Spherical to Cartesian Coordinates Transformation for Solid Harmonics Revisited

TL;DR

This paper revisits the transformation between spherical and Cartesian coordinates for solid harmonics, providing general formulas, tabulated coefficients up to ul=10, and applying the method to construct the Hartree potential via multipole expansion.

Contribution

It offers new general expressions and tabulated coefficients for coordinate transformation, enhancing computational methods in quantum chemistry.

Findings

Transformation coefficients tabulated up to ul=10

Formulas applied to construct Hartree potential

Improves accuracy and efficiency in multipole expansions

Abstract

Spherical Harmonic Gaussian type orbitals and Slater functions can be expressed using spherical coordinates or a linear combinations of the appropriate Cartesian functions. General expressions for the transformation coefficients between the two representations are provided. Values for the transformation coefficients are tabulated up to the quantum number $\ell = 10$. The formula is applied to construct the Hartree potential by an arbitrary-order multipole expansion.