Assessment of local density approximation based exchange–correlation functional for a two-dimensional spin polarized dipolar Fermi gas

In this paper we assess the accuracy of the local density approximation (LDA) based expressions for exchange and correlation functionals associated with a two-dimensional, spin-polarized, dipolar Fermi gas. For this purpose we consider and study an exactly solvable system of two interacting spin-polarized dipolar fermions confined in a two-dimensional harmonic oscillator potential. We determine the exact triplet ground state wavefunction, energy and the density by separating the time-independent Schrödinger equation in center-of-mass and relative coordinates and numerically solving the eigenvalue equation in relative coordinate. The exact Kohn–Sham orbitals, corresponding energies and exact Kohn–Sham potential are obtained from the exact density by inverting the Kohn–Sham equation. These results serve as benchmark data for assessing the accuracies of the LDA based expressions for a two-dimensional, spin-polarized, dipolar Fermi gas. We find that LDA results for total energy, interaction energy, and Kohn–Sham eigenvalues match well with the exact results for smaller values of dipole–dipole interaction strength. For higher values of interaction strength, LDA results for the total energy are underestimated. On the other hand, the LDA based interaction potentials overestimate the repulsion between the dipoles and their spatial profiles differ considerably from their exact counterparts and the discrepancy between them is higher for stronger interaction. Graphical abstract: .