In quantum chemistry, one typically represents the environment of the electrons by an electrostatic potential and then computes the electronic ground state for this potential. It is a remarkable fact in density-functional theory that this mapping from potential to ground-state density can, in principle, also be done in the other direction. However, though the inverse mapping provably exists, its detailed form is much too complicated to write down in any useful explicit form and it is also mathematically irregular. For periodic systems the task of reconstructing a potential from a density is particularly delicate, both mathematically and computationally. Small changes in the input density can lead to large changes in the reconstructed potential. In our work, we use a Moreau–Yosida regularisation approach that effectively smooths the inversion problem and makes it numerically stable. This allows us to control how errors or noise in the input density influence the final reconstructed potential.
A reliable way of doing the inversion is important for a deeper understanding of density-functional theory as well as for constructing new approximate models. In Kohn-Sham density-functional theory, the complicated interacting many-electron problem is replaced by a non-interacting problem with an effective local potential chosen to reproduce the same ground-state density. A way of reconstructing potentials from densities thus enables high-accuracy effective potentials to be computed. These can later serve as reference data for computationally cheaper approximate models.
We illustrate the framework with numerical results for one-dimensional periodic systems where the reference density comes from a Hartree-Fock wave function. The calculations show that as the regularisation parameter is decreased, the reconstructed local potentials converge, and we can obtain a well-defined local exact-exchange potential that captures the effects of the very nonlocal Hartree-Fock exchange interaction.
Published: 9th February 2026. https://pubs.aip.org/aip/jcp/article/164/6/064104/3378863/Regularized-density-potential-inversion-for