The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

A method is proposed for carrying out molecular dynamics simulations of processes that involve electronic transitions. The time dependent electronic Schrodinger equation is solved self‐consistently with the classical mechanical equations of motion of the atoms. At each integration time step a decision is made whether to switch electronic states, according to probabilistic ‘‘fewest switches’’ algorithm. If a switch occurs, the component of velocity in the direction of the nonadiabatic coupling vector is adjusted to conserve energy. The procedure allows electronic transitions to occur anywhere among any number of coupled states, governed by the quantum mechanical probabilities. The method is tested against accurate quantal calculations for three one‐dimensional, two‐state models, two of which have been specifically designed to challenge any such mixed classical–quantal dynamical theory. Although there are some discrepancies, initial indications are encouraging. The model should be applicable to a wide variety of gas‐phase and condensed‐phase phenomena occurring even down to thermal energies.

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Molecular dynamics with electronic transitions

Preface. Part I: Gathering the Tools. 1. Introduction to Quantum Physics. 2. Quantum Tests. 3. Complex Vector Space. 4. Continuous Variables. Part II: Cryptodeterminism and Quantum Inseparability. 5. Composite Systems. 6. Bell's Theorem. 7. Contextuality. Part III: Quantum Dynamics and Information. 8. Spacetime Symmetries. 9. Information and Thermodynamics. 10. Semiclassical Methods. 11. Chaos and Irreversibility. 12. The Measuring Process. Author Index. Subject Index.

Quantum Theory: Concepts and Methods

The distinction between level clustering and level repulsion is one of the quantum analogues of the classical distinction between globally regular and predominantly chaotic motion (see Figs. 1, 2, 3). In order to reveal level repulsion under conditions of global classical chaos special care may be necessary: (i) subspectra referring to different values of the quantum numbers related to symmetries must be dealt with separately and (ii) for systems with quantum localization only levels whose wavefunctions have overlapping support must be admitted. A “level” may either be an energy eigenvalue E in the case of autonomous systems or, for periodically driven systems, a quasi-energy φ, i.e. an eigenphase of the unitary Floquet operator transporting the wavevector from period to period.

Quantum signatures of chaos

Brownian motors: noisy transport far from equilibrium

This paper amends a recent statistical theory of rearrangement collisions to bring it into accord with the detailed‐balance theorem. Both classical and quantum formulations are discussed. The energy dependence of the cross sections near threshold and approximate formulas for the cross sections at arbitrary energies are derived.

On Detailed Balancing and Statistical Theories of Chemical Kinetics

We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrodinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time‐displacement operator in every order of approximation. We study the usefulness of the first‐ and second‐order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two‐state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time‐dependent force constant.

On the Exponential Form of Time‐Displacement Operators in Quantum Mechanics

Simple analytical forms for the orientation dependence of the potential between two molecules are derived from a Gaussian overlap model. Orientation‐dependent range and energy parameters are determined, which can be used with any two‐parameter atomic potential to give simple and reasonable polyatomic potentials.

Gaussian Model Potentials for Molecular Interactions

The reduced dynamics of a quantum system in contact with a reservoir is generally thought to be "completely positive"; this is certainly true if product initial conditions are used to define the dynamics. We show that with correlated initial conditions it need not be so. In this case the dynamics can properly be defined only on a subset of initial system states; extension, by linearity, to all possible initial states is trivially possible, but the extension may not be physically realizable and may not even be positive, let alone completely positive.

Reduced dynamics need not be completely positive

Several processes involving uncharged species are discussed in terms of a statistical theory of strong‐coupling collisions. Specifically, we study the Mies—Shuler—Zwanzig model for vibrational excitation of a diatomic by impulsive collisions; the reactions of K with HBr, Cl with KH, Cl with Na2, and the subsequent reaction of vibrationally excited NaCl with Na; and the breakup of electronically excited H2O to H atom and OH (2Σ+) radical (as an example of a bad failure of the statistical model). Except in the last case, rotational and vibrational distributions in the product diatomics predicted by the statistical theory agree well with experiment (where the experiment has been done); the total reactive cross sections, although a little high, compare reasonably well with the measured cross sections.

Philip Pechukas

Papers

On Detailed Balancing and Statistical Theories of Chemical Kinetics

On the Exponential Form of Time‐Displacement Operators in Quantum Mechanics

Gaussian Model Potentials for Molecular Interactions

Reduced dynamics need not be completely positive

Statistical Theory of Chemical Kinetics : Application to Neutral-Atom—Molecule Reactions