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In Part I [J. Chem. Phys. 138, 084108 (2013)] we derived a quantumtransition-state theory by taking the t->0+ (short-time) limit of a new form ofquantum flux-side time-correlation function containing a ring-polymer dividingsurface. This t->0+ limit appears to be unique in giving positive-definiteBoltzmann statistics, and is identical to ring-polymer molecular dynamics(RPMD) TST. Here, we show that quantum TST (i.e. RPMD-TST) is exact if there isno recrossing (by the real-time quantum dynamics) of the ring-polymer dividingsurface, nor of any surface orthogonal to it in the space describingfluctuations in the polymer-bead positions along the reaction coordinate. Inpractice, this means that RPMD-TST gives a good approximation to the exactquantum rate for direct reactions, provided the temperature is not too farbelow the cross-over to deep tunnelling. We derive these results by comparingthe long-time limit of the ring-polymer flux-side time-correlation functionwith that of a hybrid flux-side time-correlation function (containing aring-polymer flux operator and a Miller-Schwarz-Tromp side function), and byrepresenting the resulting ring-polymer momentum integrals as hypercubes.Together with Part I, the results of this article validate a large number ofRPMD calculations of reaction rates.

Derivation of a true (t->0+) quantum transition-state theory. II. Recovery of the exact quantum rate in the absence of recrossing