Ultrafast internal conversion without energy crossing

Ultrafast (sub-picosecond) internal conversion can occur between electronic states without energetically accessible conical intersections. For that, the molecule must remain in a region of at least weak nonadiabatic coupling, multiplying the odds of a transition, each with a small probability. This phenomenon is discussed with a simple analytical model correlating the instantaneous probability to the lifetime.

The most accepted picture of molecular events following a photoexcitation dictates that an ultrafast, sub-picosecond internal conversion between electronic states should occur at the proximity of a conical intersection, where the nonadiabatic couplings are large enough to induce the transition. 1 If, however, the molecule rests on regions of considerable energy gaps, the gap law 2 predicts that the internal conversion will be slow, competing with other time-consuming processes such as intersystem crossing and luminescence. Although such a picture tends to be correct, there may be exceptions.
Our team recently found that the S3 to S2 internal conversion of n-protonated azaindoles (n-AIH + ) was ultrafast, despite the absence of energetically accessible conical intersections. 3 For 6-AIH + , surface hopping simulations predicted an S3 lifetime of 156  40 fs, even though the energy gaps at the hopping time had a distribution with a 0.47 eV mean value and 0.13 eV standard deviation. Similarly, for 7-AIH + , the predicted S3 lifetime was 278  36 fs with energy gaps at the hopping times having also a 0.47 eV mean value and 0.22 eV standard deviation.
These results are counterintuitive as we might expect that such ultrashort lifetimes would require hoppings occurring near conical intersections. However, as we showed in that work, the lowest energy S3/S2 conical intersection was too high for both molecules. In 6-AIH + , the intersection is 1.3 eV above the S3 minimum, while in 7-AIH + , it is 1.5 eV above it.
We rationalized those results by noticing that the mean S3/S2 energy gap during the S3 dynamics was 0.6 and 0.8 eV for 6-and 7-AIH + , respectively. Thus, n-AIH + vibrating on S3 is constantly in a weak coupling region. Ethylene dynamics would be the prototypical example with its twisted-pyramidalized conical intersection funneling the excited-state population within less than 100 fs. 4 The second type-exemplified by S3-excited n-AIH + -occurs when the molecule remains at moderate gaps, where the hopping probability is small but the hopping opportunities numerous (Figure 1-right). These two regimes can also be respectively identified as the strong and weak coupling limits of Engels and Jortner. 5 This short paper further develops how ultrafast internal conversion can occur in this second scenario. We work in the frame of instantaneous probabilities, in which the probability of a nonadiabatic transition between electronic states is a function of time, such as in the fewest switches surface hopping (FSSH). 6 An alternative would be to work with global nonadiabatic frameworks, which evaluate the total probabilities that the molecule would end in each state, just like in the Landau-Zener model. 7 Another alternative would be to employ Fermi's Golden rule to get probability per unit of time (rate). 5 Nevertheless, working with instantaneous probabilities can directly connect the discussion to FSSH surface hopping simulations.

Demonstration of equation (1)
To find the lifetime of state 1, we should compute the probability pN that the system will hop to state 0 at a time ' t corresponding to '/ N t t = time steps. Under the assumption that there are no back hoppings, this probability is the product of the probability of not having hopped in any of the previous N−1 steps times the probability of hopping at step N: The probability P0(t) that the system is in state 0 at time t