Earth Mover’s Distance as a Metric to Evaluate the Extent of Charge Transfer in Excitations Using Discretized Real-Space Densities

This paper presents a novel theoretical
measure, μEMD, based on the earth mover’s
distance (EMD), 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 among
the functional-driven errors, the exact exchange content within the
functional, and the magnitude of μEMD values.