Coulomb matrix elements in multi-orbital Hubbard models

TL;DR

This paper systematically determines independent Coulomb matrix elements for multi-orbital Hubbard models across various point groups, providing formulas to express all other elements and analyzing spherical approximations.

Contribution

It introduces a comprehensive method to identify independent Coulomb parameters for $d$- and $f$-shells in multiple point groups, including correction analyses.

Findings

Derived independent Coulomb parameters for multiple point groups.

Expressed all matrix elements in terms of these parameters.

Analyzed spherical approximation and first-order corrections.

Abstract

Coulomb matrix elements are needed in all studies in solid-state theory that are based on Hubbard-type multi-orbital models. Due to symmetries, the matrix elements are not independent. We determine a set of independent Coulomb parameters for a $d$-shell and a $f$-shell and all point groups with up to $16$ elements ($O_h$, $O$, $T_d$, $T_h$, $D_{6h}$, and $D_{4h}$). Furthermore, we express all other matrix elements as a function of the independent Coulomb parameters. Apart from the solution of the general point-group problem we investigate in detail the spherical approximation and first-order corrections to the spherical approximation.