Density-potential inversion from Moreau–Yosida regularization

For a quantum-mechanical many-electron system, given a density, the Zhao–Morrison–Parr method allows to compute the effective potential that yields precisely that density. In this work, we demonstrate how this and similar inversion procedures mathematically relate to the Moreau–Yosida regularization of density functionals on Banach spaces. It is shown that these inversion procedures can in fact be understood as a limit process as the regularization parameter approaches zero. This sheds new insight on the role of Moreau–Yosida regularization in density-functional theory and allows to systematically improve density-potential inversion. Our results apply to the Kohn–Sham setting with fractional occupation that determines an effective one-body potential that in turn reproduces an interacting density.