Anharmonic Barrier Research Articles

Transition state theory for activated systems with driven anharmonic barriers.

Classical transition state theory has been extended to address chemical reactions across barriers that are driven and anharmonic. This resolves a challenge to the naive theory that necessarily leads to recrossings and approximate rates because it relies on a fixed dividing surface. We develop both perturbative and numerical methods for the computation of a time-dependent recrossing-free dividing surface for a model anharmonic system in a solvated environment that interacts strongly with an oscillatory external field. We extend our previous work, which relied either on a harmonic approximation or on periodic force driving. We demonstrate that the reaction rate, expressed as the long-time flux of reactive trajectories, can be extracted directly from the stability exponents, namely, Lyapunov exponents, of the moving dividing surface. Comparison to numerical results demonstrates the accuracy and robustness of this approach for the computation of optimal (recrossing-free) dividing surfaces and reaction rates in systems with Markovian solvation forces. The resulting reaction rates are in strong agreement with those determined from the long-time flux of reactive trajectories.

Persistence of transition-state structure in chemical reactions driven by fields oscillating in time.

Chemical reactions subjected to time-varying external forces cannot generally be described through a fixed bottleneck near the transition-state barrier or dividing surface. A naive dividing surface attached to the instantaneous, but moving, barrier top also fails to be recrossing-free. We construct a moving dividing surface in phase space over a transition-state trajectory. This surface is recrossing-free for both Hamiltonian and dissipative dynamics. This is confirmed even for strongly anharmonic barriers using simulation. The power of transition-state theory is thereby applicable to chemical reactions and other activated processes even when the bottlenecks are time dependent and move across space.

The flux-flux correlation function for anharmonic barriers

The flux-flux correlation function formalism is a standard and widely used approach for the computation of reaction rates. In this paper we introduce a method to compute the classical and quantum flux-flux correlation functions for anharmonic barriers essentially analytically through the use of the classical and quantum normal forms. In the quantum case we show that for a general f degree-of-freedom system having an index one saddle the quantum normal form reduces the computation of the flux-flux correlation function to that of an effective one-dimensional anharmonic barrier. The example of the computation of the quantum flux-flux correlation function for a fourth order anharmonic barrier is worked out in detail, and we present an analytical expression for the quantum mechanical microcanonical flux-flux correlation function. We then give a discussion of the short-time and harmonic limits.

Extremal path approach to rate constant calculations by the linearized semiclassical initial value representation

To extend the applicability of the linearized initial value representation (LIVR) method to lower temperatures and realistic potentials, a generalization to barriers other than the inverted parabola is proposed. The LIVR method calculates rate constants of chemical reactions involving quantum effects by weighting classical trajectories by the Wigner distribution function (WDF) of the Boltzmann-averaged flux operator. These calculations can be performed efficiently if the WDF is available in analytical form, which is the case for harmonic barriers only. The proposed generalization to anharmonic barriers is based on the recognition that above a critical temperature T*=ℏω/πkB, where ω is the curvature at the top of the barrier and kB is the Boltzmann constant, the WDF is dominated by an extremal trajectory. The evaluation of WDFs and thus of thermal rate constants is thereby reduced to the search for the extremal path defined by a steepest-descent condition for the WDF. This extremal trajectory is the high-temperature analogue of the instanton (bounce path), which exists for temperatures lower than T*/2. Explicit formulas are derived for the generation of realistic WDFs and barrier crossing rate constants for symmetric barriers of arbitrary shape. Approximations are introduced that will reduce the extra computational effort required for these anharmonic barriers. They are based on the fact that above the critical temperature the WDF of any anharmonic potential can be represented with good approximation in an analytical form analogous to that of the parabolic barrier by the introduction of effective parameters. Results obtained for two standard model potentials, the quartic potential and the symmetric Eckart barrier, are compared with the well-known parabolic barrier results. The formal and actual temperature limits for calculating tunneling rate constants and the extension of the method to asymmetric barriers are briefly discussed.

Phase space distribution function approach to the Kramers problem. III. Anharmonic potentials

In a recent series of papers we showed how a phase space distribution function approach could be combined with the method of reactive flux to obtain the rate of barrier crossing as a function of solvent friction in the intermediate to high friction limit. Those studies dealt with both the Markovian and non-Markovian cases, but were restricted to analytic results for parabolic barriers. Here we extend the approach to anharmonic barriers. The guiding approximation is to assume that the phase space distribution for each initial velocity, starting at the barrier top, remains Gaussian for all time, with Gaussian parameters given by time-dependent mean field equations. We expect this approximation to be accurate for short times, up to the “Ehrenfest” time; if this time exceeds the “plateau” time — the time for the distribution to reach its asymptotic partitioning — the quality of the results should be high. There are no adjustable parameters, although some reasonable criterion is needed for ending the integration of the mean field equations to prevent divergence. Numerical results for the linear cusp and the quartic potential show that the method is quite accurate for dimensionless frictions ≤1, although the accuracy degrades for higher frictions.

Effects of argon on the IR multiphoton dissociation of C 3F 6

This work reports on the infrared multiphoton dissociation of hexafluoropropene (0 to 2 mbar) in the presence of added argon (0 to 15 mbar), at a constant value of the irradiation fluence (4 J cm −2). Together with the quenching of the dissociation yield, the addition of Ar produces a modification of the qualitative behaviour of C 3F 6 with respect to the dissociation process, associated with the induction of an anharmonic barrier similar to that induced by varying the irradiation fluence. This effect is observed through the activating role of C 3F 6C 3F 6 collisions found at higher Ar pressures. The anharmonic barrier is principally vibrational, although for small values of fluence a rotational component has been detected.

Dissipative quantum systems with a potential barrier: General theory and the parabolic barrier.

We study the real time dynamics of a quantum system with potential barrier coupled to a heat-bath environment. Employing the path integral approach an evolution equation for the time dependent density matrix is derived. The time evolution is evaluated explicitly near the barrier top in the temperature region where quantum effects become important. It is shown that there exists a quasi-stationary state with a constant flux across the potential barrier. This state generalizes the Kramers flux solution of the classical Fokker-Planck equation to the quantum regime. In the temperature range explored the quantum flux state depends only on the parabolic approximation of the anharmonic barrier potential near the top. The parameter range within which the solution is valid is investigated in detail. In particular, by matching the flux state onto the equilibrium state on one side of the barrier we gain a condition on the minimal damping strength. For very high temperatures this condition reduces to a known result from classical rate theory. Within the specified parameter range the decay rate out of a metastable state is calculated from the flux solution. The rate is shown to coincide with the result of purely thermodynamic methods. The real time approach presented can be extended to lower temperatures and smaller damping.