The expressions for the 2nd-order mixed partial derivatives of Slater-Koster matrix elements at spherical coordinate singularities

TL;DR

This paper provides detailed derivations of second-order mixed partial derivatives of Slater-Koster matrix elements at coordinate singularities, crucial for accurate tight binding modeling when atomic configurations align along the z-axis.

Contribution

It offers explicit derivations of derivatives at singularities, enhancing the accuracy of tight binding models in specific atomic configurations.

Findings

Derived explicit formulas for second-order mixed derivatives at singularities.

Implemented the formulas in the DINAMO code.

Facilitates more precise atomic interaction modeling in tight binding simulations.

Abstract

In a recent publication it has been shown how to generate derivatives with respect to atom coordinates of Slater-Koster matrix elements for the tight binding (TB) modelling of a system. For the special case of a mixed second partial derivative at coordinate singularities only the results were stated in that publication. In this work, the derivation of these results is given in detail. Though it may seem rather `technical' and only applicable to a very special case, atomic configurations where the connecting vector between the two atoms involved in a two-centre matrix element is aligned along the z-axis (in the usual approach) require results for precisely this case. The expressions derived in this work have been implemented in the DINAMO code.