Abstract
We propose a new quantum transition-state theory for calculating Fermi's golden-rule rates in complex multidimensional systems. This method is able to account for the nuclear quantum effects of delocalization, zero-point energy, and tunneling in an electron-transfer reaction. It is related to instanton theory but can be computed by path-integral sampling and is thus applicable to treat molecular reactions in solution. A constraint functional based on energy conservation is introduced which ensures that the dominant paths contributing to the reaction rate are sampled. We prove that the theory gives exact results for a system of crossed linear potentials and show numerically that it is also accurate for anharmonic systems. There is still a certain amount of freedom available in generalizing the method to multidimensional systems, and the suggestion we make here is exact in the classical limit but not rigorously size consistent in general. It is nonetheless seen to perform well for multidimensional spin-boson models, where it even gives good predictions for rates in the Marcus inverted regime.
Highlights
For the atomistic simulation of reactions in solution, the semiclassical instanton method is no longer applicable, but an alternative method is provided by ring-polymer molecular dynamics (RPMD).[26–29]
The excellent accuracy achieved by RPMD rate theory can be explained by its connection to instanton theory as the instanton can be shown to be equivalent to the dominant ringpolymer configuration sampled by the RPMD scheme.[4]
In order to employ the new approach with atomistic simulations of electrontransfer reactions in solution, different sampling schemes will be required
Summary
One of the most successful approaches for deriving new quantum transition-state theories[1,2,3,4,5,6,7] is by connection to semiclassical instanton rate theory.[8,9,10] This theory can be rigorously derived as an asymptotic approximation to the exact rate constant,[11,12] and the ring-polymer representation of the instanton can be employed to study polyatomic reactions in the gas phase.[4,13,14] This makes semiclassical instanton theory a remarkably efficient and accurate method for studying quantum tunneling in molecules and clusters.[15–25]. We have rigorously derived a semiclassical instanton expression for the rate constant of this reaction.[12,47] This approach includes the important quantum effects of tunneling, delocalization, and zero-point energy within an h → 0 asymptotic approximation. Other approaches extract the quantum rate constant from a real-time dynamical simulation of the nonadiabatic reaction Examples include those based on extensions of RPMD to employ the mapping formalism,[61–67] surface hopping,[68–70] or explicit electron dynamics in the position representation[71,72] as well a related instanton theory.[73]. Like other quantum transition-state theories,[1,2,3,4,5] it should not perform realtime propagation but be based only on a constrained imaginary-time path-integral simulation. We propose a new method, called golden-rule quantum transition-state theory (GR-QTST), in which the rate constant is defined by Eq (2) with the ansatz. We shall investigate a number of model systems for which exact results are available for comparison and will apply the method to atomistic simulations in future work
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