Shape coordinates, vibration-rotation

Because of the translational invariance of the shape coordinates and Euler angles [Eqs. (13) and (25)], it follows from Eqs. (4) and (9) that translation is separated from the vibrational and rotational degrees of freedom that is, the matrix [g,y] is partitioned into an internal block gint (depending only on the shape coordinates and Euler angles) of the size (A — 3) x (A — 3) and to a translational block gtransl of the size 3x3 (depending only on the center-of-mass coordinates) as... 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

The non-separability of the vibrational modes is illustrated in Fig. 3 through the strong dependence of the JT potential energy surfaces for one of the interacting modes on the coordinates of the other mode (taken as a parameter). Whereas in Fig. 3(a) (zero displacement of the second mode) the potential surface for mode 1 exhibits the familiar Mexican-hat shape, for increasing displacements as in Figs. 3(b) and 3(c) there is an increasing distortion and the rotational symmetry is lost. ... 

When a polyatomic molecule dissociates, most vibrational modes correlate with modes of vibrational character in the products while other modes disappear, meaning that they correlate with rotation or translation of the separated fragments. As we have seen in Section 6.1.4.1 (and in Problems F and G), the transitory modes are those that determine the shape of the barrier. Taking as an example the dissociation ofC2H6 to two CH3 radicals, there are 18 vibrations ofthe parent. One vibration, the C-C stretch, becomes the reaction coordinate. Five other vibrations correlate to free rotations ofthe CH3 radicals, see Problem F. These are the torsion mode of C2H6 and the CH3 rocking modes. Twelve vibrations are conserved, meaning that they correlate to the vibrations of CH3. 

Polyatomic molecules have more than one vibrational frequency. The number can be calculated from the following. One atom in the molecule can move independently in three directions, the x, y, and z directions in a Cartesian coordinate system. Therefore, in a molecule with n atoms, the n atoms have 3n independent ways they can move. The center of mass of the molecule can move in three independent directions, x, y, and z. A nonlinear molecule can rotate in three independent ways about the x, y, and z axes, which pass through the center of mass. A linear molecule has one less degree of rotational freedom since rotation about its own axis does not displace any atoms. These translations of the center of mass and rotations can be performed with a rigid molecule and do not change its shape or size. Substracting these motions, there remain 3n — 6 degrees of freedom of internal motion for nonlinear molecules and 3n —5 for linear molecules. These... 

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