The Laplacian of the spherically averaged charge density ∇²ρ̄(r) has been computed from nonrelativistic SCF wave functions for the neutral atoms from hydrogen to uranium, and the singly positive ions, from helium to barium and lutetium to radium, in order to examine the shell structure. ∇²ρ̄(r) exhibits a number of extremal points and zeros with the absolute value of the function becoming smaller at each successive extremal point. The zeros, in particular the odd numbered zeros, are shown to exhibit good correlation with the Bohr theory of an atom while the extremal points correlate to a lesser extent. At most five shells are seen in the studied atomic cases based on the fact that the odd numbered zeros are the topological feature of ∇²ρ̄(r) most indicative of a shell.

The interpretation of the entropy sum as a correlation measure is demonstrated for isoelectronic series via an analytical expression that models the asymptotic behavior of the electronic charge density in position space and the cusp behavior in momentum space. We also develop an expression for the entropy sum in neutral atoms with an explicit dependence on the ionization energy and the atomic number. The results obtained from these relations are in qualitative agreement with the behavior observed from ab initio calculations. A connection between the entropy sum and the correlation energy is obtained for the weakly inhomogeneous electron gas and demonstrated via calculations for the helium isoelectronic series.

The conjecture by Collins that the correlation energy ${\mathit{E}}_{\mathrm{corr}}$ is proportional to the information entropy as defined by Jaynes is examined. We present numerical evidence supporting this conjecture by computing atomic information entropies from highly correlated wave functions of configuration-interaction (CI) type for members of the lithium isoelectronic series. Sequences of increasingly larger basis sets are used to show how ${\mathit{E}}_{\mathrm{corr}}^{\mathrm{CI}}$ (CI correlation energy) is related to ${\mathit{S}}_{\mathrm{Jaynes}}$. © 1996 The American Physical Society.

A similarity index based on the concept of an information distance is evaluated for a series of diatomic molecules. The results show that the momentum space information distance is a good indicator of the extent of charge separation present in a molecule. The Shannon entropy in position space is shown to decrease with increasing charge separation and as such is presented as a measure of the structure or extent of electron localization in a system. The concept of a local Shannon entropy over an atomic basin is introduced and it is shown how this quantity may be used to examine the local behavior of electron densities.

The one-electron Shannon information entropy sum is reformulated in terms of a single entropic quantity dependent on a one-electron phase space quasiprobability density. This entropy is shown to form an upper bound for the entropy of the one-electron Wigner distribution. Two-electron entropies in position and momentum space, and their sum, are introduced, discussed, calculated, and compared to their one-electron counterparts for neutral atoms. The effect of electron correlation on the two-electron entropies is examined for the helium isoelectronic series. A lower bound for the two-electron entropy sum is developed for systems with an even number of electrons. Calculations illustrate that this bound may also be used for systems with an odd number of electrons. This two-electron entropy sum is then recast in terms of a two-electron phase space quasiprobability density. We show that the original Bialynicki-Birula and Mycielski information inequality for the N-electron wave function may also be formulated in terms of an N-electron phase space density. Upper bounds for the two-electron entropies in terms of the one-electron entropies are reported and verified with numerical calculations.

The phenomenon of electron correlation in atomic systems is examined and compared from the statistical, information theoretic, and energetic perspectives. Local correlation measures, based on the correlation coefficient, information entropies, and idempotency measure, are compared to the correlation energy density. Analysis of these local measures reveals that the chemically significant valence region is responsible for the behavior of their respective global measures in contrast to the correlation energy density which has large contributions to the correlation energy from both the core and valence regions. These results emphasize the difference in the mechanisms inherent in the different perspectives, the similarity between the statistical, information entropic, and idempotency views, and provides further evidence for the use of information theoretic based quantities in studies of electron correlation.