This article shows how to evaluate rotational symmetry numbers for different molecular configurations and how to apply them to transition state theory. In general, the symmetry number is given by the ratio of the reactant and transition state rotational symmetry numbers. However, special care is advised in the evaluation of symmetry numbers in the following situations: (i) if the reaction is symmetric, (ii) if reactants and/or transition states are chiral, (iii) if the reaction has multiple conformers for reactants and/or transition states and, (iv) if there is an internal rotation of part of the molecular system. All these four situations are treated systematically and analyzed in detail in the present article. We also include a large number of examples to clarify some complicated situations, and in the last section we discuss an example involving an achiral diasteroisomer.

We present an improved version of the large-curvature tunneling method that more accurately treats the anharmonic potentials encountered along low-energy corner-cutting tunneling paths. The method is illustrated by applications to the reaction ${\mathrm{CH}}_3+{\mathrm{H}}_2\to {\mathrm{CH}}_4+\mathrm{H}$ and to the double proton transfer in formamidine hydrate.

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.