Summary and perspectives

In this work, we proposed a method to regularize the two-body off-diagonal MR DFT kernels and we presented the first application thereof to a representative case of the angular-momentum projection. The method is based on two general assumptions:

1. First, it assumes that the MR EDF is regularizable, meaning in practice that there exist a regularization scheme replacing the kernel by its smooth counterpart that can be expanded in a set of Wigner $D$-functions (16). No explicit knowledge of the regularization scheme is required.

2. Second, it assumes that the singularities, which appear in the denominator of the two-body kernel, originate from vanishing overlap, which is a result of using the GWT. This allows us to identify terms proportional to $\langle \Psi \vert\tilde\Psi\rangle^{-(1+\eta )}$, where $\eta$ comes from the direct density dependence of the generator of two-body kernel, as potentially the most dangerous ones.

The essential advantage of the method is in avoiding the necessity to explicitly remove the self-interactions [2,3]. Instead, the proposed method relies on computing a set of auxiliary integrals of non-singular kernels obtained by multiplying the original ones with an appropriately chosen power of the overlap. Provided that the GWT is the only source of singularities, the auxiliary integrals turn out to be linear functions of the true regularized matrix elements, and the entire problem can be reduced to an algebraic task of solving a set of linear equation. The method also has a certain internal flexibility, namely, it can be applied to selected parts of the EDF only. This feature is important in cases when troublesome parts of the EDF can be isolated. The example is the Coulomb exchange interaction treated within the Slater approximation.

A certain disadvantage of the method is the fact that, to achieve a desired accuracy, the auxiliary integrals require more quadrature nodes. In addition, one has to invert or perform the SVD decomposition of the auxiliary matrices $A$, see Eqs. (25) and (35)), which can be a potential source of numerical inaccuracies. Both problems become more acute with increasing power of the overlap multiplying the kernel. Nevertheless, in this exploratory study we have demonstrated that both the LR and QR schemes can be realized. Problems encountered for the calculations involving the SLy4 functional reflect the inadequacy of the scheme in applications to fractional-power density-dependent terms, which lead to non-analytic dependence of kernels on the Euler angles.

An interesting feature of our calculations is that they reveal the fact of how much the effects of using EDFs that are not derived from true interactions can be inconspicuous. Our calculations show that even the slightest departure from the true interaction is immediately detectable through the sum rules, but it can be completely invisible when looking at energies of low-lying states, which are often perfectly stable.

Finally, in principle the proposed regularization scheme can also be generalized to the generator coordinate method (GCM). For that, it would be enough that the GCM overlaps and matrix elements can be expanded on a set of appropriate orthogonal polynomials, whereupon coefficients analogous to those given in Eqs. (25) and (35) can be easily determined. In the case discussed in the present paper, or, for that matter, for any symmetry-restoration problem, such orthogonal polynomials are provided by the properties of the underlying integration over the symmetry group. However, depending on the particular case, more general implementations of our regularization scheme, within the full GCM approach, should also be possible.

This work was supported in part by the Polish National Science Centre (NCN) under Contract No. 2012/07/B/ST2/03907, by the THEXO JRA within the EU-FP7-IA project ENSAR (No. 262010), by the ERANET-NuPNET grant SARFEN of the Polish National Centre for Research and Development (NCBiR), and by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme.