  Recent work shows that strong stability and dimensionality freedom are
essential for robust numerical integration of thermostatted ring-polymer
molecular dynamics (T-RPMD) and path-integral molecular dynamics (PIMD),
without which standard integrators exhibit non-ergodicity and other pathologies
[J. Chem. Phys. 151, 124103 (2019); J. Chem. Phys. 152, 104102 (2020)]. In
particular, the BCOCB scheme, obtained via Cayley modification of the standard
BAOAB scheme, features a simple reparametrization of the free ring-polymer
sub-step that confers strong stability and dimensionality freedom and has been
shown to yield excellent numerical accuracy in condensed-phase systems with
large time-steps. Here, we introduce a broader class of T-RPMD numerical
integrators that exhibit strong stability and dimensionality freedom,
irrespective of the Ornstein-Uhlenbeck friction schedule. In addition to
considering equilibrium accuracy and time-step stability as in previous work,
we evaluate the integrators on the basis of their rates of convergence to
equilibrium and their efficiency at evaluating equilibrium expectation values.
Within the generalized class, we find BCOCB to be superior with respect to
accuracy and efficiency for various configuration-dependent observables,
although other integrators within the generalized class perform better for
velocity-dependent quantities. Extensive numerical evidence indicates that the
stated performance guarantees hold for the strongly anharmonic case of liquid
water. Both analytical and numerical results indicate that BCOCB excels over
other known integrators in terms of accuracy, efficiency, and stability with
respect to time-step for practical applications.
