  Two of the most successful methods that are presently available for
simulating the quantum dynamics of condensed phase systems are centroid
molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD). Despite
their conceptual differences, practical implementations of these methods differ
in just two respects: the choice of the Parrinello-Rahman mass matrix and
whether or not a thermostat is applied to the internal modes of the ring
polymer during the dynamics. Here we explore a method which is halfway between
the two approximations: we keep the path integral bead masses equal to the
physical particle masses but attach a Langevin thermostat to the internal modes
of the ring polymer during the dynamics. We justify this by showing
analytically that the inclusion of an internal mode thermostat does not affect
any of the desirable features of RPMD: thermostatted RPMD (TRPMD) is equally
valid with respect to everything that has actually been proven about the method
as RPMD itself. In particular, because of the choice of bead masses, the
resulting method is still optimum in the short-time limit, and the transition
state approximation to its reaction rate theory remains closely related to the
semiclassical instanton approximation in the deep quantum tunneling regime. In
effect, there is a continuous family of methods with these properties,
parameterised by the strength of the Langevin friction. Here we explore
numerically how the approximation to quantum dynamics depends on this friction,
with a particular emphasis on vibrational spectroscopy. We find that a broad
range of frictions approaching optimal damping give similar results, and that
these results are immune to both the resonance problem of RPMD and the
curvature problem of CMD.
