Analysis of transition state theory rates upon spatial coarse-graining

Spatial multiscale methods have established themselves as useful tools for extending the length scales accessible by conventional statics (i.e., zero temperature molecular dynamics). Recently, extensions of these methods, such as the finite-temperature quasicontinuum (hot-QC) or coarse-grained molecular dynamics (CGMD) methods, have allowed for multiscale molecular dynamics simulations at finite temperature. Here, we assess the quality of the long-time dynamics these methods generate by considering canonical transition rates. Specifically, we analyze the harmonic transition state theory (HTST) rates in CGMD and compare them to the corresponding HTST rate of the fully atomistic system. The ability of such an approach to reliably reproduce the HTST rate is verified through a relative error analysis, which is then used to highlight the major contributions to the error and guide the choice of degrees of freedom. We focus on the error resulting from coarsegraining, which dominates in systems with low temperature and constitutes a lower bound on the error associated with any method that employs coarse-graining. Finally, our analytical results are compared with numerical simulations for the case of a 1-D chain.