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.

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.

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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.

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