A combinatorial approach to the electron correlation problem

Starting from a path-integral 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 path-integral 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 Hartree-Fock basis, this allows for paths which include up to sixfold excitations relative to the Hartree-Fock determinant. As an illustration, we have studied the dissociation curve of the N(2) molecule in a VDZ basis, which allows comparison with full configuration-interaction calculations.