In spite of their tremendous success, the limitations of current nuclear energy density functionals (EDFs), all parameterized empirically in the form of the local Skyrme, the nonlocal Gogny or relativistic functionals, have become apparent in the past several years. In order to address these deficiencies, a current objective of low-energy nuclear theory is to build non-empirical nuclear EDFs from underlying two-, three- and possibly four-nucleon interactions and many-body perturbation theory (MBPT). In this work, the first step towards that goal is taken by calculating the HF contribution from the chiral EFT two- and three-nucleon interaction at N2LO. The density matrix expansion (DME) of Negele and Vautherin is a convenient method to map the highly non-local Hartree-Fock expression into the form of a quasi-local Skyrme-like functional with density dependent couplings. Reformulating the DME in terms of phase space averaging (PSA) techniques, we show that the resulting DME, PSA-DME, is more general and has a significantly better accuracy for spin-unsaturated systems than the original DME of Negele and Vautherin. This is achieved without compromising the accuracy of PSA-DME for spin-saturated ones. Imposing the assumption of time-reversal invariance, we apply PSA-DME to the HF energy from the chiral EFT two- and three-nucleon interaction (at N2LO) and calculate the couplings of the emerging EDF analytically using a combination of analytical and symbolic approaches. Subsequently, we perform preliminary analysis of these couplings and show that their density dependence is driven by the long-range (pion-exchange) part of the interaction. Finally, we discuss the UNEDF semi-phenomenological approach that is attempting to utilize the results of this work.
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