Shannon Entropy for the Hydrogen Atom Confined by Four Different Potentials

Martínez-Sánchez, Michael-Adán; Vargas, Rubicelia; Garza, Jorge

doi:10.3390/quantum1020018

Cited by 31 publications

(15 citation statements)

References 32 publications

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“…In confined two-electron isoelectronic series (H − , He, Li + , Be 2+ ), S has been reported for only ground states [64] by means of BLYP calculations. Some reports [52,[68][69][70][71][72][73] are available on S in penetrable confinement in atoms. For example, it was observed [68] that, in CHA, up to a certain value of r c , S r decreases with r c .…”

Section: Introductionmentioning

confidence: 99%

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Shannon Entropy in Confined He-Like Ions within a Density Functional Formalism

Majumdar

Roy

2020

Quantum Reports

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Shannon entropy in position ( S r ) and momentum ( S p ) spaces, along with their sum ( S t ) are presented for unit-normalized densities of He, Li + and Be 2 + ions, spatially confined at the center of an impenetrable spherical enclosure defined by a radius r c . Both ground, as well as some selected low-lying singly excited states, viz., 1sns (n = 2–4) 3S, 1snp (n = 2–3) 3P, 1s3d 3D, are considered within a density functional methodology that makes use of a work function-based exchange potential along with two correlation potentials (local Wigner-type parametrized functional, as well as the more involved non-linear gradient- and Laplacian-dependent Lee-Yang-Parr functional). The radial Kohn-Sham (KS) equation is solved using an optimal spatial discretization scheme via the generalized pseudospectral (GPS) method. A detailed systematic analysis of the confined system (relative to the corresponding free system) is performed for these quantities with respect to r c in tabular and graphical forms, with and without electron correlation. Due to compression, the pattern of entropy in the aforementioned states becomes characterized by various crossovers at intermediate and lower r c regions. The impact of electron correlation is more pronounced in the weaker confinement limit and appears to decay with the rise in confinement strength. The exchange-only results are quite good to provide a decent qualitative discussion. The lower bounds provided by the entropic uncertainty relation hold well in all cases. Several other new interesting features are observed.

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Section: Introductionmentioning

confidence: 99%

“…However, for small r c and depending on barrier height, S r may also increase. Apart from constant potential, it was probed [73] for confinements imposed by a dielectric continuum and by isotropic harmonic potential. It was also proposed [72] as an indicator to measure the delocalization of electron density.…”

Section: Introductionmentioning

confidence: 99%

Shannon Entropy in Confined He-Like Ions within a Density Functional Formalism

Majumdar

Roy

2020

Quantum Reports

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“…It is interesting to remark that this crossover confinement phenomenon, which is not present for the 2D‐CHA ground state, does occur at the ground state in the three‐dimensional confined hydrogen system, as was pointed out by Sen and others, although its existence is not explicitly discussed for excited states.…”

Section: Entropic Measuresmentioning

confidence: 86%

“…Most efforts have been centered around the spectroscopic properties and some density‐functional descriptors of physical and chemical quantities for the ground state of spherically confined atoms . However, not so much is known about the information‐theoretical measures of the multidimensional confined systems except for a few recent entropy‐like and complexity‐like results of the three‐dimensional confined hydrogenic atom. The aim of this work is to cover this lack of information by determining the confinement dependence of some entropy (Shannon, Fisher) and complexity (Fisher‐Shannon, lopezruiz‐mancini‐alvet (LMC) and LMC‐Rényi) measures for the 1s, 2s, 2p, and 3d quantum states of the two‐dimensional confined hydrogenic atom (2D‐CHA, in short) in both position and momentum spaces.…”

Section: Introductionmentioning

confidence: 99%

Two‐dimensional confined hydrogen: An entropy and complexity approach

Estañón

Aquino

Puertas‐Centeno

et al. 2020

Int J of Quantum Chemistry

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The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the main entropy and complexity information-theoretical measures. First, the Shannon entropy and the Fisher information, as well as the associated uncertainty relations, are computed and discussed. Then, the Fisher-Shannon, lopezruiz-mancini-alvet, and LMC-Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces. K E Y W O R D SFisher information of the two-dimensional confined hydrogen atom, Fisher-Shannon complexity measure of the two-dimensional confined hydrogen atom, LMC-Rényi complexity measure of the two-dimensional confined hydrogen atom, Shannon entropy of the two-dimensional confined hydrogen atom, two-dimensional confined hydrogen atom | INTRODUCTIONThe fundamental and practical relevance of the spherically confined quantum systems has been manifested from the early days of quantum physics [1,2] up until now. [3][4][5][6][7] They have been used as prototypes to explain numerous phenomena and systems not only in the three-dimensional world but also for nonrelativistic and relativistic D-dimensional (D ≥ 2) chemistry and physics. [8][9][10][11][12][13][14] For example, pioneered by Gerhard Ertl, the 2007 Nobel Laureate in Chemistry, and his collaborators, surface chemistry has been extensively studied and has become a key branch in chemistry. [11][12][13] In physics of materials, two-dimensional systems have been used to gain insight into the properties of semiconductors (see, eg, refs. [15, 16]), and they start to play a fundamental role in bringing strong and new forms of control to the atomic-scale limit over the dynamics of matter in the extreme confinement of electromagnetic energy by phonon polaritonics. [10] Moreover, the properties of the real fluids can be studied by means of crystalline fluids (ie, with a symmetry) with nonstandard dimensions (see, eg, ref. [17]).The idea of two-and multidimensional spherical confinement of atoms has been used not only to simulate the effect of high pressure on the static dipole polarizability in hydrogen [18] but also to model a great deal of nanotechnological objects such as quantum dots; quantum wells and quantum wires; [19,20] atoms and molecules embedded in nanocavities, for example, in fullerenes, zeolites cages, and helium droplets; [4,6,7,[21][22][23] dilute bosonic and fermionic systems in magnetic traps of extremely low temperatures; [24][25][26] and a variety of quantum-information elements. [27,28] This has provoked a fast development of a density functional theory of independent particles moving in multidimensional central potentials with various analytical forms (see, eg, refs. [5-7, 29-...

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“…This atom is the prototype that has been used to interpret numerous phenomena and systems in surface chemistry, [22][23][24] semiconductors (see, eg, Li et al [25] ), quantum dots, [26,27] atoms and molecules embedded in nanocavities (eg, fullerenes, helium droplets, …), [18,19,21,[28][29][30] dilute bosonic and fermionic systems in magnetic traps of extremely low temperatures, [31][32][33] and a variety of quantuminformation elements. [34,35] Up until now, contrary to the stationary states of the confined 3D-HA, where both the (energy-dependent) spectroscopic and the (eigenfunction-dependent) information-theoretic properties have received much attention, [27,[36][37][38][39][40][41][42][43][44][45] the knowledge of these properties for the confined 2D-HA is quite scarce. [46][47][48][49][50] Just recently, the authors have determined [51] the entropy-like (Shannon, Fisher) and complexity-like (Fisher-Shannon, LMC) measures for a few low-lying stationary states of the confined 2D-HA.…”

Section: Introductionmentioning

confidence: 99%

Crámer‐Raocomplexity of the confined two‐dimensional hydrogen

Estañón

Aquino

Puertas‐Centeno

et al. 2020

Int J of Quantum Chemistry

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The internal disorder of the confined two-dimensional hydrogenic atom is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the statistical Crámer-Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer-Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the electron distribution) is investigated in position and momentum spaces. We found that confinement does distinguish complexity of the system for all quantum states by means of these two component measures.

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Shannon Entropy for the Hydrogen Atom Confined by Four Different Potentials

Abstract: 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

Cited by 31 publications

References 32 publications

Shannon Entropy in Confined He-Like Ions within a Density Functional Formalism

Shannon Entropy in Confined He-Like Ions within a Density Functional Formalism

Two‐dimensional confined hydrogen: An entropy and complexity approach

Crámer‐Raocomplexity of the confined two‐dimensional hydrogen

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