Lagrange-distributed approximating-functional approach to wave-packet propagation: Application to the time-independent wave-packet reactant-product decoupling method

A connection is made between a recently introduced Lagrange-distributed approximating-functional and the Paley-Wiener sampling theorem. The Lagrange-distributed approximating-functional sampling is found to provide much superior results to that of Paley-Wiener sampling. The relations between discrete variable representation and Lagrange-distributed approximating functionals are discussed. The latter is used to provide an even spaced, interpolative grid representation of the Hamiltonian, in which the kinetic energy matrix has a banded, Toeplitz structure. In this paper we demonstrate that the Lagrange-distributed approximating-functional representation is an accurate and reliable representation for use in fast-Fourier-transform wave-packet propagation methods and apply it to the time-independent wave-packet reactant-product decoupling method, calculating state-to-state reaction probabilities for the two-dimensional (collinear) and three-dimensional $\mathrm{}(J=0\mathrm{})\mathrm{}$ ${\mathrm{H}+\mathrm{H}}_2$ reactions. The results are in very close agreement with those of previous calculations. We also discuss the connection between the distributed approximating-functional method and the existing mathematical formalism of moving least-squares theory.