Geometric Rényi divergence: A comparative measure with applications to atomic densities

An alternative one-parameter measure of divergence is proposed, quantifying the discrepancy among general probability densities. Its main mathematical properties include (i) comparison among an arbitrary number of functions, (ii) the possibility of assigning different weights to each function according to its relevance on the comparative procedure, and (iii) ability to modify the relative contribution of different regions within the domain. Applications to the study of atomic density functions, in both conjugated spaces, show the versatility and universality of this divergence.

Figure 1

Jensen-Rényi (JRD${}_r^{\left(q\right)}$) and geometric Rényi (GRD${}_r^{\left(q\right)}$) divergences between charge densities of He and Fr neutral atoms, for $q=0.4$ and $q=2$.

Figure 2

GRD${}_r^{\left(q\right)}(\mathrm{Ar},Z)$ and GRD${}^{\left(q\right)}(\mathrm{Ca},Z)$ in position space, for $q=0.4$ and $q=2$ of both argon and calcium with respect to neutral atoms $1\le Z\le 103$.

Figure 3

GRD${}_r^{\left(q\right)}(Z,Z^{\prime })$ of order $q=0.4$, in position space, of each alkali metal ($Z^{\prime }=3,11,19,37,55,87$) and neutral atoms $1\le Z\le 103$.

Figure 4

GRD${}_r^{\left(q\right)}$(NC) between a neutral atom (N) and its singly-charged cation (C) with $3\le Z\le 55$, and phenomenological correspondence with the atomic ionization potential (AIP) of the neutral system.

Figure 5

GRD${}_r^{\left(q\right)}(Z^{+},Z^{2+},Z^{\prime })$ among singly and doubly charged cations with nuclear charge $3\le Z\le 12$, together with a neutral system of nuclear charge $1\le Z^{\prime }\le 103$. A threshold of GRD values appears amplified in the inset.