Improved density matrix expansion for spin-unsaturated nuclei

Abstract

A current objective of low-energy nuclear theory is to build nonempirical nuclear energy density functionals (EDFs) from underlying internucleon interactions and many-body perturbation theory (MBPT). The density matrix expansion (DME) of Negele and Vautherin is a convenient method to map highly nonlocal 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 nonlocal 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 parametrizations can be built and refined.

Testing the density matrix expansion againstab initiocalculations of trapped neutron drops

“…[9] have modified the DME formalism from Negele and Vautherin [21], which provides an extremely poor description of the vector part of the density matrix [11]. The standard DME is much better at reproducing the scalar density matrices, but errors are still sufficiently large that the discrepancy with full finite-range Hartree-Fock calculations can reach the MeV per particle level.…”

“…For example, in notation introduced in Refs. [11,12], one expands the spin-scalar part (in both isospin channels) of the one-body density matrix as…”

“…The standard DME is much better at reproducing the scalar density matrices, but errors are still sufficiently large that the discrepancy with full finite-range Hartree-Fock calculations can reach the MeV per particle level. Gebremariam and collaborators [11] introduced a new phase space averaging (PSA) approach that accounts for the diffuse Fermi surface [23] and anisotropy [24] of the local momentum distribution, with no free parameters. The PSA treatment leads to substantial improvements, particularly for the vector density matrices, where relative errors in integrated quantities are reduced by as much as an order of magnitude across isotope chains [11].…”

“…Gebremariam and collaborators [11] introduced a new phase space averaging (PSA) approach that accounts for the diffuse Fermi surface [23] and anisotropy [24] of the local momentum distribution, with no free parameters. The PSA treatment leads to substantial improvements, particularly for the vector density matrices, where relative errors in integrated quantities are reduced by as much as an order of magnitude across isotope chains [11]. In the present work, we test the difference between the original Negele-Vautherin (NV) and the new PSA prescriptions only for scalar parts.…”

“…The UNEDF project aims to develop a comprehensive theory of nuclear structure and reactions utilizing the most advanced computational resources and algorithms available, including high-performance computing techniques to scale to petaflop platforms and beyond [7]. One element of the UNEDF program involves the direct injection of microscopic physics into novel energy functionals, with the DME a key tool [9][10][11][12]. Another element has led to efficient density functional theory (DFT) solvers adapted for the DME [13] and to neutrons in external traps, which allow accurate and rapid testing of candidate functionals [14].…”