7.3.2 Charge-Transfer Metrics

Standard density functionals do not yield a potential with the correct long-range Coulomb tail, owing to the self-interaction problem. Therefore, excitation energies corresponding to states that sample this tail (e.g., diffuse Rydberg states and some charge transfer excited states) are not given accurately. ²⁰⁶ Casida M. E. et al.
J. Chem. Phys.
(1998), 108, pp. 4439. Link ^{, 1309} Tozer D. J., Handy N. C.
J. Chem. Phys.
(1998), 109, pp. 10180. Link ^{, 728} Lange A., Herbert J. M.
J. Chem. Theory Comput.
(2007), 3, pp. 1680. Link The extent to which a particular excited state is characterized by charge transfer can be assessed using an a spatial overlap metric, $\mathrm{\Lambda }$, defined as ¹⁰⁰⁸ Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link ^{, 551} Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link ^{, 542} Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link

where $O_{ia}$ is the spatial overlap of occupied MO ${\psi }_i$ with virtual MO ${\psi }_a$:

The absolute value signs are necessary since the occupied and virtual MOs are orthogonal, $⟨{\psi }_i|{\psi }_a⟩$, although Q-Chem includes the option to use squares of the MOs instead:

In that case, ${\tilde{O}}_{ia}$ is used in place of $O_{ia}$ in Eq. (7.16). For the original version of the metric (using $O_{ia}$, and where ${\psi }_i$ and ${\psi }_a$ are canonical MOs), Tozer and coworkers find that $0.45\le \mathrm{\Lambda }\le 0.89$ for localized valence excitations whereas Rydberg excitations lie in the range $0.08\le \mathrm{\Lambda }\le 0.27$. ¹⁰⁰⁸ Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link Furthermore, they suggest functional-specific cutoffs for when a particular excitation may have too much charge-transfer character for TDDFT results to be trusted. Note that while Q-Chem implements the definition in Eq. (7.16) that was proposed in Ref.  1008 Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link , the normalizing denominator in this expression is inconsistent for full TDDFT calculations (i.e., those not invoking the TDA), ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link for which the normalization condition on $𝐱$ and $𝐲$ is ${\sum }_{ia}(x_{ia}^2-y_{ia}^2)=1$ rather than ${\sum }_{ia}(x_{ia}^2+y_{ia}^2)=1$.

Empirically, the $\mathrm{\Lambda }$ metric provides very good correlations with TDDFT errors, ¹⁰⁰⁸ Peach M. J. G. et al.
J. Chem. Phys.
(2008), 128, pp. 044118. Link yet its numerical value is difficult to interpret in terms of a physical charge separation. For that reason, a similar metric

was proposed, ⁴⁸³ Guido C. A. et al.
J. Chem. Theory Comput.
(2013), 9, pp. 3118. Link ^{, 482} Guido C. A., Cortona P., Adamo C.
J. Chem. Phys.
(2014), 140, pp. 104101. Link ^{, 1150} Savarese M. et al.
J. Phys. Chem. A
(2017), 121, pp. 7543. Link where

is the vector displacement between the centroids of orbitals ${\psi }_i$ and ${\psi }_a$, and $\parallel {𝐑}_{ia}\parallel$ in Eq. (7.21) is the distance between their centroids. This makes $\mathrm{\Delta }r$ an incoherent average of electron displacements (${\psi }_i\to {\psi }_a$), which has consequences for states where a coherent superposition of displacements is qualitatively important. ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link Because $\mathrm{\Delta }r$ vanishes for any centrosymmetric molecule, it was later augmented by a quantity

where

where $⟨{\psi }_r|{\widehat{r}}^2|{\psi }_r⟩$ is the second moment of orbital ${\psi }_r$. The quantity ${\sigma }_{ia}$ provides a measure of the change in orbital size between donor (${\psi }_i$) and acceptor (${\psi }_a$) MOs, and a metric

was proposed to measure charge-transfer character for TDDFT excitations. ⁴⁸² Guido C. A., Cortona P., Adamo C.
J. Chem. Phys.
(2014), 140, pp. 104101. Link It was suggested that a “trust radius” based on $\mathrm{\Gamma }$, which has dimensions of length, could be used to replace Tozer’s critical values of $\mathrm{\Lambda }$ as a detector of problematic charge-transfer transitions. ⁴⁸² Guido C. A., Cortona P., Adamo C.
J. Chem. Phys.
(2014), 140, pp. 104101. Link

A major shortcoming, however, is that none of these metrics ($\mathrm{\Lambda }$, $\mathrm{\Delta }r$, $\mathrm{\Delta }\sigma$, or $\mathrm{\Gamma }$) is invariant to unity transformation of the MOs. ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link As such, their numerical values can depend quite strongly on which orbitals $\{{\psi }_i\}$ and $\{{\psi }_a\}$ are used. In their original formulations, it was assumed that these would be canonical MOs, although it was subsequently noticed that values of $\mathrm{\Gamma }$ for Rydberg transitions were somewhat erratic but more stable (especially with regard to changes in basis set) when natural transition orbitals (NTOs, Section 7.14.3) were used instead. ⁴⁸² Guido C. A., Cortona P., Adamo C.
J. Chem. Phys.
(2014), 140, pp. 104101. Link An explanation was provided later. ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link When the transition in question is dominated by a single occupied/virtual pair of NTOs (as is often the case, although counterexamples are also easy to find), ⁵⁵¹ Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link ^{, 542} Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link then $\mathrm{\Delta }r$ and $\mathrm{\Gamma }$ approximate expectation values with respect to an excitonic wave function, and any proper expectation value must be invariant to unitary transformations of the occupied MOs and, separately, the virtual MOs. ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link For this reason, if these metrics are going to be used then it is recommended that they be evaluated in the NTO basis rather than the canonical MO basis; Q-Chem can provide both values, for each of $\mathrm{\Lambda }$, $\mathrm{\Delta }r$, $\mathrm{\Delta }\sigma$, and $\mathrm{\Gamma }$.

Although the NTO representation affords better stability with respect to changes in basis set (especially, with regard to addition of diffuse functions), ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link it remains the case that each of these metrics is formulated as an incoherent, amplitude-weighted average of ${\psi }_i\to {\psi }_a$ excitations. In a proper expectation value, the excitation amplitudes have a coherent phase relationship, as in the CIS wave function in Eq. (7.10). It is possible to find examples where the $\mathrm{\Gamma }$ metric provides a very misleading measure of charge separation, because several individual ${\psi }_i\to {\psi }_a$ excitations are associated with large values of $\parallel {𝐑}_{ia}\parallel$, yet they interfere destructively such that the net result hardly displaces any charge at all. ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link

Thus, while the metrics $\mathrm{\Lambda }$, $\mathrm{\Delta }r$, and $\mathrm{\Gamma }$ (in both the canonical and NTO representations) are included in order to help users make contact with existing literature, it is recommended that charge-transfer analysis be based on proper expectation values with respect to transition density matrices. These can be computed using the libwfa module that is described in Section 10.2.12. As documented in Table 10.1 of that section, libwfa can compute expectation values such as

or in other words the expectation value of the distance between the centroid of the excited electron and that of the hole. This constitutes a proper measure of electron–hole separation. ⁵⁴² Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link Exciton size can be measured via

which is also available from libwfa. Other metrics that were recommended in Ref.  542 Herbert J. M., Mandal A.
J. Chem. Theory Comput.
(2024), 20, pp. 9446. Link can also be computed using libwfa, including

where ${\sigma }_{\text{elec}}$ and ${\sigma }_{\text{hole}}$ represent the root-mean-square size of the excited electron and the hole, respectively.