In nuclear structure theory the two main computational methods for heavy nuclei based upon the nucleon fermionic degrees of freedom are the Hartree-Fock or energy-density-functional (EDF) method and the configuration interaction (CI) method. The EDF method is often limited to a configuration with a single Slater determinant, sometimes called the single-reference (SR) EDF method. The EDF Hamiltonian has parameters that are fitted to global properties of nuclei such as binding-energies and rms charge radii. To go beyond the SR method one can introduce a multi-reference EDF basis.

The CI method takes into account many Slater determinants. CI often uses a Hamiltonian derived from experimental single-particle energies and a microscopic nucleon-nucleon interaction. A given CI Hamiltonian is applied to a limited mass region that is related to the configurations of a few valence orbitals outside of a closed shell and the associated renormalized nucleon-nucleon interaction that is specific to that mass region. Spectra and binding energies (relative to the closed core) obtained from such calculations for two to four valence particles are in good agreement with experiment. As many valence nucleons are added the agreement with experimental spectra and binding energies deteriorates (see Fig. 36 in [1]). An important part that is missing from these CI calculations is the effective two-body interaction that comes from the three-body interaction of two valence nucleons interacting with one nucleon in the core. To improve agreement with experimental spectra one often adjusts some of the valence two-body matrix elements. The most important part of this adjustment can be traced to the monopole component of the two-body matrix elements that controls how the effective single-particle energies evolve as a function of proton and neutron number.

The figure shows Wick's theorem applied to a closed shell for the one-body kinetic energy, the two-body interaction and the three-body interaction. The part contained in the dashed box represents the closed-shell and effective one-body parts of the Hamiltonian that might be contained in an EDF approach. Up to now this has been treated phenomenologically in the framework of the Skyrme Hartree-Fock or relativistic Hartree method with some parameters (typically 6-10) fitted to global experimental data. There are efforts underway to improve the accuracy of the parameters used for EDF calculations, to extend the type of functional forms used, and to relate the parameters of these phenomenological approaches to the underlying two and three body forces between nucleons. The part contained in the solid-line box is the residual interaction used for CI calculations. The remaining term is a valence three-body interaction.

We proposed a new method for obtaining a Hamiltonian for valence nucleons outside of a doubly-closed shell [2]. The new aspect of our method is to take the monopole part of the effective two-body interaction from an EDF calculations for a closed shell plus two nucleons constrained to be in orbitals a and b restricted to spherical symmetry, and ek are the single-particle energies calculated with EDF. The EDF-monopole contains both terms in the solid-line box in figure, and also contains higher-order contributions implicit in the EDF functional including possible quadratic terms in the mass dependence of the single-particle energies. Whereas, the N3LO monopole only contains the valence two-body interaction corrected to second order. We modify the monopole part of the microscopic valence interaction to reproduce the results of Eq. [e=2]. With this modification, the CI calculations closely reproduce the EDF calculations for single-Slater determinants, even when relatively many valence nucleons are added. Thus, the CI calculations are constrained to reproduce the trends of closed-shell energies and effective single-particle energies obtained with the EDF.

When many nucleons are added, the monopole contribution goes as n(n-1)V}/2 where n is the number of valence nucleons. Thus the EDF monopole corrections become much more important as one adds many valence nucleons. When we constrain the CI to the single configuration CI calculation gives a binding energy increase of 25.05 MeV (relative to 208Pb). The EDF calculation (with the same assumption for the configuration) gives 25.24 MeV. These are close to each other due to our EDF monopole correction to the valence matrix elements. If the EDF monopole correction were not included in CI the results would differ by (45)x(0.118) = 5.3 MeV. The microscopic valance interaction on its own is too strong and gives an "over-saturation." The results for the for EDF. The difference between CI and EDF might be interpreted in terms of an effective valence three-body monopole interaction with strength DE3 = 25.24 - 25.05 = 0.19 MeV for (0h9/2)10 and DE3 = 25.24 - 25.05 = 0.14 MeV (1f7/2)8 . With DE3=n(n-1)(n-2)V3/6, V3 is on the order of 1-2 keV.


[1] L. Coraggio, A. Covello, A. Gargano, N. Itaco and T. T. S. Kuo, Prog. in Part. and Nucl. Phys. {\bf 62}, 135 (2009).

[2] B. A. Brown, A. Signoracci and M. Hjorth-Jensen, Phys. Lett. B 695, 507 (2011). link to journal