 
Summary: A combinatorial approach to the electron correlation problem
Alex J. W. Thom and Ali Alavia
Chemistry Department, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom
Received 21 July 2005; accepted 15 September 2005; published online 22 November 2005
Starting from a pathintegral formulation of quantum statistical mechanics expressed in a space of
Slater determinants, we develop a method for the Monte Carlo evaluation of the energy of a
correlated electronic system. The pathintegral expression for the partition function is written as a
contracted sum over graphs. A graph is a set of distinct connected determinants on which paths can
be represented. The weight of a graph is given by the sum over exponentially large numbers of paths
which visit the vertices of the graph. We show that these weights are analytically computable using
combinatorial techniques, and they turn out to be sufficiently well behaved to allow stable Monte
Carlo simulations in which graphs are stochastically sampled according to a Metropolis algorithm.
In the present formulation, graphs of up to four vertices have been included. In a HartreeFock basis,
this allows for paths which include up to sixfold excitations relative to the HartreeFock
determinant. As an illustration, we have studied the dissociation curve of the N2 molecule in a VDZ
basis, which allows comparison with full configurationinteraction calculations. © 2005 American
Institute of Physics. DOI: 10.1063/1.2114849
I. INTRODUCTION
It is well known that the pathintegral formulation of
quantum statistical mechanics provides, in principle, a
