Application of Potential Functional Theory to Fermions Confined by a Mexican-hat Potential

Kohn-Sham density functional theory (KSDFT) is the most used computational method for electronic structure calculations. However, the introduction of auxiliary orbitals in KSDFT increases computational complexity and orbital-free density functional theory (OFDFT) is an active field of research.

A serious drawback of OFDFT is an inability to describe discrete energy quantization. Recent work (Okun, Cancio and Burke, Phys. Rev. B 109, 195156 (2024)), applies the alternative Wentzel, Kramers, Brillouin (WKB) method to a jellium slab (a Fermi gas in two dimensions with a potential well in the third). Floor functions of WKB actions expressed as potential functionals give sub-milliHartree error for the energy of noninteracting particles, and equivalent accuracy for the density of states, including a discontinuous number staircase for single wells. Accurate expressions exist for energy and number versus chemical potential out to fourth order.

We extend this method to Mexican hat potentials. Calculations including approximate eigenvalue splitting due to binding and anti-binding across the barrier achieve reasonable accuracy for the density of states. This offers a possible method of correcting standard OFDFT calculations to include the effect of discrete energy bands due to quantum confinement.