M. M. Law, I. A. Atkinson and J. M. Hutson (eds.)
Rovibrational Bound States in Polyatomic Molecules
ISBN 0-9522736-6-7 © 1999, CCP6, Daresbury
LSDSMS (UMR 5636) - CC 014,
Université des Sciences et Techniques du Languedoc,
34095 MONTPELLIER Cedex 05, France.
Alexandra Viel and Claude Leforestier
Study of the (ro)vibrational levels of tetraatomic molecules constitutes a drastic change as compared to triatomic calculations. The reason is that the number of internal degrees of freedom doubles from 3 to 4 atoms, and as a consequence, the basis set size is roughly squared. Furthermore, quite often, the energy domain of interest lies in the middle of the spectrum. Amongst examples, one can cite
In recent years, an alternative approach called the direct method has emerged as a powerful tool. It consists of directly acting the Hamiltonian operator H on a wavefunction, without the help of the associated matrix. First implementations were made by Feit and Fleck  and Kosloff and Kosloff  through the Fast Fourier Transform method. It relies on using two different representations of the wavefunction, and has been shown able to achieve very high accuracy . Because only a few vectors have to be kept in core memory, ultra-large basis sets can be considered in a direct method.
The direct approach became very popular due to the emergence of the Discrete Variable Representation method of Light and coworkers . In this method, one emphasizes the grid representation in which the potential is diagonal. The direct approach has been used in conjunction with iterative schemes such as the Lanczos algorithm , the Chebychev time propagator  or the GMRes linear solver , to cite a few applications. The direct DVR method can result in a broad spectrum of the Hamiltonian operator when use is made of curvilinear coordinates. This is due to the presence of almost singular terms in the kinetic energy operator, such as . A DVR representation of the variable will badly behave close to or , as the correct asymptotic form of the solution is not enforced .
To cope with this problem, the direct method can also be used with a spectral representation as the primary one, such as spherical harmonics  or Wigner rotation functions . In this case, acting the Hamiltonian operator requires transforming between two or more representations by means of pseudo-spectral schemes .
Whatever is the primary representation (grid or spectral) in use, calculations can be made more efficient by preconditioning the basis set. This technique allows one to reduce the basis set size significantly in the case of a straight diagonalization approach, or to resort eventually to some perturbation scheme. For iterative approaches, preconditioning becomes essential as the convergence of such schemes strongly depends on the total width of the spectrum associated with H. We present below an adiabatic pseudo-spectral method which is currently applied to the calculation of the vibrational levels of the HFCO molecule. But first, we recall the Successive Adiabatic Reduction method of Bacic and Light which is used as the reference approach.