Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels

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 nondiagonal 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 nondiagonal 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 quasiparticle vacua. We propose a method to regularize nondiagonal energy kernels such that divergences and steps are removed from multi reference EDF energies. Such a removal is a priori quasiparticle-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 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.

Figure 1

Schematic illustration of the different bases introduced in the text. (Left) The Slater determinants $|{\Phi }_0\rangle$ and $|{\Phi }_1\rangle$ are expressed in their respective natural bases $\{|a_i\rangle \}$ and $\{|b_l\rangle \}$. The shaded areas indicate that one state of the basis $\{|b_l\rangle \}$ ($\{|a_i\rangle \}$) is a priori spread over all particle and hole states of the basis $\{|a_i\rangle \}$ ($\{|b_l\rangle \}$). (Right) The BMZ decomposition leads to two bases that are such that each transformed state $|{\tilde{b}}_i\rangle$ ($|{\tilde{a}}_i\rangle$) decomposes only onto two states $({\tilde{a}}_i,{\tilde{a}}_{\overline{ı}})$ [$({\tilde{b}}_i,{\tilde{b}}_{\overline{ı}})$] (one hole and one particle). At the same time, the latter decomposition highlights that the notion of conjugated pairs for Slater determinants relates to the association of each hole state with a particular particle state. Note that energy levels have no specific meaning in this figure; they are just meant to characterize occupied and empty levels for each of the Slater determinants.