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

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

https://iopscience.iop.org/article/10.1088/0034-4885/35/1/306/pdf

Semiclassical approximations in wave mechanics

The rotational Hamiltonian of an asymmetric‐top molecule in a given vibrational state, obtained by the usual vibrational perturbation treatment, contains more parameters than can be determined from the observed energy levels. This Hamiltonian is therefore transformed by means of a unitary transformation to a reduced Hamiltonian which is suitable for fitting to observed energies. The unitary transformation can be chosen so that the reduced Hamiltonian has the following properties: (i) It is totally symmetric in the point group D2, regardless of the symmetry of the molecule; (ii) It contains only (n+1) independent terms of total degree n in the components of the total angular momentum, for each even value of n; (iii) Its matrix elements in a symmetric‐top basis satisfy the selection rule ΔK=0, ±2. This paper is concerned mainly with the possibility of carrying out this reduction in general. However, the reduced Hamiltonian described above contains one less quartic coefficient than has been used previously, and this particular case is discussed in more detail.

Determination of Centrifugal Distortion Coefficients of Asymmetric‐Top Molecules

The simplest two-dimensional (2D) spectra show how excitation with one (variable) frequency affects the spectrum at all other frequencies, thus revealing the molecular connections between transitions. Femtosecond 2D Fourier transform (2D FT) spectra are more flexible and share some of the remarkable properties of their conceptual parent, 2D FT nuclear magnetic resonance. When 2D FT spectra are experimentally separated into real absorptive and imaginary refractive parts, the time resolution and frequency resolution can both reach the uncertainty limit set for each resonance by the sample itself. Coherent four-level contributions to the signal provide new molecular phase information, such as relative signs of transition dipoles. The nonlinear response can be picked apart by selecting a single coherence pathway (e.g., specifying the relative signs of energy level difference frequencies during different time intervals as in the photon echo). Because molecules are frozen on the femtosecond timescale, femtosecond 2D FT experiments can separate a distribution of instantaneous molecular environments and intramolecular geometries as inhomogeneous broadening. This review provides an introduction to two-dimensional Fourier transform experiments exploiting second- and third-order vibrational and electronic nonlinearities.

Two-dimensional femtosecond spectroscopy

This communication deals with the general theory of obtaining numerical electronic wave functions for the stationary states of atoms and molecules. It is shown that by taking Gaussian functions, and functions derived from these by differentiation with respect to the parameters, complete systems of functions can be constructed appropriate to any molecular problem, and that all the necessary integrals can be explicitly evaluated. These can be used in connexion with the molecular orbital method, or localized bond method, or the general method of treating linear combinations of many Slater determinants by the variational procedure. This general method of obtaining a sequence of solutions converging to the accurate solution is examined. It is shown that the only obstacle to the evaluation of wave functions of any required degree of accuracy is the labour of computation. A modification of the general method applicable to atoms is discussed and considered to be extremely practicable.

Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System

The fundamental principles of quantum mechanics

In order to put Zwaan's scheme for deriving the Kramers connection formulas on a rigorous basis, differential equations are set up governing the variation in the coefficients used to fit a linear combination of the B. W. K. type approximation functions to an exact solution of Schr\"odinger's equation in one dimension. Approximate solutions of the differential equations are worked out which lend themselves to the setting up of the connection formulas and give definite upper bounds to the errors involved in their use. The method is also used to set up the Sommerfeld phase integral quantum condition independently of the connection formulas. An upper limit to the error in the energy is worked out. A similar treatment of the problem of the transmission of matter waves through rounded potential barriers is formulated.

A Contribution to the Theory of the B. W. K. Method

The Fundamental Principles of Quantum Mechanics: With Elementary Applications

Theory of two quantum states of ${\mathrm{H}}_{2}$.\char22{}There are sixteen possible wave functions for a pair of interacting H atoms which dissociate adiabatically into a normal H atom and a two quantum H atom. These wave functions give rise to eight distinct $S$ states and four distinct $P$ states (the latter are degenerate in the fixed nuclei problem). These states may be divided into four groups of three according to the symmetry of the wave functions with respect to (a) an interchange of electron coordinates, and (b) reflection in the plane which forms the perpendicular bisector of the internuclear axis. The principles of selection for transitions between these various types of electronic state are formulated and compared with the rules of Kronig. A first order perturbation theory computation of the potential energy curves for the $P$ states shows that two of them have the form requisite for the formation of stable molecules. These two may be identified respectively with the $C$ state (upper level for Werner bands) and the $2^{3}P$ state reported by Richardson. The agreement between the computed curves and the empirical data is fair.

Edwin C. Kemble

Papers

The fundamental principles of quantum mechanics

A Contribution to the Theory of the B. W. K. Method

The Fundamental Principles of Quantum Mechanics: With Elementary Applications

The Two Quantum Excited States of the Hydrogen Molecule

The General Principles of Quantum Mechanics. Part I