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Journal of Physics B Atomic Molecular and Optical Physics

Atomic information entropies beyond the Hartree-Fock limit

Minhhuy Hô, Robin P. Sagar, Vedene H. Smith, Rodolfo O. Esquivel

Published Nov 14, 1994• 45 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/27/21/009

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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.
Pair density Shannon information measures and their NNN-dependent behaviour in diatomic molecular systems
Pair density Shannon entropies and mutual information in position and in momentum space are calculated for two series of homonuclear and heteronuclear diatomic molecules using Hartree-Fock wave functions. The pair density entropy sum increases with the quality of the basis set. Mutual information, a measure of statistical correlation, is seen to be smaller in momentum space as compared to position space. The interpretation is that the momentum pair density is closer than the position pair density to a Hartree-like reference. The NNN-dependent behaviours of the entropy and mutual information sums are examined by fitting the data to three different model behaviours. Results show that these models are capable of representing the molecular data in differing degrees. The parameters obtained from a particular model are relatively constant across different chemical series. This adds evidence to the argument of a universal NNN-dependent behaviour of the entropy sums.
Near--Hartree-Fock information entropies ${S}_{\ensuremath{\rho}}$=-F\ensuremath{\rho}(r)ln\ensuremath{\rho}(r)dr and ${S}_{\ensuremath{\gamma}}$=-F \ensuremath{\gamma}(p)ln\ensuremath{\gamma}(p)dp, where \ensuremath{\rho}(r) and \ensuremath{\gamma}(p) are one-electron densities in coordinate and momentum space, respectively, have been computed for atoms in their ground and excited states. The information entropies for the harmonic oscillator and hydrogen atom as prototype systems are also discussed. The result of this exercise is a numerical discovery that for the ground state, ${S}_{\ensuremath{\rho}}$+${S}_{\ensuremath{\gamma}}$ shows its minimum value. This study for atoms in their ground states, using wave functions of single-zeta, double-zeta, and near--Hartree-Fock quality, also unearths a startling feature that the entropy sum, ${S}_{\ensuremath{\rho}}$+${S}_{\ensuremath{\gamma}}$, increases with an enhancement in the ground-state wave-function quality.
The Fisher-Shannon plane which embodies the Fisher information measure in conjunction with the Shannon entropy is tested in its ability to quantify and compare the informational behavior of the process of atomic ionization. We report the variation of such an information measure and its constituents for a comprehensive set of neutral atoms, and their isoelectronic series including the mononegative ions, using the numerical data generated on 320 atomic systems in position, momentum, and product spaces at the Hartree-Fock level. It is found that the Fisher-Shannon plane clearly reveals shell-filling patterns across the periodic table. Compared to position space, a significantly higher resolution is exhibited in momentum space. Characteristic features in the Fisher-Shannon plane accompanying the ionization process are identified, and the physical reasons for the observed patterns are described.
In this work, we investigate the reliability of information-theoretic measures based on the electron-density and shape-function, specifically Shannon and Rényi entropies, as descriptors of electronic correlation. By establishing a rigorous decomposition of these entropic measures into additive and nonadditive contributions, supported on a Mulliken-like atomic partition of molecules, we systematically analyze the asymptotic behavior of the entropies at the infinite-internuclear-distance limit to assess the problem of static correlation and extensivity. Our algebraic and numerical analysis reveals several flaws in the use of these density-based descriptors. We demonstrate that for minimal-basis and different theoretical levels, the Shannon and Rényi entropies fail to encode the amount of static correlation conveyed by the underlying wavefunction. Conversely, shape-function Shannon entropies and Rényi entropies (for α≠1α\neq 1α=1) violate extensivity. In larger basis sets, uncorrelated Hartree-Fock densities consistently overestimate entropy compared to sufficiently correlated (e.g., full-valence-CAS) densities. Moreover, the entropies for insufficiently correlated methods violate extensivity. These findings indicate that electron-density-based measures are insufficient for capturing static correlation, suggesting that robust entropic descriptors should be constructed from higher-dimensional Hilbert-space objects.
A systematic investigation of three different electron-electron entanglement measures, namely the von Neumann, the linear and the occupation number entropy at full configuration interaction level has been performed for the four helium-like systems hydride, helium, Li(+) and Be(2+) using a large number of different basis sets. The convergence behavior of the resulting energies and entropies revealed that the latter do in general not show the expected strictly monotonic increase upon increase of the one-electron basis. Overall, the three different entanglement measures show good agreement among each other, the largest deviations being observed for small basis sets. The data clearly demonstrates that it is important to consider the nature of the chemical system when investigating entanglement phenomena in the framework of Gaussian type basis sets: while in case of hydride the use of augmentation functions is crucial, the application of core functions greatly improves the accuracy in case of cationic systems such as Li(+) and Be(2+). In addition, numerical derivatives of the entanglement measures with respect to the nucleic charge have been determined, which proved to be a very sensitive probe of the convergence leading to qualitatively wrong results (i.e., the wrong sign) if too small basis sets are used.
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.