Robin P. Sagar

The Laplacian of the charge density and its relationship to the shell structure of atoms and ions

The Laplacian of the spherically averaged charge density ∇2ρ(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. ∇2ρ(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 ∇2ρ(r) most indicative of a shell.

Molecular similarity based on information entropies and distances

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.

Local correlation measures in atomic systems

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.

Information uncertainty-type inequalities in atomic systems

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.

A numerical study of molecular information entropies

Abstract Molecular information entropies are computed by means of a three-dimensional numerical integration from wavefunctions expanded in a variety of Gaussian basis sets at different levels. The results substantiate the use of the entropy sum as a measure of basis set quality. This sum is also shown to be sensitive to electron correlation. The previously observed trends for atomic systems computed from Slater-type orbitals are seen to be present in the results from wavefunctions expanded in Gaussian-type orbitals.

Atomic information entropies beyond the Hartree-Fock limit

Atomic information entropies are computed from configuration interaction wavefunctions for members of the lithium isoelectronic series. For each member of the series, a sequence of wavefunctions built from increasingly larger basis sets is generated which satisfies a density convergence criterion, and exhibits a monotonic behaviour with respect to the density generated from the wavefunctions. Studies of the behaviour of the entropies along a sequence for a particular atom includes trends which have been shown to be present at the Hartree-Fock level. Results suggest that the position and momentum entropies are sensitive to small changes in density and thus can be useful indicators of the quality of the density. The sum of the entropies is also shown to be a useful indicator when density differences are larger. The nature of the information entropy-energy relationship is examined.

Mutual information and correlation measures in atomic systems

Mutual information is introduced as an electron correlation measure and examined for isoelectronic series and neutral atoms. We show that it possesses the required characteristics of a correlation measure and is superior to the behavior of the radial correlation coefficient in the neon series. A local mutual information, and related local quantities, are used to examine the local contributions to Fermi correlation, and to demonstrate and to interpret the intimate relationship between correlation and localization.

Shannon entropy of chemical changes: SN2 displacement reactions

The Shannon entropies along the intrinsic reaction coordinates (IRC) of two SN2 reactions were calculated at the RHF/6-31++G** level. The resulting entropic profiles were compared with the corresponding energy profiles. The Shannon entropy profiles in position and momentum space, as well as their sum, show interesting features about the bond forming and breaking process that are not apparent from the conventional reaction energy profile.

On the determination of atomic shell boundaries

Integrations of the charge density between the critical points of ∇2ρ(r) show that these points are not suitable as definitions of atomic shell boundaries. A comparison with the corresponding scheme based on D(r) favors the latter. Minima in D(r) and corresponding shell populations are tabulated for 55≤Z≤92.

Mutual information and electron correlation in momentum space

Mutual information and information entropies in momentum space are proposed as measures of the nonlocal aspects of information. Singlet and triplet state members of the helium isoelectronic series are employed to examine Coulomb and Fermi correlations, and their manifestations, in both the position and momentum space mutual information measures. The triplet state measures exemplify that the magnitude of the spatial correlations relative to the momentum correlations depends on and may be controlled by the strength of the electronic correlation. The examination of one- and two-electron Shannon entropies in the triplet state series yields a crossover point, which is characterized by a localized momentum density. The mutual information density in momentum space illustrates that this localization is accompanied by strong correlation at small values of p.