Simple regularization scheme for multireference density functional theories

Background: Extensions of single-reference (SR) energy-density functionals (EDFs) to multireference (MR) applications involve using the generalized Wick theorem (GWT), which leads to singular energy kernels that cannot be properly integrated to restore symmetries, unless the EDFs are generated by true interactions.

Purpose: We propose a new method to regularize the MR EDFs, which is based on using auxiliary quantities obtained by multiplying the kernels with appropriate powers of overlaps.

Methods: Regularized matrix elements of two-body interactions are obtained by integrating the auxiliary quantities and then solving simple linear equations.

Results: We implement the new regularization method within the self-consistent Skyrme-Hartree-Fock approach and we perform a proof-of-principle angular-momentum projection (AMP) of states in odd-odd nucleus ${}^{26}\mathrm{Al}$. We show that for EDFs generated by true interactions, our regularization method gives results identical to those obtained within the standard AMP procedure. We also show that for EDFs that do not correspond to true interactions, it gives stable and converging results that are different from unstable and nonconverging standard AMP values.

Conclusions: The new regularization method proposed in this work may provide us with a relatively inexpensive and efficient tool to generalize SR EDFs to MR applications, thus allowing for symmetry restoration and configuration mixing performed for typical nuclear EDFs, which most often do not correspond to true interactions.

Figure 1

Convergence of the lowest $I=0$ energy (top) and Skyrme-energy sum-rule residuals (bottom) in function of the highest angular momentum $I_{\text{max}}$ included in the calculations. Open and full circles represent results obtained using the standard AMP method and our LR method, respectively. Calculations were performed for the true Skyrme interaction ${\mathrm{SV}}_{\mathrm{T}}$ (that is, with the tensor EDF terms included).

Figure 2

Same as in Fig. 1 but for the original Skyrme functional SV, that is, with the tensor EDF terms neglected.

Figure 3

Same as in Fig. 1 but for the original SIII Skyrme functional. Panels (b) and (c) show sum-rule residuals of the Skyrme and Coulomb energies, respectively. The Slater approximation of the Coulomb exchange energy was used.

Figure 4

Energies $E_{I=0}$ calculated using the standard AMP (open circles) and LR (full circles) methods, as functions of the number of mesh points $N$. Calculations were performed for the original SIII Skyrme functional. The maximum angular momentum of $I_{\text{max}}=20$ was used.

Figure 5

Same as in Fig. 1 but for the SLy4 Skyrme functional. Full squares represent results obtained using our QR method.