Thermal quantum time-correlation functions are of fundamental importance in
quantum dynamics, allowing experimentally-measurable properties such as
reaction rates, diffusion constants and vibrational spectra to be computed from
first principles. Since the exact quantum solution scales exponentially with
system size, there has been considerable effort in formulating reliable
linear-scaling methods involving exact quantum statistics and approximate
quantum dynamics modelled with classical-like trajectories. Here we review
recent progress in the field with the development of methods including Centroid
Molecular Dynamics (CMD), Ring Polymer Molecular Dynamics (RPMD) and
Thermostatted RPMD (TRPMD). We show how these methods have recently been
obtained from `Matsubara dynamics', a form of semiclassical dynamics which
conserves the quantum Boltzmann distribution. We also rederive t->0+ quantum
transition-state theory (QTST) in the Matsubara dynamics formalism showing that
Matsubara-TST, like RPMD-TST, is equivalent to QTST. We end by surveying areas
for future progress.Timothy J. H. Hele wishes to thank Jesus College, Cambridge for a Research Fellowship