Electronic structure reading group - A Moreau–Yosida-Based Kohn–Sham Inversion Scheme

Abstract:

Density-functional theory (DFT) is central to efficient calculations in chemistry, materials
science, and solid-state physics. While formally exact, the universal density functional is, in
general, unknown and must therefore be approximated. Most applications use the Kohn–Sham
formulation of DFT to compute ground-state densities, but the inverse Kohn–Sham problem –
recovering the non-interacting potential from a given density – is equally fundamental. Kohn–
Sham inversion has been proposed as a route to obtaining accurate exchange-correlation poten-
tials and improving functional approximations. In this talk, I will present a rigorous inversion
scheme based on a Moreau–Yosida-regularised formulation of DFT, which enables the extraction
of the exchange-correlation potential as a strict mathematical limit. This regularisation not only
stabilises the inversion but also provides a framework for deriving error bounds, thereby bridg-
ing theory and computation. Using periodic homogeneous Sobolev spaces, the theoretical
framework harmonises with the physics of periodic systems, and the structure of the inversion
scheme allows it to be implemented in existing electronic structure codes, such as DFTK. This
enables the inversion to be performed on realistic systems, such as bulk silicon, gallium arsenide,
potassium chloride, and sodium chloride. I will in particular highlight the study of error propa-
gation from the input density to the resulting potential and the consequences of various choices
for the guiding functional.