Earth Mover's Distance as a metric to evaluate the extent of charge transfer in excitations using discretized real-space densities

TL;DR

This paper introduces a new Earth Mover's Distance-based metric, $MD$, to quantify charge transfer in electronic excitations using real-space densities, applicable across various computational methods.

Contribution

It proposes a novel $MD$ metric for evaluating electronic excitation charge transfer, compatible with most electronic structure calculations, and demonstrates its effectiveness and diagnostic potential.

Findings

$MD$ effectively characterizes all excitation types.

$MD$ correlates with functional errors and exchange content.

The metric can predict functional failure in excited state calculations.

Abstract

This paper presents a novel theoretical measure, $\mu^{\text{EMD}}$, 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 $\mu^{\text{EMD}}$ 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 $\mu^{\text{EMD}}$ values.