Entropic properties of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi></mml:math>-dimensional Rydberg systems

Accurate values of physical quantities serve as the stepping stone for further researches. Consequently, we provide benchmark values of Shannon, Rényi, Tsallis entropies, and Onicescu information energy for ground state helium. With the highly correlated Hylleraas wave functions, our calculations fully considered the effect of electron correlation. Presented numerical results converge with increasing size of basis set, fulfill analytic relations between the quantities, and satisfactorily agree with those in the literature. In particular, we present these information‐theoretic quantities with high accuracy, and it is believed that the reported data would be a valuable reference for further research on information‐theoretic quantities of atomic and molecular systems.

Section: Resultssupporting

confidence: 88%

“…This behavior is also reported in the literature. [28,70,75,76] Furthermore, for α > 1, Rényi entropy is larger than Tsallis entropy and have similar slopes. On the other hand, for α < 1, Tsallis entropy is larger and increases much more drastically than Rényi entropy.…”

Section: Subadditivity Of Shannon Entropy In Position Spacementioning

confidence: 91%

Benchmark calculations of Rényi, Tsallis entropies, and Onicescu information energy for ground state helium using correlated Hylleraas wave functions

2019

“…Here, we restrict ourselves to a few references pertaining to H atom. Some of these for an unconfined free H atom (FHA) are:

I_{r}, I_{p}, I

in 3D, in D‐dimension, upper bounds of S , R , S in 3D, in D‐dimension, R , T in 3D, in D ‐dimension . Relativistic effects on the information measures of FHA are also examined .…”

Section: Introductionmentioning

confidence: 99%

“…Recently, R and T were studied for Rydberg hydrogenic states within a strong Laguerre asymptotic approximation. [31,33] But, their exact solutions are as yet unknown; moreover these were reported only in r space and mostly for l 5 0 states. And to the best of our knowledge, E has not yet been explored at all whatsoever.…”

mentioning

confidence: 99%

Information‐entropic measures in free and confined hydrogen atom

Mukherjee

Roy

2018

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 Rrnormalα,Rpnormalβ, Trnormalα,Tpnormalβ,Er,Ep are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as (35,3) 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 Sboldr,Rboldrnormalα 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.

“…[25][26][27] The entropic properties of electronic systems (which crucially depend on the states' eigenfunctions) quantify the various facets of the spatial extension or multidimensional spreading of the electronic charge. Nowadays, there is an increasing interest on the dimensional dependence of the entropic properties for the stationary states of the multidimensional quantum systems [12,13,[28][29][30][31][32][33][34][35][36][37][38] in order to contribute to the emergent informational representation of the quantum systems which extends and complements the standard energetic representation. This is basically because of the increasing relevance of entropic properties in numerous biological, chemical, and physical phenomena [39][40][41][42][43][44] as well as in electronic correlations [45][46][47][48] and chemical reactions.…”

mentioning

confidence: 99%

The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Dehesa

Belega

Toranzo

et al. 2019

There exists an increasing interest on the dimensionality dependence of the entropic properties for the stationary states of the multidimensional quantum systems in order to contribute to its emergent informational representation, which extends and complements the standard energetic representation. Nowadays, this is specially so for high-dimensional systems as they have been recently shown to be very useful in both scientific and technological fields. In this work, the Shannon entropy of the discrete stationary states of the high-dimensional harmonic (ie, oscillator-like) and hydrogenic systems is analytically determined in terms of the dimensionality, the potential strength, and the state's hyperquantum numbers. We have used an informationtheoretic methodology based on the asymptotics of some entropy-like integral functionals of the orthogonal polynomials and hyperspherical harmonics which control the wave functions of the quantum states, when the polynomial parameter is very large; this is basically because such a parameter is a linear function of the system's dimensionality. Finally, it is shown that the Shannon entropy of the D-dimensional harmonic and hydrogenic systems has a logarithmic growth rate of the type D log D when D ! ∞. K E Y W O R D S high-dimensional harmonic systems, high-dimensional hydrogenic systems, high-dimensional quantum systems, logarithmic integrals of orthogonal polynomials-asymptotics, Shannon entropy of high-dimensional quantum systems 1 | INTRODUCTION Multidimensional quantum systems consisting of many particles are a major challenge in Quantum Chemistry and Physics, as their behavior can be determined only with immense computational power. The physical solutions of their associated wave equations, and consequently all the chemical and physical properties of these systems, crucially depend on the space dimensionality D. Scientists are trying to discover elegant notions and techniques to simplify the problem (see, eg, ref. [1]). The 1986 Nobel Prize in Chemistry winner Dudley Robert Herschbach et al have developed [2][3][4][5] a very useful strategy to study the atomic and molecular systems, the D-dimensional scaling method, where the D is the basic variable. The pseudoclassical or highdimensional (D ! ∞) limit is the starting point of this strategy. Indeed, this method requires to solve a finite many-electron problem in the (D ! ∞)-limit, and then perturbation theory in 1D is used to have an approximate result for the standard dimension (D = 3), obtaining at times a quantitative accuracy comparable to the self-consistent Hartree-Fock calculations. Most important here is that the electronic structure for the (D ! ∞)-limit is beguilingly simple and exactly computable for any atom and molecule. For finite D but very large, the electrons are confined to harmonic oscillations about the fixed positions attained in the (D ! ∞)-limit. Indeed, at this high-dimensional limit, the electrons of a many-electron system assume fixed positions relative to