Calculation of Free-Energy Barriers with TD-DFT: A Case Study on Excited-State Proton Transfer in Indigo

The performance of time-dependent density functional theory (TD-DFT) for the calculation of excited states of molecular systems has been the subject of many benchmark studies. Often, these studies focus on excitation energies or, more recently, excited-state equilibrium geometries. In this work, we take a different angle by instead exploring how well TD-DFT reproduces experimental free-energy barriers of a well-known photochemical reaction: the excited-state proton transfer (ESPT) in indigo. Specifically, by exploiting the possibility of using TD-DFT to locate and compute free energies of first-order saddle points in excited states, we test the performance of several popular density functionals in reproducing recently determined experimental free-energy barriers for ESPT in indigo and in an N-hexyl substituted derivative thereof. Through the calculations, it is found that all of the tested functionals perform quite well, uniformly overestimating the experimental values by 1.4-3.5 (mean error) and 2.5-5.5 kcal mol^-1 (maximum error) only. Given that these errors are not larger than those typically observed when barriers for ground-state proton transfer reactions are calculated in ground-state DFT, the results highlight the potential of TD-DFT to enable accurate modeling of ESPT reactions based on free energies and explicit localization of transition states.