Stable, Fast, and Accurate Kohn-Sham Inversion in Gaussian Basis for Open Shell Molecular and Condensed Phase Systems via Density Matrix Penalization

Abstract

Here we present a density matrix based KS inversion method formulated entirely within a Gaussian basis representation to optimize a KS potential matrix that reproduces a target electron density. Inverse Kohn-Sham (KS) density functional theory (DFT) aims to determine the effective local KS potential that reproduces a target electron density, and is important both for electronic structure analysis and for the development of orbital based correction methods. In finite Gaussian basis implementations, however, conventional inverse KS-DFT approaches such as the Zhao-Morrison-Parr (ZMP) method often become poorly constrained and inefficient, because the real space penalty potential is projected onto a limited number of Gaussian basis matrix elements, which can strongly coarse-grain its spatial variation. In the present method, the density matrix mismatch is defined in a Lowdin orthogonalized basis, which yields a penalty energy invariant under unitary rotations in that basis. The corresponding penalty potential contribution to the KS Hamiltonian is derived analytically in the original nonorthogonal Gaussian basis. Across a wide range of penalty strengths, the self consistent field (SCF) optimization remains robust and efficient for various open shell systems, while progressively tightening the penalty drives the electron density into accurate agreement with the target. Benchmarks on molecules and condensed phase systems show that the method achieves substantially smaller attainable density deviations than the conventional ZMP method. The method provides a fast and accurate route to KS inversion in finite Gaussian basis sets and may also be useful for future orbital based correction schemes.