Stochastic perturbation theory: a low-scaling approach to correlated electronic energies

We discuss a stochastic implementation of M?ller-Plesset (MP) theory based upon the concept of a "graph," a set of connected Slater determinants. We show how contributions from an arbitrary level, MPn, of perturbation theory can be expressed diagrammatically in terms of graphs, and that these may be stochastically sampled to give a good estimate of the energy. We show this to be the case for Ne, Ar, N2, and H2O molecules. N-molecule chains of He atoms and H2 molecules at equilibrium and stretched geometries show an effective scaling of ON(2.6)] and ON(5.6)] for MP2 and MP3 theories.