Chemical Reaction Rates
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, increasing at small friction and eventually falling off when the friction constant becomes sufficiently large to ensure Boltzmann equilibrium. Extending Kramers' theory to the quantum regime has proven a challenging task, particularly so at low temperatures where quantum effects can be very significant.
We have presented the first fully quantum mechanical approach for calculating thermal rate constants in dissipative environments. The rate is 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 our path integral 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.
Other recent work focused on impurity diffusion in crystalline silicon. Hydrogen impurities are ubiquitous in silicon crystals and understanding of their kinetics is important in semiconductor devices. Theoretical understanding of hydrogen-silicon interactions is particularly interesting in view of recent reports that use of deuterium rather than hydrogen in the passivation of silicon surfaces makes the latter more resistant toward electron bombardment, thereby increasing the lifetime of a device by up to two orders of magnitude.
We recently investigated the mechanism of hydrogen and deuterium migration in crystalline silicon, in particular the significance of quantum effects, using our path integral 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.
Related articles:
- Quasi-adiabatic propagator path integral methods: exact quantum rate constants for condensed phase reactions.
- Tunneling dynamics in dissipative curve-crossing problems.
- Quantum rates for a double well coupled to a dissipative bath: accurate path integral results and comparison with approximate theories.
- Path integral calculation of quantum nonadiabatic rates in model condensed phase reactions.
- Path integral study of hydrogen and deuterium diffusion in crystalline silicon.
- Effects of frequency variation in modes orthogonal to the reaction path on condensed phase rate constants.
- Dissipative tunneling in a bath of two-level systems.
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