Bent triatomic molecules vibrational modes

Figure 4.14 Local vibrational quantum numbers of bent triatomic molecules. Also shown are the relative displacements of the atoms in the different modes.

Figure 4.19 Normal-mode vibrational quantum numbers for a bent triatomic molecule. Contrast the results for water, which is (cf. Table 4.6) near the local-mode limit with that for S02, which is near the normal-mode limit.

Figure 2.2 presents a schematic diagram of the possible vibrational modes for diatomic, linear triatomic, and bent triatomic molecules. The vibrational term symbol provides an ordered list of quantum numbers... 

The anharmonic potential energy is usually easier to represent in internal coordinates than in normal mode coordinates. However, what restricts the use of internal coordinates is the complicated expression for the vibrational/rotational kinetic energy in these coordinates (Pickett, 1972). It is difficult to write a general expression for the vibrational/rotational kinetic energy in internal coordinates and, instead, one usually considers Hamiltonians for specific molecules. For a bent triatomic molecule confined to rotate in a plane, the internal coordinate Hamiltonian is (Blais and Bunker, 1962) ... 

Figure 34. Local-mode coupling (according to the three-dimensional algebraic model) in a bent triatomic molecule for the first two vibrational polyads.

The most common bent triatomic molecule that you encounter daily is H2O. Like SO2, H2O belongs to the C2V point group and possesses three modes of vibration, all of which are IR and Raman active. These are illustrated in Fig. 3.13a which shows a calculated IR spectrum of gaseous H2O. (An experimental spectrum would also show rotational fine structure.) In contrast, the IR spectrum of liquid water shown in Fig. 3.13b is broad and the two absorptions around 3700 cm are not resolved. The broadening arises from the presence of hydrogen bonding between water molecules (see Section 10.6). In addition, the vibrational wavenumbers in the liquid and gas phase spectra are shifted with respect to one another. 

Theory of vibrational spectroscopy considers the energy of a vibrating molecule and the selection rules governing absorption and emission processes. We calculate the number of normal modes of vibration for linear and non-linear (bent) molecules, and view a computer simulation of the normal modes for both linear and bent triatomic molecules. 

For two substances in the same phase, and with similar molar masses, the substance with the more complex molecular stracture has the greater standard entropy. [Compare the standard entropies of 03(g) and F2(g).] The more complex a molecular stracture, the more different types of motion the molecule can exhibit. A diatomic molecule such as F2, for example, exhibits only one type of vibration, whereas a bent triatomic molecule such as O3 exhibits three different types of vibrations. Each mode of motion contributes to the total number of available energy levels within which a system s energy can be dispersed. Figure 18.3 illustrates the ways in which the F2 and O3 molecnles can rotate and vibrate. 

The large-dimension limit has recently resolved at least some of the difficulties of the molecular model. The molecule-like structure falls out quite naturally from the rigid bent triatomic Lewis configuration obtained in the limit D — oo [5], and the Langmuir vibrations at finite D can be analyzed in terms of normal modes, which provide a set of approximate quantum numbers [6,7]. These results are obtained directly from the Schrodinger equation, in contrast to the phenomenological basis of some of the earlier studies. When coupled with an analysis of the rotations of the Lewis structure, this approach provides an excellent alternative classification scheme for the doubly-excited spectrum [8]. Furthermore, an analysis [7] of the normal modes offers a simple explanation of the connection between the explicitly molecular approaches of Herrick and of Briggs on the one hand, and the hyperspherical approach, which is rather different in its formulation and basic philosophy. 

The three fundamental modes of vibration are shown in Figure 4.12. In the case of a triatomic molecule, it is simple to deduce that the three modes of vibration are composed of two stretching modes (symmetric and asymmetric) and a bending mode. However, for larger molecules it is not so easy to visualize the modes of vibration. We return to this problem in the next section. The three normal modes of vibration of SO2 all give rise to a change in molecular dipole moment and are therefore IR active. A comparison of these results for CO2 and SO2 illustrates that vibrational spectroscopy can be used to determine whether an X3 or XY2 species is linear or bent. 

The normal modes of vibration of linear and bent X Y2 molecules are illustrated in Figure 5.3, with the infrared bands of some common linear and bent triatomic... 

Here R, is some vibrational state, v, are the vibrational quantum numbers for the (3iV —6) normal modes which specify this state, d, is the degeneracy of the (th normal mode, and the and are constants analogous to those of the diatomic case. The problem of extrapolating observed rotational constants to equilibrium values is very much more demanding than in the diatomic case. Even when, as in the great majority of studies, the y constants are assumed zero, the a constants must be measured with comparable accuracy for all normal modes of the molecule. Even for a linear triatomic molecule, this means the measurement of three a constants for a bent... 

Diatomic Molecules. For triatomic molecules there are terms involving Coriolis interactions between vibrational modes in the correction from Bg to Bz, and the theory of the preceding sections is required. For linear or for bent symmetrical XY, molecules the rotational constants of a single isotopic species are sufficient to define the molecular structure completely, and thus for such molecules spectroscopic methods are superior in precision to the electron diffraction method. Electron diffraction studies serve as a test not only of the electron diffraction technique but also of the comparability between and r%. Some examples are given in Table 4. Besides the molecules CS., CIO, and CO, for which details are given, a... 

For polyatomic molecules, there are several normal modes of vibration that must be considered, as well as the bond angle that determines whether triatomics are linear or bent. The information in Table 28-3 for seven triatomics and one polyatomic was obtained from the same sources as Table 28-2 for diatomics, as well as Reed and Gubbins (1973, p. 74). Degenerate vibrational energy levels are indicated in parentheses and, in some cases, the characteristic rotational temperature is included under the molecular formula. 

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