Efficient calculation of highly excited vibrational energy levels of floppy molecules: The band origins of H+3 up to 35 000 cm−1

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Abstract
Recent testing of a discrete variable representation (DVR) Lanczos product‐basis method to calculate polyatomic vibrational energy levels [M. J. Bramley and T. Carrington, J. Chem. Phys. 99, 8519 (1993)] suggested that, for increasingly floppy molecules, its efficiency will be increasingly competitive with that of contracted‐basis explicit‐diagonalization methods if one can overcome the problem of poor Lanczos convergence caused by kinetic energy singularities. This may be accomplished through the realization that nondirect product finite basis representations (FBRs) (and the related DVRs) can be used efficiently in dynamics calculations for which the rate‐determining step is the evaluation of Hamiltonian matrix–vector products, as is the case with Lanczos recursion [J. W. Tromp and G. C. Corey, J. Chem. Phys. (to be submitted); D. Lemoine and G. C. Corey, J. Chem. Phys. (to be published)]. A synthesis of these two procedures provides a near‐optimally efficient variational vibrational method for molecules for which good basis contraction schemes cannot be designed, and for which the inevitable coordinate singularities require ideally a nondirect product basis. To substantiate this claim, we have performed hybrid DVR/FBR Lanczos calculations of vibrational energies of the classic floppy triatomic molecule H+3 up to near dissociation with unprecedently good convergence and unprecedently low computational cost.