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.