Geminal-based statistics for the energies of many-electron molecular systems

In 1959, Bopp developed a lower bound to atomic and molecular ground-state energies by summing the lowest eigenvalues of the two-particle reduced Hamiltonian, ²K̂. His approximation is accurate only for very small systems (fewer than about four electrons), with the results degenerating rapidly for larger problems. In this paper, we extend and improve Bopp’s work by introducing a flexible distribution function, guided by familiar Fermi-Dirac statistics, in order to generate occupation numbers for the energy levels of ²K̂. The distribution function and the resulting energy are parametrized by a correlation temperature T. For a given system, characteristic temperatures may be identified that yield the true energy or any other benchmark energy of the system. Using a geometric argument and the empirical properties of the energy vs temperature curve, the two-electron statistics are investigated as a predictive tool for a variety of small atoms and molecules.