Linear bending coordinate

Atomic units (me = 1, qe = 1, h = 1) are used throughout this chapter.] The coefficients T, T2, and To are assumed to be in general analytical functions of the bending coordinate p. The term Tz represent the operator describing the rotation of the molecule around the (principal) axis z corresponding to the smallest moment inertia—this axis coincides at the linear nuclear arrangement with the molecular axis. Now Tz can be written in the form... 

In a very recent publication [1], we have presented a new model for the rotation-vibration motion of pyramidal XY3 molecules, based on the Hougen-Bunker-Johns (henceforth HBJ) approach [2] (see also Chapter 15, in particular Section 15.2, of Ref. [3]). In this model, inversion is treated as a large-amplitude motion in the HBJ sense, while the other vibrations are assumed to be of small amplitude they are described by linearized stretching and bending coordinates. The rotation-vibration Schrddinger equation is solved variationally to determine rotation-vibration energies. The reader is referred to Ref. [1] for a complete description of the theoretical and computational details. 

Dumont and Bougeard (68, 69) reported MD calculations of the diffusion of n-alkanes up to propane as well as ethene and ethyne in silicalite. Thirteen independent sets of 4 molecules per unit cell were considered, to bolster the statistics of the results. The framework was held rigid, but the hydrocarbon molecules were flexible. The internal coordinates that were allowed to vary were as follows bond stretching, planar angular deformation, linear bending (ethyne), out-of-plane bending (ethene), and bond torsion. The potential parameters governing intermolecular interactions were optimized to reproduce infrared spectra (68). 

Finally, the normal coordinates need not be linear combinations of the internal extension coordinates. In the results reported in this chapter, we use Simons-Parr-Finlan (SPF) (68) or Morse coordinates (2) for describing the stretching degrees of freedom. The normal coordinates are then defined as the appropriate linear combination of these coordinates. When we expand the coordinate dependent terms of the Hamiltonian in a Taylor series to a given order, the normal coordinates based on the SPF or Morse coordinates lead to a more accurate representation of both the model potential and the G matrix elements than do expansions based on the usual internal extension coordinates. In the case of the SPF coordinates, these expansions are exact at fourth order. Likewise, an appropriate choice of bending coordinates can also provide a more rapid convergence of these terms (49). 

Fig. 3.10. CCI3 bending normal coordinates (schematic) for chloroform transformed by the CJ operation. The Ai bending coordinate Q3 is transformed into H-1 times itself by the CJ operation. Each of the doubly degenerate bending coordinates, g6 and Qeb, is transformed by the CJ operation into a linear combination of both coordinates. Top views of the doubly degenerate coordinates are illustrated where the central arrow represents both carbon and hydrogen displacements.

Fig. 3.11. The doubly degenerate bending coordinates Qf and of chloroform (schematic) each transformed by every symmetry operation of the group. As was detailed in Fig. 3.10, each a and h member of the degenerate set is transformed into a linear combination of the members of the set.

One convenient set of equations to calculate the elements of the L tensor has been given for the five basic types of internal coordinates (stretch, bend, linear bend, out-of-plane, and torsion). These equations can also be generalized. For example, calculation of the L tensor elements for the bond stretching coordinate can be accomplished through equations almost identical to those given in equation (37). 

Fig,4.30. Transverse modes of the linear chain described by bending coordinates a... 

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