Analytical Shannon information entropies for all discrete multidimensional hydrogenic states

The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work, the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers (n, μ₁, μ₂, …, μ _D−1), the system's dimensionality D and the nuclear charge Z in an analytical, compact form. Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform to a relevant class of specific states of the D-dimensional hydrogenic system characterized by the hyperquantum numbers μ₁ = μ₂ … = μ _D−1 = n − 1, including the ground state.     

Information theory of D‐dimensional hydrogenic systems: Application to circular and Rydberg states

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     

Information theoretical measures from cumulative and survival densities in quantum systems

Entropic uncertainty and statistical correlation measures, based on survival and cumulative densities, are explored in some representative quantum systems. We illustrate how the cumulative residual entropy in the quantum well system recovers the correct classical behavior for larger quantum numbers while the Shannon entropy does not. Two interacting and noninteracting oscillators are used to examine two‐particle entropies and their related correlation measures. The joint cumulative residual entropy does distinguish between symmetric and antisymmetric wave functions in interacting systems as the interaction is turned on. The joint Shannon entropy does not distinguish between the symmetries even in the presence of interaction. Conversely, the joint Shannon entropy and joint cumulative residual entropy are both unable to distinguish between the symmetries for certain states of the noninteracting oscillators. As measures of statistical correlation, the mutual information and the cross cumulative residual entropy yield similar behaviors as a function of the strength of the interparticle interaction.     

Two-dimensional confined hydrogen: An entropy and complexity approach

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.     

A Closed-Form Relation for Dimension-Dependent Two-Electron Matrix Elements of the Interelectronic Distance

The evaluation of matrix elements of two electron atoms is fundamental for the study of the electronic properties of those systems. We add to this knowledge by presenting an explicit expression for the matrix elements of the inverse of the interelectronic distance of two-electron atoms in any spatial dimension D. The basis functions used are the D-dependent hydrogenic wavefunctions {1s ²,2p ²,3d ²,4f ²,5g ²,...,21y ²,...}, extending and including, in this way, the results of the previous basis set {1s ²,2p ²,3d ²,4f ²}. The methodology used does not employ Fourier integral transforms as in previous works but hypergeometric transformation formulas.     

Effect of magnetic field on diamagnetic susceptibility of two interacting electrons in a quantum dot

We have estimated the energy levels of the low‐lying states as a function of magnetic field when two electrons are introduced in a quantum dot (QD). Oscillator strength of interacting electrons for different magnetic field strengths has been calculated. There is no appreciable change in oscillator strength for stronger confinements for all the magnetic field strengths. We present the shift of diamagnetic susceptibility of a hydrogenic donor impurity in GaAs/Ga_1?xAl_xAs QD systems for the ground and low lying excited states. The effect of magnetic field on diamagnetic susceptibilities is estimated by two different methods and it has been found that values obtained from both the methods resemble each other. The diamagnetic shift is in good agreement with the other investigators. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

A spherically averaged representation of the atomic one-particle reduced density matrix

Summary A new way of representing the one-particle reduced density matrix (ODM) of closed-shell atoms in a spherically averaged manner is presented, and connections of this representation to the radial density distributionD(R) and the isotropic reciprocal form factorB(s) are shown. In this representation, certain characteristics of the angular nodal structure of the natural orbitals (NOs) are preserved. Examples of hydrogenic orbitals and near-Hartree-Fock wave functions for some closed-shell atoms are given.     

Correlation entropy of the H2 molecule

Correlation of a quantum many-body state makes the one-particle density matrix nonidempotent. Therefore, the Shannon entropy of the natural occupation numbers measures the correlation strength on the one-particle level. Here, it is shown how this general idea of a correlation entropy must be adapted for two-electron systems in view of conservation laws which mix Slater determinants even in the noninteracting limit. Results are presented for the correlation entropy s of H₂ as a function of the nucleus-nucleus separation R. In the ground state, the entropy of the spatial factor of the wave function maximizes 1.7 bohr beyond the Coulson-Fischer separation. The role of the correlation entropy in density functional theory is also discussed. © 1997 John Wiley & Sons, Inc.     

Exploiting the interdependence of maximum entropy parameters in eigenvalue problems

We exploit the interrelation among the parameters embedded in the maximum entropy ansatz to develop a scheme for obtaining accurate estimates of the ground‐state energy and wave function of systems for which the potential is represented by a rational function. Our scheme reduces an N‐parameter optimization problem to a two‐parameter one, leading to considerable simplification of the prevalent strategy. An indirect route for the study of excited states is also sketched. Test calculations on hydrogenic systems subject to strong or superstrong radial magnetic fields with and without electric field reveal the advantages of our approach. Additional studies on 1‐D anharmonic oscillators affirm its workability and generality. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Information entropies of many-electron systems

The Boltzmann–Shannon (BS ) information entropy S_ρ = ∫ ρ(r)log ρ(r)dr measures the spread or extent of the one-electron density ρ(r), which is the basic variable of the density function theory of the many electron systems. This quantity cannot be analytically computed, not even for simple quantum mechanical systems such as, e.g., the harmonic oscillator (HO ) and the hydrogen atom (HA ) in arbitrary excited states. Here, we first review (i) the present knowledge and open problems in the analytical determination of the BS entropies for the HO and HA systems in both position and momentum spaces and (ii) the known rigorous lower and upper bounds to the position and momentum BS entropies of many-electron systems in terms of the radial expectation values in the corresponding space. Then, we find general inequalities which relate the BS entropies and various density functionals. Particular cases of these results are rigorous relationships of the BS entropies and some relevant density functionals (e.g., the Thomas–Fermi kinetic energy, the Dirac–Slater exchange energy, the average electron density) for finite many-electron systems. © 1995 John Wiley & Sons, Inc.     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

Quantumness of correlations and Maxwell's demon in molecular excitations created by neutron scattering

Nonclassical correlations known as entanglement, quantum discord, quantum deficit, measurement‐induced disturbance, quantum Maxwell's demon, etc., may provide novel insights into quantum‐information processing, quantum‐thermodynamics processes, open‐system dynamics, quantum molecular dynamics, and general quantum chemistry. We study a new effect of quantumness of correlations accompanying collision of two distinguishable quantum systems A and B, the latter being part of a larger (interacting) system B + D. In contrast to the common assumption of a classical environment or “demon” D, the quantum case exhibits striking new qualitative features. Here, in the context of incoherent inelastic neutron scattering from H‐atoms which create molecular excitations (vibration, rotation, translation), we report theoretical and experimental evidence of a new phenomenon: a considerably reduced effective mass of H, or equivalently, an anomalous momentum‐transfer deficit in the neutron‐H collision. These findings contradict conventional theoretical expectations even qualitatively, but find a straightforward interpretation in the new theoretical frame under consideration. © 2015 Wiley Periodicals, Inc.     

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

In this work, we study the position and momentum information entropies of multiple quantum well systems in fractional Schrödinger equations, which, to the best of our knowledge, have not so far been studied. Through a confining potential, their shape and number of wells (NOW) can be controlled by using a few tuning parameters; we present some interesting quantum effects that only appear in the fractional Schrödinger equation systems. One of the parameters denoted by the L_d can affect the position and momentum probability densities if the system is fractional (1 < α < 2). We find that the position (momentum) probability density tends to be more severely localized (delocalized) in more fractional systems (ie, in smaller values of α). Affecting the L_d on the position and momentum probability densities is a quantum effect that only appears in the fractional Schrödinger equations. Finally, we show that the Beckner Bialynicki-Birula-Mycieslki (BBM) inequality in the fractional Schrödinger equation is still satisfied by changing the confining potential amplitude V_conf, the NOW, the fractional parameter α, and the confining potential parameter L_d .     

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.     

A relationship between atomic correlation energy and Tsallis entropy

A new relationship between electron correlation energy and Tsallis entropy is presented. This relationship is a generalization of previous equations which correlate the atomic correlation energy and the Shannon entropy. The results, relatively to the atoms with atomic number 2 < Z < 29, put in evidence the crucial role of the p‐parameter in terms of representation of the long‐range interaction contribution in the correlation energy. Moreover, the p‐values, which reproduce the experimental values of the correlation energy, indicate that the atomic wave functions are more localized with respect to those calculated in the limit of p → 1. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Approach of the associated Laguerre functions to the su(1,1) coherent states for some quantum solvable models

Using second‐order differential operators as a realization of the su(1,1) Lie algebra by the associated Laguerre functions, it is shown that the quantum states of the Calogero‐Sutherland, half‐oscillator and radial part of a 3D harmonic oscillator constitute the unitary representations for the same algebra. This su(1,1) Lie algebra symmetry leads to derivation of the Barut‐Girardello and Klauder‐Perelomov coherent states for those models. The explicit compact forms of these coherent states are calculated. Also, to realize the resolution of the identity, their corresponding positive definite measures on the complex plane are obtained in terms of the known functions. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Entropic measures of Rydberg‐like harmonic states

The Shannon entropy, the desequilibrium and their generalizations (Rényi and Tsallis entropies) of the three‐dimensional single‐particle systems in a spherically symmetric potential V(r) can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this article, we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillator‐like systems. The angular and radial contributions to these entropic measures are analytically expressed in terms of the quantum numbers which characterize the corresponding quantum states and, for the radial part, the oscillator strength. In the latter case, we use some recent powerful results of the information theory of the Laguerre polynomials and spherical harmonics which control the oscillator‐like wavefunctions.     

On the shell-confined atom problem

The properties of spherically symmetric, one-electron atomic systems are analyzed for the systems where the electron is placed in the region between two impenetrable concentric spheres of radii R_int and R_ext. The changes in the structure of the energy levels of low-lying electron states and the corresponding variations in radial wave functions are analyzed for different types of changes in R_int and R_ext. Special attention is paid to the changes in electric dipole polarizability for problems with varied dimension, D, of the space. The form and the quality of some simple classical estimates for polarizability are also studied for various space dimensions.     

Application of Shannon-like diversity measures to cell-based chemistry spaces

The use of multi-dimensional “chemistry spaces” to represent large compound collections has become widespread in pharmaceutical research. In such spaces compounds are treated as points. Points in close proximity represent similar compounds, while distant points represent dissimilar compounds. Assessing the diversity of a compound collection, thus, is tantamount to characterizing the distribution of points in chemistry space. To facilitate many procedures such as selecting subsets of compounds for screening, for compound acquisition and designing combinatorial libraries, chemistry spaces have been partitioned into sets of non-overlapping, multi-dimensional cells, which are generated by dividing each axis into a number of equal-sized bins. This leads to a lattice of (N_bins)^N_dim{(N_{bins})^{N_{\rm dim}}} cells, where N _bins is the number of bins on each axis and N _dim is the dimensionality of the space. One diversity measure that is typically used in cell-based chemistry spaces is identical in form to Shannon entropy, D_Ncpd^cpd{D_{N_{cpd}}^{cpd}} A normalized measure of this Shannon entropy given by, D_rel^cpd{D_{rel}^{cpd}} enables comparison between compound collections that occupy different number of occupied cells. Although D_rel^cpd{D_{rel}^{cpd}} characterizes the uniformity and “spreadout” of the corresponding compound collection, it treats cells as positionally independent. Some of the positional information lost can be recaptured by another diversity measure, which is also related in form to Shannon entropy. This new measure D_Nbin^cell (l){D_{N_{bin}}^{cell} (\lambda)} characterizes the distribution of occupied cells along each axis of chemistry space. The normalized measure á D_rel^cell ñ{\left\langle {D_{rel}^{cell}}\right\rangle} over all axes is given then by the average. Examples illustrating the applicability of these two Shannon-like measures to compound collections are presented.