A pseudospectral algorithm for the computation of transitional-mode eigenfunctions in loose transition states. II. Optimized primary and grid representations

Abstract

A highly optimized pseudospectral algorithm is presented for effecting the exact action of a transitional-mode Hamiltonian on a state vector within the context of iterative quantum dynamical calculations (propagation, diagonalization, etc.). The method is implemented for the benchmark case of singlet dissociation of ketene. Following our earlier work [Chem. Phys. Lett. 243, 359 (1995)] the action of the kinetic energy operator is performed in a basis consisting of a direct product of Wigner functions. We show how one can compute an optimized (k,Ω) resolved spectral basis by diagonalizing a reference Hamiltonian (adapted from the potential surface at the given center-of-mass separation) in a basis of Wigner functions. This optimized spectral basis then forms the working basis for all iterative computations. Two independent transformations from the working basis are implemented: the first to the Wigner representation which facilitates the action of the kinetic energy operator and the second to an angular discrete variable representation (DVR) which facilitates the action of the potential energy operator. The angular DVR is optimized in relation to the reference Hamiltonian by standard procedures. In addition, a scheme which exploits the full sparsity of the kinetic energy operator in the Wigner representation has been devised which avoids having to construct full-length vectors in the Wigner representation. As a demonstration of the power and efficiency of this algorithm, all transitional mode eigenstates lying between the potential minimum and 100 cm−1 above threshold have been computed for a center-of-mass separation of 3 Å in the ketene system. The performance attributes of the earlier primitive algorithm and the new optimized algorithm are compared.