Derivation of a true (t → 0+) quantum transition-state theory. I. Uniqueness and equivalence to ring-polymer molecular dynamics transition-state-theory

Abstract

Surprisingly, there exists a quantum flux-side time-correlation function which has a non-zero t → 0+ limit, and thus yields a rigorous quantum generalization of classical transition-state theory (TST). In this Part I of two articles, we introduce the new time-correlation function, and derive its t → 0+ limit. The new ingredient is a generalized Kubo transform which allows the flux and side dividing surfaces to be the same function of path-integral space. Choosing this common dividing surface to be a single point gives a t → 0+ limit which is identical to an expression introduced on heuristic grounds by Wigner in 1932, but which does not give positive-definite quantum statistics, causing it to fail while still in the shallow-tunnelling regime. Choosing the dividing surface to be invariant to imaginary-time translation gives, uniquely, a t → 0+ limit that gives the correct positivedefinite quantum statistics at all temperatures, and which is identical to ring-polymer molecular dynamics (RPMD) TST. We find that the RPMD-TST rate is not a strict upper bound to the exact quantum rate, but a good approximation to one if real-time coherence effects are small. Part II will show that the RPMD-TST rate is equal to the exact quantum rate in the absence of recrossing.

Non-oscillatory flux correlation functions for efficient nonadiabatic rate theory

“…For example, for adiabatic dynamics, a dividing surface in nuclear-configuration space is usually defined and the TST assumption is that trajectories will never cross this dividing surface more than once. For the classical Born-Oppenheimer flux-side correlation function 55,56 and a particular generalization of its quantum equivalent, 53 it can be shown that the non-recrossing assumption leads to a step-like shape and therefore that the TST rate is proportional to its t → 0 + limit.…”

“…V, the corresponding distribution will be localized about the crossing region. Like other quantum TSTs, 53 our formulation will not necessarily be variational, i.e. the exact rate may be smaller or larger than the TST rate.…”

“…59 Of the set of possible correlation functions, it is the symmetric version which is most commonly computed by exact wave-function 10,27 or real-time path-integral methods. 12 The Kubo-transformed version is often the most similar to the correlation functions calculated by classical mechanics, 61 and it is a generalization of the Kubo-transformed flux-side function 53 which is approximated by standard Born-Oppenheimer RPMD rate theory. 51 The flux-correlation-function formulation of Miller et al 27 has been successfully applied to numerous reactions and is frequently used as a starting point for further developments in quantum rate theory.…”

“…It is obvious that in the adiabatic limit, where the Born-Oppenheimer approximation is valid, such a theory should reduce to something equivalent to classical rate theory, classical TST or ring-polymer TST, 51,52 which is known to be exact in the absence of recrossing. 53,54 In this limit, nonadiabatic dynamical approaches 40,41 can be applied to compute the small amount of dividing-surface recrossing due to nonadiabatic effects. However, this approach breaks down in the weak-coupling limit where the transmission coefficient is extremely small and therefore inefficient to calculate.…”

Non-equilibrium dynamics from RPMD and CMD

“…Following Hele and Althorpe, 98 we transform to normal modes and do the integration over the N−1 delta functions…”