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

In this review we discuss a recently introduced method of calculating hydro gen tunnelling rates and tunnelling splittings in medium and large molecule s. It is a non-empirical, direct-dynamics method that uses ab initio quantu m-chemical output as input data for the calculation of dynamic properties b y means of the instanton approach. This approach is based on the recognitio n that there is a single path that dominates the tunnelling rate. This so-c alled instanton trajectory is the path that minimizes the classical action. Although it is very difficult to calculate this trajectory for multidimens ional systems, it will be shown that the corresponding instanton action, wh ich is the quantity of practical interest, can be obtained with sufficient accuracy to reproduce experimental observations without the explicit evalua tion of the instanton trajectory. In this approximation scheme the instanto n action is calculated from the one-dimensional action through the introduc tion of appropriate correction terms for all modes coupled to the tunnellin g 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 tran sition state. Nonlinear couplings of large-amplitude transverse modes are a lso briefly discussed. The reaction coordinate is identified with the norma l mode with imaginary frequency in the transition state and not with the mi nimum-energy path used in variational transition-state theory. The multidim ensional potential-energy surface is formulated in terms of the normal coor dinates of the transition state. Formulas and computer codes are presented which allow direct evaluation of mode-specific tunnelling splittings as wel l as of proton transfer rate constants across symmetric or asymmetric barri ers as a function of temperature. Results are presented for these rates and splittings that can be critically compared with experimental data. Mode-sp ecific splittings are discussed for 9-hydroxyphenalenone and tropolone, two large molecules for which excellent experimental data are available. Tunne lling rate constants are discussed for aziridine, oxiranyl and dioxolanyl, three medium-size molecules that undergo inversion by a tunnelling mechanis m, and for porphine, a large molecule for which an abundance of high-qualit y proton-transfer data has been reported. All of these systems are handled successfully by the method. The calculations are performed with the DOIT (d ynamics of instanton tunnelling) code, which is available on the internet. This dynamics code is very efficient compared to other available codes base d on transition-state theory with tunnelling corrections and takes only a f raction of the computer time required for the computation of the quantum-ch emical input data.