Density analysis for estimating the degree of on-site correlation on transition-metal atoms in extended systems

In the context of the modified Becke-Johnson (mBJ) potential, we recently underlined that $\bar{g}$, the average of $\left\vert\nabla\rho\right\vert/\rho$ in the unit cell, has markedly different values in transition-metal oxides and pure transition metals [Tran et al., J. Appl. Phys. 126, 110902 (2019)]. However, since $\bar{g}$ is a constant it is not able to provide local information about a particular atom in the system. Furthermore, while $\overline{g}$ can be used only for periodic bulk solids, a local (i.e., position-dependent) version would allow us to consider also low-dimensional systems and interfaces. Such a local function has been proposed by Rauch et al. [J. Chem. Theory Comput. 16, 2654 (2020)] for the local mBJ potential. Actually, a local version of $\overline{g}$, or of another similar quantity like the reduced density gradient $\overline{s}$, could also be used in the framework of other methods. Here, we explored the idea to use such a local function $\tilde{g}$ (or $\tilde{s}$), defined as the average of $g$ (or $s$) over a certain region around a transition-metal atom, to estimate the degree of on-site correlation on this atom. We found a large difference in our correlation estimators between non-correlated and correlated materials, proving its usefulness and reliability. Our estimators can subsequently be used to determine whether or not a Hubbard $U$ on-site correction in the DFT+$U$ method should be applied to a particular atom. This is particularly interesting in cases where the degree of correlation of the transition-metal atoms is not clear, like interfaces between correlated and non-correlated materials or oxygen-covered metal surfaces. In such cases, our estimators could also be used for an interpolation of $U$ between correlated and non-correlated atoms.