Information-theoretic measures of atomic and molecular systems

Resumen

This Thesis is a contribution to the debate on the implications of the information theory of quantum systems for the description of numerous quantum phenomena of the electronic structure. It contains several applications of the basic concepts, techniques and relations of the information theory to selected problems of atomic and molecular structure and chemical reactivity. Such treatment gives rise to the information-entropic representation of the atomic and molecular states, which complements the familiar energy-representation of the density-functional and wave-function-based theories. The most important role in this treatment is played by the concepts of information, complexity and divergence. The Thesis is structured in three Parts and two Appendices. Each part is composed by four Chapters, which are deeply self-contained with their own Introduction and Conclusions. They correspond to one (Chapters 3, 6 and App. A), two (Chapters 4, 6, 10, 11) and three (Chapters 2, 7, 12) scientific publications. In the following we briefly outline the contents of the Thesis. A more detailed introduction and motivation can be found at the beginning of the respective parts and chapters. Part I, which is entitled "Entropic functionals of some selected quantum systems'', is devoted to the study of some physical and chemical systems and processes in terms of different information-theoretic measures. This Part starts with the presentation and discussion of the concept of "information'' in Chapter 1, where some of the information measures that will be used throughout this Thesis are defined (i.e. the standard deviation, the Shannon, Renyi and Tsallis entropies, the Fisher information as well as the entropic moments), their properties and distinctive characteristics are pointed out. Then, in Chapter 2, we carry out the analytical determination of the single information-theoretic measures and their associated uncertainty relations for multidimensional hydrogenic systems. The spreading properties of the ground and excited states of d-dimensional hydrogenic systems by means of these information measures are investigated in terms of their quantum numbers. Emphasis on the circular and Rydberg states is made. In Chapter 3, the Fisher information is used to study a one- and a two-step simple chemical reactions; namely, a typical nucleophilic substitution reaction and the radical abstraction reaction involving a free radical (atomic hydrogen) as an intermediate reactive, respectively. We will show that this measure seems to be very useful in order to analyze the course of the chemical reaction because, due to its local character, the Fisher information is able to detect the relevant points in the reaction path (such as transition state or bond breaking/forming regions) which are not so clear from the energy profile. Finally, in Chapter 4, the multidimensional extremization problems of the Shannon, Tsallis and Fisher information measures subject to a radial expectation value as main constraint are analyzed and applied to all ground-state neutral atoms from Hydrogen to Lawrencium. In addition, the existence conditions for the d-dimensional Maximum Entropy problem problem are found, which extends a number of results previously encountered by various authors for the one-dimensional case. Part II, which is called "Complexity measures of atomic and molecular systems'', is devoted to the study of the complexity. Although there is no general agreement about what complexity is, there exist various technical notions of this quantity which have been shown to be very useful for the quantum mechanical interpretation of numerous physical and chemical phenomena of atomic and molecular systems. This Part begins in Chapter 5 with a discussion of the concept of complexity and its description by means of three product-like measures of complexity (the LMC shape, the Fisher-Shannon and the Cram\'er-Rao complexities) and their generalizations: the Shape-Renyi and the Fisher-Renyi complexities. Then, in Chapter 6, we explore the analytical properties of these measures as well as the associated uncertainty-like inequalities. Moreover, the generalized complexities defined in the previous chapter are employed in order to analyze atomic electron densities. In Chapter 7, we study both analytically and numerically the Fisher-Shannon and LMC shape complexities of the multidimensional hydrogenic systems, emphasizing the realistic hydrogenic atoms in position space. In addition we study the relativistic effects of Klein-Gordon type in the aforementioned measures of complexity. Finally, in Chapter 8, we perform a numerical study of the complexity measures and the information planes of some molecular systems, finding interesting trends in the behaviour of molecular complexities when these quantities are interpreted according to the molecular structure and composition, reactivity, etc. Part III, entitled "Divergence measures of atomic systems and processes'', is devoted to the study of the divergence measures. In the previous Parts I and II we have focused in the quantification of the information features of a given systems and their connection with its physical or chemical properties. In this Part we will analyze the similarity or dissimilarity between two or more systems in terms of the divergence measures. We start with the elucidation of the concept of divergence and other related information-theoretic indices in Chapter 9, giving the definition of the measures that will be used in this Part, i.e, the quadratic distance, the quantum similarity index as well as Jensen-like and Fisher divergences. Then, in Chapter 10, we study the dissimilarity or divergence between neutrals and/or ions throughout the Periodic Table by using the Jensen-Shannon and Fisher divergence. These quantities are found to show the complex organization and the shell-filling patterns throughout the Periodic Table. In Chapter 11 we propose an extension of the Jensen-Shannon divergence by means of the use of the Shannon entropy generalizations, i.e., the Renyi and Tsallis entropies, from which the Jensen-Renyi and the Jensen-Tsallis divergences, respectively, are defined. Application of these measures to the study of the dissimilarity among neutral atoms are also carried out. Finally, in Chapter 12 we present other applications of the divergence measures. We start analyzing the interelectronic repulsion by means of the computation of the divergence between atomic densities with the Hartree-Fock and Bare-Coulomb-Field models. Then, we use the generalization of the Jensen-like divergences for a set of distributions in order to provide a very useful tool for quantifying the information content of a composite system with respect to that of their constituents.