Examples illustrating the use of CT numbers Ω A→B in complicated cases of photochemical reactions involving transition metal complexes have been considered in Ref. 23. Here, we consider a relatively simple example, 4-(dimethylamino)benzonitrile (DMABN), which has something of a history within TD-DFT. Its spectroscopy which consists primarily of the usual 1 L a and 1 L b states, often discussed in the context of polycyclic aromatic hydrocarbons (PAHs). Setting aside detailed symmetry considerations, these two states have roughly perpendicular transition moments (along axes "a" and "b"). The former is primarily a HOMO → LUMO excitation, with significant ionic character in PAHs, while 1 L b is a mixture of (HOMO -1) → LUMO and HOMO → (LUMO + 1). TD-DFT calculations often afford an unbalanced treatment of these two states, which are quite close in energy in the case of DMABN. The DMABN molecule is a canonical example of the phenomenon of dual fluorescence, or the appearance of two fluorescence bands whose intensity ratio is highly sensitive to solvent polarity. Other donor-π-acceptor (or "push-pull") systems also exhibit this behavior, and examples such as push-pull porphyrins and thiophene-based push-pull polymers have been widely studied as potential photosensitizers for solar cells. Often, TD-DFT calculations have been used in an attempt to establish design principles. Other categories of push-pull systems may be useful as dopants to produce devices with novel optoelectronic properties, including photoswitchable molecules, and molecules that exhibit TADF without the use of heavy metals. Dual fluorescence represents an exception to Kasha's rule, which states that emission typically occurs in a single band originating from the lowest excited state, insofar as radiationless internal conversion from higher-lying excited states is usually rapid and efficient. The dependence on solvent polarity has long been been interpreted in terms of excited-state dynamics that access a twisted intramolecular CT (TICT) state, characterized by rotation of the -N(CH 3 ) 2 group out of the phenyl plane. In this picture, the TICT state is stabilized in polar solvents, relative to the "locally excited" (LE) or 1 ππ * state, and is the origin of the longer-wavelength fluorescence band. This interpretation has been questioned, however, in both DMABN and similar donor-π-acceptor systems. What is not in dispute is that the S 1 and S 2 states of DMABN exhibit different degrees of CT upon vertical excitation. In the gas phase, S 1 is the LE state and S 2 is the CT state, as evidenced by a dipole moment that is ≈ 6 D larger in S 2 than S 1 , whose dipole moment is ≈ 3 D larger than that of the ground state. This interpretation is furthered by examining the CT numbers and GLIs for both states, which are provided in Fig. based on Luzanov's definition (l A→B ), normalized as percentages to sidestep issues with the normalization of Eq. (4.3). These quantities suggest that the S 0 → S 1 transition is characterized by a single large CT number corresponding to electron transfer from the amino lone pair into the phenyl ring, yet the GLI analysis suggests that 73% of the excited electron is localized on the phenyl ring, consistent with the idea that S 1 is the ππ * state. For the S 2 state, the CT numbers provide clear evidence of amino → phenyl → cyano electron transfer, with smaller fractions of the excited electron is localized on the amino and phenyl groups as compared to S 1 , and a larger fraction transferred to the cyano moiety.