Page 70 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

Reduced Density Matrix versus Wave Function: Recent Developments

C. VALDEMORO
Instituto de Ciencia de Materiales, Serrano 123, 28006 Madrid, Spain

1. Introduction

Much of the great interest that the Reduced Density Matrices (RDM) theory has
arisen since the pioneer works of Dirac [1], Husimi [2] and Löwdin [3], is due to the
simplification they introduce by averaging out a set of the variables of the many body
system under study. For all practical purposes, the averaging with respect to N-1
or N-2 electron variables which is carried out in the 1-RDM or 2-RDM respectively,
does not imply any loss of the necessary information. The reason for this is that
the operators representing the N-electron observables are sums of operators which
depend only on one or two electron variables.
The RDM’s are therefore much simpler objects than the N-electron Wave Function
(WF) which depends on the variables of N electrons. Unfortunately, the search for
the N-representability conditions has not been completed and this has hindered the
direct use of the RDM’s in Quantum Chemistry. In 1963 A. J. Coleman [4] defined
the N-representability conditions as the limitations of an RDM due to the fact that
it is derived by contraction from a matrix represented in the N-electron space. In
other words, an antisymmetric N-electron WF must exist from which this RDM
could have been derived by integrating with respect to a set of electron variables.
The research for finding these conditions, has been intense and fruitful [5-13]. Thus,
although an exact procedure for determining directly an N-representable 2-RDM
has not been found, many mathematical properties of these matrices are now known
and several methods for approximating RDM’s and for employing them have been
developed [14-19].
To study the electronic structure of small systems within the framework of the RDM
formalism is a good strategy to adopt, but where it is of the foremost importance is
in the study of the electronic structure of very large systems. In this latter case, to
work within the framework of an N-electron WF does not seem the best approach
to take even now that large and fast computers are available. It seems clear to me
that it would be advantageous to approach the study of these large systems within a
theoretical framework having a quantum statistical character. Since the RDM’s are
statistical objects their formalism would fit in a natural way in such a framework.
The aim of this paper is to review the work done by our group in this direction
in the last ten years. The reader wishing to have a broader outlook of this vast
and fascinating field of research is referred to the Proceedings of the A. J. Coleman
Symposium on Reduced Density Matrices and Density Functionals [20]. In this book
the opening contribution is by A. J. Coleman himself, where he masterly describes
the history of the Reduced Density Matrix (RDM) research from 1929 up to 1987.
55
Y. Ellinger and M. Defranceschi (eds.). Strategies and Applications in Quantum Chemistry, 55–75.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.