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 averages over coherent superposition states. As a result, these metrics afford markedly different values in various common representations including canonical molecular orbitals, Boys-localized orbitals, and natural transition orbitals (NTOs). Numerical values of these charge-transfer metrics can be unstable with respect to basis-set expansion and may afford nonsensical values in the presence of extremely diffuse basis functions. In contrast, metrics based on well-defined expectation values are stable, representation-invariant, and physically interpretable. Although the NTO representation improves the stability of these ad hoc charge-transfer diagnostics, it remains the case that an incoherent average can only be connected to an expectation value in the absence of superposition. For the NTO representation 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 a single pair of NTOs. Counterexamples are readily found where this is not the case.