Wavelets and Basis Set Optimization for Molecular and Other Few-Body Quantum Calculations

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

0070879 Littlejohn Quantum mechanics is the fundamental theory which describes the behavior of atoms, molecules and nuclei. This work concerns relatively simple, or ``few-body,'' systems. In some cases it is possible to use quantum theory and computer calculations to predict the outcome of molecular reactions (one example of an application) which are difficult or impossible to test in the laboratory, with ultimately an important impact on biology, medicine and environmental sciences, among other fields. The application of quantum theory involves finding certain wave functions, which in practice are expressed in terms of more elementary waves (the basis wave functions). The problem of finding an optimal basis is an old one in quantum theory, but in recent years there has arisen a host of new ideas relevant to this question from a variety of fields, including pure and applied mathematics, physics, chemistry and engineering. The object of this work is to integrate these new developments, to search for and exploit fundamental, unifying principles, and to apply them to basis set selection in few-body quantum physics. The new ideas include the following. The first is wavelets, a relatively recent development in applied mathematics, which has already had an important impact on the processing, transmission and storage of signals and data. A second concerns phase space or semiclassical methods, which are called ``microlocal analysis'' in the mathematics literature. These methods are characterized by a consideration of position and momentum together, an unusual and relatively unfamiliar point of view in quantum mechanics where the Heisenberg uncertainly principle limits the simultaneous knowledge of these quantities. A third concerns the mathematics of abstract, higher dimensional spaces in few-body quantum problems, the so-called ``geometrical'' methods. Such methods have been a very active area of mathematics in recent years and are well known in certain areas of physics. They have not, however, been exploited much in few-body quantum mechanics, although they are vital for understanding the nature of the ``internal'' spaces of few-body problems. Finally, this work will take the corpus of methods which are in current use in atomic, molecular and nuclear physics for basis set selection, compare them, integrate them, search for generalizations and improvements, and apply them.