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Tunneling Rates

Tunneling is a process by which particles (actually, waves) can penetrate and proceed through barriers which have a too high energy for classical transmission. Tunneling is strongest for light particles and thin and low barriers. In chemical reactions, tunneling is most relevant for hydrogen and deuterium, due to their low mass. While for classical chemical reactions, the reaction rate decreases with decreasing temperature, the tunneling rate is independent of the temperature. For high temperature, the over-the-barrier mechanism is always faster than the tunneling process. At low temperature, tunneling is faster. In the intermediate range, temperature-assisted tunneling occurs which is particularly interesting and challenging to calculate. A recent review on the calculation of tunneling rates is given in [36], a more light-weighted discussion (in German) in [13].

We use instanton theory to calculate tunneling rates. On the one hand, it is accurate enough to calculate rates in the deep tunneling regime [25], on the other hand it is efficient enough to treat systems with a few tens of atoms. We even applied it to systems with many thousand atoms (combined with the QM/MM approach), 78 of which were allowed to tunnel [29].

An instanton represents the most likely tunneling path at a given temperature. It is a saddle point in the space spanned by closed Feynman paths. We adapted our geometry optimizer DL-FIND to search for instantons. Using that, we achieved near-quadratic convergence behavior [20]. A variable integration grid resulted in a significant reduction in the number of grid points (control points) required [22].

In a focus on astrochemistry [18] we used instanton theory to investigate the formation of H2 in space. The figure below shows the delocalization of atoms during the addition of a hydrogen atom to a benzene molecule. We showed how tunneling can explain the deuterium enrichment in interstellar methanol [25] and and demonstrated that atoms heavier than hydrogen (carbon, in fact) can participate in the tunneling [24]. We also used instanton theory to provide a well-defined measure of the tunneling path length in the decay of carbenes [32] and to demonstrate how tunneling synchronizes the proton movement in Grotthuss chains [38]. Meanwhile most of our effort goes into improving the method in terms of computational efficiency.

Our effort concerning the simulation of quantum mechanical tunneling of atoms will be broadened in the coming years within the ERC project TUNNELCHEM.

The instanton (quantum transition state) of the addition of a hydrogen atom to benzene at a temperature of 20 K. The incoming H atom is delocalized and, thus, shows clear tunneling behavior. However, also the benzene skeleton, mainly the carbon atom to which the hydrogen is to be added is delocalized.