Conclusions

In summary, in this work, for the first time we constructed the finite-range, higher-order in derivatives pseudopotential, for which we built all terms up to sixth order N$^3$LO, and then we derived the corresponding nonlocal nuclear N$^3$LO EDF by calculating the HF average energy over a nuclear Slater determinant. The proposed pseudopotential can thus be regarded as a generator of the EDF, which has several advantages for calculations beyond the simple HF level. First, the use of a regulator introduces a natural cut-off at high momenta, preventing divergences of pairing energies or correlation energies in the (Q)RPA limit. Then, for multi-reference EDF calculations, the link with a pseudopotential guaranties the absence of poles in energy kernels. Moreover, the fact that the pseudopotential does not depend on density guaranties that average energies are not polluted by self-interaction or self-pairing.

The ability of this pseudopotential to generate attractive pairing has not been checked yet - this is of course of crucial importance for future developments. However, the fact that at NLO it has all the features of the Gogny interaction gives us hope that this will indeed be the case. Let us also mention that all the central terms of the pseudopotential and the corresponding mean fields have already been implemented in the solver HFODD [20,21,22] and numerical studies of finite nuclei are under way. Moreover, a preliminary study has demonstrated that the NLO functional can provide binding energy of doubly-magic nuclei with a reasonable accuracy [17].

This work has been supported in part by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme and by the Polish National Science Center under Contract No. 2012/07/B/ST2/03907. The research of FR is supported by the Natural Science and Engineering Research Council (NSERC).