| Multi-reference calculations along the lines of the Generator Coordinate Method or the restoration of broken
symmetries within the nuclear Energy Density Functional (EDF) framework are becoming a standard tool in nuclear
structure physics. These calculations rely on the extension of a single-reference energy functional, of the Gogny
or the Skyrme types, to non-diagonal energy kernels. There is no rigorous constructive framework for this
extension so far. The commonly accepted way proceeds by formal analogy with the expressions obtained when applying
the generalized Wick theorem to the non-diagonal matrix element of a Hamilton operator between two product states.
It is pointed out that this procedure is ill-defined when extended to EDF calculations as the generalized Wick
theorem is taken outside of its range of applicability. In particular, such a procedure is responsible for the
appearance of spurious divergences and steps in multi-reference EDF energies, as was recently observed in
calculations restoring particle number or angular momentum. In the present work, we give a formal analysis of the
origin of this problem for calculations with and without pairing, i.e.\\ constructing the density matrices from
either Slater determinants or quasi-particle vacua. We propose a method to regularize non-diagonal energy kernels such that
divergences and steps are removed from multi-reference EDF energies. Such a removal is a priori quasi-particle-basis dependent. A special feature of the method we use to proceed to the actual regularization is that it singles out one basis among all possible ones. The regularization method is applicable to calculations based on any symmetry restoration or generator coordinate but is limited to EDFs depending only on \\emph{integer} powers of the normal and anomalous density matrices. Eventually, the method is formally illustrated for particle number restoration and is specified to configuration mixing calculations based on Slater determinants. |
