Antisymmetry in the three-nucleon-interaction matrix elements

The inclusion of three-body forces to the effective nucleon-nucleon interaction has been frequently applied in nuclear self-consistent Hartree-Fock calculations. Thus, a dependence of the effective force on the nucleon density is exhibited, stimulating great interest in this manner. Antisymmetrized three-body matrix elements are evaluated, and applied with a zero-range three-body force within a finite basis of spherical single-particle states. The imposed antisymmetry ignores the existence of matrix elements of three identical nucleons. It provides for a better understanding of the structure of the three-body contribution to the Hartree-Fock ground state energy and for the exact relation between the three-body force and a density dependent two-body force. The discussion is given as well for spherical as for deformed nuclei. Problems concerning possible overbinding and violation of the spin stability by the zero-range three-body interaction are examined and represented in the angular momentum picture