Energy levels, rotational species

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

Whereas the gas lasers described use energy levels characteristic of individual atoms or ions, laser operation can also employ molecular energy levels. Molecular levels may correspond to vibrations and rotations, in contrast to the electronic energy levels of atomic and ionic species. The energies associated with vibrations and rotations tend to be lower than those of electronic transitions thus the output wavelengths of the molecular lasers tend to He farther into the infrared. 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

The eigenfunctions of J2, Ja (or Jc) and Jz clearly play important roles in polyatomic molecule rotational motion they are the eigenstates for spherical-top and symmetric-top species, and they can be used as a basis in terms of which to expand the eigenstates of asymmetric-top molecules whose energy levels do not admit an analytical solution. These eigenfunctions IJ,M,K> are given in terms of the set of so-called "rotation matrices" which are denoted Dj m,K ... 

Microwave spectra. Rotational energy levels for pyridine using isotopic species. Definition of bond lengths and angles, dipole moment... 

Energy barriers for internal rotation have been derived, especially during the 1950s, by analyzing (68M12 68M13) microwave spectra of molecules. The method works with molecules with a permanent dipole moment and in the gas phase. Limitations are dictated by the molecular size. The barriers are obtained from rotational energy levels of the molecule as a whole, perturbed by the internal rotor. When different conformers are present in the sample and their interconversion is slower than microwave absorption (barriers smaller than 20 kJ mol can be measured), the spectrum is just a superposition of the lines of the separate species which can be qualitatively and quantitatively determined. 

There is another optical method which studies these energy modifications and produces a spectrum that contains almost the same information as that obtained in the mid IR Raman. In this technique, a solution of the sample in a solvent such as water is irradiated by intense, monochromatic laser light in the visible region. The composition of the beam diffused by species present in the sample is analysed at 90c to the incident beam. In this process, bands called Stokes fines are observed beside the incident beam, at greater wavelengths. If the differences between these bands and the wavelength of the incident beam are expressed as wavenumbers, the values obtained correspond to the difference in rotational and vibrational energy levels obtained by absorption spectroscopy (Fig. 10.24). 

Although we are mainly interested in adsorbed molecules, spectra often contain contributions from gas-phase species, and therefore some knowledge of gas-phase spectra is essential. Molecules in the gas phase have rotational freedom, and as a consequence the vibrational transitions are accompanied by rotational transitions. For a rigid rotor that vibrates as a harmonic oscillator, the expression for the available energy levels is ... 

Figure 9.34. Laser magnetic resonance spectrum of CH in its a 4 state recorded in parallel polarisation (AMj = 0) with the 166.6 /un laser line of CH2F2. The rotational transition is N = 2 <—, and the quintet fine structure may be understood by reference to the energy level diagram in figure 9.33. The lines marked with an asterisk arise from an impurity species the doublet splittings of the CH lines are due to proton hyperfine interaction [69].

Furthermore, the NH3 molecules consist of two nuclear spin modifications of total spin 3/2 (parallel) and 1/2 (antiparallel). The Fermi statistics of the three hydrogen nuclei divide the NH3 rotational energy levels into an ortho-and para-species, respectively, depending on whether AT is a multiple of 3 or not. Transitions between the two modifications are strongly forbidden. As a further consequence, for K = 0 only alternating levels occur. 

Fig. 19. Rotational energy levels of HjO divided into para- and ortho-species and sorted according to Kc. The strongest dipole transitions are indicated by thin lines. Heavy arrows indicate some microwave and millimeter wave transitions, together with the transition frequency in MHz. Double arrows indicate transitions of potential astrophysical interest which may appear in maser emission. It may be noted that these four transitions are the only way out of the series of lines with KC = J...

The infrared and especially microwave spectra of methylamine and its deuterated species have been studied in considerable detail [see paper for further references]. The potential barriers to internal rotation and inversion are both relatively high [Table 6 internal rotation barrier is 684 cm in the ground state of CH3NH2] but the splittings of the energy levels are measurable. 

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