Hartree-Fock Approximation for the $Ab initio$ No-Core Shell Model

The spherical Hartree-Fock approximation is applied to the $abinitio$ no-core shell model, with a realistic effective nucleon-nucleon interaction in order to investigate the range of its utility. Hartree-Fock results for binding energies, one-body density distributions and occupation probabilities are compared with results from exact diagonalization in similar model spaces. We show that this mean field approximation, especially with second order corrections, is able to provide some useful approximatons for $^4He$ and $^{16}O$. We also explore the physical insights provided by the Hartree-Fock results for single-particle properties such as spin-orbit splittings. We find single particle state ordering consistent with the phenomenological shell model