Understanding density driven errors for reaction barrier heights

Delocalization errors, such as charge-transfer and some self-interaction errors, plague computationally-efficient and otherwise-accurate density functional approximations (DFAs). Evaluating a semi-local DFA non-self-consistently on the Hartree-Fock (HF) density is often recommended as a computationally cheap remedy for delocalization errors. For sophisticated meta-GGAs like SCAN, this approach can achieve remarkable accuracy. When this HF-DFT (or DFA@HF) significantly improves over the DFA, it is often presumed that the HF density is more accurate than the self-consistent DFA density. By applying the metrics of density-corrected density functional theory (DFT), we show that HF-DFT works for barrier heights by making a localizing charge transfer error or density over-correction, thereby producing a somewhat-reliable cancellation of density- and functional-driven errors for the energy. A quantitative analysis of the charge transfer errors in a few transition states confirms this trend. We do not have the exact functional and exact densities that are needed to evaluate the exact density- and functional-driven errors for the large BH76 database of barrier heights. Instead, we have identified and used three non-local proxy functionals (the SCAN 50% global hybrid, the range-separated hybrid LC-$\omega$PBE, and SCAN-FLOSIC) and their self-consistent densities. These functionals yield reasonably accurate self-consistent barrier heights, and their self-consistent total energies are nearly piecewise linear in fractional electron number - two important points of similarity to the exact functional. We argue that density-driven errors of the energy in a self-consistent density functional calculation are second-order in the density error, and that large density-driven errors arise primarily from incorrect electron transfers over length scales larger than the diameter of an atom.