Discretized propagators, Hartree, and Hartree–Fock equations, and the Hohenberg–Kohn theorem

The question of how electron exchange can be realistically included in discretized propagator treatments of simple quantum mechanical systems is investigated by showing how the many-body Hartree and Hartree–Fock approximations can be formulated in terms of discretized propagators. The Hartree approximation takes a surprisingly simple form suggestive of Thomas–Fermi theories in high dimensional spaces. For the Hartree–Fock approximation, the effect of nonlocal potential energy operators on the short-time propagators must be addressed. These nonlocal operators make the Hartree–Fock results considerably more complicated. However, in each case we indicate how the universal energy functional of the density implied by the Hohenberg–Kohn theorem may be obtained.