The Impact of Giulio Racah on Crystal- and Ligand-field Theories

TL;DR

This paper examines Giulio Racah's foundational contributions to crystal- and ligand-field theories, highlighting the development of symmetry-adapted weak-field models and their applications in quantum information.

Contribution

It introduces a generalized symmetry-adapted weak-field model for ligand-field theory based on Racah's work and explores novel applications of SU(2) bases in quantum information.

Findings

Development of a generalized weak-field model for ligand-field theory.

Use of SU(2) bases adapted to cyclic groups for quantum information.

Review of Wigner-Racah algebra for finite and compact groups.

Abstract

This paper focuses on the impact of Racah on crystal- and ligand-field theories, two branches of molecular physics and condensed matter physics (dealing with ions embedded in aggregates of finite symmetry). The role of Racah and some of his students in developing a symmetry-adapted weak-field model for crystal-field theory is examined. Then, we discuss the extension of this model to a generalized symmetry-adapted weak-field model for ligand-field theory. Symmetry considerations via the use of the Wigner-Racah algebra for chains of type SU(2) > G is essential for these weak-field models. Therefore, the basic ingredients for the Wigner-Racah algebra of a finite or compact group are reviewed with a special attention paid to the SU(2) group in a SU(2) > G basis. Finally, as an unexpected application of nonstandard SU(2) bases, it is shown how SU(2) bases adapted to the cyclic group allow to build bases of relevance in quantum information.