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Physica Scripta

Rigorous bounds to information entropies for atomic systems

Jianmin Tao, Guobao Li, Jian-Min Li

Published Sep 1, 1997• 7 citations
Publisher

Abstract

Rigorous bounds to atomic information entropies and their sum in position and momentum spaces have been derived in terms of radial and momentum expectation values ⟨ r n ⟩ and ⟨ p n ⟩. Numerical studies on atomic systems show that the bounds presented in terms of ⟨ r ⟩ and ⟨ p ⟩, the first moments of the position and momentum - space densities are sharper than those given by Gadre and Bendale in terms of ⟨ r 2 ⟩ and ⟨ p 2 ⟩, the second moments of the position and momentum-space densities. Within the BKCM procedure, several relationships between the information entropies and the average electron densities ⟨ρ⟩ and ⟨γ⟩ in position and momentum spaces have been established.

DOI
10.1088/0031-8949/56/3/010

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Rigorous bounds to atomic information entropies and their sum in position and momentum spaces have been derived in terms of radial and momentum expectation values ⟨ r n ⟩ and ⟨ p n ⟩. Numerical studies on atomic systems show that the bounds presented in terms of ⟨ r ⟩ and ⟨ p ⟩, the first moments of the position and momentum - space densities are sharper than those given by Gadre and Bendale in terms of ⟨ r 2 ⟩ and ⟨ p 2 ⟩, the second moments of the position and momentum-space densities. Within the BKCM procedure, several relationships between the information entropies and the average electron densities ⟨ρ⟩ and ⟨γ⟩ in position and momentum spaces have been established.
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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.
General model-independent relationships among radial expectation values of the one-particle densities in position and momentum spaces for any quantum-mechanical system are obtained. They are derived from the Stam uncertainty principle and the recently reported lower bounds to the Fisher information entropy of both densities. The results are usually expressed in terms of some uncertainty products of the system. The accuracy of the bounds is numerically analyzed for neutral atoms within a Hartree-Fock framework.
Rigorous and universal bounds on frequency moments of one-particle densities in terms of radial expectation values in the conjugate space are obtained. The results, valid for any d-dimensional quantum-mechanical system, are derived by using Rényi-like position-momentum inequalities in an information-theoretical framework. Especially interesting are the upper bounds on the Dirac exchange and Thomas-Fermi kinetic energies, as well as the disequilibrium or self-similarity of both position and momentum distributions. A variety of bounds for these functionals in a given space are known, but most usually in terms of quantities defined within the same space. Very few results including a density functional on one space, and expectation values on the conjugate one, are found in the literature. A pioneering bound on the disequilibrium in terms of the kinetic energy is improved in this work. A numerical study of the aforementioned relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility and the height of the peak of the Compton profile, among others.
Rigorous and universal relationships among radial expectation values of any DDD-dimensional quantum-mechanical system are obtained, using R\'enyi-like position-momentum inequalities in an information-theoretical framework. Although the results are expressed in terms of four moments (two in position space and two in the momentum one), especially interesting are the cases that provide expressions of uncertainty in terms of products $\ensuremath{\langle}{r}^{a}\ensuremath{\rangle}{}^{1/a}\ensuremath{\langle}{p}^{b}\ensuremath{\rangle}{}^{1/b}$, widely considered in the literature, including the famous Heisenberg relationship $\ensuremath{\langle}{r}^{2}\ensuremath{\rangle}\ensuremath{\langle}{p}^{2}\ensuremath{\rangle}\ensuremath{\geqslant}{D}^{2}/4$. Improved bounds for these products have recently been provided, but are always restricted to positive orders a,b>0a,b>0a,b>0. The interesting part of this work are the inequalities for negative orders. A study of these relationships is carried out for atomic systems in their ground state. Some results are given in terms of relevant physical quantities, including the kinetic and electron-nucleus attraction energies, the diamagnetic susceptibility, and the height of the peak of the Compton profile, among others.
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.