We demonstrate that the ring-polymer molecular dynamics (RPMD) method is equivalent to an automated and approximate implementation of the "Im F" version of semiclassical instanton theory when used to calculate reaction rates in the deep-tunneling regime. This explains why the RPMD method is often reliable in this regime and also shows how it can be systematically improved. The geometry of the beads at the transition state on the ring-polymer potential surface describes a finite-difference approximation to the "instanton" trajectory (a periodic orbit in imaginary time beta variant Planck's over 2pi on the inverted potential surface). The deep-tunneling RPMD rate is an approximation to the rate obtained by applying classical transition-state theory (TST) in ring-polymer phase-space using the optimal dividing surface; this TST rate is in turn an approximation to a free-energy version of the Im F instanton rate. The optimal dividing surface is in general a function of several modes of the ring polymer, which explains why centroid-based quantum-TSTs break down at low temperatures for asymmetric reaction barriers. Numerical tests on one-dimensional models show that the RPMD rate tends to overestimate deep-tunneling rates for asymmetric barriers and underestimate them for symmetric barriers, and we explain that this is likely to be a general trend. The ability of the RPMD method to give a dividing-surface-independent rate in the deep-tunneling regime is shown to be a consequence of setting the bead-masses equal to the physical mass.

This review discusses recent quantum scattering calculations on bimolecular chemical reactions in the gas phase. This theory provides detailed and accurate predictions on the dynamics and kinetics of reactions containing three atoms. In addition, the method can now be applied to reactions involving polyatomic molecules. Results obtained with both time-independent and time-dependent quantum dynamical methods are described. The review emphasises the recent development in time-dependent wave packet theories and the applications of reduced dimensionality approaches for treating polyatomic reactions. Calculations on over 40 different reactions are described.

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The semiclassical instanton expression for the tunneling splitting between two symmetric wells is rederived, starting from the ring-polymer representation of the quantum partition function. This leads to simpler mathematics by replacing functional determinants with matrix determinants. By exploiting the simple Hückel-like structure of the matrices, we derive an expression for the instanton tunneling splitting in terms of a minimum on the potential surface of a linear polymer. The latter is a section cut out of a ring polymer, consisting of an infinite number of beads, which describes a periodic orbit on the inverted potential surface. The approach is straightforward to generalize to multiple dimensions, and we demonstrate that it is computationally practical by carrying out instanton calculations of tunneling splittings in HO(2) and malonaldehyde in full dimensionality.

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or 'classical Wigner approximation') results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their 'Matsubara' values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of 2 at 0 (which gives back the standard LSC-IVR approximation), but in the normalmode derivatives corresponding to the lowest Matsubara frequencies. The resulting 'Matsubara' dynamics is inherently classical (since all terms O( 2 ) disappear from the Matsubara Liouvillian in the limit N → ∞), and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum timecorrelation function converges with respect to the number of modes, and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.