This thesis is concerned with the development of a new reformulation of the Path Integral method in terms of graphs rather than paths. In it we describe the reformulation of the Path Integral method in a Slater determinant space. We have developed a method of resumming paths in this space into graphs, objects on which one can represent many paths. The signs of these graphs are very well behaved, with a significant fraction being positive definite. The method in this form, however, is unsuitable for large systems as it has a high scaling. To overcome this problem, we recast the method in a form suitable for Monte Carlo evaluation; this process is not
possible for most correlated quantum chemical methods, which must instead resort to significant levels of approximation.

We present some calculations on both the neon atom, and some small molecules showing the ability of the full graph sums to describe molecular dissociation, even
in the very strongly correlated N2 molecule. We then demonstrate the accuracy of this Monte Carlo against the full sums, as well as in comparison to Coupled Cluster
methods, on a test set of twenty closed shell molecules. The scaling of the Monte Carlo with basis size is also considered. We also detail some of the more technical
aspects required in the implementation of this method, particularly in the generation of graphs, along with the derivation of an eigenvector-based formulation of the graph sums.