Importance of Orbital Invariance in Quantifying Electron-Hole Separation and Exciton Size

A fundamental tenet of quantum mechanics is that properties should be independent of representation. In self-consistent field methods such as density functional theory, this manifests as a requirement that properties be invariant with respect to unitary transformations of the occupied molecular orbitals and (separately) the unoccupied molecular orbitals. Various ad hoc measures of excited-state charge separation that are commonly used to analyze time-dependent density-functional calculations violate this requirement, as they are based on incoherent averages of excitation amplitudes rather than expectation values involving coherent superpositions. As a result, these metrics afford markedly different values in various common representations, including canonical molecular orbitals, Boys-localized orbitals, and natural orbitals. Numerical values can be unstable with respect to basis-set expansion and may afford nonsensical results in the presence of extremely diffuse basis functions. In contrast, metrics based on well-defined expectation values are stable, representation-invariant, and physically interpretable. Use of natural transition orbitals improves the stability of the incoherent averages, but numerical values can only be interpreted as expectation value in the absence of superposition. To satisfy this condition, the particle and hole density matrices must each be dominated by a single eigenvector so that the transition density is well described by a single pair of natural transition orbitals. Counterexamples are readily found where this is not the case. Our results suggest that ad hoc charge-transfer diagnostics should be replaced by rigorous expectation values, which are no more expensive to compute.