Algebraic Methods In Quantum Chemistry And Physics

Algebraic Methods in Quantum Chemistry and Physics provides straightforward presentations of selected topics in theoretical chemistry and physics, including Lie algebras and their applications, harmonic oscillators, bilinear oscillators, perturbation theory, numerical solutions of the Schrödinger equation, and parameterizations of the time-evolution operator. The mathematical tools described in this book are presented in a manner that clearly illustrates their application to problems arising in theoretical chemistry and physics. The application techniques are carefully explained with step-by-step instructions that are easy to follow, and the results are organized to facilitate both manual and numerical calculations. Algebraic Methods in Quantum Chemistry and Physics demonstrates how to obtain useful analytical results with elementary algebra and calculus and an understanding of basic quantum chemistry and physics.

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the non-Abelian Berry's geometric factor, super- and BRST symmetries, non-commutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The book aims at being a guide to advanced differential geometric and topological methods in quantum mechanics. Their main peculiarity lies in the fact that geometry in quantum theory speaks mainly the algebraic language of rings, modules, sheaves and categories. Geometry is by no means the primary scope of the book, but it underlies many ideas in modern quantum physics and provides the most advanced schemes of quantization.

On the occasion of the fourth International Conference on Industrial and Applied Mathematics!, we decided to organize a sequence of 4 minisymposia devoted to the mathematical aspects and the numerical aspects of Quantum Chemistry. Our goal was to bring together scientists from different communities, namely mathematicians, experts at numerical analysis and computer science, chemists, just to see whether this heterogeneous set of lecturers can produce a rather homogeneous presentation of the domain to an uninitiated audience. To the best of our knowledgde, nothing of this kind had never been tempted so far. It seemed to us that it was the good time for doing it, both . because the interest of applied mathematicians into the world of computational chemistry has exponentially increased in the past few years, and because the community of chemists feels more and more concerned with the numerical issues. Indeed, in the early years of Quantum Chemistry, the pioneers (Coulson, Mac Weeny, just to quote two of them) used to solve fundamental equations modelling toy systems which could be simply numerically handled in view of their very limited size. The true difficulty arose with the need to model larger systems while possibly taking into account their interaction with their environment. Hand calculations were no longer possible, and computing science came into the picture.

This book provides an introduction to the use of algebraic methods and sym bolic computation for simple quantum systems with applications to large order perturbation theory. It is the first book to integrate Lie algebras, algebraic perturbation theory and symbolic computation in a form suitable for students and researchers in theoretical and computational chemistry and is conveniently divided into two parts. The first part, Chapters 1 to 6, provides a pedagogical introduction to the important Lie algebras so(3), so(2,1), so(4) and so(4,2) needed for the study of simple quantum systems such as the D-dimensional hydrogen atom and harmonic oscillator. This material is suitable for advanced undergraduate and beginning graduate students. Of particular importance is the use of so(2,1) in Chapter 4 as a spectrum generating algebra for several important systems such as the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This approach provides an interesting and important alternative to the usual textbook approach using series solutions of differential equations.

This text on the use of electron correlation effects in the description of the electronic structure of atoms, molecules, and crystals is intended for graduate students in physical chemistry and physics. Modern theories of electronic structure and methods of incorporating electron correlation contributions are developed using a diagrammatic and algebraic formulation, and the methods developed in the text are illustrated with examples from molecular and solid state quantum mechanics. A brief Introduction is followed by chapters on operator algebra, the independent-particle model, occupation-number formalism, and diagrams. Additional topics include the configuration-interaction method, the many-body perturbation theory, and the coupled-cluster method.

In this volume we have collected some of the contributions made to the Twelfth European Workshop on Quantum Systems in Chemistry and Physics (QSCP-XII) in 2007. The workshop was held at Royal Holloway College, the most westerly campusof the University of London,and situated just a stone’s throw from Windsor Great Park. The workshop, which ran from 30 August to 5 September, continued the series that was established by Roy McWeeny in April 1996 with a meeting held at San Miniato, near Pisa. The purpose of the QSCP workshops is to bring together, in an informal atmosphere and with the aim of fostering collaboration, those chemists and physicists who share a common ?eld of interest in the theory of the quantum many-body problem. Quantum mechanics provides a theoretical foundation for our understandingof the structure,propertiesanddynamicsof atoms, moleculesandthe solid state, in terms of their component particles: electrons and nuclei. The study of ‘Quantum Systems in Chemistry and Physics’ therefore underpins many of the emerging?elds in twenty-?rstcenturyscience andtechnology:nanostructure,smart materials, drug design – to name but a few. Members of the workshop were keen to discuss their research and engage in collaboration centred upon the development of fundamental and innovative theory which would lead to the exploration of new concepts. The proceedings of all of the workshops, which have been held annually since 1996, have been published both to disseminate the latest developments within the wider community and to stimulate further collaboration.

The Advances in Chemical Physics series provides the chemical physics and physical chemistry fields with a forum for critical, authoritative evaluations of advances in every area of the discipline. Filled with cutting-edge research reported in a cohesive manner not found elsewhere in the literature, each volume of the Advances in Chemical Physics series serves as the perfect supplement to any advanced graduate class devoted to the study of chemical physics.

At the end of the workshop on "New Theoretical Concepts in Physical Chemistry", one of the participants made an attempt to present a first impression of its achievements from his own personal standpoint. Appar ently his views reflected a general feeling, so that the organizers thought they would be suitable as a presentation of the proceedings for future readers. That is the background from which this foreword was born. The scope of the workshop is a very broad one. There are contribu tions from mathematics, physics, crystallography, chemistry and biology; the problems are approached either by means of axiomatic and rigorous methods, or at an empirical phenomenological level. This same diversifi cation can be found in the new basic concepts presented. Some arise from pure theoretical investigation in C*-algebra or in quantum probability theory; others from an analysis of very complex experimental data like nuclear energy levels, or processes on the frontier between classical and quantum physics; others again have their origin in the discovery of new ordered structures like the icosahedral crystal phases, or the knots of DNA molecules; others follow from the application of ideas like frac tals or chaos to new fields like spectral theory or chemical reactions. It is to be expected that readers will have to face the same sort of difficulties as did the participants in understanding such diverse languages, in applying themselves to subjects possibly far from their own experience, and in grasping highly sophisticated new concepts.