John C. Light

Generalized discrete variable approximation in quantum mechanics

The formal definition of the generalized discrete variable representation (DVR) for quantum mechanics and its connection to the usual variational basis representation (VBR) is given. Using the one dimensional Morse oscillator example, we compare the ‘‘Gaussian quadrature’’ DVR, more general DVR’s, and other ‘‘pointwise’’ representations such as the finite difference method and a Simpson’s rule quadrature. The DVR is shown to be accurate in itself, and an efficient representation for optimizing basis set parameters. Extensions to multidimensional problems are discussed.

An R matrix approach to the solution of coupled equations for atom–molecule reactive scattering

We describe in detail a new method for solving the coupled linear differential equations which appear in the theoretical statement of many types of scattering phenomena. Combining the speed advantage of propagation methods with the inherent stability of R‐matrix methods, this method is fast and is unaffected by exponential growth of closed channels. We describe the propagation of the R‐matrix in terms of a collinear reactive scattering problem, and provide numerical results for several systems, clearly demonstrating the speed, stability, and accuracy of this method.

On Detailed Balancing and Statistical Theories of Chemical Kinetics

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.

Unitary quantum time evolution by iterative Lanczos reduction

A general unitary time evolution method for wave packets defined on a fixed L2 basis is developed. It is based on the Lanczos reduction of the full N×N Hamiltonian to a p‐dimensional subspace defined by the application of H p−1 times to the initial vector. Unitary time evolution in the subspace is determined by exp{−iHpt}, retaining accuracy for a time interval τ, which can be estimated from the Lanczos reduced Hamiltonian Hp. The process is then iterated for additional time intervals. Although accurate results over long times can be obtained, the process is most efficient for large systems over short times. Time evolution employing this method in one‐ (unbounded) and two‐dimensional (bounded) potentials are done as examples using a distributed Gaussian basis. The one‐dimensional application is to direct evaluation of a thermal rate constant for the one‐dimensional Eckart barrier.

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

We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrodingers 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.

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

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.

On distributed Gaussian bases for simple model multidimensional vibrational problems

Distributed Gaussian bases (DGB) are defined and used to calculate eigenvalues for one and multidimensional potentials. Comparisons are made with calculations using other bases. The DGB is shown to be accurate, flexible, and efficient. In addition, the localized nature of the basis requires only very low order numerical quadrature for the evaluation of potential matrix elements.

Discrete variable representations and sudden models in quantum scattering theory

Abstract An exact formalism in which the scattering problem may be described by sets of coupled equations labeled either by basis functions or quadrature points is presented. Use of each frame and the simply evaluated unitary transformation which connects them results in an efficient procedure for performing quantum scattering calculations. Two approximations are compared with the IOS.

Highly excited vibrational levels of ‘‘floppy’’ triatomic molecules: A discrete variable representation—Distributed Gaussian basis approach

A novel, efficient, and accurate quantum method for the calculation of highly excited vibrational levels of triatomic molecules is presented. The method is particularly well suited for applications to ‘‘floppy’’ molecules, having large amplitude motion, on potential surfaces which may have more than one local minimum. The discrete variable representation (DVR) for the angular, bend coordinate is combined with the distributed (real) Gaussian basis (DGB) for the expansion of other, radial coordinates. The DGB is tailored to the potential, covering only those regions where V(r)<EMAX. The DVR permits a contraction of the primitive Gaussian basis to a small eigenfunction basis at each of the discretized values of the angular coordinate. It is shown for the floppy two‐mode LiCN/LiNC system (fixed CN distance) that N lowest vibrational levels (N=131) can be converged to within 1 cm−1 (the lowest 117 to 0.1 cm−1) using only 3N basis functions. This appears to reduce the computational effort by a factor of 10–40 o...

Accurate localized and delocalized vibrational states of HCN/HNC

Results of the first accurate quantum calculation of the delocalized, large amplitude motion vibrational (J=0) levels of HCN/HNC, lying above the isomerization barrier, are presented. The recently developed DVR‐DGB quantum method [Z. Bacic and J. C. Light, J. Chem. Phys. 85, 4594 (1986)] is employed in this work. A model, empirical surface by Murrell et al. is used. All modes are included; the energy level calculation does not involve any approximations. Over a hundred vibrational levels are calculated accurately for this model surface. A number of them lie above the isomerization barrier; some are extensively delocalized over both HCN and HNC minima. Analysis shows that for HCN/HNC the threshold for significant delocalization is determined by the height of the vibrationally adiabatic bending barrier. In addition, the nearest neighbor level spacing distribution is obtained and compared to that of LiCN/LiNC. Various computational aspects of the DVR‐DGB approach, which is applicable to any triatomic molecul...