0^1\left({\xi }_{\mathrm{normals}}\right)}true(\frac{F_1^1\left({\xi }_{\mathrm{normals}}\right)E_1^1\left({\xi }_{\mathrm{normalb}}\right)}{F_1^1\left({\xi }_{\mathrm{normalb}}\right)E_1^1\left({\xi }_{\mathrm{normals}}\right)}{x}_{\mathrm{normals}},\frac{F_1^2\left({\xi }_{\mathrm{normals}}\right)E_1^2\left({\xi }_{\mathrm{normalb}}\right)}{F_1^2\left({\xi }_{\mathrm{normalb}}\right)E_1^2\left({\xi }_{\mathrm{normals}}\right)}{y}_{\mathrm{normals}},\frac{F_1^3\left({\xi }_{\mathrm{normals}}\right)E_1^3\left({\xi }_{\mathrm{normalb}}\right)}{F_1^3\left({\xi }_{\mathrm{normalb}}\right)E_1^3\left({\xi }_{\mathrm{normals}}\right)}{z}_{\mathrm{normals}}true).$$ Similarly, this image approximation has two sources of errors.…”

Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding

“…We can repeat the procedure above for ellipsoids, so that the Coulomb and reaction potentials are expanded in the internal ellipsoidal harmonics ${\mathbb{E}}_n^{m}false(\mathrm{boldr}false)$, and the solvent potential is expanded in the external ellipsoidal harmonics ${\mathbb{F}}_n^{m}false(\mathrm{boldr}false)$. The exact boundary conditions are 88 : $$\frac{C_n^m}{{\epsilon }_{\text{protein}}}+R_n^m\frac{E_n^{m}false(afalse)}{F_n^{m}false(afalse)}=S_n^m$$ $$\frac{C_n^m}{{\epsilon }_{\text{water}}}+R_n^m\frac{{\epsilon }_{\text{protein}}}{{\epsilon }_{\text{water}}}\frac{1\frac{\partial E_n^m(\lambda )}{\partial \lambda }{\mid }_a}{1\frac{\partial F_n^m(\lambda )}{\partial \lambda }{\mid }_a}=S_n^m$$ where the integer pair ( n , m ) identifies a single harmonic just as for spherical harmonics, the functions $E_n^{m}false(\lambda false)$ and $F_n^{m}false(\lambda false)$ are the first-kind and second-kind Lamè functions for that harmonic, and the argument a is the ellipsoid’s longest semi-axis. To derive a BIBEE model in ellipsoidal harmonics, we perform a similar procedure of considering the modified boundary condition as was done for the sphere.…”