Inequalities for exchange Slater integrals

TL;DR

This paper explores inequalities governing exchange Slater integrals, extending Racah's classical results to derive new relations that aid in analyzing complex atomic spectra and energy level classification.

Contribution

It generalizes Racah's technique to derive additional inequalities involving multiple exchange integrals, enhancing tools for spectral analysis.

Findings

Derived two new inequalities involving three and four exchange integrals.

Generalized Racah's method with complex calculations.

Potential applications in spectral regularity detection and energy level classification.

Abstract

The variations of exchange Slater integrals with respect to their order $k$ are not well known. While direct Slater integrals $F^k$ are positive and decreasing when the order increases, this is not stricto sensu the case for exchange integrals $G^k$. However, two inequalities were published by Racah in his seminal article "Theory of complex spectra. II". In this article, we show that the technique used by Racah can be generalized, albeit with cumbersome calculations, to derive further relations, and provide two of them, involving respectively three and four exchange integrals. Such relations can prove useful to detect regularities in complex atomic spectra and classify energy levels.