J. E. Harriman

EDUCATION

• B.S. 1959, University of Wisconsin-Madison
• Ph.D. 1962, Harvard University

PUBLICATIONS & AWARDS

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RESEARCH DESCRIPTION

Professor Harriman is no longer taking students.

The goal of research in this group is to gain a better understanding of the electronic structure of molecules by use of reduced density matrices. Methods include formal extensions of the theory, calculations of good quality for specific systems and the graphical presentation of results, and investigations of simple models.

State-of-the-art electronic wave functions may involve millions of coefficients, which never get out of the computer. Fortunately, such a wave function contains much more information than is needed. Reduced density matrices and densities in coordinates, momenta, or both provide a way of concentrating our attention on the relevant information. Densities provide less information than density matrices, and the relationships among these quantities is of interest. Not only is the formal theory of these objects of interest itself, but an analysis in terms of them can aid us in the calculation and understanding of properties such as correlation of various types, spin density distributions, etc.

In a formal sense, densities and density matrices are elements of vector spaces, we can investigate their geometric properties. Some of the relationships among these spaces depend on the basis set used. Others are determined by symmetry. One of our goals is to learn as much as we can of the properties of reduced density matrices and densities that are determined by symmetry, basis set, and general requirements of quantum mechanics, so that when we look at results for a particular system we can concentrate on those aspects specific to it, and thus providing information about it. It has long been a goal of density matrix theory to calculate the simpler reduced density matrix directly, without involving the more complicated wave function. This has been difficult because of the need to impose appropriate boundary conditions: the “n-representability problem.” Recent methods involving contracted Schrodinger equations or constraints seem promising and we are interested in justification of or problems with these methods.

In addition, many of the formal results of density matrix theory with a basis set are directly applicable to magnetic resonance theory and to formally similar problems arising in laser experiments. We are interested in applying what we have learned to these problems.