Vibrations angular

These new wave functions are eigenfunctions of the z component of the angular momentum iij = —with eigenvalues = +2,0, —2 in units of h. Thus, Eqs. (D.l 1)-(D.13) represent states in which the vibrational angular momentum of the nuclei about the molecular axis has a definite value. When beating the vibrations as harmonic, there is no reason to prefer them to any other linear combinations that can be obtained from the original basis functions in... 

In the configuration space spanned by (411,1 2, 73), we may then define the vibrational angular momentum k through its classical components, that is,... 

Thus, I and m quantize the vibrational angular momentum and its z component. 

The case of water is particularly convenient because the required high Ka states may be detected in the solar absorption spectrum. However, it is difficult to observe the necessary high vibrational angular momentum states in molecules, which can only be probed by dispersed fluorescence or stimulated emission techniques. On the other hand, it is now possible to perform converged variational calculations on accurate potential energy surfaces, from which one could hope to verify the quantum monodromy and assess the extent to which it is disturbed by perturbations with other modes. Examples of such computed monodromy are seen for H2O in Fig. 2 and LiCN in Fig. 12. 

The functions (6.94) are the linear combinations of (6.84) that correspond to definite values of the vibrational angular momentum the absolute value of the vibrational angular momentum about the molecular axis is Ih. 

Spherical tops and nonlinear symmetric tops also have degenerate vibrational modes. For a linear molecule, there is no rotational angular momentum about the symmetry axis, only vibrational angular... 

Asymmetric tops have no degenerate vibrational modes (this will be shown in Chapter 9) hence we need not worry about vibrational angular momentum for them. 

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). 

Quantum mechanically, there is also the possibility of nuclear vibrational angular momentum for these degenerate modes. We denote the normal coordinates for the doubly degenerate vibrations by Qx and Qy. From Fig. 6.2, we have... 

The functions (6.85) each correspond to a specific pair of the normal-mode vibrational quantum numbers v2a and v2b, whereas the correct zeroth-order functions (6.87) do not. Hence the quantum numbers v2a and v2b are not particularly physically significant. The quantum numbers of physical significance are v2 = v2a -I- v2b, which (together with c, and t>3) specifies which degenerate vibrational level we are dealing with, and the vibrational angular-momentum quantum number. 

A linear molecule in a vibrational level with vibrational angular momentum quantum number / can be viewed as a symmetric top with Ih replacing K h as the absolute value of the component of nuclear angular momentum about the symmetry axis. Replacing A with / in (5.67), we have as the rotational energy of a linear molecule for a degenerate vibrational level... 

S. A. Rice Have you studied any excited ion states with degenerate vibrations to ascertain the coupling of the vibrational angular momentum to the electron angular momentum ... 

The infrared spectra of the cycloproparenes are characterized by a weak band that appears between 1660 and 1690 cm"1 due to a combination of the aromatic double bond stretch with the three-membered ring skeletal vibration. Angular fused cyclo-propa[a]naphthalene (10) has the highest value (1687 cm"1) while for linear 11 it is more usual at 1673 cm"1, and cyclopropa[6]anthracene falls between these at 1678 cm"1. The... 

Vt = 2-cojr/h. Finally, returning to equation (54), Ja and ja are the components of the total angular momentum and the vibrational angular momentum about the a axis, respectively, Ja being given by... 

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