Zero-point tunneling splittings in compounds with multiple hydrogen bonds calculated by the rainbow instanton method

Zero-point tunneling splittings are calculated, and the values are compared with the experimentally observed values for four compounds in which the splittings are due to multiple-proton transfer along hydrogen bonds. These compounds are three binary complexes, namely, the formic acid and benzoic acid dimer and the 2-pyridone-2-hydroxypyridine complex, in which the protons move in pairs, and the calix[4]arene molecule, in which they move as a quartet. The calculations make use of and provide a test for the newly developed rainbow approximation for the zero-temperature instanton action which governs the tunneling splitting (as well as the transfer rate). This approximation proved to be much less drastic than the conventional adiabatic and sudden approximations, leading to a new general approach to approximate the instanton action directly. As input parameters the method requires standard electronic-structure data and the Hessians of the molecule or complex at the stationary configurations only; the same parameters also yield isotope effects. Compared to our earlier approximate instanton method, the rainbow approximation offers an improved treatment of the coupling of the tunneling mode to the other vibrations. Contrary to the conventional instanton approach based on explicit evaluation of the instanton trajectory, both methods bypass this laborious procedure, which renders them very efficient and capable of handling systems that thus far have not been handled by other theoretical methods. Past results for model systems have shown that the method should be valid for a wide range of couplings. The present results for real compounds show that it gives a satisfactory account of tunneling splittings and isotope effects in systems with strong coupling that enhances tunneling, thus demonstrating its applicability to low-temperature proton dynamics in systems with multiple hydrogen bonds.