Scientific background

Solving the Schroedinger equation for the nuclear motions in small polyatomic 
molecules has been and will remain one of the most challenging 
computational problems.  Accurate solutions for the vibrational states 
of molecules which play a significant role in combustion and atmospheric 
chemistry as well as in biology are highly demanded since the recent advances 
in high resolution spectroscopy [1]. Unequivocally, the progress in experiment
and theory of vibrational spectroscopy of the last twenty years is immense. 

Grid methods for the solution of the Schroedinger equation are nowadays of 
the most powerful and exploited tools both for the time independent 
(here mainly known as Discrete Variable Representation - DVR [2-4]) and 
time dependent pictures.  Concerning the last one, it was the introduction 
of the Fourier Pseudospectral (PS) method which provided the necessary 
efficiency to compete with the traditional variational techniques and to make 
feasible the task [5,6]. Lately, the ideas of the Chebychev expansion of the
time propagator for solving the time dependent Schroedinger equation have
been extended to the energy domain, thus, producing the filtering techniques
for diagonalizing the Hamiltonian matrix [7,8]. This allowed the extraction
of accurate wave functions in triatomic molecules at high energies with 
large densities of states.

The present computational algorithms and computer technology have made 
vibrational calculations on triatomic molecules relatively a routine job, 
even on small size computers [9], and feasible the calculations with the
rotational degrees of freedom included [10] and on tetratomic 
molecules (six degrees of freedom) [11]. However, as the size of molecule
increases or the density of vibrational states increases
the goal of solving the Schroedinger equation remains unsatisfied.

It is interesting that in our effort to surmount the above difficulties old
techniques are revisited. For example, local methods such as the 
Finite Difference (FD) algorithms were the first to apply in problems of 
Molecular Dynamics [12]. The results obtained that time were rather 
unsatisfactory and the appearance of global methods such as the PS ones 
(DVR or Fourier) dominated in solving the vibrational problem in polyatomic 
molecules.

However, new developments in the theory of Finite Differences [13] have led us 
to examine their efficiency in solving the Schroedinger equation.
Thanks to new computer algorithms for computing the necessary weights
in FD schemes, it was shown that the FD method may lead to a considerable
saving in computer memory and time because of the local representation
of the wave function [14,15].

Old ideas based on the hydrodynamic formulation of quantum mechanics going
back to de Broglie and Bohm have been revived in a new scheme known as
Quantum Trajectory Method (QTM) [16]. Again, the advances are due to the
progress in numerical solutions of computational fluid dynamics [17]. 
These ideas replace the well known wave equation with trajectory - type 
(quantum) equations.

The molecular vibrational problem is not only that of solving the Schroedinger
equation. Even if that was possible, the huge amount of data would be 
useless without being able to assign the vibrational states and extract
their physical meaning. Assigning the vibrationally highly excited states is
also a difficult task because of the non linear mechanical behaviour of the
molecule. Progress in this direction has been achieved by developing
methods based on effective Hamiltonians, semiclassical theories and periodic
orbits [18,19]. It turns out that up to now there has been no effort to 
face both problems (computation and assignment of vibrational states) 
simultaneously within the same theory.

Motivation and objectives

The CECAM workshop on Spectroscopy and Computational Challenges in 
Vibrationally Highly Excited Polyatomic Molecules aims into examining 
techniques for solving the Schroedinger equation in conjunction to the
interpretation of vibrational spectra in highly excited polyatomic
molecules. It is believed, that instead of computing the whole spectrum
in a range of energies the computation of those parts of the spectrum
related to experimental or other type of theoretical information will 
help in circumventing the difficulties in computer memory and time. 

In the list of participants a few leading experimentalists are included.
By bringing together theorists and experimentalists we hope to trace the
problems better and stimulate new research projects.

Through the presentations of the participants the currently used algorithms 
will be reviewed. During the discussion sessions ideas of designing 
algorithms which take in account previous experimental or theoretical
information will be worked out.


References

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{3}{C. Iung and C. Leforestier, {J. Chem. Phys.} { 102}, 8453 (1995).}
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{5}{R. Kosloff, in Dynamics of Molecules and Chemical Reactions, R. E. Wyatt and J. Z. H. Zhang eds. Marcel Dekker Inc., New York, 1996.}
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{12}{E. A. McCullough and R. E. Wyatt, { J. Chem. Phys.} { 51}, 1253 (1969).}
{13}{B. Fornberg and D. M. Sloan, { Acta Numerica} (1994), pp. 203-267.}
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{17}{T. Belytschko, Y. Krongauz, D. Organ M. Fleming and P. Krysl, { Comput. Methods Appl. Mech. Eng.} { 139}, 3 (1996).}
{18}{S. C. Farantos, { Comp. Phys. Comm.} {108}, 240 (1998).}
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