Predicting the rate of a chemical reaction is one of the most fundamental questions in chemistry. Since the first half of this century a large amount of theoretical effort has been devoted toward developing approximate theories of rate processes. One of the most prominent advances was made by Kramers, who considered the problem of barrier crossing for a classical particle in a metastable potential subject to frictional forces and showed that the rate depends nonmonotonically on the friction. Extending Kramers’ theory to the quantum regime has proven a challenging task, particularly so at low temperatures where quantum effects can be very significant.
Using the QuAPI approach, we presented in the 1990s the first fully quantum mechanical approach for calculating thermal rate constants in dissipative environments. The rate was obtained from the time integral of Miller’s flux-flux autocorrelation function which is evaluated using our path integral representation of the propagator. As shown by our test calculations on model proton transfer reactions the method yields well-converged results in practically all regimes of chemical interest, from thermal activation to deep tunneling and over a wide range of friction.
We have used the QuAPI methodology to provide benchmark results for reaction rates in generic dissipative environments. The first of our studies employed a double well potential as a model for adiabatic proton transfer and obtained the rate constant over a wide of temperature and friction. The double well problem, which reduces practically to a two-level system at low temperature, combines the dissipative tunneling features of the latter with phenomena associated with bound motion in the reactant (or product) well. The computed rate constant displayed a Kramers turnover as a function of friction at high temperature and large quantum corrections at temperatures below the characteristic “crossover”. Our results established definitively the degree of accuracy of several analytic and numerical approximations, including classical and quantum Grote-Hynes theories, semiclassical transition state theory estimates, classical and quantum turnover theories and the centroid density approximation. We concluded that no analytical theory applies successfully to the low-temperature regime and that the centroid theory largely underestimates the rate at small values of the dissipation. Since the centroid density approximation is essentially a quantum transition state theory, positive dynamical corrections constitute a purely nonclassical effect.
Another study employed a dissipative two-surface system as a model for nonadiabatic reactions. Our results confirmed the existence of a broad golden rule plateau that spans several orders of magnitude in the friction constant, as well as the rate enhancement at low friction due to quantum resonances predicted from a semiclassical analysis by Onuchic and Wolynes. In this regime, as well as in the adiabatic case at weak friction, coherence effects manifest themselves a step structures in the flux correlation function.
Although the above models offer valuable insight into the dynamics of condensed phase proton transfer, an closer look at most reactive processes indicates that the frequency of several vibrational degrees of freedom changes along the reaction path. By introducing zero point contributions to the effective barrier height, variable frequency degrees of freedom orthogonal to a reaction coordinate can affect the rate dynamics very significantly. Depending on their symmetry and coupling strength, competition between zero point energy effects, reaction path bottlenecks and corner cutting can lead to sizable positive or negative corrections to vibrationally adiabatic models. Our calculations on a model potential surface established quantitative criteria for the magnitude of such effects.
We have investigated the mechanism of hydrogen and deuterium migration in crystalline silicon, in particular the significance of quantum effects, using the QuAPI methodology to calculate the diffusion rate. Our results indicate that tunneling makes significant contributions to this rate at or below room temperature, although the crossover to the non-activated regime does not occur until much lower temperatures are reached. In addition, a reverse isotope effect that arises from zero point energy effects was observed, in semi-quantitative agreement with the prediction of the vibrationally adiabatic approximation.
In separate theoretical work, we have recast the flux formulation of reaction rate theory in a non-equilibrium form, which involves the expectation value of the reactive flux subject to an initial condition that corresponds to the reactant density. In the common case of slow reactive processes, the non-equilibrium expression reaches a plateau regime only slightly slower than the equilibrium flux form. In addition, the time integral of the non-equilibrium flux expression yields the reactant population as a function of time, allowing characterization of the dynamics in cases where there is no clear separation of time scales and thus a plateau regime cannot be identified. Thus, the non-equilibrium flux offers a unified approach to the kinetics of slow and fast chemical reactions. In addition, the non-equilibrium initial density is easier to compute and directly amenable to a variety of quantum-classical methods.
Beyond the harmonic bath treatment, we have performed all-atom simulations of the ferrocene-ferrocenium charge transfer reaction in organic solvents using the QCPI methodology, which combines quantum mechanics with classical trajectories in a rigorous way that treats the interaction between quantum and classical degrees of freedom in full detail and without any assumptions. Depending on the solvent, the reaction can be ultrafast or rather slow. The ultrafast reaction displays prominent deviations from exponential kinetics.
- A. Bose and N. Makri, “Non-equilibrium reactive flux: A unified framework for slow and fast reaction kinetics”, J. Chem. Phys. 147, 152723 (2017).
- P. L. Walters and N. Makri, “Quantum-classical path integral simulation of ferrocene-ferrocenium charge transfer in liquid hexane”, J. Phys. Chem. Lett. 6, 4959-4965 (2015).
- K. Forsythe and N. Makri, “Dissipative tunneling in a bath of two-level systems”, Phys. Rev. B 60, 972-978 (1999).
- K. M. Forsythe and N. Makri, “Effects of frequency variation in modes orthogonal to the reaction path on condensed phase reaction rates”, J. Mol. Structure 466, 103-110 (1999).
- K. M. Forsythe and N. Makri, “Path integral study of hydrogen diffusion in crystalline silicon”, J. Chem. Phys. 108, 6819-6828 (1998); K. M. Forsythe and N. Makri, “Erratum on Path integral study of hydrogen diffusion in crystalline silicon”, J. Chem. Phys. 110, 6082 (1999).
- E. Sim and N. Makri, “Path integral simulation of charge transfer dynamics in photosynthetic reaction centers”, J. Phys. Chem. B 101, 5446-5458 (1997).
- M. Topaler and N. Makri, “Path integral calculation of quantum nonadiabatic rates in model condensed phase reactions”, J. Phys. Chem. 100, 4430-4436 (1996).
- N. Makri, E. Sim, D. E. Makarov and M. Topaler, “Long-time quantum simulation of the primary charge separation in bacterial photosynthesis”, Proc. Natl. Acad. Sci. U.S.A. 93, 3926-3931 (1996).
- M. Topaler and N. Makri, “Quantum rates for a double well coupled to a dissipative bath: accurate path integral results and comparison with approximate theories”, J. Chem. Phys. 101, 7500-7519 (1994).
- D. E. Makarov and N. Makri, “Tunneling dynamics in dissipative curve crossing problems”, Phys. Rev. A 48, 3626-3635 (1993).
- M. Topaler and N. Makri, “Quasi-adiabatic propagator path integral methods: exact quantum rate constants for condensed phase reactions”, Chem. Phys. Lett. 210, 285-293 (1993).