Multipole characteristics of the open-shell electron eigenstates

TL;DR

This paper introduces a multipole-based framework to analyze the crystal-field effects on open-shell electron states, emphasizing the roles of asphericity and multipolar interactions in splitting phenomena.

Contribution

It redefines the second moment of sublevels using multipole strengths and matrix elements, providing a new way to understand crystal-field effects based on electron state asphericity and environment.

Findings

The second moment can be expressed via multipole strengths and matrix elements.

Asphericity of the ion and environment determines crystal-field splitting strength.

Tabulated asphericities reveal the multipolar structure and susceptibility of electron states.

Abstract

The second moment of the sublevels within the initial state | \alpha SLJ > which constitutes a natural and adequate measure of the crystal-field (CF) effect can be redefined as sigma^{2}=1/(2J+1)\sum_{k} S_{k}^{2} A_{k}^{2}, where S_{k}=[1/(2k+1)\sum_{q}|B_{kq}|^2]^{1/2} is the so-called 2^{k}-pole CF strength, whereas A_{k}= < \alpha SLJ||C^{(k)}||\alpha SLJ > the reduced matrix element of the k-rank spherical tensor operator. Therefore, the CF effect depends on the sum of products of the two factors representing the identical multipole components of two different charge distributions. The term A_{k} expresses the asphericity of the central ion open-shell, whereas the term S_{k} the asphericity of its surroundings. When these two distributions do not fit each other the observed CF splitting can be unexpectedly weak even for considerable values of the total S=(\sum_{k}S_{k}^{2})^{1/2} and A=(\sum_{k}A_{k}^{2})^{1/2}. The tabulated quantities of the A_{k}(|\alpha SLJ >), as the 2^{k}-pole type asphericities, are the intrinsic characteristics of the electron states revealing their multipolar structure and hence their potential susceptibility to CF splitting separately for each effective multipole.