Symbolic computation of the Hartree-Fock energy from a chiral EFT three-nucleon interaction at N$^2$LO

TL;DR

This paper introduces a symbolic computational method using Mathematica to derive a Skyrme-like energy density functional from chiral EFT three-nucleon interactions at N$^2$LO, facilitating quasi-local approximations of the Hartree-Fock energy.

Contribution

It presents the first symbolic approach to derive a Skyrme-like functional from chiral EFT three-nucleon interactions, enabling extensions to other interactions.

Findings

Derivation of Hartree-Fock energy in symbolic form.

Simplification using scalar/vector-isoscalar/isovector parts.

Methodology for extending to other three-nucleon interactions.

Abstract

We present the first of a two-part Mathematica notebook collection that implements a symbolic approach for the application of the density matrix expansion (DME) to the Hartree-Fock (HF) energy from a chiral effective field theory (EFT) three-nucleon interaction at N$^2$LO. The final output from the notebooks is a Skyrme-like energy density functional that provides a quasi-local approximation to the nonlocal HF energy. In this paper, we discuss the derivation of the HF energy and its simplification in terms of the scalar/vector-isoscalar/isovector parts of the one-body density matrix. Furthermore, a set of steps is described and illustrated on how to extend the approach to other three-nucleon interactions.