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 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 Fisher–Shannon and LMC shape complexities and the Shannon–disequilibrium, Fisher–Shannon and Fisher–disequilibrium information planes, which consist of two localization–delocalization factors, are computed in both position and momentum spaces for the one-particle densities of 90 selected molecules of various chemical types, at the CISD/6-311++G(3df,2p) level of theory. We found that while the two measures of complexity show general trends only, the localization–delocalization planes clearly exhibit chemically significant patterns. Several molecular properties (energy, ionization potential, total dipole moment, hardness, electrophilicity) are analyzed and used to interpret and understand the chemical nature of the composite information–theoretic measures above mentioned. Our results show that these measures detect not only randomness or localization but also pattern and organization.

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.