A bent triatomic

To illustrate its use, let us consider the 2s atomic orbital of the O atom in water. 

Apply each symmetry operation of the C2V point group in turn. Applying the E operator leaves the 2s atomic orbital unchanged rotation about the C2 axis leaves the atomic orbital unchanged reflections through the cr and planes leave the 2s atomic orbital unchanged. These results correspond to the following row of characters  

This matches the row of characters for symmetry type Bi in the C2V character table, and the 2p orbital therefore possesses bi symmetry. The 2py orbital is left unchanged by the E operator and by reflection through the a. (pz) plane, but rotation about the C2 axis and reflection through the 

To determine the composition of the aj LGO of the H H fragment in H2O, we multiply each character in the above row by the corresponding character for the Ai representation in the 2 character table, i.e. 

The H2O molecule has C2v symmetry (Fig. 5.14) and we now show how to use this information to develop an MO picture of the bonding in H2O. Part of the C2v character table is shown below  

The inclusion of the xz and yz terms in the last two columns of the character table specifies that the H2O molecule is taken to lie in the yz plane, i.e. the z axis coincides with the principal axis (Fig. 5.14). The character table has several important features. 

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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 9.3. Orbital geometry and nomenclature for a bent triatomic molecule. Solid contours indicate regions where the wave function is positive broken contours indicate regions where the wave function is negative.

This expression has been analyzed for the dissociation of ICN. The initial thermal distribution corresponds to large j. A distribution peaked around j 25 was obtained, in good agreement with experimental data. The analysis has been generalized to describe the case of a bent triatomic molecule (53,5A). Moreover, these authors consider the scalar coupling which corresponds to indirect photodissociation. 

In this paper we will present some examples of the application of resonance Raman spectroscopy to the study of transition metal diatomics. The application of Raman spectroscopy to matrix-isolated metal clusters was first reported by Schulze et al. (] ). Having observed only a single line in the Raman spectrum of Ag3, Schulze concluded that the molecule was linear since a bent triatomic and an equilateral triangular geometry would have, in principle, 3 and 2 Raman-active modes. The evidence, however, is not conclusive since many Czy molecules have very weak asymmetric stretches in the Raman ( ) (for example, the V3 mode of O3 is undetectable in the Raman (3 )). Moreover, the bend (V2) of Ag3 is expected to be a very low-frequency mode, perhaps lower than one can feasibly detect in a matrix Raman experiment. 

For a bent triatomic symmetrical molecule with a bending angle a, the frequencies of the symmetric and anti-symmetric vibrations are given by ... 

FIGURE 20.9 The three types of vibrational motion that are possible for a bent triatomic molecule. Arrows show the displacement of each atom during each type of vibration. 

The first example is quite simple, but serves to introduce the idea. The structure of water has already been discussed. You will recall that it is a bent triatomic molecule based on a tetrahedron, with the two lone pairs of the oxygen pointed towards the unoccupied comers of the tetrahedron. Given this structure, what is the shape of the closely related compound, hydrogen peroxide, H202 ... 

In particular, ifz = r3j and z2 = r32 (in which case det[c] 3 = 1), the volume element for a bent triatomic molecule in bond-angle coordinates is obtained as... 

FIGURE 2 Schematic plot of harmonic bending constant faa against interatomic distance r for a bent triatomic molecule. 

To summarize, in sodium azide there is experimental and theoretical evidence to associate the 730-nm optical absorption with an F center. The association of colloidal particles with the 520 nm band produced by irradiation and subsequent heating is plausible. An F2 center and a bent triatomic radical have been offered to account for the 610-nm absorption band. The Fj center is less tenable because one would not expect an infrared absorption to be associated... 

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.

It was shown before that a bent triatomic molecule undergoes two A and one B2 vibrations. Inspection of the C2V character table revels that z spans Aj and y spans B2. Therefore, all the vibrations are allowed. This is not to say that all three bands will appear. The magnitude of the transitions dipole moments may be so small that a transition may not be observed. 

Consider a bent triatomic molecule such as water. Fig. 2.1. Each atom is described by a set of three cartesian coordinates, which form an array of nine numbers Xi,i— 1 to 9, and each displacement during a vibration is called 5xi. The atomic mass corresponding to each coordinate is called tm. Defining mass-weighted coordinates qi and writing the total kinetic energy T for the motions during the displacements yields ... 

Next, consider a reaction between an atom and a diatomic molecule, A + BC. Reactions can differ also in their energetic requirements, but to focus attention on the steric requirements with the energetic effects being equal we take the barrier height Eq to be the same as in the previous reaction. To have minimal steric requirements let us take the transition state, ABC, to be bent. This choice allows A to approach BC within a cone. Because BC has an internal structure, the partition function for the reactants becomes Q = Q Q QvQ - The transition state is a bent triatomic. It has three vibrations, one of which is the reaction coordinate. (As we saw in Section 5.1, this is the asymmetric stretch vibration.) The bent transition state has three planes of rotation, = Q QIQ - Accordingly, for reasons that will become immediately apparent, we write k T) as... 

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. 

Fi9Ure 18.3 In addition to translational motion, molecules exhibit both vibrations, in which the atoms positions relative to one another change, and rotations, in which the molecule rotates about its center of mass, (a) A diatomic molecule such as fluorine only exhibits one type of vibration. A bent, triatomic molecule such as ozone exhibits three types of vibration, (b) A diatomic molecule exhibits two (hffeient rotations, whereas a bent, triatomic molecule exhibits three different rotations. (Note that rotation of F2 about the x axis would cause no change in the positions of either atom in the molecule.)... 

Figure 7.22 (a) Internal coordinates of a bent triatomic molecule, (b) Space-fixed and body-fixed... 

Ozone, O3, an allotropic form of oxygen, is a bent triatomic molecule. 

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