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

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