Eli Pollak

Theory of activated rate processes: A new derivation of Kramers’ expression

The generalized Langevin equation of motion for a particle trapped in a one‐dimensional well with a barrier height V0 and coupled to a dissipative medium is modeled by a harmonic bath. Using the properties of the bath and a normal mode analysis we prove that the reactive frequency defined by Grote and Hynes for averaged motion across the barrier is actually a renormalized effective barrier frequency. We then show that the Kramers–Grote–Hynes expression for the rate of escape over the barrier is just the continuum limit of the usual gas phase harmonic transition state theory expression.

Theory of activated rate processes for arbitrary frequency dependent friction: Solution of the turnover problem

An analytical theory is formulated for the thermal (classical mechanical) rate of escape from a metastable state coupled to a dissipative thermal environment. The working expressions are given solely in terms of the quantities entering the generalized Langevin equation for the particle dynamics. The theory covers the whole range of damping strength and is applicable to an arbitrary memory friction. This solves what is commonly known as the Kramers turnover problem. The basic idea underlying the approach is the observation that the escape dynamics is governed by the unstable normal mode coordinate—and not the particle system coordinate. An application to the case of a particle moving in a piecewise harmonic potential with an exponentially decaying memory‐friction is presented. The comparison with the numerical simulation data of Straub, Borkovec, and Berne [J. Chem. Phys. 84, 1788 (1986)] exhibits good agreement between theory and simulation.

Transition states, trapped trajectories, and classical bound states embedded in the continuum

We show that the best choice of transition state, for the atom exchange reaction in a classical collinear collision of an atom with a diatomic, is a classical bound state embedded in the continuum: a periodic vibration of the triatomic system across the interaction region of the potential surface. These unstable bound states also serve as limit sets of the trapped trajectories that form the boundary of reactivity bands in molecular collisions, and we comment on the implications of this result for calculation of product state distributions. Numerical calculations of transition states are presented for the collinear H+H2 and F+H2 reactions.

A new quantum transition state theory

An old challenge in rate theory is the formulation of a quantum thermodynamic theory of rates which gives accurate estimates but does not demand any real time propagation. In this paper we attempt to answer the challenge by extending an idea suggested by Voth, Chandler and Miller [J. Phys. Chem. 93, 7009 (1989)]. A new quantum expression for the rate is derived by replacing the exact time dependent dynamics with the analytically known dynamics of a parabolic barrier and utilizing the symmetrized thermal flux operator. The new rate expression is exact for a parabolic barrier, and leads by derivation rather than by ansatz to a phase space integration of a Wigner thermal flux distribution function. The semiclassical limit is similar but not identical to Miller’s semiclassical transition state theory. Numerical computations on the symmetric and asymmetric one dimensional Eckart barrier give results which are equal to or greater than the exact ones, as expected from a transition state theory. In contrast to ot...

Classical transition state theory: A lower bound to the reaction probability

We derive a rigorous lower bound to the microcanonical reaction probability in classical collinear atom–diatom collisions. This lower bound complements the upper bound provided by transition state theory, and the information needed to calculate the bound is acquired automatically in the search for the periodic orbit dividing surfaces that are possible transition states for the reaction. Numerical calculations for F+H2 and H+Cl2 over a wide energy range show that the lower bound provides the best available estimate of the reaction probability, short of a full dynamical calculation.

Hamiltonian theory for vibrational dephasing rates of small molecules in liquids

A technique is developed for solving the generalized Langevin equation (GLE) describing anharmonic oscillators in the weak coupling limit. The GLE is rewritten as a Hamiltonian with a nonlinear system coupled to an infinite bath of harmonic oscillators. A normal mode transformation followed by a perturbation technique is used to obtain the fluctuating system frequency. When the method is applied to a single oscillator with cubic anharmonicity, both the classical and quantal dephasing rates are shown to be equal to the well‐known result of Oxtoby. The technique is also applied to a system with more than one vibrational degree of freedom (linear triatomic molecules) to obtain the dephasing rates for the symmetric and antisymmetric normal modes. The effects of system anharmonicity on frequency shifts are investigated.

Classical transition state theory is exact if the transition state is unique

Under mild conditions on the long‐range behavior of the potential, we show that classical transition state theory is exact at energy E for a collinear atom–diatom reaction if there is only one candidate for transition state at energy E—that is, only one periodic vibration of energy E across the interaction region.

A simple classical prediction of quantal resonances in collinear reactive scattering

Abstract Quantal collinear reactive scattering computations have shown that in the vicinity of thresholds of reactant or product vibrational states, one finds resonances in the state to state reaction probability. We find that these resonances can be explained classically in terms of energy transfer between adiabatic reactant and product channels. This transfer is attributable to resonant periodic orbits, resonating between reactants and products. The classical condition for a quantal resonance is that the action of the orbit over one period be an integer (in units of h ) and that the energy at which this occurs be lower than the adiabatic barrier heights of the resonating states. These conditions suffice for a prediction of the location of the quantal resonance within a 1% accuracy!

Activated rate processes : generalization of the Kramers-Grote-Hynes and Langer theories

The variational transition state theory approach for dissipative systems is extended in a new direction. An explicit solution is provided for the optimal planar dividing surface for multidimensional dissipative systems whose equations of motion are given in terms of coupled generalized Langevin equations. In addition to the usual dependence on friction, the optimal planar dividing surface is temperature dependent. This temperature dependence leads to a temperature dependent barrier frequency whose zero temperature limit in the one dimensional case is just the usual Kramers–Grote–Hynes reactive frequency. In this way, the Kramers–Grote–Hynes equation for the barrier frequency is generalized to include the effect of nonlinearities in the system potential. Consideration of the optimal planar dividing surface leads to a unified treatment of a variety of problems. These are (a) extension of the Kramers–Grote–Hynes theory for the transmission coefficient to include finite barrier heights, (b) generalization of ...

A new possibility of chemical bonding: vibrational stabilization of IHI

Abstract The IHI system has four vibrationally bonded collinear bound states. They are located in the saddle point region of a minimum-free potential energy surface.