Integration over spin-angular variables in atomic physics

TL;DR

This paper reviews and compares methods for calculating matrix elements in atomic physics, introducing an efficient LS coupling approach that leverages symmetry, second quantization, and graphical techniques to handle correlation effects in atoms and ions.

Contribution

It presents a novel, efficient method for computing matrix elements in LS coupling using symmetry properties, second quantization, and graphical techniques.

Findings

The method effectively accounts for correlation effects in atoms and ions.

Comparison of various methods highlights advantages and shortcomings.

The approach is applicable to any atom or ion in the periodic table.

Abstract

A review of methods for finding general expressions for matrix elements (non-diagonal with respect to configurations included) of any one- and two-particle operator for an arbitrary number of shells in an atomic configuration is given. These methods are compared in various aspects, and the advantages or shortcomings of each particular method are discussed. Efficient method to find the abovementioned quantities in LS coupling is presented, based on the use of symmetry properties of operators and matrix elements in three spaces (orbital, spin and quasispin), second quantization in coupled tensorial form, graphical technique and Wick's theorem. This allows to efficiently account for correlation effects practically for any atom and ion of periodical table.