Triatomic molecules vibrational motion

We find it convenient to reverse the historical ordering and to stait with (neatly) exact nonrelativistic vibration-rotation Hamiltonians for triatomic molecules. From the point of view of molecular spectroscopy, the optimal Hamiltonian is that which maximally decouples from each other vibrational and rotational motions (as well different vibrational modes from one another). It is obtained by employing a molecule-bound frame that takes over the rotations of the complete molecule as much as possible. Ideally, the only remaining motion observable in this system would be displacements of the nuclei with respect to one another, that is, molecular vibrations. It is well known, however, that such a program can be realized only approximately by introducing the Eckart conditions [38]. 

Figure 4.3 Vibrational modes of a nonlinear triatomic molecule such as H20. Arrows indicate motion in the plane of the paper, + is towards and - away from the observer, (a) symmetric stretching, (b) asymmetric stretching, (c) out-of-plane wagging, (d) out-of-plane twisting, (e) in-plane scissoring, (f) in-plane rocking.

For linear triatomic molecules 3 N - 5 = 4 and we expect form vibrational modes instead of three as shown in the fig. Out of the four there are two vibrations, one in the plane of the paper and the other in which the oxygen atoms move simultaneously into and out of the plane. The two sorts of motion are termed degenerate and so we have only three vibrations. 

There are several important effects associated with degenerate vibrational levels. We begin by considering the doubly degenerate v2a and v2b vibrational modes of a linear triatomic molecule. In these modes, the three atoms each vibrate in a plane perpendicular to the molecular axis, the vibrations being either in the xz or the yz plane, where the z axis is the molecular axis (Fig. 6.2). Classically, if both these modes are excited, the vibrations may give rise to vibrational angular momentum about the internuclear axis for example, if v2a and v2b are of equal amplitude and differ in phase by 90°, then the resultant motion of each nucleus is a circle about the molecular axis, as shown in Fig. 6.8 (see Problem 6.18). 

Transient two-photon ionization experiments on trimer systems were, of course, motivated by a need for time-resolved verification of the pseudorotation motion, which can be considered as a superposition of the asymmetric stretch (Qx) and the bending vibration (Qy) [12]. The triatomic molecule with its three modes is quite different from an isolated oscillating dimer, which vibrates in its single mode until eventually it radiates or predissociates. The interplay of vibrational modes in a trimer system can be considered as the prototype of IVR. 

One of the earliest models is the quasi-diatomic model (10-13). This model is based on the assumption that the normal modes describing the state(s) of the photofragments are also the normal modes of the precursor molecule. This means, for example, that in the photodissociation of a linear triatomic molecule ABC A + BC (e.g., photodissociation of ICN - I + CN), the diatomic oscillator BC is- assumed to be a normal mode vibration in the description of the initial state of the triatomic molecule ABC. This means that the force constant matrix describing the vibrational motion of the molecule ABC can be written in the form (ignoring the bending motion) ... 

Equation 2.17 is of the form A = PDP-1. The 9x9 Hessian for a triatomic molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization into a P matrix whose columns are direction vectors for the vibrations whose force constants are given by the k matrix. Actually, columns 1, 2 and 3 of P and the corresponding k, k2 and k3 of k refer to translational motion of the molecule (motion of the whole molecule from one place to another in space) these three force constants are nearly zero. Columns 4, 5 and 6 of P and the corresponding k4, k5 and k6 of k refer to rotational motion about the three principal... 

The central concept of mode-selective chemistry is illustrated in Fig. 1, which depicts the ground and excited state potential energy surfaces of a hypothetical triatomic molecule, ABC. One might wish, for example, to break selectively the bond between atoms A and B to yield products A+BC. Alternatively, one might wish to activate that bond so that in a subsequent collision with atom D the products AD+BC are formed. To achieve either goal it is necessary to cause bond AB to vibrate, thereby inducing motion along the desired reaction coordinate. 

Figure 3-48. Bending motion and a sampler of potential energy functions. Top bending vibration of a linear triatomic molecule, where r is the instantaneous distance between the end atoms and re is the equilibrium distance of the linear configuration (r

To compensate for the above, the number of theoretical normal vibrations may be reduced by two inherent factors of the molecule. Some vibrations may be degenerate. For example, a Unear triatomic molecule should, by theory, have four vibrational modes. However, the deformational mode of carbon dioxide (see Fig. 2, A, iii) is not uniquely defined, since the motions could take place either in the plane of the paper or in a plane perpendicular to it. If a molecule is highly symmetrical, it is probable that certain vibrations will not be accompanied by a change in the dipole moment, thus the frequency will be forbidden in the infrared. ... 

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. 

As a linear triatomic molecule, carbon dioxide has four degrees of vibrational motion. These are the symmetrical stretch (vj), the asymmetrical stretch (V3), and the bending mode (V2). The later vibration is doubly degenerate and can be described in two directions perpendicular to the interatomic axis. 

The emerging picture is one in which the quantum-mechanical equivalents of the constants of motion for the two valence electrons in these atoms are like those associated with the near-rigid rotations, bending vibrations, and stretching vibrations we normally associate with linear triatomic molecules. These new results bring into question the range of validity of the nearly-independent-particle model, the quantum-mechanical counterpart of Bohr s planetary model, for atoms with more than one valence electron. 

Whereas a diatomic molecule has only one mode of vibration which corresponds to- a stretching motion, a non-linear B—type triatomic molecule has three modes, two of which correspond to stretching motions, with the other corresponding to a bending motion. A linear-type triatomic molecule has four modes, where two of which have the ame frequency and are said to be degenerate. 

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