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 $g_1$, $g_2$, ..., $g_N$. 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.

References in corpus (10)

• Expansions for nearly Gaussian distributions
• Canonical and Microcanonical Distributions for Fermi Systems
• Calculations of canonical averages from the grand canonical ensemble
• Partition Functions in Statistical Mechanics, Symmetric Functions, and Group Representations
• Jensen-Feynman approach to the statistics of interacting electrons
• Generating functions for canonical systems of fermions
• Effect of third- and fourth-order moments on the modeling of Unresolved Transition Arrays
• Alternative Mathematical Technique to Determine LS Spectral Terms
• Statistics of electric-quadrupole lines in atomic spectra
• System-Size Dependence in Grand Canonical and Canonical Ensembles

Cited by in corpus (3)

• The number of populated electronic configurations in a hot dense plasma
• Angular momentum distribution in a relativistic configuration: Magnetic quantum number analysis
• Distribution of the total angular momentum in relativistic configurations