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Physical Review A

Some novel characteristics of atomic information entropies

Shridhar Ramachandra Gadre, Stephen B. Sears, Subhas J. Chakravorty, Rajeev D. Bendale

Published Nov 1, 1985• 290 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.32.2602

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Near--Hartree-Fock information entropies S\ensuremathρ{S}_{\ensuremath{\rho}}S\ensuremathρ​=-F\ensuremath{\rho}(r)ln\ensuremath{\rho}(r)dr and S\ensuremathγ{S}_{\ensuremath{\gamma}}S\ensuremathγ​=-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\ensuremathρ{S}_{\ensuremath{\rho}}S\ensuremathρ​+S\ensuremathγ{S}_{\ensuremath{\gamma}}S\ensuremathγ​ 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\ensuremathρ{S}_{\ensuremath{\rho}}S\ensuremathρ​+S\ensuremathγ{S}_{\ensuremath{\gamma}}S\ensuremathγ​, increases with an enhancement in the ground-state wave-function quality.
The universal property for the information entropy S = a + h In Ζ is verified for atoms using a systematic study with Roothaan-Hartree-Fock (RHF) wave functions with atomic number Ζ — 2 — 54. The above relation was proposed previously for atoms, nuclei, atomic clusters and correlated atoms in a trap. Kullback-Leibler relative entropy Κ and Jensen-Shannon divergence J are employed to compare RHF with Thomas-Fermi (TF) density of atoms as well as another phenomenological density obtained by Sagar et al. Two-body density distributions in position- and momentum-space are used to calculate and compare the corresponding information entropies for correlated and uncorrelated nuclei and bosonic systems (correlated atoms in a trap). It is seen that short-range correlations (SRC) increase the values of S. One-body information entropy entropy S\ is compared with two-body information entropy and a conjecture is made for TV-body information entropy SN- The entropy Κ and the divergence J are also used to evaluate the information distance between correlated and uncorrelated densities both at the one- and the two-body levels for nuclei and trapped Bose gases.
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The position-space entropy Sρ and the momentum-space entropy Sγ are two increasingly important quantities in the study of the structure and scattering phenomena of atomic and molecular systems. Here, an information-theoretic method which makes use of the Bialynicki–Birula and Mycielski’s inequality is described to find rigorous upper and lower bounds to these two entropies in a compact, simple and transparent form. The upper bounds to Sρ are given in terms of radial expectation values <rα≳ and/or the mean logarithmic radii <ln r≳ and <(ln r)2≳, whereas the lower bounds depend on the momentum expectation values <pα≳ and/or the mean logarithmic momenta <ln p≳ and <(ln p)2≳. Similar bounds to Sγ are also shown in a parallel way. A near Hartree–Fock numerical analysis for all atoms with Z≤54 shows that some of these bounds are so tight that they may be used as computational values for the corresponding quantities. The role of the mean logarithmic radius <ln r≳ and the mean logarithmic momentum <ln p≳ in the improvement of accuracy of the aforementioned bounds is certainly striking.
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.
The asymptotics of the Boltzmann-Shannon information entropy as well as the Renyi entropy for the quantum probability density of a single-particle system with a confining (i.e., bounded below) power-type potential V(x)=x2k{V(x)=x}^{2k}V(x)=x2k with $k\ensuremath{\in}N$ and $x\ensuremath{\in}R,$ is investigated in the position and momentum spaces within the semiclassical (WKB) approximation. It is found that for highly excited states both physical entropies, as well as their sum, have a logarithmic dependence on its quantum number not only when k=1k=1k=1 (harmonic oscillator), but also for any fixed k. As a by-product, the extremal case $\stackrel{\ensuremath{\rightarrow}}{k}\ensuremath{\infty}$ (the infinite well potential) is also rigorously analyzed. It is shown that not only the position-space entropy has the same constant value for all quantum states, which is a known result, but also that the momentum-space entropy is constant for highly excited states.