If we do not make any truncations in the possible sets of Slater determinants (many-body states) we can make by distributing \( A \) nucleons among \( n \) single-particle states, we call such a calculation for Full configuration interaction theory

If we make truncations, we have different possibilities

• The standard nuclear shell-model. Here we define an effective Hilbert space with respect to a given core. The calculations are normally then performed for all many-body states that can be constructed from the effective Hilbert spaces. This approach requires a properly defined effective Hamiltonian
• We can truncate in the number of excitations. For example, we can limit the possible Slater determinants to only \( 1p-1h \) and \( 2p-2h \) excitations. This is called a configuration interaction calculation at the level of singles and doubles excitations, or just CISD.
• We can limit the number of excitations in terms of the excitation energies. If we do not define a core, this defines normally what is called the no-core shell-model approach.

What happens if we have a three-body interaction and a Hartree-Fock basis?