Nonlinear molecules vibrational wave function

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

This phenomenon of vibronic coupling can be treated very effectively by using group theoretical methods. As will be shown in Chapter 10, the vibrational wave function of a molecule can be written as the product of wave functions for individual modes of vibration called normal modes, of which there will be 3n - 6 for a nonlinear, /i-atomic molecule. That is, we can... 

As mentioned in the introduction, the above discussion of the small-, large-, and intermediate-molecule limits of electronic relaxation processes can also be utilized with very minor modifications to discuss the phenomena of intramolecular vibrational relaxation in isolated polyatomic mole-cules. ° Figure 4 is still applicable to this situation. The basis functions are now taken to be either pure harmonic vibrational states, some local-mode vibrational eigenfunctions, or some alternative nonlinear mode-type wave-functions. In the following the nomenclature of vibrational modes is utilized, but its interpretation as normal or local can be chosen to suit the circumstances at hand. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

It can be shown (Chapter 14) that the 3N — 6 internal degrees of freedom of motion of a nonlinear molecule correspond to 37V — 6 independent normal modes of vibration. In each normal mode of vibration all the atoms in the molecule vibrate with the same frequency and all atoms pass through their equilibrium positions simultaneously. The relative vibrational amplitudes of the individual atoms may be different in magnitude and direction but the center of gravity does not move and the molecule does not rotate. If the forces holding the molecule together are linear functions of the displacement of the nuclei from their equilibrium configurations, then the molecular vibrations will be harmonic. In this case each cartesian coordinate of each atom plotted as a function of time will be a sine or cosine wave when the molecule performs one normal mode of vibration (see Fig. 1.1). 

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