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arXiv:1801.05172 · International Journal of Quantum Chemistry

Information‐entropic measures in free and confined hydrogen atom

Neetik Mukherjee, Amlan K. Roy

Published Feb 10, 2018• 76 citations
arXivHTMLPDF

Abstract

Abstract Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.

DOI
10.1002/qua.25596

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Abstract Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.
Abstract Information‐based uncertainty measures like Rényi entropy (R), Shannon entropy (S) and Onicescu energy (E) are employed to understand the influence of radial confinement in an isotropic harmonic oscillator (IHO). The transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of oscillation frequency (ω) and confinement radius (). For a given quantum state, accurate results are provided by applying the respective exact analytical wave function in the r‐space. The p‐space wave functions are produced from Fourier transforms of radial functions. Pilot calculations are done taking order of entropic moments () as in r‐ and p‐spaces. A detailed, systematic analysis is performed for a 3D confined harmonic oscillator (CHO) with respect to state indices , and . It has been found that the CHO acts as a bridge between particles in a spherical box and the 3D free IHO. In the low‐ region, increases while decrease with the rise in . At moderate , there exists an interaction between two competing factors: i) radial confinement (localization) and ii) accumulation of radial nodes with growth of (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.
Abstract The information‐theoretic measure of confined hydrogen atom has been investigated extensively in the literature. However, most of them were focused on the ground state and accurate values of information entropies, such as Shannon entropy, for confined hydrogen are still not determined. In this work, we establish the benchmark results of the Shannon entropy for confined hydrogen atom in a spherical impenetrable sphere, in both position and momentum spaces. This is done by examining the bound state energies, the normalization of wave functions, and the scaling property with respect to isoelectronic hydrogenic ions. The angular and radial parts of Shannon entropy in two conjugate spaces are provided in detail for both free and confined hydrogen atom in ground and several excited states. The entropies in position space decrease logarithmically with decreasing the size of confinement, while those in momentum space increase logarithmically. The Shannon entropy sum, however, approaches to finite values when the confinement radius closes to zero. It is also found that the Shannon entropy sum shares same trend for states with similar density distributions. Variations of entropy for nodeless bound states are significantly distinct form those owning nodes when changing the confinement radius.
Spatial confinements induce localization or delocalization on the electron density in atoms and molecules, and the hydrogen atom is not the exception to these results. In previous works, this system has been confined by an infinite and a finite potential where the wave-function exhibits an exact solution, and, consequently, their Shannon entropies deliver exact results. In this article, the Shannon entropy in configuration space is examined for the hydrogen atom submitted to four different potentials: (a) infinite potential; (b) Coulomb plus harmonic oscillator; (c) constant potential; and (d) dielectric continuum. For all these potentials, the Schrödinger equation admitted an exact analytic solution, and therefore the corresponding electron density has a closed-form. From the study of these confinements, we observed that the Shannon entropy in configuration space is a good indicator of localization and delocalization of the electron density for ground and excited states of the hydrogen atom confined under these circumstances. In particular, the confinement imposed by a parabolic potential induced characteristics that were not presented for other confinements; for example, the kinetic energy exhibited oscillations when the confinement radius is varied and such oscillations coincided with oscillations showed by the Shannon entropy in configuration space. This result indicates that, when the kinetic energy is increased, the Shannon entropy is decreased and vice versa.
In this study, the information‐theoretic measures in both the position and momentum spaces for the pseudoharmonic potential using Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy, and Onicescu information energy are investigated analytically and numerically. The results obtained are applied to some diatomic molecules. The Renyi and Tsallis entropies are analytically obtained in position space using Srivastava–Niukkanen linearization formula in terms of the Lauricella hypergeometric function. Also, they are obtained in the momentum space in terms of the multivariate Bell polynomials of Combinatorics. We observed that the Fisher information increases with n in both the position and momentum spaces, but decreases with for all the diatomic molecules considered. The Shannon entropy also increases with increasing n in the position space and decreases with increasing . The variations of the Renyi and Tsallis entropies with are also discussed. The exact and numerical values of the Onicescu information energy are also obtained, after which the ratio of information‐theoretic impetuses to lengths for Fisher, Shannon, and Renyi are obtained. © 2015 Wiley Periodicals, Inc.
Abstract Confinement of atoms inside various cavities has been studied for nearly eight decades. However, the Compton profile (CP) for such systems has not yet been investigated. Here we construct the CP for a H atom radially confined inside a hard spherical enclosure, as well as in a free condition . Some exact analytical relations for the CP’s of circular or nodeless states of free atoms is presented. By means of a scaling idea, this has been further extended to the study of a H-like atom trapped inside an impenetrable cavity. The accuracy of these constructed CPs has been confirmed by computing various momentum moments. Apart from that, several information theoretical measures, like Shannon entropy ( S ) and Onicescu energy ( E ) have been exploited to characterize these profiles. Exact closed-form expressions are derived for S and E using the ground state CP in free H-like atoms. A detailed study reveals that, increase in confinement inhibits the rate of dissipation of kinetic energy. At a fixed ℓ , this rate diminishes with a rise in n . However, at a certain n , this rate accelerates with progress in ℓ . Similar analysis on the respective free counterpart displays an exactly opposite trend as that in a confined system. However, in both free and confined environments, the CP generally gets broadened with rise in Z . Representative calculations are done numerically for low-lying states of the confined systems, taking two forms of position-space wave functions: (a) exact (b) highly accurate eigenfunctions through a generalized pseudospectral method. In essence, CPs are reported for confined H atoms (and isoelectronic series) and investigated adopting an information-theoretic framework.