A note on the accuracy of spin-densities from Kohn-Sham Density Functional Theory

Abstract

Quantifying the magnetic properties of open-shell molecules is a common task in chemistry and is increasingly performed in silico using Kohn-Sham density functional theory (KS-DFT). Previous work demonstrates that the predictive accuracy of a few functionals for one such property - hyperfine coupling constants (HFCCs) - is possible, implying that such approximations must yield accurate spin-densities at the nucleus. However, the ability of such functionals to globally predict accurate spin-densities of comparable quality to rigorous ab initio coupled cluster theory, for example, is dubious, despite this being a fundamental quantity for KS-DFT. This work intends to explore the matter by evaluating moments of the spin-density, $\langle r^n \rangle = \int ρ(r)r^n dτ$, $n=-2,\cdots,2,3$, for second-, third-, and fourth-row atoms to compare various KS-DFT functionals against CCSD. Our results broadly indicate that the tested functionals experience significant deviations with respect to CCSD spin-densities in regions up to 1 Bohr away from the nuclei, with most errors occurring in the immediate vicinity of the nucleus. We find evidence of extreme errors by some functionals for individual $α$/$β$ spin-densities, although several ultimately end up benefiting from significant error cancellation once the corresponding global spin-density is formed. Nevertheless, a comparison between CAM-B3LYP and the Quantum Theory Project (QTP)-family of DFT functionals based on Correlated Orbital Theory conditions across all error metrics demonstrates that QTP00 more accurately reproduces the global spin-density as well as HFCCs, generally offering results in better agreement with CCSD. In line with previous work, we also corroborate the success of PBE0 and the TPSS-family of functionals for HFCCs, further finding that both approximations generally yield spin-densities that are amongst the best.