A comparison of exact and model exchange-correlation potentials for molecules

Accurate exchange-correlation (XC) potentials for 3-dimensional systems -- via solution of the \emph{inverse} density functional theory (DFT) problem -- are now available to test the quality of DFT approximations. Herein, the \emph{exact} XC potential for five molecules -- hydrogen molecule at three different bond-lengths, lithium hydride, and water -- are computed using accurate ground-state densities from full configuration interaction (CI) calculations. These potentials are then compared to model XC potentials obtained from DFT calculations with commonly used non-local (B3LYP, HSE06, SCAN0, and M08-HX) and local/semi-local (SCAN, PBE, PW92) XC functionals. While relative errors in the ground-state densities from these models are order $\mathcal{O}(10^{-3}-10^{-2})$, much larger errors in the model XC potentials are found, $\mathcal{O}(10^{-1}-10^0)$, in both the $L_2$ norm of the potential as well as its gradients. These errors are exacerbated in strongly correlated situations, such as the stretched $\text{H}_2$ molecule. Among the model XC functionals under consideration, SCAN0 offers the best quantitative and qualitative agreement with the exact XC potential, underlining the significance of satisfying the exact conditions as well as the incorporation of non-local effects in the construction of XC functionals. Overall, this work indicates that tests against the exact XC potential will provide a promising new direction for building more accurate XC functionals for DFT.