Density-potential inversion from Moreau–Yosida regularization

Abstract

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.

The Structure of Density-Potential Mapping. Part I: Standard Density-Functional Theory

“…But to avoid confusion we will not say that in a regularized setting the HK theorem “holds” even though a unique and well-defined (quasi)­density-potential mapping exists. It is interesting to note that the popular Zhao–Morrison–Parr method for density-potential inversion already implicitly employs Moreau–Yosida regularization and a limit procedure ε → 0 …”

The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields

“…Yet, although the strategy is very beneficial in order to get differentiable functionals, a unique quasidensity-potential mapping, and for setting up a well-defined Kohn–Sham scheme, it has not evolved into a practical method as of yet. On the other hand this form of regularization relates closely to the Zhao–Morrison–Parr method for density-potential inversion which clearly has a practical purpose …”

“…On the other hand this form of regularization relates closely to the Zhao−Morrison−Parr method 52 for density-potential inversion which clearly has a practical purpose. 53 In addition to the above outlined approach of achieving functional differentiation in CDFT, ref 31 also demonstrated the construction of a well-defined Kohn−Sham iteration scheme labeled "MYKSODA". Although implemented only for a toy model (a one-dimensional quantum ring), presented MYKSO-DA is an algorithm for calculations in the full setting of groundstate CDFT employing a Moreau−Yosida-regularized functional.…”

“…On the other hand this form of regularization relates closely to the Zhao–Morrison–Parr method 52 for density-potential inversion which clearly has a practical purpose. 53 …”

Exchange energies with forces in density-functional theory

“…We note that the theoretical setting of an exact regularization procedure is available that renders the involved functionals differentiable [19][20][21] and that this surprisingly links to the Zhao-Morrison-Parr method for mapping ρ(r) ↦ v(r). 22 Importantly, in this work we highlight that also the exchange-only energy is non-differentiable with respect to densities, thus allowing local-exchange potentials only in the form of generalized constructions such as the optimized effective potential (OEP), or, alternatively, leading to an additional vector potential for exchange effects. This vector potential naturally appears in a forcebased approach and acts semi-locally on the wave function.…”