Exact Constraint of Density Functional Approximations at the Semiclassical Limit

We introduce the semiclassical limit to electronic systems by taking the limit $\hbar\rightarrow 0$ in the solution of Schr\"odinger equations. We show that this limit is closely related to one type of strong correlation that is particularly challenging from conventional multi-configurational perspective but can be readily described through semiclassical analysis. Furthermore, by studying the performance of density functional approximations (DFAs) in the semiclassical limit, we find that mainstream DFAs have erroneous divergent energy behaviors as $\hbar \rightarrow 0$, violating the exact constraint of finite energy. Importantly, by making connection of the significantly underestimated DFA energies of many strongly correlated transition-metal diatomic molecules to their rather small estimated $\hbar_{\text{eff}}$, we demonstrate the usefulness of our semiclassical analysis and its promise for inspiring better DFAs.