Mutual information and electron correlation in momentum space

Mutual information and information entropies in momentum space are proposed as measures of the nonlocal aspects of information. Singlet and triplet state members of the helium isoelectronic series are employed to examine Coulomb and Fermi correlations, and their manifestations, in both the position and momentum space mutual information measures. The triplet state measures exemplify that the magnitude of the spatial correlations relative to the momentum correlations depends on and may be controlled by the strength of the electronic correlation. The examination of one- and two-electron Shannon entropies in the triplet state series yields a crossover point, which is characterized by a localized momentum density. The mutual information density in momentum space illustrates that this localization is accompanied by strong correlation at small values of p.     

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.     

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 and Fisher information for screened Kratzer potential

In this paper, Shannon entropy and Fisher information is studied for the screened Kratzer potential model and compared with the screened Coulomb in three dimensions. Our results showed similar higher-order characteristic behavior for position and momentum space. Our numerical results showed that increases in the accuracy of predicting particle location occurred in the position space. Our result shows that the sum of the position and momentum entropies satisfies the lower-bound Berkner, Bialynicki-Birula, and Mycieslki inequality. The Stam-Cramer-Rao inequalities relation for Fisher information and the expectation values were also satisfied for the different eigenstates.     

Theoretic quantum information entropies for the generalized hyperbolic potential

The Shannon entropy (S) and the Fisher Information (I) entropies are investigated for a generalized hyperbolic potential in position and momentum spaces. First, the Schrodinger equation is solved exactly using the Nikiforov-Uvarov-Functional Analysis method to obtain the energy spectra and the corresponding wave function. By Fourier transforming the position space wave function, the corresponding momentum wave function was obtained for the low-lying states corresponding to the ground and first excited states. The positions and momentum of Shannon entropy and Fisher Information entropies were calculated numerically. Finally, the Bialynicki-Birula and Mycielski and the Stam-Cramer-Rao inequalities for the Shannon entropy and Fisher Information entropies, respectively, were tested and were found to be satisfied for all cases considered.     

Position and momentum information‐theoretic measures of the pseudoharmonic potential

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.     

Quantum information entropy of modified Hylleraas plus exponential Rosen Morse potential and squeezed states

In this article, some information theoretic concepts are analyzed for modified Hylleraas plus exponential Rosen Morse potential in position and momentum space. The angular and radial contributions of the information density are graphically demonstrated for different states. The entropy densities have asymmetric shape which depends on the values of quantum numbers. The information entropy is analytically obtained for ground state of the potential whereas the numerical calculations have been performed for the higher states and Bialynicki‐Birula and Mycielski inequality is tested for various states using different parameters of the potential. It is shown that the information entropy is reduced, both in position and momentum space, for careful selection of some parameters. Further, it is found that there exist eigenstates exhibiting squeezing in information entropy of modified Hylleraas plus exponential Rosen Morse and Eckart potential. Interestingly, in case of Eckart potential, the squeezed states are obtained in position as well as momentum space and are attempted to saturate for some values of the parameters.     

Influence of electronic correlation in monoelectronic density in p-space

The radial molecular monoelectronic density and their orbital contributions have been calculated in the momentum space. For these purposes, densities for the ground state of several atoms and molecules, using a cc-pVTZ basis set at HF level, as well as some post-HF and DFT methods are computed. The difference between the radial monoelectronic density computed with each method and that using the HF wave function is used as a tool to study the influence of the electronic correlation in the momentum space. Densities obtained with post HF calculations show a similar behavior around p = 1.0 and 2.0, that are different from the DFT results. Radial momentum densities (p-densities) are more influenced by the electronic correlation than the exchange part of the DFT methods. CISD p-density is more affected than DFT p-density when the intermolecular distance increases. An analysis of the powers of moments calculated with different methods has been carried out. Contribution to the Serafin Fraga Memorial Issue.     

Atomic quantum similarity indices in position and momentum spaces

Quantum similarity for atoms is investigated using electron densities in position and momentum spaces. Contrary to the results in position space, the analysis in the momentum space shows how the momentum density carries fundamental information about periodicity and structure of the system and reveals the pattern of Mendeleev's table. A global analysis in the joint r-p space keeps this result.     

Information entropy, information distances, and complexity in atoms

Shannon information entropies in position and momentum spaces and their sum are calculated as functions of Z(2 < or = Z < or = 54) in atoms. Roothaan-Hartree-Fock electron wave functions are used. The universal property S = a + b ln Z is verified. In addition, we calculate the Kullback-Leibler relative entropy, the Jensen-Shannon divergence, Onicescu's information energy, and a complexity measure recently proposed. Shell effects at closed-shell atoms are observed. The complexity measure shows local minima at the closed-shell atoms indicating that for the above atoms complexity decreases with respect to neighboring atoms. It is seen that complexity fluctuates around an average value, indicating that the atom cannot grow in complexity as Z increases. Onicescu's information energy is correlated with the ionization potential. Kullback distance and Jensen-Shannon distance are employed to compare Roothaan-Hartree-Fock density distributions with other densities of previous works.     

Maximization of atomic information-entropy sum in configuration and momentum spaces

The maximum entropy procedure (MEP ) of Jaynes has been extended to the case involving constraints in complementary spaces. It has been rigorously shown that the sum of information entropies in position and momentum spaces is invariant to uniform scaling of the electron coordinates. A new MEP procedure requires that this sum of entropies must be maximized subject to the known constraints in both spaces. A specific application of this maximization procedure for synthesizing atomic-electron densities in coordinate and momentum spaces has been outlined.     

Basis set quality. II. Information theoretic appraisal of various s- orbitals

The expectation values 〈r^k〉 (?2 ? k ? 4, k = 10), values of the charge density ρ(r) at selected points, and coefficients in the MacLaurin expansion of ρ(r) are used to test the quality of 71 orbital basis sets used for the atomic helium Hartree–Fock problem. These tests in position space are complementary to the momentum space tests previously carried out [Int. J. Quantum Chem. 21 , 419 (1982)]. Information theoretic measures with respect to either or both position and momentum space properties are subsequently defined and the orbitals are ranked accordingly. These measures indicate that, for a given orbital, momentum space properties are more poorly predicted than position space ones. Moreover an improvement in the virial ratio does not necessarily lead to a more balanced orbital with respect to position and momentum space properties. Basis sets containing Slater-type orbitals are again found to be rather accurate. The exponentially damped rational function is confirmed to be the outstanding two-parameter unconventional orbital.     

On-top pair-density interpretation of spin density functional theory,with applications to magnetism

The on-top pair density P(r, r) gives the probability that one electron will be found on top of another at position r. We find that the local spin density (LSD) and generalized gradient (GGA) approximations for exchange and correlation predict this quantity with remarkable accuracy. We show how this fact and the usual sum-rule arguments explain the success of these approximations for real atoms, molecules, and solids, where the electron spin densities do not vary slowly over space. Self-consistent LSD or GGA calculations make realistic predictions for the total energy E, the total density n(r), and the on-top pair density P(r,r), even in those strongly “abnormal” systems (such as stretched H₂) where these approximations break symmetries and yield unrealistic spin magnetization densities m(r). We then suggest that ground-state ferromagnetic iron is a “normal” system, for which for LSD or GGA m(r) and the related local spin moment are trustworthy, but that iron above the Curie temperature and antiferromagnetic clusters at all temperatures are abnormal system for which the on-top pair density interpretation is more viable than the standard physical interpretation. As an example of a weakly abnormal system, we consider the four-electron ion with nuclear charge Z → ∞ © 1997 John Wiley & Sons, Inc.     

Measuring localization-delocalization phenomena in a quantum corral

The standard deviations and Shannon information entropies of the probability densities for a particle in a quantum corral are compared and contrasted to determine their effectiveness in measuring particle (de)localization. We illustrate how the two measures emphasize different aspects of the underlying distributions which can lead to inconsistent interpretations. Among these, we show that the Shannon entropy is able to distinguish between the presence of an attractive or repulsive effective potential in the radial Schrödinger equation while the standard deviation does not. The analysis of this radial model is then extended to momentum space where the dependence of the measures, entropic sum and uncertainty product on the effective potential, is examined.     

Atomic shells from the electron localizability in momentum space

The electron localizability indicator in momentum space is proposed as a functional of the same‐spin momentum pair density. This functional yields a discrete distribution of values, which are proportional to the charge needed to form a fixed very small fraction of a same‐spin electron pair. It resolves all atomic shells for the examined atoms (Li–Kr) with reasonable occupation numbers, especially in the valence region. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Diatomic interactions in momentum space. The effect of polarization and floating functions

The method of momentum density for interatomic interactions is used to investigate the pictures and roles of the polarization and floating functions in momentum (p-) space. Referring to the previous results from the minimal LCAO (Finkelstein-Horowitz) momentum density, we quantitatively discuss the effect of these functions for the bonding process in the ground state of H ₂ ⁺ system. The essence of the polarization and floating effects is found to be a modulation of the oscillation in the two-center part of the momentum density. The polarization function introduces a term with a phase and the floating function enlarges the period of the oscillation. An increased migration of the density from the one-center to the two-center part is also important. As a result, both the functions contribute to emphasize the contraction and expansion of momentum density observed previously. However, the floating function disturbs the density distribution in high momentum region, reflecting the destruction of cusps in position (r-) space. We point out an error in the pioneer work of Duncanson.     

Momentum density interpretation of the H(1s)–H+ long-range force

The recently proposed method of using momentum densities for interatomic interactions is applied to the long-range force between the ground-state hydrogen atom and the proton, and the results are compared with those from using the position density based on the electrostatic Hellmann–Feynaman theorem. A new physical interpretation of the long-range force is obtained, which is complementary to that in position space. It is found that some perturbative changes in the position density do not accompany changes in the momentum density.     

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 .