Umbrella Sampling for Excited States Using a Semiempirical Method

Umbrella sampling is widely used to explore ground-state reaction pathways that exhibit high energy barriers and to identify stable molecular conformations. In this work, we extend umbrella sampling to the excited states by using the energy gap between electronic states as a collective variable, not only to locate intersection seams but also to enable the systematic exploration of excited-state relaxation pathways leading to conical intersections. Based on this approach, we present a complete workflow for studying excited-state relaxation mechanisms in detail. As a case study, we apply this method to a push-pull stilbene derivative, 4-(N,N-dimethylamino)-4 '-nitrostilbene, using the semiempirical multireference configuration interaction (MRCI) based on the orthogonalization model 2 (OM2) method to simulate photoisomerization. Combining potential energy surface scans, nonadiabatic dynamics simulations, and excited-state umbrella sampling, we identify and characterize five distinct conical intersections for the trans and cis isomers. Umbrella sampling allows assessing the accessibility of these conical intersections by determining the free energy barriers that separate them from the excited-state minimum. Moreover, it offers insight into the evolution of free energy and entropy along the relaxation pathways, highlighting key thermodynamic features as the system approaches the conical intersections.