The transformation of irreducible tensor operators under spherical functions

TL;DR

This paper develops a new method for transforming irreducible tensor operators under spherical functions, providing a systematic approach to calculate tensor product matrix elements, with applications to two-electron systems.

Contribution

It introduces a novel technique for calculating irreducible coupled tensor product matrix elements using Racah algebra and spherical coordinate transformations.

Findings

New transformation procedure for tensor operators

Explicit rotation matrix parametrization in spherical coordinates

Application to two-electron matrix element calculations

Abstract

The irreducible tensor operators and their tensor products employing Racah algebra are studied. Transformation procedure of the coordinate system operators act on are introduced. The rotation matrices and their parametrization by the spherical coordinates of vector in the fixed and rotated coordinate systems are determined. A new way of calculation of the irreducible coupled tensor product matrix elements is suggested. As an example, the proposed technique is applied for the matrix element construction for two electrons in a field of a fixed nucleus.