Quantum information measuresProperties and analysis of structure and dynamics of multielectronic systems

Resumen

This Thesis is a contribution to the research on the Information Theory of multielectronic quantum systems. It focuses on numerous and diverse appli- cations of a range of information-theoretic measures to specific atomic and molecular structures. Relativistic and non relativistic multielectronic atomic systems, the relativistic quantum oscillator, and molecular systems are the principal systems under study in this Thesis. A variety of information- theoretical measures, including complexity and divergence measures, are employed in order to establish how these quantities can be related to dif- ferent physicochemical properties of the systems under consideration. This connections will allow us to reach a better understanding of those systems and propose an alternative methododology of their analysis, which could be applied to other systems in the future. This Thesis is composed by seven Chapters. The first one is devoted to the description and enumeration of the informational measures used throughout this Thesis. The following five Chapters are focused on the anal- ysis of these mentioned information-theoretic measures applied to atomic systems (Chapters two, three, four and five), molecular systems (Chap- ter six) and the relativistic quantum oscillator or Dirac oscillator (Chapter seven). Each of these Chapters has a specific introduction, which provides the scientific context and motivation of the work. Let us now summarize the contents of this Thesis. Chapter 1, namely “Theoretical foundations: Information-theoretic mea- sures”, details the definition, properties, scientific context and recent appli- cations of the different information-theoretical measures employed in the development of this thesis. The measures are classified in three different sections: Fundamental information measures (which comprise the Shannon entropy, the entropic moments and the Fisher information), complexity mea- sures (which comprise the Fisher-Shannon complexity and the LM C com- plexity) and divergence measures (enclosing the Kullback-Leibler divergence, the Jensen-Shannon divergence, the Fisher divergence, the Jensen-Fisher di- vergence, the quantum similarity measure, the quantum similarity index and the generalized quantum similarity index). Chapter 2, which is called “Fisher-like atomic divergences: Mathemat- ical grounds and physical applications” is devoted to the study and com- parison of the Fisher divergence and the Jensen-Fisher divergence, which is applied to neutral and ionic multielectronic atomic densities in position and momentum space. We focus in the different descriptions provided by both divergence measures and the connections found with the physical properties of the atomic systems studied, such as the atomic shell structure, ioniza- tion potential and nuclear charge, arising due to the local character of both measures. Chapter 3, which is referred to as “Jensen-Shannon and Kullback-Leibler 1divergences as quantifiers of relativistic effects in neutral atoms” is dedi- cated to the study of relativistic effects in multielectronic atomic systems in position space. In order to do that, the Kullback-Leibler and the Jensen- Shannon divergences were employed to compare relativistic and non-relativistic atomic densities, analyzing their value and comparing it with the atomic shell structure and atomic charge. Chapter 4, entitled “Generalized quantum similarity in atomic systems: A quantifier of relativistic effects”, is devoted to the analysis of the contri- bution that different regions of the atomic density have in relativistic effects. The generalized quantum similarity index is the information-theoretic mea- sure employed to compare relativistic and non-relativistic multielectronic atomic densities. Its main property of regulating which different region of the density domain is considered on the comparison, made it perfect to this precise purpose. Chapter 5, named “Electron pair densities: An information-theoretical approach”, is dedicated to the analysis of electron pair atomic densities. Many and diverse information-theoretic measures are employed to achieve such objective, namely, Shannon entropy, disequilibrium, LM C complex- ity, Jensen-Shannon divergence and the quantum similarity index. Those quantities are employed to compare both monoelectronic and electron pair atomic densities in position and momentum spaces, for both neutral and ionic systems, which allow us to establish the similarities and differences between both approaches to the atomic density. Chapter 6, which is titled “A molecular analysis using an information- theoretical approach”, is devoted to the analysis of molecular systems using different divergence measures. The main purpose of this chapter is deter- mine a connection between differences in information-theoretic measures and chemical and physical properties of molecules under consideration. The complexity of this study is discussed and a more restricted approach is con- sidered in order to achieve realistic results. We employ the well-studied Jensen-Shannon and Jensen-Fisher divergence measures, which allow us to establish a direct comparison between a reference molecule and the rest of the group considered. Then we match these quantities to physical and chem- ical properties of the molecules, namely hardness and number of electrons. Chapter 7, which is called “The relativistic harmonic oscillator”, is ded- icated to the discussion of the relativistic harmonic oscillator and the study of the properties of the radial part of its density by means of information- theoretic measures. Shannon entropy, disequilibrium, Fisher information, LM C complexity and Fisher-Shannon complexity are calculated for differ- ent values of the quantum numbers, showing a connection between both quantities and the radial density structure of the system.