Vibrational modes representations

Single surface calculations with a vector potential in the adiabatic representation and two surface calculations in the diabatic representation with or without shifting the conical intersection from the origin are performed using Cartesian coordinates. As in the asymptotic region the two coordinates of the model represent a translational and a vibrational mode, respectively, the initial wave function for the ground state can be represented as. 

Similarly, it can be shown that the nanotube modes at the T-point obtained from the zone-folding eqn by setting Ai = 1), where 0 < ri < N/2, transform according to the , irreducible representation of the symmetry group e. Thus, the vibrational modes at the T-point of a chiral nanotube can be decomposed according to the following eqn... 

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

The major vibrational modes observed for isoquinoline are listed in Table II. The assignments made by Wait et al. (22) are also included. These authors made their assignments from considerations of the higher symmetry parent species, instead of the Cs symmetry group they demonstrated that the assignments arising from this representation are reasonable. 

Fig. 13. Top Schematic representation of the two components of the Jahn-Teller-active vibrational mode for the E e Jahn-Teller coupling problem for octahedral d9 Cu(II) complexes. Bottom Resulting first-order Mexican hat potential energy surface for showing the Jahn-Teller radius, p, and the first-order Jahn-Teller stabilization energy, Ejt.

Fig. 35. Schematic representation of the first-order potential energy surface for T0e vibronic coupling. The two components of the eg vibrational mode are shown on the left.

By trial and error, the only solution to this equation is found to be eight one-dimensional and four two-dimensional representations, as listed in the character table. There is no standard order for listing the classes. The irreducible representations should, however, always be listed in the order given in Section 9.12 this order determines the numbering of the vibrational modes (see Section 9.9). The significance of the symbols x, y, z, Rx, Ry, Rz will be explained in Section 9.9. 

As noted earlier, point groups with no threefold or higher proper or improper axis have only one-dimensional representations hence a necessary condition for a molecule to have degenerate vibrational modes is that it possess a Cn or an S axis with n> 3. Asymmetric tops have no degenerate vibrational modes. 

To find the contributions of the internal coordinates, C—H bond lengths and HCH angles, to these vibrational modes we first use the set of four C—H bond lengths as the basis for a representation, obtaining TCH shown below. 

Consider next the water molecule. As we have seen, it has a dipole moment, so we expect at least one IR-active mode. We have also seen that it has CIt, symmetry, and we may use this fact to help sort out the vibrational modes. Each normal mode of iibratbn wiff form a basis for an irreducible representation of the point group of the molecule.13 A vibration will be infrared active if its normal mode belongs to one of the irreducible representation corresponding to the x, y and z vectors. The C2 character table lists four irreducible representations A, Ait Bx, and B2. If we examine the three normal vibrational modes for HzO, we see that both the symmetrical stretch and the bending mode are symmetrical not only with respect to tbe C2 axis, but also with respect to the mirror planes (Fig. 3.21). They therefore have A, symmetry and since z transforms as A, they are fR active. The third mode is not symmetrical with respect to the C2 axis, nor is it symmetrical with respect to the ojxz) plane, so it has B2 symmetry. Because y transforms as Bt, this mode is also (R active. The three vibrations absorb at 3652 cm-1, 1545 cm-1, and 3756 cm-, respectively. 

As a second example of the use of character tables in the analysis of IR and Raman spectra, we tum to BCIj with D3h symmetry. Because it has four atoms, we expect six vibrational modes, three of which will be stretching modes (because there are three bonds) and three of which will be bending modes. Table 3.5 shows the derivation of r,ol for the molecule s twelve degrees of freedom. Application of the reduction equation and subtraction of the translational and rotational representations gives... 

We see that the ax fundamental vibrations of BC13 transform as A, A2, and 2E. Each representation describes two vibrational modes of equal energy. Thus the 2E notation refers to four different vibrations, two of one energy and two of another. The a mode is Raman active, the A2 is IR active, and the modes are both Raman and IR active. 

In carrying out the procedure for a tetrahedral species, it is convenient to let four vectors on the central atom represent the hybrid orbitals we wish to construct (Fig. 3.26). Derivation of the reducible representation for these vectors involves performing on them, in turn, one symmetry operation from each class in the Td point group. As in the analysis of vibrational modes presented earlier, only those vectors that do not move will contribute to the representation. Thus we can determine the character for each symmetry operation we apply by simply counting the number of vectors that remain stationary. The result for AB4 is the reducible representation, I",. 

Draw the structure of the planar BrONO molecule and determine the irreducible representations for iLs vibrational modes. Which modes arc IR active and which ones are R.iman active (Wilson. W W Christe. K O. hhtre. Cheat. 1987. 2h. IS73 )... 

In other words, the irreducible representation / of the vibrational mode has to be contained in the direct product of those (i"j and / ) of the electronic states [26], For the intra-state, or JT, couplings, the selection rule (1) involves the symmetrised direct product (J3)2 for the degenerate electronic state and leads to the well-known result ... 

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