Calculation of triatomic vibrational eigenstates: Product or contracted basis sets, Lanczos or conventional eigensolvers? What is the most efficient combination?

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Abstract
Numerous practical methods have been described for exact quantum calculations of vibrational eigenstates (energy levels and wave functions) for three‐ and four‐atom molecules. Many descriptions are accompanied by bold claims of efficiency. Such claims are, unfortunately, difficult to test in the absence of fair comparisons on a single computer. The efficiency of these calculations depends above all (once the most appropriate coordinate system has been chosen) on clever choices of (i) the multidimensional basis set, and (ii) the Hamiltonian matrix eigensolver. In the first category come techniques such as the discrete variable representation (DVR) and basis contraction (also known as sequential adiabatic reduction or diagonalization truncation). In the second category, the Lanczos recursion is being increasingly applied. In a recent study taking the HCN/HNC molecule as a test case [R. A. Friesner, J. A. Bentley, M. Menou, and C. Leforestier, J. Chem. Phys. 99, 324 (1993)], reductions in computational effort of one to three orders of magnitude were found for a method combining basis contraction and Lanczos recursion, compared to one widely considered to be state of the art in which the Hamiltonian matrix is diagonalized conventionally [Z. Bačić and J. C. Light, J. Chem. Phys. 86, 3065 (1987)]. We have investigated this finding by developing a computer program which permits choosing both between direct product and two kinds of contracted basis (all derived from DVRs), and between Lanczos and conventional eigensolvers. It has been applied to the calculation of vibrational frequencies both of HCN/HCN up to 12 000 cm−1 and of H2O up to 22 000 cm−1, with a strict convergence criterion of 1 cm−1 in each case. We find the conclusions of Friesner et al. to be exaggerated: while a contracted/Lanczos method is consistently most efficient, other combinations, even the rather simple direct‐product Lanczos [M. J. Bramley and T. Carrington, J. Chem. Phys. 99, 8519 (1993)], are never as much as a factor of 5 more costly.