Conical intersections (CIs), at which two potential-energy surfaces (PESs) touch, are known to be of crucial importance in photochemistry as well as certain thermal reactions. Hence, they are sometimes considered the analogue of transition states in nonadiabatic reactions, where energetically close electronic states render the Born–Oppenheimer approximation invalid. However, no accurate rate theories exist for complex chemical reactions proceeding via CIs...until now.

Semiclassical golden-rule instanton theory constitutes a rigorous approximation to Fermi's golden rule and has been shown to provide excellent agreement with experimental results for nonadiabatic reactions without CIs. Here, we extend the theory to describe the intricate interplay of nonadiabatic and nuclear quantum effects like zero-point energy, quantum tunneling and geometric phase at CIs in the nonadiabatic regime, where the degeneracy is lifted slowly away from the CI.

Such a situation is found in the diabatic trapping of a charge in the bis(methylene) adamantyl (BMA) cation. We show our instanton rates to compare favorably with exact quantum-mechanical results for a BMA model system and combine it with on-the-fly electronic-structure calculations to study the molecule in full dimensionality.

The cornerstone of the theory is the "optimal tunneling pathway" or instanton, a trajectory that hops from the reactant to the product state. This path travels below the potential-energy barrier providing an intuitive picture of tunneling. In contrast to previous studies, we find not only one but several instantons that traverse, bypass and wind around the CI (see Figure). Interestingly, the winding paths give a negative contribution to the total rate. This rate reduction caused by the geometric phase competes with the speed-up from carbon tunneling, which is illustrated by the different instanton pathways.

(Left) Geometry of the BMA cation at the minima and the CI along with the location of the positive charge. (Right) PESs of BMA projected on the two-dimensional branching space. The lines illustrate various instantons that traverse (magenta), bypass (cyan) and wind around (black) the CI.

Dr. Eric R. Heller

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