1307.3020 (Stuart C. Althorpe et al.)

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

Stuart C. Althorpe, Timothy J. H. Hele

In Part I [J. Chem. Phys. 138, 084108 (2013)] we derived a quantum transition-state theory by taking the short-time (t->0+) limit of a new form of quantum flux-side time-correlation function containing a ring-polymer dividing surface. This quantum TST appears to be unique (in the sense that no other known short-time limit gives positive-definite Boltzmann statistics) and, remarkably, is identical to ring-polymer molecular dynamics (RPMD) TST. Here, we show that quantum TST (i.e. RPMD-TST) is exact if there is no recrossing of the ring-polymer dividing surface, nor of any surface orthogonal to it in ring-polymer space (by which we mean the space obtained by ring-polymerizing a classical reaction coordinate). In practice, this means that RPMD-TST will give a good approximation to the exact quantum rate if the amount of such recrossing is small. We derive these results by comparing the long-time limit of the ring-polymer flux-side time-correlation function with that of a hybrid flux-side time-correlation function (containing a ring-polymer flux operator and a Miller-Schwarz-Tromp side function), and by representing the resulting ring-polymer momentum integrals as hypercubes.

View original: http://arxiv.org/abs/1307.3020