# An improved density matrix expansion for spin-unsaturated nuclei

## TL;DR

This paper improves the density matrix expansion method for spin-unsaturated nuclei, achieving higher accuracy in reproducing non-local exchange energies, which can enhance the development of microscopic energy density functionals in nuclear physics.

## Contribution

The authors reformulate the density matrix expansion for the vector part, incorporating phase-space averaging and anisotropy considerations, leading to a few-percent accuracy in semi-magic nuclei.

## Key findings

- Phase-space averaging significantly improves the vector density matrix.
- The reformulated DME achieves a few-percent accuracy for semi-magic nuclei.
- The improved DME can serve as a foundation for microscopic energy functionals.

## Abstract

A current objective of low-energy nuclear theory is to build non-empirical nuclear energy density functionals (EDFs) from underlying inter-nucleon interactions and many-body perturbation theory (MBPT). The density matrix expansion (DME) of Negele and Vautherin is a convenient method to map highly non-local Hartree-Fock expressions into the form of a quasi-local Skyrme functional with density-dependent couplings. In this work, we assess the accuracy of the DME at reproducing the non-local exchange (Fock) contribution to the energy. In contrast to the scalar part of the density matrix for which the original formulation of Negele and Vautherin is reasonably accurate, we demonstrate the necessity to reformulate the DME for the vector part of the density matrix, which is needed for an accurate description of spin-unsaturated nuclei. Phase-space averaging techniques are shown to yield a significant improvement for the vector part of the density matrix compared to the original formulation of Negele and Vautherin. The key to the improved accuracy is to take into account the anisotropy that characterizes the local-momentum distribution in the surface region of finite Fermi systems. Optimizing separately the DME for the central, tensor and spin-orbit contributions to the Fock energy, one reaches a few-percent accuracy over a representative set of semi-magic nuclei. With such an accuracy at hand, one can envision using the corresponding Skyrme-like energy functional as a microscopically-constrained starting point around which future phenomenological parameterizations can be built and refined.