Abstract
Parametrizing a curved surface with flat triangles in electrostatics problems creates a diverging electric field. One way to avoid this is to have curved areal elements. However, charge density integration over curved patches appears difficult. This paper, dealing with spherical triangles, is the first in a series aiming to solve this problem. Here, we lay the ground work for employing curved patches for applying the surface charge method to electrostatics. We show analytically how one may control the accuracy by expanding in powers of the the arc length (multiplied by the curvature). To accommodate not extremely small curved areal elements, we have provided enough details to include higher order corrections that are needed for better accuracy when slightly larger surface elements are used.
Highlights
Of general importance at the molecular level, electrostatic interactions are crucial for biomolecular systems which consist of many charged molecules embedded in the polar solvent water
The implicit solvent methods [3, 4] are in principle less computationally intensive when larger systems are considered; their application is limited to systems where fine details of solute-solvent interactions do not play a major role
This is because we are mainly interested in applying the classical formalism here to the explicit solvent model where we are allowed to make the probe sphere small enough to avoid these potential singularities
Summary
Of general importance at the molecular level, electrostatic interactions are crucial for biomolecular systems which consist of many charged molecules embedded in the polar solvent water. The formalism introduced in [5] allows one to find a solution with arbitrary accuracy for an arbitrary number of interacting dielectric spheres with point charges at their centers To extend this formalism to model molecules, one need to account for the non-spherical nature of chemical bonds by either introducing permanent and inducible higher multipoles [7] or expanding the scope of dielectric bodies to include non-spherical shapes. The former extension [7] is appropriate when the separation distances among the molecules are large and details of geometry play a lesser role; the latter extension may better represent regions of significant electronic density and might be suitable for shorter separations. Some relevant mathematical formulas/derivations are provided in the appendices
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