Wavefunctions polyatomic vibrational

Degeneracies of the SCF states are an obvious cause for breakdown of the approximation in the form discussed in the previous sections. We discuss now an extension of the method that applies to such cases, that is, to resonances and near-resonances between SCF modes. Just as the vibrational SCF method is an adaptation of the Hartree approximation from electronic structure calculations, so is the generalization discussed here an application of the configuration interaction (Cl) method, which uses for the wavefunctions a linear combination of the strongly interacting SCF states. Quantum Cl for polyatomic vibrations was introduced by Bowman and co-workers,7-21 the semi-classical version is due to Ratner et al.33... 

The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 

The harmonic oscillator energies and wavefunctions comprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomic molecule are often characterized in terms of individual bond-stretching and angle-bending motions each of which is, in turn, approximated harmonically. This results in a total vibrational wavefunction that is written as a product of functions one for each of the vibrational coordinates. 

In asymmetric top molecules (and in all polyatomic molecules when nontotally symmetric vibrations are excited), K is no longer a good quantum number. The rotational wavefunctions may be then expressed in the JKM basis as the linear superpositions... 

Using perturbation theory, Watson" has derived an expression, similar to eqn (20.31) but with rotation included, for a polyatomic molecule s vibration/ rotation energy levels. The analytic form for a prolate symmetric top has been given." This approach assumes the molecule has good vibration/rotation quantum numbers and, if this is not the case or if more accurate values for the energy levels are required, the variational method may also be used to determine vibrational/rotational energy levels. For this approach the wavefunction for a given vibration-rotation level is written as a linear combination of basis function j/iRjK -... 

In order to illustrate the approximations involved when trying to mix quantum and classical mechanics we consider a simple system with just two degrees of freedom r and R. The R coordinate is the candidate for a classical treatment - it is, e.g., the translational motion of an atom relative to the center of mass of a diatomic or polyatomic molecule. This motion is slow compared to the vibrational motion of the diatom - here described by the r coordinate. Thus if we treat the latter quantum mechanically we could introduce a wavefunction t) and expand this in eigenstates of the molecule, i.e., eigenstates to the hamiltonian operator Hq for the isolated molecule. Thus we have... 

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