Molecular rotations and vibrations

Vibrational spectroscopy can help us escape from this predicament due to the exquisite sensitivity of vibrational frequencies, particularly of the OH stretch, to local molecular environments. Thus, very roughly, one can think of the infrared or Raman spectrum of liquid water as reflecting the distribution of vibrational frequencies sampled by the ensemble of molecules, which reflects the distribution of local molecular environments. This picture is oversimplified, in part as a result of the phenomenon of motional narrowing The vibrational frequencies fluctuate in time (as local molecular environments rearrange), which causes the line shape to be narrower than the distribution of frequencies [3]. Thus in principle, in addition to information about liquid structure, one can obtain information about molecular dynamics from vibrational line shapes. In practice, however, it is often hard to extract this information. Recent and important advances in ultrafast vibrational spectroscopy provide much more useful methods for probing dynamic frequency fluctuations, a process often referred to as spectral diffusion. Ultrafast vibrational spectroscopy of water has also been used to probe molecular rotation and vibrational energy relaxation. The latter process, while fundamental and important, will not be discussed in this chapter, but instead will be covered in a separate review [4],... 

The Section on Molecular Rotation and Vibration provides an introduction to how vibrational and rotational energy levels and wavefunctions are expressed for diatomic, linear polyatomic, and non-linear polyatomic molecules whose electronic energies are described by a single potential energy surface. Rotations of "rigid" molecules and harmonic vibrations of uncoupled normal modes constitute the starting point of such treatments. 

The coupling functions 1 and still depend on the molecular vibrational and rotational degrees of freedom as well as the relative molecule-perturber separation, R. Since the experiments imply that the physical origin of the collision-induced intersystem crossing resides in long-range attractive interactions, we may adopt a semiclassical approximation where the quantum-mechanical variables for the relative translation is replaced by a classical trajectory, R(l), for the relative molecule-perturber motion. The internal dynamics is then influenced by the time-dependent interactions f s[ (0] and Fj-j-fR(r)], which are still functions of molecular rotational and vibrational variables. For simplicity and for illustrative purposes we consider only the pair of coupled levels S and T and a pure triplet level T, which represents the molecular state after the collision. Note T may differ in rotational and/or vibrational quantum... 

In the limit of co = 2nf oo no dynamic process in medium can foUow the field the electric polarization P = x E vanishes (i.e. dielectric susceptibility X 0) and the displacement vector D = (1 + 4jtx )E coincides with E, that is = 1 + 47tx 1. With decreasing frequency, fast electronic processes have enough time to follow the field and, at optical fi-equencies, e = (n is refraction index) shows peculiarities related to electronic absorption bands (normal and abnormal dispersion). With further decreasing frequency other processes such as molecular rotations and vibrations begin to contribute to the electric polarization and s = again increases, see Fig. 7.3. 

When studying the expansion in powers of k, traditionally one first separates off the center of mass motion and expresses the remaining Hamiltonian for relative motion of nuclei and electrons in suitable nuclear coordinates describing molecular rotations and vibrations. The resulting expressions are rather complex and depend on the particular choice of coordinates made, see, for instance. Refs. 43-45. On the other hand, one also arrives at the correct analysis in powers of k without first separating off the center of mass motion. This has been demonstrated by many authors, see, for instance. Refs. 3, 29 and 46-48. 

Therefore, in contrast to the generally sharper signals of IR spectra (Fig. 1-3), the UV-VIS spectra of most liquids, solutions, and solids show broad bands with indistinct shoulders (Fig. 1-4), and their fine structure is suppressed. This is due to inter-molecular interactions (including interactions with the solvent molecules), which hinder molecular rotation and vibration. In the gas phase the different absorption peaks (Fig. 1-5) can sometimes be resolved. 

TABLE 9.1 Ground state molecular rotational and vibrational constants for selected diatomic molecules. Values for the vibrational constants are based on varying numbers of anharmonic terms. In this table Dq is the v = 0 rotational distortion constant. Missing values indicate the constant was not measured in the corresponding experiment... 

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