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The Journal of Chemical Physics

Tight rigorous bounds to atomic information entropies

Juan Carlos Angulo, J. S. Dehesa

Published Nov 1, 1992• 68 citations
Publisher

Abstract

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.

DOI
10.1063/1.463710

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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.
Rigorous relationships among radial and logarithmic expectation values of an N-body system are obtained by means of information-theoretic methods. Especially interesting are the uncertainty expressions in terms of the product ((r alpha ) 1 alpha /(p beta ) 1 beta /, which generalize the well known relation (r 2 ))(p 2 )>or=9N 2 /4. Additionally, an inequality involving the uncertainty in the logarithmic radius and the logarithmic momentum is shown. For illustration, the accuracy of the inequalities found is numerically analysed in a Hartree-Fock framework for atomic systems. Finally, it is pointed out how these inequalities are improved in atoms by taking into account other physical quantities or by considering some semiempirical results on atomic information entropies.
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.
An uncertainty-type lower bound [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] to the information-entropy sum in complementary spaces has recently been reformulated by Gadre et al. [Phys. Rev. A 32, 2602 (1985)] in terms of the respective one-particle probability densities. This bound has been exploited to derive rigorous upper as well as lower bounds to the information entropies and their sum in terms of the corresponding second moments of their distributions. Thus the present work establishes a direct connection, as suggested by Sears, Parr, and Dinur [Israel J. Chem. 19, 165 (1980)], between the quantum-mechanical kinetic energy and information entropy in position space. It has also been demonstrated that given at least one arbitrary moment-type constraint in each space, it is possible to derive an upper bound to the information entropy sum in complementary spaces.
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.
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.
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.