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A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 10 13 Hz to approximately 10 14 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm −1 and wavelengths of approximately 30 to 3 μm.
In CO 2, the bending vibration ν 2 (667 cm −1) has symmetry Π u.The first excited state of ν 2 is denoted 01 1 0 (no excitation in the ν 1 mode (symmetric stretch), one quantum of excitation in the ν 2 bending mode with angular momentum about the molecular axis equal to ±1, no excitation in the ν 3 mode (asymmetric stretch)) and clearly transforms according to the irreducible ...
Each normal mode of molecular vibration has a symmetry which forms a basis for one irreducible representation of the molecular symmetry group. [19] For example, the water molecule has three normal modes of vibration: symmetric stretch in which the two O-H bond lengths vary in phase with each other, asymmetric stretch in which they vary out of ...
In some systems, such as silicon and germanium, the decay of the anti-symmetric stretch mode of interstitial oxygen involves the symmetric stretch mode with a strong isotope dependence. For example, it was shown that for a natural silicon sample, the lifetime of the anti-symmetric vibration is 11.4 ps.
The symmetry of a vibrational mode is deduced from the depolarization ratio ρ, which is the ratio of the Raman scattering with polarization orthogonal to the incident laser and the Raman scattering with the same polarization as the incident laser: = Here is the intensity of Raman scattering when the analyzer is rotated 90 degrees with respect ...
This mode is called symmetric. The general solution is a superposition of the normal modes where c 1 , c 2 , φ 1 , and φ 2 are determined by the initial conditions of the problem. The process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics .
All 3N degrees of freedom have symmetry relationships consistent with the irreducible representations of the molecule's point group. [1] A linear molecule is characterized as possessing a bond angle of 180° with either a C ∞v or D ∞h symmetry point group. Each point group has a character table that represents all of the possible symmetry ...
The asymmetric stretching vibration, of B 2 symmetry in the point group C 2v is a normal vibration. The H-O-H bending mode origin is at 1595 cm −1 (ν 2 , 6.269 μm). Both symmetric stretching and bending vibrations have A 1 symmetry, but the frequency difference between them is so large that mixing is effectively zero.