Quantum chemical dynamics in two dimensions

An instanton theory for finding both tunnel splittings and incoherent tunneling rates in two-dimensional potentials is presented. The exact two-dimensional extremal trajectory is numerically calculated, and then the prefactor is expressed via its stability parameter. The method does not require any specific form of the potential and permits one to describe the rate constant at temperatures ranging from that of the low-temperature plateau to the classical activated transitions. The symmetric double well and the metastable well coupled to vibration are studied as examples, and, for the latter case, the instanton results are compared with recent numerical complex scaling method data, showing excellent agreement. The method is applied to inter- and intramolecular hydrogen transfer reactions. It is shown that in the latter case (tunnel splitting in malonaldehyde and in hydrogenoxalate anion) the tunneling trajectory is far from both the cutting-corner straight line and from the minimum energy path.