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 diatomic molecule examples where this position-dependent DFT+U(R) provides a significant two- to four-fold improvement over DFT+U predictions, when compared to accurate correlated quantum chemistry and experimental references. DFT+U(R) reduces errors in binding energies, frequencies, and equilibrium bond lengths by applying the linear-response, position-dependent U(R) at each configuration considered. This extension is most relevant where variations in U are large across the points being compared, as is the case with covalent diatomic molecules such as transition-metal oxides. We thus provide a tool for deciding whether a standard DFT+U approach is sufficient by determining the strength of the dependence of U on changes in coordinates. We also apply this approach to larger systems with greater degrees of freedom and demonstrate how DFT+U(R) may be applied automatically in relaxations, transition-state finding methods, and dynamics.