Accurate potential energy surfaces for transition-metal complexes with DFT+U(R)

We introduce an improvement to the Hubbard U augmented density functional approach known as DFT+$U$ that incorporates variations in the value of self-consistently calculated, linear-response $U$ with changes in geometry. This approach overcomes the one major shortcoming of previous DFT+$U$ studies, i.e. the use of an averaged Hubbard U when comparing energies for different points along a potential energy surface is no longer required. While DFT+$U$ is quite successful at providing accurate descriptions of localized electrons (e.g. $d$ or $f$) by correcting self-interaction errors of standard exchange correlation functionals, we show several examples from diatomic molecules to porphyrins to surface science applications where this position-dependent DFT+$U(\mathbf{R})$ provides a significant two- to four-fold improvement over DFT+$U$ predictions. DFT+$U(\mathbf{R})$ reduces errors in binding energies, frequencies, and equilibrium bond lengths by applying the linear-response, position-dependent $U(\mathbf{R})$ at each point. We also propose a metric for whether a standard DFT+$U$ approach is sufficient by determining the strength of the dependence of $U$ on changes in coordinates.