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

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 g1, g2, …, gN. 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. All these formulas can be expressed as summations of products of binomial coefficients. 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 analysed 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 analysed distribution in terms of a normal distribution multiplied by a sum of Hermite polynomials are given. These Gram–Charlier expansions are tested at various orders and for various examples of supershells. When few terms are kept they are shown to provide simple and efficient approximations of the distribution, even for moderate values of the number of subshells, though such expansions diverge when higher order terms are accounted for. 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.