Extraction of shell densities from the electron-pair density of atoms

Applying a property of the conditional probability density function, we report that good approximations to the electron shell densities can be extracted from the total electron-pair density of atoms. Numerical tests on the 53 atoms He through Xe show that the quantum similarity index, bounded by 0 (total dissimilarity) and 1 (complete similarity), is never smaller than 0.98 between the two radial shell densities, one extracted from the radial electron-pair density and the other obtained as an occupation number weighted sum of the Hartree–Fock subshell densities. Approximate but theoretical justification is given for the numerical results.