Analytical and numerical expressions for the number of atomic configurations contained in a supershell

TL;DR

This paper introduces three new explicit formulas for counting electronic configurations in atomic supershells, along with analytical expressions for distribution moments and cumulants, applicable to atomic physics and combinatorial problems.

Contribution

The paper presents novel generating-function-based formulas for configuration counts and distribution moments, including recursion relations and cumulant expressions, advancing combinatorial analysis in atomic physics.

Findings

Derived three explicit formulas for configuration counts.

Provided analytical expressions for distribution moments and cumulants.

Validated the formulas with applications to atomic supershells and fermion distributions.

Abstract

We present three explicit formulas for the number of electronic configurations in an atom, i.e. the number of ways to distribute $Q$ electrons in $N$ subshells of respective degeneracies $g_1$, $g_2$, ..., $g_N$. The new expressions are obtained using the generating-function formalism. The first one contains sums involving multinomial coefficients. The second one relies on the idea of gathering subshells having the same degeneracy. A third one also collects subshells with the same degeneracy and leads to the definition of a two-variable generating function, allowing the derivation of recursion relations. Concerning the distribution of population on $N$ distinct subshells of a given degeneracy $g$, analytical expressions for the first moments of this distribution are given. The general case of subshells with any degeneracy is analyzed through the computation of cumulants. A fairly simple expression for the cumulants at any order is provided, as well as the cumulant generating function. Using Gram-Charlier expansion, simple approximations of the analyzed distribution in terms of a normal distribution multiplied by a sum of Hermite polynomials are given. The Edgeworth expansion has also been tested. Its accuracy is equivalent to the Gram-Charlier accuracy when few terms are kept, but it is much more rapidly divergent when the truncation order increases. While this analysis is illustrated by examples in atomic supershells it also applies to more general combinatorial problems such as fermion distributions.