Electron Density Errors and Density-Driven Exchange-Correlation Energy Errors in Approximate Density Functional Calculations

Since its formal introduction, density functional theory has achieved many successes in the fields of molecular and solid-state chemistry. According to its central theorems, the ground state of a many-electron system is fully described by its electron density, and the exact functional minimizes the energy at the exact electron density. For many years of density functional development, it was assumed that the improvements in the energy are accompanied by the improvements in the density, and the approximations approach the exact functional. In a recent analysis ( Medvedev et al. Science 2017 , 355 , 49 - 52 .), it has been pointed out for 14 first row (Be-Ne) atoms and cations with 2, 4, or 10 electrons that the nowadays popular flexible but physically less rigorous approximate density functionals may provide large errors in the calculated electron densities despite the accurate energies. Although far-reaching conclusions have been drawn in this work, the methodology used by the authors may need improvements. Most importantly, their benchmark set was biased toward small atomic cations with compressed, high electron densities. In our paper, we construct a molecular test set with chemically relevant densities and analyze the performance of several density functional approximations including the less-investigated double hybrids. We apply an intensive error measure for the density, its gradient, and its Laplacian and examine how the errors in the density propagate into the semilocal exchange-correlation energy. While we have confirmed the broad conclusions of Medvedev et al., our different way of analyzing the data has led to conclusions that differ in detail. Finally, seeking for a rationale behind the global hybrid or double hybrid methods from the density's point of view, we also analyze the role of the exact exchange and second-order perturbative correlation mixing in PBE-based global hybrid and double hybrid functional forms.