Quantum numbers nonlinear molecules

Because spherical symmetry in the potential is lost, there is no good quantum number in molecules equivalent to the atomic 1. In nonlinear molecules, A also ceases to be a good quantum number. 

The rotation-vibration interaction of Section 4.32 produces different effects in nonlinear molecules than those discussed in the previous section. In nonlinear molecules the quantum numbers are vavhvcKJM >. The connection between the group quantum numbers Ico , co2> xi > 2 -A 3/ > and the usual quantum numbers is given by Eq. (4.85). The different effect can be traced to the different nature of the rotational spectrum. In lowest order, the spectrum of a bent molecule is given by Eq. (4.107) and Figure 4.21. The rotation-vibration interaction introduces terms with selection rules... 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

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. 

For the experienced practitioner of atomic physics there appears to be an enigma right at this point. What does nonlinear chaos theory have to do with linear quantum mechanics, so successful in the classification of atomic states and the description of atomic dynamics The answer, interestingly, is the enormous advances in atomic physics itself. Modern day experiments are able to control essentially isolated atoms and molecules to unprecedented precision at very high quantum numbers. Key elements here are the development of atomic beam techniques and the revolutionary effect of lasers. Given the high quantum numbers, Bohr s correspondence principle tells us that atoms are best understood on the basis of classical mechanics. The classical counterpart of most atoms and molecules, however, is chaotic. Hence the importance of understanding chaos in atomic physics. 

We have found it convenient to calculate Qj(e) classically and to express the coordinates contributing to Q,(e) in terms of action-angle variables. (Approximate quantum corrections can be made.11) The actions are the classical counterparts of the quantum numbers juj2, k2, k2, k, /, and J previously mentioned, and we shall simply denote the actions by the same symbols to simplify the notation. When the ith fragment is an atom the numbers jh k and k are absent, while if one of the fragments is a linear molecule, its k, is absent. For two nonlinear fragments we have, in units of ft = l,11 for a... 

Many nonlinear molecules can be treated as symmetric top rotors in which two of the moments of inertia are equal. The moment of inertia about the synunetiy axis is 7, while the two other moments of inertia are 7 = ly A symmetric top can be visualized as a rotating cylinder. For a given J, the cylinder can rotate in a total of 27 -I- 1 orientations, each with a different K quantum number which determines its projection along the symmetry axis. Figure 7.9 shows the case of prolate and oblate tops rotating with K J and K = 0. 

In the harmonic-oscillator approximation, the quantum-mechanical energy levels of a polyatomic molecule turn out to be vib 2, (v, + )hv,, where the v s are the frequencies of the normal modes of vibration of the molecule and v, is the vibrational quantum number of the ith normal mode. Each v, takes on the values 0,1,2,... independently of the values of the other vibrational quantum numbers. A linear molecule with n atoms has 3n — 5 normal modes a nonlinear molecule has 3n — 6 normal modes. (See Levine, Molecular Spectroscopy, Chapter 6 for details.)... 

The only atom-nonlinear molecule system whose excited states have been studied in any detail is Ar-H20. This system is actually quite weakly anisotropic the anisotropy of the potential splits and shifts the H2O free-rotor levels, but the free-rotor quantum numbers are... 

This mechanism of nonlinear resonances acting in sequence has been used to account for IVR in a variety of molecules, including benzene and substituted benzenes (51), alkanes (52), peroxides (53), and other molecules. In addition, the quantum mechanical correspondence to sequential nonlinear resonance has been developed (54-56). Most recently, very high level quantum dynamical calculations of IVR processes have been presented (57), involving extremely large numbers of quantum states, and these calculations have reinforced the utility of the basic classical picture presented in Fig. 2. 

If another X group is added onto the aniline molecule, the An—X2 dissociation rate is more likely to be determined by the statistical dissociation step because with each additional nonlinear monomer the number of van der Waals modes increases by six. Because the van der Waals modes are extremely anharmonic and coupled to each other, a proper RRKM calculation should use anharmonic densities and sums. However, these are not yet generally available for the systems of interest. In all cases it is best to use the quantum density of states (i.e., RRKM) and not the classical approximation of it (RRK). With a binding energy of say 480 cm and six oscillators, the average energy per van der Waals mode is 60 cm. Since these frequencies typically vary between 20 and about 400 cm, it is evident that the average number of quanta excited per mode is only about 1 or 2, which does not correspond to the classical limit. 

A significant increase in the fraction of polyene fragments with a large number of conjugated bonds was observed when pulsed light was used [40]. A photochemical autocatalytic reaction in a condensed medium can have a quantum yield which is nonlinear with respect to light intensity, as a result of the nonequilibrium state of the active molecules situated in the same diffusional cage as the catalyst molecules. The lifetime of the nonequilibrium excitation depends on catalyst diffusion in the medium. 

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