A new method for the calculation of bound state eigenvalues and eigenfunctions of the SchrOdinger equation is presented. The Fourier grid Hamiltonian method is derived from the discrete Fourier transform algorithm. Its implementation and use is extremely simple, requiring the evaluation of the potential only at certain grid points and yielding directly the amplitude of the eigenfunctions at the same grid points. "When one has a particular problem to work out in quantum mechanics, one can minimize the labor by using a representation in which the representatives of the most important abstract quantities occurring in the problem are as simple as possible." P.A.M. Dirac, from The Principles of Quantum Mechanics, 1958.

We show how to extract S matrix elements for reactive scattering from just the real part of an evolving wave packet. A three-term recursion scheme allows the real part of a wave packet to be propagated without reference to its imaginary part, so S matrix elements can be calculated efficiently. Our approach can be applied not only to the usual time-dependent Schrödinger equation, but to a modified form with the Hamiltonian operator Ĥ replaced by f(Ĥ), where f is chosen for convenience. One particular choice for f, a cos−1 mapping, yields the Chebyshev iteration that has proved to be useful in several other recent studies. We show how reactive scattering can be studied by following time-dependent wave packets generated by this mapping. These ideas are illustrated through calculation of collinear H+H2→H2+H and three-dimensional (J=0)D+H2→HD+D reactive scattering probabilities on the Liu–Siegbahn–Truhlar–Horowitz (LSTH) potential energy surface.

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The time-dependent quantum-mechanical description of molecular photodissociation processes is briefly reviewed. A new easily implementable method for the calculation of partial cross-sections to produce specific fragment quantum states is presented. The equivalence of the partial cross-sections calculated using these time-dependent quantum-mechanical methods to those calculated using standard time-independent quantum theory is explicitly demonstrated. Sample calculations using a model potential-energy surface for a system having physical parameters corresponding to the H,S molecule are presented. The power of the method is clearly demonstrated by explicitly showing, for this model system, how a single time-dependent calculation yields the partial photodissociation cross-sections for all photon energies. We furthermore point out the suitability of modern parallel computing techniques in connection with such methods.

We study, within a helicity decoupled quantum approximation, the total angular momentum J dependence of reaction probabilities for the reaction A recently developed real wave packet O(1D) ] H 2 ] OH ] H. approach is employed for the quantum calculations. The ab initio based, ground electronic potential (X 3 1A@) energy surface of Ho et al. (T-S. Ho, T. Hollebeeck, H. Rabitz, L. B. Harding and G. C. Schatz, J. Chem. Phys., 1996, 105, 10472) is assumed for most of our calculations, although some calculations are also performed with a recent surface due to Dobbyn and Knowles. We Ðnd that the low J reaction probabilities tend to be, on average, slightly lower than the high J probabilities. This e †ect is also found to be reproduced in classical trajectory calculations. A new capture model is proposed that incorporates the available quantum data within an orbital angular momentum or l-shifting approximation to predict total cross sections and rate constants. The results agree well with classical trajectory results and the experimental rate constant at room temperature. However, electronically non-adiabatic e †ects may become important at higher temperature.