New sum rules for Wigner 3jm symbols: application to expectation values of hydrogenic ions

TL;DR

This paper derives new sum rules for Wigner 3jm symbols involving additional weighting factors, facilitating the calculation of expectation values of powers of radius in hydrogenic ions, relevant for spectral and Stark effect analyses.

Contribution

It introduces novel sum rules for Wigner 3jm symbols that connect expectation values in different coordinate systems for hydrogenic ions.

Findings

New sum rules involving $[j(j + 1)]^k$ terms

Enables calculation of $r^k$ expectation values in parabolic coordinates

Applicable to modeling rotational spectra and Stark effect

Abstract

We present new sum rules for $3jm$ coefficients, which involve, in addition to the usual weighting factor $(2j + 1)$ where $j$ is an angular momentum, the quantity $[j(j + 1)]^k$ with $k \ge 1$. The sum rules appear for instance in the statistical modeling of rotational spectra within the theory of moments, and enable one to deduce the expectation values of $r^k$ (used in the theory of Stark effect for hydrogenic ions) in parabolic coordinates from the expectation values of $r^k$ in spherical coordinates.