The correlation term, accounting for correlations going beyond the simple product state, is an integral part of the DFT. Since nuclei are self-bound systems, many-body correlations due to spontaneous symmetry-breaking effects are of particular importance. A large part of those correlations can indeed be included by considering symmetry-breaking product states. Within the mean-field approach, one can understand many physical observables by directly employing broken-symmetry states; however, for finite systems, a quantitative description often does require symmetry restoration. For this purpose, one can apply a variety of theoretical techniques, in particular projection methods, the generator coordinate method (GCM), the random phase approximation (RPA), and various approximations performed on top of self-consistent mean fields [61,62,63].
In this context, it is important to recall that the realistic energy density functional does not have to be related to any given effective Hamiltonian. This creates a problem if a symmetry is spontaneously broken. While the projection can be carried out in a straightforward manner for energy functionals that are related to a Hamiltonian, the restoration of spontaneously broken symmetries of a general density functional still poses a conceptional dilemma that needs to be properly addressed [64,65,66].
Since the correlation term is a part of the functional, it should be treated as such during the variational procedure and during the fitting process in which the functional's coupling constants are determined. So far, perhaps with the exception of the center-of-mass term (see Sec. 4.3.1 below), such an ambitious program has not been carried out. In the near future, one hopes to work out approximate expressions for the correlation term that would capture the essence of results of microscopic calculations performed on top of self-consistent mean fields. In this way, the hope is to develop the tractable parametrization of the correlation energy in terms of local densities that would allow an explicit inclusion of dynamical effects into the energy functional.