Commutation Relations for Double Tensors of Two Equivalent D Electrons

TL;DR

This paper derives the commutation relations for double tensors of two equivalent d electrons using Clebsch-Gordan and Racah coefficients, identifying a Lie algebra structure to facilitate calculations in atomic and nuclear physics.

Contribution

It introduces a method to calculate double tensor commutation relations and identifies the underlying Lie algebra as B2, enabling easier matrix element computations.

Findings

Derived commutation relations for double tensors of two d electrons.

Identified the Lie algebra B2 structure in the relations.

Facilitated matrix element calculations using the Wigner-Eckart theorem.

Abstract

We apply the Clebsch-Gordan and Racah coefficients to calculate the double tensors for two equivalent d electrons. We also obtain the commutation relations for these double tensors and choose certain quantum numbers, which produce a subgroup. From the root vectors of the commutation relations, we identify them with Lie algebra B2. Once we have the correct Lie algebra, it is feasible to use the Wigner-Eckart theorem to find matrix elements for transition states among atomic spectra or nuclear shell models.