Approximation decoupling of rotation and vibrations

After the Eckart conditions are introduced, all the coordinates, Le. the components of the vectors Rcm treated as independent. 

Since the Coriolis term is small, in the first approximation we may decide to neglect it. Also, when assuming small vibrational amplitudes which is a reasonable approximation in most cases, we may replace r by the corresponding equilibrium positions in the rotational term of eq. (634) x r Y  

Ma(io X OaY, which is analogous to eq. (637). After these two approximations have been made the kinetic energy represents the sum of the three independent 

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Quasi-Rigid Model-Simplifying by Eckart Conditions Approximation Decoupling of Rotations and Vibration Spherical, Sjfminetric, and Asymmetric Tops Separation of Translational, Rotational, and Vibrational Motions... 

The T-matrix calculated according to the prescriptions above is computed in the Bom-Oppenheimer approximation and depends parametrically on the internal vibrational coordinates of the molecule. The Bom-Oppenheimer approximation decouples the nuclear and electronic degrees of freedom just as it does in the bound state case, and therefore some additional assumptions have to be made to use the fixed-nuclei quantities in approximate calculations of vibrational or rotational excitation cross sections. The simplest approximation for vibrational excitation cross sections is to take advantage of the parametric dependence of the T-matrix on the vibrational degrees of freedom of the molecule and write... 

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

The classical phase-space averages for bound modes in Eq. (11) are replaced by quantum mechanical sums over states. If one assumes separable rotation and uses an independent normal mode approximation, the potential becomes decoupled, and onedimensional energy levels for the bound modes may be conveniently computed. In this case, the quantized partition function is given by the product of partition functions for each mode. Within the harmonic approximation the independent-mode partition functions are given by an analytical expression, and the vibrational generalized transition state partition function reduces to... 

However, a complete set of molecular energy levels needed for calculation of the partition function (Eq. (1.16)) is not available in most cases. The arising problem can be simplified through the approximation that the different types of motion such as vibration, rotation, and electronic excitations are on a different timescale and therefore are unaffected by each other and can be treated as decoupled motions. This leads to a separation of Q into factors that correspond to separate partition functions for electronic excitations, translation, vibration, external molecular rotation, and hindered and free internal rotation ... 

Different approaches to the calculation of vibrational corrections to response properties can be found in the work of Sauer and Pack (2000), Ruud et al. (2000), and Kongsted and Christiansen (2006). The Boltzmann averaging procedure for conformationally flexible molecule has been critically reviewed by Crawford and Allen (2009). Mort and Autschbach (2008) have proposed an approach based on a decoupling of hindered rotations from the remaining (high-frequency) vibrational modes, which allows for a separate calculation of the hindered rotations without invoking the harmonic approximation. 

The system Hamiltonian can be approximated, as in the ARRKM theory, by decoupling the diatom vibrational motion from overall rotational motion of the molecule and from the van der Waals bond stretching. With this approximation. 

More recently KUBPERMANN, SCHATZ and BAER /77c/ developed a method for an accurate treatment of complanar collisions of an atom A with a diatomic molecule BC This method was then extended by SCHATZ and KUPPERMANN /77d/ to atom-diatom collisions in a three-dimensional physical space, making use of the "tumbling-decoupling approximation /41b/ In this treatment the collision is conveniently described in a body-fixed coordinate system. Together with the quantum number v of BC-vibration, two quantum numbers j and J are introduced for the rotation of BC-molecule and the overall rotation of the system ABC, respectively. Two corresponding qiiantum numbers m and n. are associated with the projections of the BC-angular momentum... 

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