Proton tunnelling in polyatomic molecules: A direct-dynamics instanton approach

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Abstract
In this review we discuss a recently introduced method of calculating hydrogen tunnelling rates and tunnelling splittings in medium and large molecules. It is a non-empirical, direct-dynamics method that uses ab initio quantum-chemical output as input data for the calculation of dynamic properties by means of the instanton approach. This approach is based on the recognition that there is a single path that dominates the tunnelling rate. This so-called instanton trajectory is the path that minimizes the classical action. Although it is very difficult to calculate this trajectory for multidimensional systems, it will be shown that the corresponding instanton action, which is the quantity of practical interest, can be obtained with sufficient accuracy to reproduce experimental observations without the explicit evaluation of the instanton trajectory. In this approximation scheme the instanton action is calculated from the one-dimensional action through the introduction of appropriate correction terms for all modes coupled to the tunnelling mode. These coupled transverse modes are taken to be harmonic oscillators; the couplings are assumed to be linear and derived from the displacements of the transverse modes between the equilibrium configuration and the transition state. Nonlinear couplings of large-amplitude transverse modes are also briefly discussed. The reaction coordinate is identified with the normal mode with imaginary frequency in the transition state and not with the minimum-energy path used in variational transition-state theory. The multidimensional potential-energy surface is formulated in terms of the normal coordinates of the transition state. Formulas and computer codes are presented which allow direct evaluation of mode-specific tunnelling splittings as well as of proton transfer rate constants across symmetric or asymmetric barriers as a function of temperature. Results are presented for these rates and splittings that can be critically compared with experimental data. Mode-specific splittings are discussed for 9-hydroxyphenalenone and tropolone, two large molecules for which excellent experimental data are available. Tunnelling rate constants are discussed for aziridine, oxiranyl and dioxolanyl, three medium-size molecules that undergo inversion by a tunnelling mechanism, and for porphine, a large molecule for which an abundance of high-quality proton-transfer data has been reported. All of these systems are handled successfully by the method. The calculations are performed with the DOIT (dynamics of instanton tunnelling) code, which is available on the internet. This dynamics code is very efficient compared to other available codes based on transition-state theory with tunnelling corrections and takes only a fraction of the computer time required for the computation of the quantum-chemical input data.

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