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Physical Review A

Relativistic global and local divergences in hydrogenic systems: A study in position and momentum spaces

J. Antolín, Juan Carlos Angulo, S. Mulas, Sheila López‐Rosa

Published Oct 27, 2014• 9 citations
Publisher

Abstract

Relativistic effects in one-particle densities of hydrogenic systems are quantified by means of global and local density functionals: the Jensen-Shannon and the Jensen-Fisher divergences, respectively. The Schr\"odinger and Dirac radial densities are compared, providing complementary results in position and momentum spaces. While the electron cloud gets compressed towards the origin in the Dirac case, the momentum density spreads out over its domain, and the raising of minima in position space does not occur in the momentum space. Regarding the dependence on the nuclear charge and the state quantum numbers for all divergences here considered, as well as their mutual interconnection, accurate powerlike laws $y\ensuremath{\approx}C{x}^{a}$ are found systematically. The parameters ${C,a}$ defining the respective dependences are extremely sensitive to the closeness of the system to the ground and/or the circular state. Particularly interesting are the analyses of (i) the plane subtended by the Jensen-Shannon and Jensen-Fisher divergences, in a given space (position or momentum), and (ii) either of the above two divergences in the position-momentum plane. These kinds of results show the complementary role of global and local divergences and that of both conjugate spaces.

DOI
10.1103/physreva.90.042511

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Relativistic effects in one-particle densities of hydrogenic systems are quantified by means of global and local density functionals: the Jensen-Shannon and the Jensen-Fisher divergences, respectively. The Schr"odinger and Dirac radial densities are compared, providing complementary results in position and momentum spaces. While the electron cloud gets compressed towards the origin in the Dirac case, the momentum density spreads out over its domain, and the raising of minima in position space does not occur in the momentum space. Regarding the dependence on the nuclear charge and the state quantum numbers for all divergences here considered, as well as their mutual interconnection, accurate powerlike laws y\ensuremath≈Cxay\ensuremath{\approx}C{x}^{a}y\ensuremath≈Cxa are found systematically. The parameters C,a{C,a}C,a defining the respective dependences are extremely sensitive to the closeness of the system to the ground and/or the circular state. Particularly interesting are the analyses of (i) the plane subtended by the Jensen-Shannon and Jensen-Fisher divergences, in a given space (position or momentum), and (ii) either of the above two divergences in the position-momentum plane. These kinds of results show the complementary role of global and local divergences and that of both conjugate spaces.
Position and momentum information measures are evaluated for the ground state of the \emph{relativistic} hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and Rényi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large rrr, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of ppp. Both features yield divergent Rényi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.
The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction towards the origin of the electronic charge in hydrogenic systems and the nodal disappearance (because of the raising of all the nonrelativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far from the nucleus. In this work we quantify these effects by means of single (Fisher information) and composite [Fisher-Shannon complexity and plane, L\'opez-Ruiz, Mancini, and Calbet (LMC) complexity] information-theoretic measures. While the Fisher information measures the gradient content of the density, the (dimensionless) composite information-theoretic quantities grasp twofold facets of the electronic distribution: The Fisher-Shannon complexity measures the combined balance of the gradient content and the total extent of the electronic charge, and the LMC complexity quantifies the disequilibrium jointly with the spreading of the density in the configuration space. Opposite to other complexity notions (e.g., computational and algorithmic complexities), these two quantities describe intrinsic properties of the system because they do not depend on the context but are functionals of the electron density. Moreover, they are closely related to the intuitive notion of complexity because they are minimum for the two extreme (or least complex) distributions of perfect order and maximum disorder.
The analytic information theory of quantum systems includes the exact determination of their spatial extension or multidimensional spreading in both position and momentum spaces by means of the familiar variance and its generalization, the power and logarithmic moments, and, more appropriately, the Shannon entropy and the Fisher information. These complementary uncertainty measures have a global or local character, respectively, because they are power-like (variance, moments), logarithmic (Shannon) and gradient (Fisher) functionals of the corresponding probability distribution. Here we explicitly discuss all these spreading measures (and their associated uncertainty relations) in both position and momentum for the main prototype in D-dimensional physics, the hydrogenic system, directly in terms of the dimensionality and the hyperquantum numbers which characterize the involved states. Then, we analyze in detail such measures for s-states, circular states (i.e., single-electron states of highest angular momenta allowed within an electronic manifold characterized by a given principal hyperquantum number), and Rydberg states (i.e., states with large radial hyperquantum numbers n). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
The charge spreading of the ground and excited states of Klein-Gordon particles moving in a Coulomb potential is quantitatively analysed by means of ordinary moments and the Heisenberg measure as well as by using the most relevant information-theoretic measures of global (Shannon entropic power) and local (Fisher information) types.The dependence of these complementary quantities on the nuclear charge Z and the quantum numbers characterizing the physical states is carefully discussed.The comparison of relativistic Klein-Gordon and non-relativistic Schrödinger values is made.Nonrelativistic limits at large principal quantum number n and for small values of Z are also reached.Contents 1. Introduction 2 2. Radial expectation values and Heisenberg's measure 5 3. Shannon and Fisher information measures 8 4. Conclusions 12