This paper presents a novel theoretical measure, μEMD, based on
the Earth Mover's Distance, for quantifying the density shift caused by
electronic excitations in molecules. As input, the EMD metric uses only the
discretized ground and excited state electron densities in real space,
rendering it compatible with almost all electronic structure methods used to
calculate excited states. The EMD metric is compared against other popular
theoretical metrics for describing the extent of electron-hole separation in a
wide range of excited states (valence, Rydberg, charge-transfer, etc). The
results showcase the EMD metric's effectiveness across all excitation types and
suggest that it is useful as an additional tool to characterize electronic
excitations. The study also reveals that μEMD can function as a
promising diagnostic tool for predicting the failure of pure
exchange-correlation functionals. Specifically, we show statistical
relationships between the functional-driven errors, the exact exchange content
within the functional, and the magnitude of μEMD values.Comment: 31 pages, 7 figure