Symmetry of Molecular Vibrations

Vibrational movement in polyatomic molecules is much more complicated than that of the diatomics. That is because much greater number of possible degrees of freedom of a polyatomic molecule. For instance, if there are $N$ nuclei we need $3N$ coordinates to describe their motion. However, if we want to study the vibrational motion of a molecule, we are not interested in the translational motion of the system as a whole, which can be described completely by tree coordinates of the molecular center of mass, $X_c, Y_c, Z_c$. Therefore, $3N-3$ coordinates are sufficient for fixing the relative positions of all $N$ nuclei with respect to the molecular center of mass.

The motion relative the center of mass includes the rotation of the molecule. The rotation alone can also be described by 3 coordinates, which are usually the two polar angles which fix a certain direction in the molecule and the angle of rotation about that direction. Thus, $3N-6$ coordinates are left for describing the relative motion of the nuclei with fixed orientation of the system as a whole, in other words, we have $3N-6$ vibrational degrees of freedom. However, for linear molecules two coordinates, for instance, the two angles of the internuclear axis, are sufficient for describing the rotation and therefore for linear molecules we have $3N-5$ vibrational degrees of freedom.

When the molecule is in its equilibrium configuration, the coordinates of nucleus $i$ in the molecular $x, y, z$ coordinate frame are written $x_i^e, y_i^e, z_i^e$ and, at a displacement configuration, the Cartesian vibrational displacement coordinates are given by:

$$\Delta x_i = (x_i - x_i^e)\:\:\:\:\:\:\:\Delta y_i = (y_i - y_i^e)\:\:\:\:\:\:\:\Delta z_i = (z_i - z_i^e)$$ (1)

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