Diatomic Molecules Rotations

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Simultaneously with the oscillations with respect to the internuclear axis, a diatomic molecule rotates as a whole with a frequency ft (not to be confused with the quantum number of the projection of the total momentum, as in Section 1.2) around an axis which is perpendicular to the direction of the internuclear axis, exhibiting a total angular momentum J see Fig. 1.3. Hence in the laboratory coordinate system, with respect to which the light wave possesses a certain polarization, electrons participate simultaneously in two types of motion dipole oscillations with respect to the internuclear axis, and rotation of the molecule as a whole. 

As mentioned in section B. 1., only one rotational constant, (B + C), per molecule is available from the observed a-type spectrum (AK = 0, AJ = 1), and the results are given in Table 1. (B + C)/2 is essentially an effective diatomic molecule rotational constant, and therefore depends most strongly on the distance between the monomers of the complex. The rotational constants in Table 1 were best fitted with a model in which O. .. H—O distances were 2.67 A and O. .. H—N distances were 2.71 A. These O. .. H—O distances are somewhat shorter than the 2.73 A distance in formic acid dimer and the 2.76A distances in acetic acid dimer68. However, they are... 

FIGURE 20.3 A diatomic molecule rotates about its center of gravity. The center of gravity is located by the distances r-[ and rx, determined from the condition = mxr -... 

ISe. Classical Calculation of Heat Capacities.— For a diatomic molecule two types of rotation are possible, as seen above, contributing RT per mole to the energy. Since there are two atoms in the molecule, i.e., n is 2, there is only one mode of vibration, and the vibrational energy should be RT per mole. If the diatomic molecules rotate, but the atoms do not vibrate, the total energy content E will be the sum of the translational and rotational energies, i.e., RT + RT = RT, per mole hence,... 

The information contained in a diatomic molecule rotation-vibration-electronic wavefunction is enormous. But this is dwarfed by the information content of a time-evolving wavefunction that originates from a non-eigenstate pluck. A simplified, reduced-dimension representation, rather than an exact numerical description, is prerequisite to visualization and understanding. The concepts and techniques presented in this book, developed explicitly for diatomic molecule spectra and dynamics, are applicable to larger molecules. Indeed, any attempt... 

Centrifugal distortion of a symmetrical diatomic molecule. Rotation at an angular velocity co generates a centrifugal force F on each atom of mass m, leading to an extension Sr of the bond. This generates a contrary force kt>r, where k is the force constant. The moment of inertia increases from 2m r/2 f to 2m(r/2 + 8r/2f. 

The above results for the quantization of angular momentum have been derived for a single particle, but the same results hold for two masses (say, a diatomic molecule) rotating around their center of mass, when / is the moment of inertia of the system. This opens the way to the quantum mechanical treatment of rotating molecules. 

The spectra of polyatomic molecules are more complicated than those of atoms or diatomic molecules. As with diatomic molecules, rotational transitions can occur without vibrational or electronic transitions, vibrational transitions can occur without electronic transitions but are generally accompanied by rotational transitions, and electronic transitions are accompanied by both vibrational and rotational transitions. 

Consider a diatomic molecule, rotated around an axis passing through the CM (Figure 1.17). The problem is to express the Ml of this molecule and the corresponding kinetic energy of rotation through its parameters, which are supposed to be known (for instance, from the reference hterature). hi the figure, the masses of two atoms of the molecule are marked by letters m and m2, and the letter d denotes an interatomic distance... 

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