The formal definition of the generalized discrete variable representation (DVR) for quantum mechanics and its connection to the usual variational basis representation (VBR) is given. Using the one dimensional Morse oscillator example, we compare the ‘‘Gaussian quadrature’’ DVR, more general DVR’s, and other ‘‘pointwise’’ representations such as the finite difference method and a Simpson’s rule quadrature. The DVR is shown to be accurate in itself, and an efficient representation for optimizing basis set parameters. Extensions to multidimensional problems are discussed.

Generalized discrete variable approximation in quantum mechanics

H+3 potential energies and multipole moments are calculated from a full CI with a 10s, 4p, 2d GTO hydrogen basis. 69 calculated energy points with energies of up to 25 000 cm−1 above the minimum are fitted by a power series expansion in terms of a Morse‐type coordinate with a mean square error of less than 1 cm−1. Rotationless vibrational states with energies of up to 12 000 cm−1 above equilibrium are calculated variationally for ten isotopomers. The resulting band origins for the seven analyzed fundamental transitions show a mean deviation of less than 2 cm−1. For the other predicted frequencies, the errors are expected to be below 0.1% also. The equilibrium bond length of H+3 is predicted to be 0.8732(2) A.

Ab initio calculation of near‐equilibrium potential and multipole moment surfaces and vibrational frequencies of H+3 and its isotopomers

A partitioning of the generalized triatomic hamiltonian of the preceding paper is developed which allows the calculation of highly-excited rotational states, without approximation, in a two-step variational procedure. Iterative diagonalization techniques are found to be particularly useful for the second variational step. The rotationally-excited states of H2D+ are studied with J ⩽ 20, well into the region where the ground and excited state manifolds overlap. Comparison of results for two different ab initio potentials and convergence considerations suggest that pure rotational transition frequencies obtained from our results should be accurate to about 1 cm-1 for J ∼ 15.

Highly rotationally excited states of floppy molecules: H2D+ with J ⩽ 20

The development of a new Morse-oscillator based rotation-vibration Hamiltonian for H3+

The potential surface for H+3, obtained from Hylleraas configuration interaction calculations by Frye et al. [J. Chem. Phys. 92, 4948 (1990)], has been fitted to a seventh‐degree polynomial in Morse–Dunham coordinates to study the low‐lying vibrational states of H+3, D+3, H2D+, D2H+, H2T+, and T2H+. Vibrational levels of up to two quanta excitations are reported and compared with experiments and other theoretical calculations. There are indications that adiabatic corrections may be important in correcting the small errors in the calculated vibrational quanta. The exothermicity of the proton‐transfer reaction between H2 and H+2 predicted is 1.685 eV, to be compared with the experimental value of 1.7 eV. The calculated dissociation energy for H+3 is 4.337±0.002 eV, 355 cm−1 smaller than the experimental value of 4.381±0.021 eV. No source of discrepancy from theoretical side can be identified.

Vibrational analysis of a Hylleraas‐configuration interaction potential for H+3

The H3 + ab initio potential energy surface and its analytical representation has been re-examined to promote more accurate vibration-rotation calculations. We present here a new and accurate 78 point PNO-CI grid which predicts the minimum energy geometry to be an equilateral triangle of side, r (H-H) = 1·6525 a 0. and energy of -1·34188 E h. Of the analytical representations only the sixth order Simons-Parr-Finlan (SPF) and Ogilvie force fields were found to satisfy our fitting criteria, resulting in associated errors of less than 11·1 kJ mol-1. Our calculated vibrational band origins are in good agreement with those of Carney and Porter, with the differences of the lowest-lying vibration states essentially reflecting the analytical characteristics of the potential energy surfaces used.

The vibration spectrum of H3

Vibration spectrum of H3+: A model hamiltonian

A combination of known methods have been spliced together in order to calculate accurate vibrational energies and wavefunctions. The algorithm is based on the Rayleigh-Ritz variational procedure in which the trial wavefunction is a linear combination of configuration products of one-dimensional basis functions. The Hamiltonian is that due to Carney and Porter (1976). The kernel of the algorithm consists o( the one-dimensional basis functions, which are the finite element solutions of the associated one-dimensional problems.

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M. Hamilton

Papers

The vibration spectrum of H3

Vibration spectrum of H3+: A model hamiltonian

A numerical variational method for calculating accurate vibrational energy separations of small molecules and their ions