S. López-Rosa

Insight into the informational-structure behavior of the Diels-Alder reaction of cyclopentadiene and maleic anhydride

The course of the Diels-Alder reactions of cyclopentadiene and maleic anhydride were studied. Two reaction paths were modelled: endo- and exo-selective paths. All structures within the transient region were characterized and analyzed by means of geometrical descriptors, physicochemical parameters and information-theoretical measures in order to observe the linkage between chemical behavior and the carriage of information. We have shown that the information-theoretical characterization of the chemical course of the reaction is in complete agreement with its phenomenological behavior in passing from reactants to products. In addition, we were able to detect the main differences between the two reaction mechanisms. This type of informational analysis serves to provide tools to help understand the chemical reactivity of the two simplest Diels-Alder reactions, which permits the establishment of a connection between the quantum changes that molecular systems exert along reaction coordinates and standard physicochemical phenomenology. In the present study, we have shown that every reaction stage has a family of subsequent structures that are characterized not solely by their phenomenological behavior but also by informational properties of their electronic density distribution (localizability, order, uniformity). Moreover, we were able to describe the main differences between endo-adduct and exo-adduct pathways. With the advent of new experimental techniques, it is in principle possible to observe the structural changes in the transient regions of chemical reactions. Indeed, through this work we have provided the theoretical concepts needed to unveil the concurrent processes associated with chemical reactions.

Fisher information of D-dimensional hydrogenic systems in position and momentum spaces

The spreading of the quantum-mechanical probability distribution density of D-dimensional hydrogenic orbitals is quantitatively determined by means of the local information-theoretic quantity of Fisher in both position and momentum spaces. The Fisher information is found in closed form in terms of the quantum numbers of the orbital.

Fisher Information Study in Position and Momentum Spaces for Elementary Chemical Reactions.

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.

Fisher and Jensen-Shannon divergences: Quantitative comparisons among distributions. Application to position and momentum atomic densities.

The Fisher divergence (FD) and Jensen-Shannon divergence (JSD) are used in this work with the aim of providing quantitative measures of the discrepancies between two arbitrary D-dimensional distribution functions, the FD being of local character and the JSD of global one. In doing so, the concepts of Fisher information and Shannon entropy associated to a distribution are the essential quantities for building up these comparative functionals. This kind of relative measures are here applied to the study of the one-particle densities in both conjugated spaces (position and momentum) of neutral atoms, discussing the results as compared to those provided by other previous functional measures. It is clearly shown how these divergences provide relevant information on the atomic shell structure, up to a level which depends on the considered space and measure.

Configuration complexities of hydrogenic atoms

AbstractThe Fisher-Shannon and Cramer-Rao information measures, and the LMC-like or shape complexity (i.e., the disequilibrium times the Shannon entropic power) of hydrogenic stationary states are investigated in both position and momentum spaces. First, it is shown that not only the Fisher information and the variance (then, the Cramer-Rao measure) but also the disequilibrium associated to the quantum-mechanical probability density can be explicitly expressed in terms of the three quantum numbers (n,l,m) of the corresponding state. Second, the three composite measures mentioned above are analytically, numerically and physically discussed for both ground and excited states. It is observed, in particular, that these configuration complexities do not depend on the nuclear charge Z. Moreover, the Fisher-Shannon measure is shown to quadratically depend on the principal quantum number n. Finally, sharp upper bounds to the Fisher-Shannon measure and the shape complexity of a general hydrogenic orbital are given in terms of the quantum numbers.

Entropy and Complexity Analyses of D -dimensional Quantum Systems

This chapter briefly reviews the present knowledge about the analytic information theory of quantum systems with non-standard dimensionality in the position and momentum spaces. The main concepts of this theory are the power and entropic moments, which are very fertile largely because of their flexibility and multiple interpretations. They are used here to study the most relevant information-theoretic one-element (Fisher, Shannon, Renyi, Tsallis) and some composite two-elements (Fisher-Shannon, LMC shape and Cramer-Rao complexities) measures which describe the spreading measures of the position and momentum probability densities farther beyond the standard deviation. We first apply them to general systems, then to single particle systems in central potentials and, finally, to hydrogenic systems in D-dimensions.

Information-theoretic measures of hyperspherical harmonics

The multidimensional spreading of the hyperspherical harmonics can be measured in a different and complementary manner by means of the following information-theoretic quantities: the Fisher information, the average density or first-order entropic moment, and the Shannon entropy. They give measures of the volume anisotropy of the eigenfunctions of any central potential in the hyperspace. Contrary to the Fisher information, which is a local measure because of its gradient-functional form, the other two quantities have a global character because they are powerlike (average density) and logarithmic (Shannon’s entropy) functionals of the hyperspherical harmonics. In this paper we obtain the explicit expression of the first two measures and a lower bound to the Shannon entropy in terms of the labeling indices of the hyperspherical harmonics.

Phenomenological description of a three-center insertion reaction: an information-theoretic study

Information-theoretic measures are employed to describe the course of a three-center chemical reaction in terms of detecting the transition state and the stationary points unfolding the bond-forming and bond-breaking regions which are not revealed in the energy profile. The information entropy profiles for the selected reactions are generated by following the intrinsic-reaction-coordinate (IRC) path calculated at the MP2 level of theory from which Shannon entropies in position and momentum spaces at the QCISD(T)/6-311++G(3df,2p) level are determined. Several complementary reactivity descriptors are also determined, such as the dipole moment, the molecular electrostatic potential (MEP) obtained through a multipole expansion (DMA), the atomic charges and electric potentials fitted to the MEP, the hardness and softness DFT descriptors, and several geometrical parameters which support the information-theoretic analysis. New density-based structures related to the bond-forming and bond-breaking regions are proposed. 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.

Quantum entanglement of helium-like systems with varying-Z: compact state-of-the-art CI wave functions

In this work we have performed state-of-the-art configuration-interaction (CI) calculations to determine the linear and von Neumann entanglement entropies for the helium-like systems with varying nuclear charge Z in the range . The focus of the work resides on determining accurate entanglement values for 2-electron systems with the lowest computational cost through compact CI-wave functions. Our entanglement results for the helium atom fully agree with the results obtained with higher quality wave functions of the Kinoshita type (Dehesa [5]). We find that the correlation energy is linearly related to the entanglement measures associated with the linear and von Neumann entropies of the single-particle reduced density matrizes, which sheds new light on the physical implications of entanglement in helium-like systems. Moreover, we report CI-wave-function-based benchmark results for the entanglement values for all members of the helium isoelectronic series with an accuracy similar to that of Kinoshita-type wave functions. Finally, we give parametric expressions of the linear and von Neumann entanglement measures for two-electron systems as Z varies from 1 to 10.

Complexity of D-dimensional hydrogenic systems in position and momentum spaces

The internal disorder of a D-dimensional hydrogenic system, which is strongly associated to the non-uniformity of the quantum-mechanical density of its physical states, is investigated by means of the shape complexity in the two reciprocal spaces. This quantity, which is the product of the disequilibrium or averaging density and the Shannon entropic power, is mathematically expressed for both ground and excited stationary states in terms of certain entropic functionals of Laguerre and Gegenbauer (or ultraspherical) polynomials. We emphasize the ground and circular states, where the complexity is explicitly calculated and discussed by means of the quantum numbers and dimensionality. Finally, the position and momentum shape complexities are numerically discussed for various physical states and dimensionalities, and the dimensional and Rydberg energy limits as well as their associated uncertainty products are explicitly given. As a byproduct, it is shown that the shape complexity of the system in a stationary state does not depend on the strength of the Coulomb potential involved.