Shannon-information entropy sum in the confined hydrogenic atom

The appearance of critical points in the Shannon entropy sum as a function of confinement radius, in ground and excited state confined hydrogenic systems, is discussed. We illustrate that the Coulomb potential in tandem with the hard sphere confinement are responsible for these points. The positions of these points are observed to vary with the intensity of the potential. The effects of the Coulomb potential on the system are further probed, by examining the differences between the densities of the confined atom and those of the particle confined in a spherical box, for the same confinement radius. These differences are quantified by using Kullback-Leibler and cumulative residual Kullback-Leibler distance measures from information theory. These measures detect that the effects of the Coulomb potential are squeezed out of the system as the confinement radius decreases. That is, the confined atom densities resemble the particle in a box ones, for smaller confinement radii. Furthermore, the critical points in the entropy sum lie in the same regions where there are changes in the distance measures, as the atom behaves more particle in a spherical box-like. The analysis is further complemented by examination of the derivative of the entropy sum with respect to confinement radius. This study illustrates the inhomogeneity in the magnitudes of the derivatives of the entropy sum components and their dependence on the Coulomb potential. A link between the derivative and the entropic force is also illustrated and discussed. Similar behaviors are observed when the virial ratio is compared to the entropic power one, as a function of confinement radius.     

Benchmark values of Shannon entropy for spherically confined hydrogen atom

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.     

Shannon entropy as a predictor of avoided crossing in confined atoms

Avoided crossing is one of the unique spectroscopic features of a confined atomic system. Shannon information entropy of the ground state and some of the excited states of confined H atom as a predictor of avoided crossing is studied in this work. This is accomplished by varying the strength of the confinement and examining structure properties like ionization energy and Shannon information entropy. Along with the energy level repulsion at the avoided crossing, Shannon information entropy is also exchanged between the involved states. This work also addresses a question: In addition to that regarding localization, what other property of the system can be extracted from Shannon entropy? Insightful connection is discovered between Shannon entropy and the average value of confinement potential, Coulomb potential, and kinetic energy.     

The confined helium atom: An information–theoretic approach

In this article, we study the helium atom confined in a spherical impenetrable cavity by using informational measures. We use the Ritz variational method to obtain the energies and wave functions of the confined helium atom as a function of the cavity radius $r_0$. As trial wave functions we use one uncorrelated function and five explicitly correlated basis sets in Hylleraas coordinates with different degrees of electronic correlation. We computed the Shannon entropy, Fisher information, Kullback–Leibler entropy, Tsallis entropy, disequilibrium and Fisher–Shannon complexity, as a function of $r_0$. We found that these entropic measures are sensitive to electronic correlation and can be used to measure it. As expected these entropic measures are less sensitive to electron correlation in the strong confinement regime ( a.u.).     

Confinement of hydrogen atom with Dirac equation

The problem of a particle confined in a spherical cavity is studied with the Dirac equation. A hard confinement is obtained by forcing the large component to vanish at the cavity radius. It is shown that the small component cannot vanish simultaneously at this radius. In the case of a confined hydrogen atom, the energies are given by an implicit equation. For some values of the radius, explicit analytical expressions of the energy exist like in the nonrelativistic case. Very accurate energies and wave functions are obtained with the Lagrange-mesh method with few mesh points. To this end, two differently regularized Lagrange-Jacobi bases associated with the same mesh are used for the large and small components. The importance of relativistic effects is discussed for hydrogen-like ions. The validity of this definition of hard confinement is discussed with a soft-confinement model studied with the R-matrix method.     

Static field ionization of the spherically confined hydrogen atom

The ionization of the hydrogen atom confined in a spherical potential well and subjected to a static electric field is studied, using the diffusion Monte Carlo (DMC) method. Atomic ionization within a potential well is found to be a stationary, gradual, and reversible process. The value of the electric field at the onset of ionization is of the order of 0.1 atomic units, and depends on the symmetry of the atomic wave function and on the confinement dimension. By decreasing the confinement sphere, the difference between the bound and ionized states disappears, showing that strict confinement leads to pressure ionization of the atom. The off-center case is studied characterizing the potential energy surface (PES), and the transition between field-induced and pressure-induced ionization is confirmed. Except for very weak fields, the minimum of the PES is reached when the proton is in contact with the boundary of the well.     

Reactivity dynamics of a confined molecule in presence of an external magnetic field

Reactivity dynamics and stability of a confined hydrogen molecule in presence of an external magnetic field has been studied using quantum fluid density functional theory. Dynamic profiles of various reactivity parameters such as hardness, electrophilicity, magnetizability, phase volume, entropy, etc. have been studied within a confined environment. Responses in the reactivity parameters as well as the associated electronic structure principles validate the stability of the confined H₂ molecule in ground and excited states in presence of an external magnetic field. Confinement to the system has been imposed by the Dirichlet type boundary condition. Confinement and excitation act in opposite directions. Ground state type dynamics is obtained on simultaneous electronic excitation and confinement. © 2014 Wiley Periodicals, Inc.     

Accurate energy eigenvalues and eigenfunctions for the two‐dimensional confined hydrogen atom

A study of the two‐dimensional hydrogen atom confined within a circle of impenetrable walls is presented. The potential inside the box is Coulomb type, whereas outside it is infinite. The energy eigenvalues and some radial wave function properties are computed with high accuracy for different box sizes. We derive the polarizability in the Kirkwood approximation, calculate the Fermi contact term as a function of the confinement radius, and investigate the filling order of the one‐electron states. When the electronic configuration of many electrons is constructed by means of the Aufbau principle, the model predicts the inversion 2s–3d levels in the N atom. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Ionization probability of the hydrogen atom suddenly released from confinement

The problem of the stability of a confined atom when it is extracted from the confining cavity has been investigated, modeled by a spherical hard wall potential. The ionization probability when the atom is released from confinement has been obtained. The dependence of the ionization probability on the confinement radius and on the quantum numbers of the initial confined state has been studied. The probability density function of the ionization energy of the ejected electron has been obtained for the different cases considered. The oscillatory structure of this distribution function, with a principal maximum located in the neighborhood of the energy of the initial state and minima very close to zero has been elucidated. The sudden approximation has been applied and the analytic continuation method has been used to calculate the different stationary states.     

Information‐entropic measures in free and confined hydrogen atom

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 (r_c) for the latter. In a CHA, at small r_c, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate r_c, 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.     

Fine structure in the hydrogen atom boxed in a spherical impenetrable cavity

The spectrum of the hydrogen atom confined in a spherical impenetrable box of radius R_c has been investigated by many authors up to date, but not at the level of relativistic corrections. It is well known that, as R_c diminishes, all energy levels and the pressure increase very rapidly, whereas the polarizability goes to zero. In this report, we have computed the relativistic corrections that underlie the fine structure of the confined hydrogen atom, as a function of R_c. Such corrections correspond to relativistic kinetic energy, spin‐orbit coupling and the Darwin term, which are calculated in the frame of time‐independent perturbation theory, for which, use was made of the exact confined hydrogen atom wave functions. We show that for a confinement radius of 0.5 au the relativistic corrections increase up to three orders of magnitude with respect to those corresponding to the free atom. As R_c decreases, the kinetic energy correction and the spin‐orbit coupling for become negative whereas their absolute value and the Darwin term, which is positive, increase very rapidly.     

Electron‐density delocalization in many‐electron atoms confined by penetrable walls: A Hartree–Fock study

The electronic structure of several many‐electron atoms, confined within a penetrable spherical box, was studied using the Hartree–Fock (HF) method, coupling the Roothaan's approach with a new basis set to solve the corresponding one‐electron equations. The resulting HF wave‐function was employed to evaluate the Shannon entropy, , in configuration space. Confinements imposed by impenetrable walls induce decrements on when the confinement radius, R_c, is reduced and the electron‐density is localized. For confinements commanded by penetrable walls, exhibits an entirely different behavior, because when an atom starts to be confined, delivers values less than those observed for the free system, in the same way that the results presented by impenetrable walls. However, from a confinement radius, shows increments, and precisely in these regions, the spatial restrictions spread to the electron density. Thus, from results presented in this work, the Shannon entropy can be used as a tool to measure the electron density delocalization for many‐electron atoms, as the hydrogen atom confined in similar conditions.     

An introduction to analysis of Rényi complexity ratio of quantum states for central potential

Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for six diatomic molecules (CO, NO, N₂, CH, H₂, and ScH) and for other quantum systems.     

Characteristic features of Shannon information entropy of confined atoms

The Shannon information entropy of 1-normalized electron density in position and momentum space Sr and Sp, and the sum ST, respectively, are reported for the ground-state H, He+, Li2+, H-, He, Li+, Li, and B atoms confined inside an impenetrable spherical boundary defined by radius R. We find new characteristic features in ST denoted by well-defined minimum and maximum as a function of confinement. The results are analyzed in the background of the irreducible lower bound stipulated by the entropy uncertainty principle [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)]. The spherical confinement model leads to the ST values which satisfy the lower bound up to the limits of extreme confinements with the interesting new result displaying regions over which a set of upper and lower bounds to the information entropy sum can be locally prescribed. Similar calculations on the H atom in 2s excited states are presented and their novel characteristics are discussed.     

Edges of two-dimensional electron systems under strong magnetic fields

Two-dimensional electron systems, which exist e.g. at interlaces between two different semiconductors, exhibit interesting physical properties under strong magnetic fields. In interpreting the quantum Hall effect the role of one-dimensional system edges begins to be taken into account.

The electron structure, connected with Landau quantization of 2D electron states under magnetic field, has been studied in the vicinity of system edges. Model systems with abrupt confinement barriers exhibit electron dispersions with edge plateaus above the barrier tops, accompanied by regions of substantially reduced gaps between neighbouring Landau branches. Selfconsistent results for smoothly confined systems provide alternating channels of compressible and incompressible Fermi liquids along the system edges. Recent investigation illustrates the transition between the two limiting confinement barrier cases.

In order to evaluate the Hall conductivity, the Kubo formula has been adopted in a straightforward manner to two-dimensional stripes confined by arbitrary barriers. Total deviation of the Hall conductivity from the integer values is given by the product of two factors: the geometrical factor is inversely proportional to the sample width and the edge factor is proportional to the derivative of the electron dispersion at the Fermi level and is thus governed by the shape of the confinement barrier. The deviations have been evaluated for model systems of various widths and a qualitative agreement with recent experimental data for quantum wires has been found.

The formulas provide also current densities and this enables to investigate spatial distributions of the electron current across the Hall stripes. Application to the abruptly confined model shows that the quantized part of the total current takes place within the interior of the stripe whereas the edge currents distribution is affected by the confinement barrier.     

Modeling exact exchange potential in spherically confined atoms

In this work, local exchange potentials corresponding to the Hartree–Fock (HF) electron density have been obtained using the Zhao–Morrison–Parr method for a number of closed‐shell confined atoms and ions. The exchange potentials obtained and the resulting density were compared with those given by the Becke–Johnson (BJ) model potential. It is demonstrated that introducing a scaling factor to the BJ potential allows improving the quality of the resulting density. The optimum scaling factor increases with decreasing confinement radius. The performance of Karasiev and Ludeña's SCα‐LDA method as well as of the Becke‐88 exchange potential for reproducing the HF electron densities in confined atoms has been also examined. © 2015 Wiley Periodicals, Inc.     

Repulsive impurity doped quantum dot subjected to oscillatory confinement potential: Role of dopant strength and dopant location on time-evolution

We explore the pattern of time evolution of eigenstates of a repulsive impurity doped quantum dot. The quantum dot is 2-dimensional and contains one electron which is harmonically confined. We have considered Gaussian impurity centers. A static transverse magnetic field is also present. Under a periodically fluctuating confinement potential, the system reveals a long time dynamics. The investigation points to a typical value of impurity potential strength at which the excitation is maximum. This typical value has also been found to be strongly dependent on dopant location. The rate of transition between the eigenstates depends delicately on several impurity dependent factors modulated by the oscillating confinement potential and explains the excitation maximization quite elegantly.     

Quantum dot with N interacting electrons confined in a power-law external potential

The ground-state physical properties, such as electron density, chemical potential, and total energy, of a two-dimensional quantum dot with N interacting electrons confined in a power-law external potential are numerically determined by the Thomas-Fermi approximation. The effect of the confining potential on properties such as electron density and chemical potential is examined for both interacting and non-interacting systems. It is shown that the results of the calculations are in excellent agreement with those given in the literature. The results indicate that interactions and the shape of the confinement affect the density and thus the ground-state properties of the electrons significantly.     

Confined systems and the modified virial theorem from semiclassical considerations

We show that two simple semiclassical strategies, one based on the Wilson–Sommerfeld rule and the other on the uncertainty principle, yield the exact modified form of the virial theorem for confined systems. An alternative, easier quantum mechanical route to arrive at this result is also sketched. Pilot calculations on confined oscillators reveal decisive trends. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Determination of the rod-wire transition length in colloidal indium phosphide quantum rods

Colloidal InP quantum rods (QRs) having controlled diameters and lengths are grown by the solution-liquid-solid method, from Bi nanoparticles in the presence of hexadecylamine and other conventional quantum dot surfactants. These quantum rods show band-edge photoluminescence after HF photochemical etching. Photoluminescence efficiency is further enhanced after the Bi tips are selectively removed from the QRs by oleic acid etching. The QRs are anisotropically 3D confined, the nature of which is compared to the corresponding isotropic 3D confinement in quantum dots and 2D confinement in quantum wires. The 3D-2D rod-wire transition length is experimentally determined to be 25 nm, which is about 2 times the bulk InP exciton Bohr radius (of approximately 11 nm).