Stavros C. Farantos^* and Raul Guantes ⁺

Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, PO Box 1527, Iraklion 71110, Crete, GREECE

^*Also at Department of Chemistry, University of Crete, Iraklion 711 10, Crete, Greece.
⁺Current address: Instituto de Matematicas y Fisica Fundamental, Consejo Superior de Investigaciones Cientificas, Serrano 123, 28006 Madrid, Spain.
 

The problem of interpreting vibrational spectra of polyatomic molecules is twofold;  first, we must find the solutions of the Schroedinger equation, and second, we must assign the energy levels with quantum numbers. In spite of the impressive advances  in solving the Schroedinger equation for triatomic and tetratomic  molecules limitations remain especially for excited molecules. Significant is also the progress in the physical interpretation of vibrational spectra thanks to the links with non-linear mechanics and semiclassical theories [1].
Recently we have addressed the problem of solving the time dependent and time independent Schroedinger equation via Finite Differences (FD). The view of  considering global Pseudospectral methods (Sinc and Fourier)  as the infinite order limit of local Finite Difference methods [2], and vice versa,  Finite Difference as a certain sum acceleration of the Pseudospectral methods  has been exploited to investigate high order Finite Difference algorithms for solving the Schroedinger equation in Molecular Dynamics. Radial and angular variables are treated on the same ground with centered equi-spaced grids as well as non uniform Legendre and Chebyshev grids [3-5]. Results from 1D to 5D systems will be presented which demonstrate the stability and  accuracy of High Order Finite  Difference approximations of wave functions compared to global Pseudospectral  techniques.

[1] H. Ishikawa, R. W. Field, S. C. Farantos, M. Joyeux, J. Koput, C. Beck and R. Schinke,
     Annual Review of Physical Chemistry, 50, 443 (1999).
[2] B. Fornberg, "A Practical Guide to Pseudospectral Methods", Cambridge Monographs on
     Applied and Computational Mathematics  (Cambridge Univ.  Press, 1, 1998).
[3] R. Guantes and S. C. Farantos, J. Chem. Phys. 111, 10827 (1999).
[4] R. Guantes, A. Nezis and S. C. Farantos, J. Chem. Phys. 111, 10836 (1999).
[5] R. Guantes and S. C. Farantos, J. Chem. Phys. 113, Dec. 18, 2000.