Efficient methods for calculating the number of states, levels and lines in atomic configurations

Exact or statistical methods for determining the distribution of the MJ values (projection of total angular momentum J) in an electron configuration are presented. This distribution, noted P(MJ), is used to calculate the allowed values of J and the number of electric-dipolar (E1) lines between two configurations. First, the difficulty to account for the Pauli exclusion principle for equivalent electrons is stressed. Showing the limit of the usual exact approach, a very efficient recursive technique is proposed for determining exactly the distribution P(MJ). Second, the statistical approach of Bauche and Bauche-Arnoult [J. Phys. B Atom. Mol. Opt. Phys. 20 (1987) 1659] is extended in order to account for configurations with a high-ℓ spectator. In this case, identical consecutive values may exist in the center of P(MJ), which can neither be modeled by a Gaussian nor by a Gram–Charlier type function. It is shown that the Generalized Gaussian function, with the exponent constrained by the kurtosis (reduced fourth-order centered moment) of P(MJ), is more suited in these situations. A new analytical formula for the evaluation of the number of E1 lines with a larger range of applicability is then proposed.