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Agencia Andaluza del Conocimiento of the Junta de Andalucia (Spain), Grant/Award Number: PY20-00082; European Regional Development Fund (FEDER), Grant/Award Number: PID2020-113390GB-I00; Agencia Estatal de Investigacion (Spain), Grant/Award Number: FIS2017-89349P; Basque Government and UPV/EHU, Grant/Award Number: IT1249-19, The Jacobi polynomials Poa,ss THORN n oxTHORN conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function o1 xTHORNao1thornxTHORN ss, a,ss > 1, on the interval 1/21, thorn1. The spreading of its associated probability density (i.e., the Rakhmanov density) over the support interval has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic Lq-norms (Shannon and Renyi entropies) and the monotonic complexity-like measures of Cramer-Rao, Fisher-Shannon, and LMC (LopezRuiz, Mancini, and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non-handy way; specially when the degree or the parameters oa, ss THORN are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity-like properties of the Jacobi polynomials in the two extreme situations: (n!8; fixed a, ss) and (a!8; fixed n, ss). These two asymptotics are relevant per se and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly, and quasi-exactly solvable quantum- mechanical potentials which model numerous atomic and molecular systems., Junta de Andalucia PY20-00082, European Commission PID2020-113390GB-I00, Agencia Estatal de Investigacion (Spain) FIS2017-89349P, Basque Government IT1249-19, UPV/EHU IT1249-19

The quantum entanglement of the two components of a hydrogenic system with dimensionality $$D\ge 2$$ is investigated for the ground and excited states from first principles, that is, in terms of the Coulomb potential parameters (the dimensionality and the nuclear charge) and the state’s hyperquantum numbers. To quantify this multidimensional entanglement, we use an heuristic quantifier and a practical genuine entanglement measure which are closely related to the variance and disequilibrium of the system in momentum space, respectively. Then, our interest is focused on the multidimensional entanglement of highly excited (Rydberg) states, obtaining at the leading order a simple dependence on the dimensionality and the principal hyperquantum number n which characterizes the state. Applications to various specific low-lying and high-lying hydrogenic states are shown. In particular, it is rigorously shown that the momentum disequilibrium and the entanglement for the Rydberg multidimensional states follow a scaling law of $$n^2\log \,n$$ and $$n^{2(D-1)}$$ type for two-dimensional and D-dimensional ( $$D>2$$ ) systems, respectively.

This work was partially supported by the Agencia Estatal de Investigacion (Spain) and the European Regional Development Fund (FEDER) under the grant PID2020-113390GB-I00., In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the Lq-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant., Agencia Estatal de Investigacion (Spain), European Commission PID2020-113390GB-I00

The spreading properties of the stationary states of the quantum multidimensional harmonic oscillator are analytically discussed by means of the main dispersion measures (radial expectation values) and the fundamental entropy-like quantities (Fisher information, Shannon and Renyi entropies, disequilibrium) of its quantum probability distribution together with their associated uncertainty relations. They are explicitly given, at times in a closed compact form, by means of the potential parameters (oscillator strength, dimensionality, D) and the hyperquantum numbers $$(n_r,\mu _1,\mu _2,\ldots ,\mu _{D-1})$$ which characterize the state. Emphasis is placed on the highly excited Rydberg (high radial hyperquantum number $$n_r$$ , fixed D) and the high-dimensional (high D, fixed hyperquantum numbers) states. We have used a methodology where the theoretical determination of the integral functionals of the Laguerre and Gegenbauer polynomials, which describe the spreading quantities, leans heavily on the algebraic properties and asymptotical behavior of some weighted $${\mathfrak {L}}_{q}$$ -norms of these orthogonal functions.

This work has been done in the framework of the Grant FIS2017-89349P of the Agencia Estatal de Investigacion (Spain)) and the European Regional Development Fund (FEDER), and the Grant FQM-207 of the Agencia de Innovacion y Desarrollo de Andalucia., The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Renyi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state's angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies-Thakkar, Lieb-Thirring, Redheffer-Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies., Agencia Estatal de Investigacion (Spain) FIS2017-89349P, European Commission FIS2017-89349P, Agencia de Innovacion y Desarrollo de Andalucia FQM-207

The quasi-homogeneous systems have important properties and they have been studied from various points of view. In this work, we provide the classification of quasi-homogeneous systems on the basis of the weight vector concept, especially in terms of the minimum weight vector, which is proved to be unique for any quasi-homogeneous system. Later we obtain the exact number of different forms of non-homogeneous quasi-homogeneous systems of arbitrary degree, proving a nice relation between this number and Euler’s totient function. Finally, we provide software implementations for some of the above results, and also for the algorithm, recently published by Garcia et al., that generates all the quasi-homogeneous systems.

The Shannon entropy, the desequilibrium and their generalizations (Renyi 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.

The relative structural location of a selected group of 27 sulfonamide-like molecules in a chemical space defined by three information theory quantities (Shannon entropy, Fisher information, and disequilibrium) is discussed. This group is composed of 15 active bacteriostatic molecules, 11 theoretically designed ones, and para-aminobenzoic acid. This endeavor allows molecules that share common chemical properties through the molecular backbone, but with significant differences in the identity of the chemical substituents, which might result in bacteriostatic activity, to be structurally classified and characterized. This is performed by quantifying the structural changes on the electron density distribution due to different functional groups and number of electrons. The macroscopic molecular features are described by means of the entropy-like notions of spatial electronic delocalization, order, and uniformity. Hence, an information theory three-dimensional space (IT-3D) emerges that allows molecules with common properties to be gathered. This space witnesses the biological activity of the sulfonamides. Some structural aspects and information theory properties can be associated, as a result of the IT-3D chemical space, with the bacteriostatic activity of these molecules. Most interesting is that the active bacteriostatic molecules are more similar to para-aminobenzoic acid than to the theoretically designed analogues.

In this work the momentum spreading of a multidimensional hydrogenic system in highly excited (Rydberg) states is quantified by means of the Rényi and Shannon entropies of its momentum probability density. These quantities, which rest at the core of numerous fields from atomic and molecular physics to quantum technologies, are determined by means of a methodology based on the strong degree-asymptotics of a modified L q -norm of the Gegenbauer polynomials which control the wavefunctions of these states in momentum space. The leading term of these entropic quantities is found from first principles, i.e. by means of the Coulomb potential parameters (space dimensionality, the nuclear charge) and the states’s hyperquantum numbers, in a rigorous simple manner. It is shown that they fulfill a logarithmic growth scaling law with the principal hyperquantum number n which characterizes the Rydberg state.

We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the three main intrinsic measures of complexity (Cramer-Rao, Fisher-Shannon, LMC) and some of their generalizations., Comment: Version accepted in Physics Letters A

The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue \textit{per se} and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowest-lying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and R\'enyi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a $D$-dimensional quadratic potential, $D > 1$) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the R\'enyi uncertainty measures of the $D$-dimensional harmonic system for all ground and excited quantum states directly in terms of $D$, the potential strength and the hyperquantum numbers., Comment: Accepted in EPJ-ST

In this work we undertake a pioneer information-theoretical analysis of 18 selected amino acids extracted from a natural protein, bacteriorhodopsin (1C3W). The conformational structures of each amino acid are analyzed by use of various quantum chemistry methodologies at high levels of theory: HF, M062X and CISD(Full). The Shannon entropy, Fisher information and disequilibrium are determined to grasp the spatial spreading features of delocalizability, order and uniformity of the optimized structures. These three entropic measures uniquely characterize all amino acids through a predominant information-theoretic quality scheme (PIQS), which gathers all chemical families by means of three major spreading features: delocalization, narrowness and uniformity. This scheme recognizes four major chemical families: aliphatic (delocalized), aromatic (delocalized), electro-attractive (narrowed) and tiny (uniform). All chemical families recognized by the existing energy-based classifications are embraced by this entropic scheme. Finally, novel chemical patterns are shown in the information planes associated with the PIQS entropic measures.

Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named ( p , β , λ ) -Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function ρ characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of ρ , and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the ( p , β , λ ) -Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential., The authors are very grateful to the CNRS (Steeve Zozor) and the Junta de Andalucía and the MINECO–FEDER under the grants FIS2014–54497 and FIS2014–59311P (Jesús Sánchez-Dehesa) for partial financial support.

The basic variables of the information theory of quantum systems (e.g., frequency or entropic moments, Renyi and Tsallis entropies) can be expressed in terms of $$L_{q}$$ norms of general hypergeometrical polynomials. These polynomials are known to control the radial and angular parts of the wavefunctions of the quantum-mechanically allowed states of numerous physical and chemical systems. The computation of the $$L_{q}$$ norms of these polynomials is presently an interesting issue per se in the theory of special functions; moreover, these quantities are closely related to the frequency moments and other information-theoretic properties of the associated Rakhmanov probability density. In this paper we calculate the unweighted and weighted $$L_{q}$$ -norms $$(q=2k, k\in {\mathbb {N}})$$ of general hypergeometric real orthogonal polynomials (Hermite, Laguerre and Jacobi) and some entropy-like integrals of Bessel polynomials, in terms of $$q$$ and the parameters of the corresponding weight function by using their explicit expression and second order differential equation. In addition, the asymptotics $$(q\rightarrow \infty )$$ of the unweighted $$L_{q}$$ norms of the Jacobi polynomials is determined by the Laplace method.

The quantification of the spreading of the orthogonal polynomials p"n(x) can be investigated by means of the Renyi entropies R"q[@r],q being a positive integer number, of the associated Rakhmanov probability densities, @r(x)[email protected](x)p"n^2(x), where @w(x) is the corresponding weight function. The Renyi entropies are closely related to the L"q-norms of the polynomials. In this manuscript, the L"q-norms and the associated Renyi entropies of the real hypergeometric orthogonal polynomials (i.e., Hermite, Laguerre, and Jacobi polynomials) and the generalized Hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of Lauricella and Srivastava-Daoust types which are evaluated at some specific values of 2q variables. These functions depend on 4q+1 and 6q+2 parameters, respectively, which are determined by the order q, the degree n of the polynomial, and the parameters of the orthogonality weight function @w(x). The key idea is based on some extended linearization formulas for these polynomials. These results open the way to determine the Renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials.

An information-theoretical analysis of the SN2 exchange reaction for CH3Cl + F− is performed in both position and momentum spaces by means of the following functionals of the one-particle density: Shannon entropy (S), disequilibrium (D), and Fisher information (I). We have shown that the information-theoretical characterization of the chemical course of the reaction is in complete agreement with its phenomenological behavior in the passage from reactants to products. These kinds of analyses permit to reveal all the concomitant physical processes involved in the reaction: charge transferring, bond breaking, electrostatic equalization, bond forming, and electrostatic repulsion. It is worth emphasizing that information theory concepts, such as localization, order, and uniformity, offers a unique advantage as a complementary reactivity theory to provide a full description of chemical reactions, revealing chemical aspects of reactions which are not present in the energy-based interpretative approach. © 2013 Wiley Periodicals, Inc.

The Cramer–Rao product of the Fisher information F [ ρ ] and the variance 〈 x 2 〉 ≡ ∫ x 2 ρ ( x ) d x of a probability density ρ ( x ) , defined on a domain Ω ⊂ R D , is found to have a minimum value reached by the density associated with the ground state of the harmonic oscillator in Ω , when Ω is an unbounded domain. If Ω is bounded, the minimum value of the Fisher information is achieved by the ground state of the quantum box described itself by this domain.

In this letter, we present an Information-Theoretic Space generated from the Shannon entropy, the Fisher information, and the disequilibrium measures along with their corresponding Fisher–Shannon and LMC complexity measures, which unveils the unique physical, chemical and biological aspects of a great diversity of many-electron systems, ranging from neutral and ionized atomic systems and simple molecules to much more complex systems such as amino acids and pharmacological molecular ensembles. This endeavour is achieved upon the physical ground that atoms and molecules can be described through the basic information-theoretic notions of delocalization, order, uniformity and complexity, thus revealing that a universal three-dimensional information-theoretic chemical space for all systems of Nature there might exist.

The R\'enyi entropies $R_{p}[\rho]$, $p>0,\neq 1$ of the highly-excited quantum states of the $D$-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability $\rho_n(\vec{r})$ density of the Rydberg wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the asymptotics $n\to\infty$ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the R\'enyi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers ($n,l$), the parameter order $p$ and the universe dimensionality $D$ for all possible cases $D\ge 1$. We find that (a) the R\'enyi entropy power decreases monotonically as the order $p$ is increasing and (b) the disequilibrium (closely related to the second order R\'enyi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of $D$., Comment: Accepted in The European Physical Journal B (EPJ B)

The fundamental information-theoretic measures (the Renyi R p [ ρ ] and Tsallis T p [ ρ ] entropies, p > 0 ) of the highly-excited (Rydberg) quantum states of the D -dimensional ( D > 1 ) hydrogenic systems, which include the Shannon entropy ( p → 1 ) and the disequilibrium ( p = 2 ), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability ρ ( r → ) densities associated to the Rydberg hydrogenic wavefunctions Ψ n , l , { μ } ( r → ) , which are closely connected to the L p -norms of the associated Laguerre polynomials. Then, we determine the ( n → ∞ )-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality D , the nuclear charge and the hyperquantum numbers ( n , l , { μ } ) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states.

In this work, we have investigated in position (r) and momentum (p) spaces the concurrent phenomena occurring at the vicinity of the transition state (TS) (the so-called transition region) of selected chemical reactions (such as the hydrogenic abstraction and the exchange hydrogenic reactions) by means of a broad set of single information-theoretic functionals of the one-particle density (such as the disequilibrium (D), exponential entropy (L), Fisher information (I) and the power entropy (J)) and composite information-theoretic measures which includes various information planes (such as the I-D, D-L, and I-J planes) and complexities of the Fisher-Shannon and Lopez-Mancini-Calbet (LMC) types. The analysis of the single functionals and the information planes revealed that these information-theoretical elements can identify all the chemically significant regions, not only the reactant/product regions (R/P) and the TS but also those that are not present in the energy profile, such as the bond cleavage energy region (BCER), the bond breaking/forming regions (B-B/F) and the charge transfer complex. Moreover, the analysis of the complexities shows that in position as well as in the joint (r-p) spaces, the energy profile of the abstraction reaction bears the same information-theoretical features of the LMC and FS measures. Finally, it is shown why most of the chemical features of interest (such as e.g., BCER and B-B/F) are lost in the energy profile, being only revealed when particular information-theoretical aspects of localizability (L or J), uniformity (D) and disorder (I) are considered. © 2012 Wiley Periodicals, Inc.

We investigate the complexity of the hydrogenic identity S N 2 exchange reaction by means of information-theoretic functionals such as disequilibrium (D), exponential entropy (L), Fisher information (I), power entropy (J) and joint information-theoretic measures, i.e., the I–D, D–L and I–J planes and the Fisher–Shannon (FS) and Lopez-Mancini-Calbet (LMC) shape complexities. The several information-theoretic measures of the one-particle density were computed in position (r) and momentum (p) spaces. The analysis revealed that the chemically significant regions of this reaction can be identified through most of the information-theoretic functionals or planes, not only the ones which are commonly revealed by the energy, such as the reactant/product (R/P) and the transition state (TS), but also those that are not present in the energy profile such as the bond cleavage energy region (BCER), the bond breaking/forming regions (B–B/F) and the charge transfer process (CT). The analysis of the complexities shows that the energy profile of the identity S N 2 exchange reaction bears no simple behavior with respect to the LMC and FS measures. Most of the chemical features of interest (BCER, B–B/F and CT) are only revealed when particular information-theoretic aspects of localizability (L or J), uniformity (D) and disorder (I) are considered.

We determine families of spherically symmetrical D -dimensional quantum potential functions V ( r ) having ground-state wavefunctions that exhibit, either in configuration space or in momentum space, the form of an isotropic q -Gaussian. These wavefunctions admit a maximum-entropy description in terms of S q power-law entropies. We show that the potentials with a ground state of the q -Gaussian form in momentum space admit the Coulomb potential − 1 / r as a particular instance. Furthermore, all these potentials behave asymptotically as the Coulomb potential for large r for all values of the parameter q such that 0 q 1 .

The utility of the Fisher information measure is analyzed to detect the transition state, the stationary points of a chemical reaction, and the bond breaking/forming regions of elementary reactions such as the simplest hydrogen abstraction and the identity SN2 exchange ones. This is performed by following the intrinsic reaction path calculated at the MP2 and QCISD(T) levels of theory with a 6-311++G(3df, 2p) basis set. Selected descriptors of both position and momentum space densities are utilized to support the observations, such as the molecular electrostatic potential (MEP), the hardness, the dipole moment, along with geometrical parameters. Our results support the concept of a continuum of transient of Zewail and Polanyi for the transition state rather than a single state, which is also in agreement with reaction force analyses.

In this work, we have investigated the complexity of the hydrogenic abstraction reaction by means of information functionals such as disequilibrium (D), exponential entropy (L), Fisher information (I), power entropy (J) and joint information-theoretic measures, i.e. the I–D, D–L and I–J planes and the Fisher–Shannon and Lopez–Mancini–Calbet (LMC) shape complexities. The analysis of the information-theoretical functionals of the one-particle density was computed in position (r) and momentum (p) space. The analysis revealed that all of the chemically significant regions can be identified from the information functionals and most of the information-theoretical planes, i.e. the reactant/product regions (R/P), the transition state (TS), including those that are not present in the energy profile such as the bond cleavage energy region (BCER), and the bond breaking/forming regions (B–B/F). The analysis of the complexities shows that, in position as well as in the joint space, the energy profile of the abstractio...

The Fisher information has a locality property which allows one to quantify the gradient content of the quantum-mechanical states of a physical system, so grasping the distinctive oscillatory character of the corresponding wavefunctions. We discuss the uncertainty relation fulfilled by the position and momentum Fisher information for general systems, and the tighter uncertainty relation satisfied by the eigenfunctions of arbitrary central potentials. Moreover, we show the usefulness of these two quantities together with the measure of the steric effect for the explanation of the origin of the internal rotation barrier between the eclipsed and staggered conformers of ethane.

On the basis of a density-based quantification of the steric effect [Liu, S. B. J. Chem. Phys.2007, 126, 244103], the origin of the internal rotation barrier between the eclipsed and staggered conformers of ethane is systematically investigated in this work from an information-theoretical point of view by using the Fisher information measure in conjugated spaces. Two kinds of computational approaches are considered in this work: adiabatic (with optimal structure) and vertical (with fixed geometry). The analyses are performed systematically by following, in each case, the conformeric path by changing the dihedral angle from 0 to 180° . This is calculated at the HF, MP2, B3LYP, and CCSD(T) levels of theory and with several basis sets. Selected descriptors of the densities are utilized to support the observations. Our results show that in the adiabatic case the eclipsed conformer possesses a larger steric repulsion than the staggered conformer, but in the vertical cases the staggered conformer retains a larger steric repulsion. Our results verify the plausibility for defining and computing the steric effect in the post-Hartree-Fock level of theory according to the scheme proposed by Liu.

The macroscopic properties of a quantum system strongly depend on the spreading of the physical eigenfunctions (wavefunctions) of its Hamiltonian operador over its confined domain. The wavefunctions are often controlled by classical or hypergeometric-type orthogonal polynomials (Hermite, Laguerre and Jacobi). Here we discuss the spreading of these polynomials over its orthogonality interval by means of various information-theoretic quantities which grasp some facets of the polynomial distribution not yet analyzed. We consider the information-theoretic lengths closely related to the Fisher information and R\'enyi and Shannon entropies, which quantify the polynomial spreading far beyond the celebrated standard deviation., Comment: 17 pages

The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean-square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q is a natural number) is also found in terms of the polynomials parameters by means of two error-free computing approaches; one makes use of the Lauricella functions, which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n>>1)., 15 pages, 5 figures, accepted in Journal of Computational and Applied Mathematics

Complexity theory embodies some of the hardest, most fundamental and most challenging open problems in modern science. The very term complexity is very elusive, so that the main goal of this theory is to find meaningful quantifiers for it. In fact we need various measures to take into account the multiple facets of this term. Here some biparametric Cr\'amer-Rao and Heisenberg-R\'enyi measures of complexity of continuous probability distributions are defined and discussed. Then, they are applied to the blackbody radiation at temperature T in a d-dimensional universe. It is found that these dimensionless quantities do not depend on T nor on any physical constants. So, they have an universal character in the sense that they only depend on the spatial dimensionality. To determine these complexity quantifiers we have calculated their dispersion (typical deviations) and entropy (R\'enyi entropies and the generalized Fisher information) constituents. They are found to have a temperature-dependent behavior similar to the celebrated Wien's displacement law of the dominant frequency $\nu_{max}$ at which the spectrum reaches its maximum. Moreover, they allow us to gain insights into new aspects of the d-dimensional blackbody spectrum and about the quantification of quantum effects associated with space dimensionality.

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

The problem of finding the information-extremizer distribution with a set of given constraints (partial information) has a relevant role from both physical and chemical points of view, specially when working within a density functional theory framework. Beyond the variance, there exist different measures of information susceptible of being extremized, such as the Fisher information and the Shannon and Tsallis entropies. Each one possesses its own properties which make their use more or less convenient according to the systems and/or the process we are dealing with. In this work, we analyze the main information measures of the electron densities of neutral atoms throughout the periodic table, in the two conjugated position and momentum spaces. It is shown how these measures display shell-filling patterns, within a level which depends on the information measure and the space considered. Additionally, the values of all these measures for the solution of various atomic information extremization problems (MaxEnt, MaxTent, MinInf), using radial expectation values as constraints, are analytically obtained, numerically evaluated and also interpreted and discussed in terms of physical characteristics of the atomic systems, such as periodicity and shell structure.

Given a nontrivial Borel measure on ℝ, let pn be the corresponding orthonormal polynomial of degree n whose zeros are λj(n), j=1,…,n. Then for each j=1,…,n, $$\vec{\Psi}_{j}^{2}\stackrel {\mathrm {def}}{=}\bigl(\Psi_{1j}^{2},\dots,\Psi_{nj}^{2}\bigr)$$ with $$\Psi_{ij}^{2}=p_{i-1}^{2}\bigl(\lambda_{j}^{(n)}\bigr)\Biggl(\sum_{k=0}^{n-1}p_{k}^{2}\bigl(\lambda_{j}^{(n)}\bigr)\Biggr)^{-1},\quad i=1,\dots,n,$$ defines a discrete probability distribution. The Shannon entropy of the sequence {pn} is consequently defined as $$\mathcal{S}_{n,j}\stackrel {\mathrm {def}}{=}-\sum_{i=1}^{n}\Psi_{ij}^{2}\log\bigl(\Psi_{ij}^{2}\bigr).$$ In the case of Chebyshev polynomials of the first and second kinds, an explicit and closed formula for \(\mathcal{S}_{n,j}\) is obtained, revealing interesting connections with number theory. In addition, several results of numerical computations exemplifying the behavior of \(\mathcal{S}_{n,j}\) for other families are presented.

The inequality , with L being the grand orbital quantum number, and its conjugate relation for (r2, p−2) are shown to be fulfilled in the D-dimensional central problem. Their use has allowed us to improve the Fisher-information-based uncertainty relation (IρIγ≥ const) and the Cramer–Rao inequalities (r2Iρ ≥ D2; p2Iγ ≥ D2). In addition, the kinetic energy and the radial expectation value r2 are shown to be bounded from below by the Fisher information in position and momentum spaces, denoted by Iρ and Iγ, respectively.

Heisenberg-like and Fisher-information-based uncertainty relations which extend and generalize previous similar expressions are obtained for $N$-fermion $d$-dimensional systems. The contributions of both spatial and spin degrees of freedom are taken into account. The accuracy of some of these generalized spinned uncertainty-like relations is numerically examined for a large number of atomic and molecular systems.

In this work we use the extremization method of various information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy) for $d$-dimensional quantum systems, which complementary describe the spreading of the quantum states of natural systems. Under some given constraints, usually one or two radial expectation values, this variational method allows us to determine an extremum-entropy distribution, which is the \textit{least-biased} one to characterize the state among all those compatible with the known data. Then we use it, together with the spin-dependent uncertainty-like relations of Daubechies-Thakkar type, as a tool to obtain relationships between the position and momentum radial expectation values of the type $\langle r^{\alpha}\rangle^{\frac{k}{\alpha}}\langle p^k\rangle\geq f(k,\alpha,q,N), q=2s+1$, for $d$-dimensional systems of $N$ fermions with spin $s$. The resulting uncertainty-like products, which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system, are shown to often improve the best corresponding relationships existing in the literature., Comment: Accepted for publication in Journal of Physics A

The classical hypergeometric polynomials {pn(x)}n=0∞, which are orthogonal with respect to a weight function ω(x) defined on a real interval, are analyzed in the Cramer–Rao information plane, that is the plane defined by both Fisher information and variance of the probability density ρn(x)=pn(x)2ω(x). The Rakhmanov density ρn(x) of these polynomials, which describes the probability density of the quantum states for various physical prototypes in an exact manner and for numerous physical systems to a very good approximation, is discussed in detail.

We investigate the correlations between different bipartitions of an exactly solvable one-dimensional many-body Moshinsky model consisting of Nn "nuclei" and Ne "electrons". We study the dependence of entanglement on the inter-particle interaction strength, on the number of particles, and on the particle masses. Consistent with kinematic intuition, the entanglement between two subsystems vanishes when the subsystems have very different masses, while it attains its maximal value for subsystems of comparable mass. We show how this entanglement feature can be inferred by means of the Born-Oppenheimer Ansatz, whose validity and breakdown can be understood from a quantum information point of view., Comment: Accepted in Eur. Phys. J. D (2014)

The Fisher information of single-particle systems with a central potential, which is a gradient functional of their quantum-mechanical probability density, is studied in detail in the position and momentum spaces. It is found that this local information-theoretic quantity can be expressed in a simple closed form via the radial expectation values (〈 p 2 〉, 〈 r −2 〉) in position space, and (〈 r 2 〉, 〈 p −2 〉) in momentum space. Applications to various prototype systems (hydrogen and harmonic oscillator) are shown. Furthermore, a new uncertainty relation which involves the Fisher information in the two complementary spaces, is proposed at the same level than the variance-based Heisenberg and entropic uncertainty relations.

Fisher's information and Shannon's entropy are two complementary information measures of a probability distribution. Here, the probability distributions which characterize the quantum-mechanical states of a hydrogenic system are analyzed by means of these two quantities. These distributions are described in terms of Laguerre polynomials and spherical harmonics, whose characteristics are controlled by the three integer quantum numbers of the corresponding states. We have found the explicit expression for the Fisher information, and a lower bound for the Shannon entropy with the help of an isoperimetric inequality.

Atomic avoided crossings in strong magnetic and electric fields are usually identified by the energy level repulsion and by the sharp extremal value of the Shannon's entropy of the two involved states. In this work the proposal of a new information-theoretic parameter, the continuous Fisher information, for the prediction and characterization of these nonlinear phenomena is done. For just around the irregular region it is found that Fisher's informations of the two states cross over and show that the states exchange their informational character. Moreover, the state Fisher information sum keeps constant through the avoided crossing. This is illustrated for some hydrogenic excited states in strong magnetic fields as well as in parallel magnetic and electric fields.

11 pages, no figures.-- MSC2000 codes: 81V55, 33C45, 81Q10.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001). MR#: MR1985711 (2004f:33026) Zbl#: Zbl 1017.81048 The expansion of a large class of functions in series of linearly varying Laguerre polynomials, i.e., Laguerre polynomials whose parameters are linear functions of the degree, is found by means of the hypergeometric functions approach. This expansion formula is then used to obtain the Brown–Carlitz generating function (which gives a characterization of the exponential function) and the connection formula for these polynomials. Finally, these results are employed to connect the bound states of the quantum–mechanical potentials of Morse and Pöschl–Teller, which are frequently used to describe molecular systems. This work has been partially supported by the Junta de Andalucía, under the research Grants FQM0207 (J.S.D. and J.S.R.) and FQM0229 (P.L.A.); the Spanish MCyT projects BFM2001-3878-C02 (J.S.R., J.S.D. and P.L.A.) and BFM2000-0206-C04-01 (J.S.R.); and the European Union project INTAS 2000-272 (J.S.D. and J.S.R.). Publicado

The spreading of the quantum probability density for the highly-excited states of a single-particle system with an exponential-type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann–Shannon information entropy. This problem boils down to the calculation of the asymptotics of the entropy-like integrals of the modified Bessel function of the second kind (also called the Mcdonald function or Basset function). The dependence of the two physical entropies on the large quantum number n is given in detail. It is shown that the semiclassical (WKB) position–space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single-particle systems with any power-type potential of the form V(x)=x2k, x∈R and k∈N. The momentum–space entropy, calculated with a method based on the properties of the Mcdonald function, is rigorously found to have a behavior of the form −ln ln n, in strong contrast with the...

The spreading of the position and momentum probability distributions for the stable free oscillations of a circular membrane of radius l is analyzed by means of the associated Boltzmann–Shannon information entropies in the correspondence principle limit (n → ∞, m fixed), where the numbers (n, m), n ∈ ℕ and m ∈ ℤ, uniquely characterize an oscillation of this two-dimensional system. This is done by solving the short-wave asymptotics of the physical entropies in the two complementary spaces, which boils down to the calculation of the asymptotic behavior of certain entropic integrals of Bessel functions. It is rigorously shown that the position and momentum entropies behave as 2 ln (l) + ln (4π) - 2 and ln (n) - 2 ln (l) + ln (2π3) when n → ∞, respectively. So the total entropy sum has a logarithmic dependence on n and it does not depend on the membrane radius. The former indicates that the ordering of short-wavelength oscillations is exactly identical for the entropic sum and the single-particle energy. The latter holds for all oscillations of the membrane because of the uniform scaling invariance of the entropy sum.

In this letter we analyze the effect of the spin dimensionality of a physical system in two mathematical formulations of the uncertainty principle: a generalized Heisenberg uncertainty relation valid for all antisymmetric N-fermion wavefunctions, and the Fisher-information- based uncertainty relation valid for all antisymmetric N-fermion wavefunctions of central potentials. The accuracy of these spin-modified uncertainty relations is examined for all atoms from Hydrogen to Lawrencium in a self-consistent framework., 9 pages, 2 figures

The complexity measures of the Cr\'amer-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density $\rho_n(x)=\omega(x) p_n^2(x)$ of the polynomials $p_n(x)$ orthogonal with respect to the weight function $\omega(x)$, $x\in (a,b)$, are used to quantify various two-fold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways. Their explicit (Cr\'amer-Rao) and asymptotical (Fisher-Shannon, LMC) values are given for the three systems of orthogonal polynomials. Then, these complexity-type mathematical quantities are numerically examined in terms of the polynomial's degree $n$ and the parameters which characterize the weight function. Finally, several open problems about the generalised hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted $L_q$-norms of Laguerre and Jacobi polynomials are pointed out., Comment: 15 pages, 6 figures

The primary dynamical Dirac relativistic effects can only be seen in hydrogenic systems without the complications introduced by electron-electron interactions in many-electron systems. They are known to be the contraction towards the origin of the electronic charge in hydrogenic systems and the nodal disappearance (because of the raising of all the nonrelativistic minima) in the electron density of the excited states of these systems. In addition we point out the (largely ignored) gradient reduction of the charge density near and far from the nucleus. In this work we quantify these effects by means of single (Fisher information) and composite [Fisher-Shannon complexity and plane, L\'opez-Ruiz, Mancini, and Calbet (LMC) complexity] information-theoretic measures. While the Fisher information measures the gradient content of the density, the (dimensionless) composite information-theoretic quantities grasp twofold facets of the electronic distribution: The Fisher-Shannon complexity measures the combined balance of the gradient content and the total extent of the electronic charge, and the LMC complexity quantifies the disequilibrium jointly with the spreading of the density in the configuration space. Opposite to other complexity notions (e.g., computational and algorithmic complexities), these two quantities describe intrinsic properties of the system because they do not depend on the context but are functionals of the electron density. Moreover, they are closely related to the intuitive notion of complexity because they are minimum for the two extreme (or least complex) distributions of perfect order and maximum disorder.

The connection problem is considered in a hypergeometric function framework for (i) the two most general families of polynomials belonging to the Askey scheme (Wilson and Racah), and (ii) some generalized Laguerre and Jacobi polynomials falling outside that scheme (Sister Celine, Cohen and Prabhakar–Jain), which are relevant to the study of quantum-mechanical systems and include as particular cases, the generalizations of the classical families with Sobolev-type orthogonality.In addition, using the same method three new linearization-like formulae for the Gegenbauer polynomials are also derived: a linearization formula that generalizes the m = n case of Dougall’s formula, the analogue of the m = n case of Nielsen’s inverse linearization formula for Hermite polynomials, and a connection formula for the squares.Closed analytical formulae for the corresponding connection and linearization coe:cients are given in terms of hypergeometric functions of unit argument, which at times can be further simpli;ed and expressed as single hypergeometric terms. c � 2001 Published by Elsevier Science B.V.

22 pages, no figures.-- PACS nrs.: 02.10.De, 02.30.Hq.-- MSC2000 code: 33C45. MR#: MR1820610 (2003c:33014) Zbl#: Zbl 0945.33006 The linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a continuous variable, which consists of the expansion of the product of two arbitrary HPs in series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of the coefficients of the second-order differential equations satisfied by the involved polynomials. General expressions for the expansion coefficients are given in integral form, and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several generalized linearization formulae involving these families. The connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions, which often reduce to a single function of the same type; when possible, these functions are evaluated in closed form. In some cases, sign properties of the coefficients such as positivity or non-negativity conditions are derived as a by-product from their resulting explicit representations. This work was partially supported by research grants from the Dirección General de Enseñanza Superior (DGES) of Spain (project codes PB95-1205 for AMF and JSD, and PB96-0120 for JSR), the European Union (INTAS-93-219-ext, for JSR, AMF and JSD), and the Junta de Andalucía (FQM0207 for JSR and JSD, FQM0229 for PLA and AMF). Publicado