On the variational computation of a large number of vibrational energy levels and wave functions for medium-sized molecules

In a recent publication [J. Chem. Phys. 127, 084102 (2007)], the nearly variational DEWE approach (DEWE denotes Discrete variable representation of the Watson Hamiltonian using the Eckart frame and an Exact inclusion of a potential energy surface expressed in arbitrarily chosen coordinates) was developed to compute a large number of (ro)vibrational eigenpairs for medium-sized semirigid molecules having a single well-defined minimum. In this publication, memory, CPU, and hard disk usage requirements of DEWE, and thus of any DEWE-type approach, are carefully considered, analyzed, and optimized. Particular attention is paid to the sparse matrix-vector multiplication, the most expensive part of the computation, and to rate-determining steps in the iterative Lanczos eigensolver, including spectral transformation, reorthogonalization, and restart of the iteration. Algorithmic improvements are discussed in considerable detail. Numerical results are presented for the vibrational band origins of the (CH4)-C-12 and (CH2D2)-C-12 isotopologues of the methane molecule. The largest matrix handled on a personal computer during these computations is of the size of (4 center dot 10(8))x(4 center dot 10(8)). The best strategy for determining vibrational eigenpairs depends largely on the actual details of the required computation. Nevertheless, for a usual scenario requiring a large number of the lowest eigenpairs of the Hamiltonian matrix the combination of the thick-restart Lanczos method, shift-fold filtering, and periodic reorthogonalization appears to result in the computationally most feasible approach.