## Assigning Vibrational Frequencies

Calculation of the vibrational frequency spectrum of a nonlinear molecule containing N centers leads to 3N-6 true vibrational normal modes (linear molecules: 3N-5). As these are usually not localized motions of a small part of the molecule, assignment of the individual modes can, particularly in larger systems, be somewhat difficult. Several strategies can be used to facilitate the assignment. These strategies will be illustrated using two small model systems: acetylene (C_{2}H_{2}) and methanol (CH_{3}OH). Optimization and frequency calculations have been done in both cases at the HF/6-31G(d) level of theory.

### 1) Assignment using symmetry properties

Acetylene is a highly symmetric molecule of D_{(infinite)h} symmetry. The overall seven (3N-5) vibrational frequencies calculated for acetylene are therefore labeled according to four of the irreducible representations of this point group. The groups of type "PI" designate pi-type, degenerate irreps while the label "SG" designates irreps of sigma-type. The latter are not degenerate. The modifiers "G" and "U" designate irreps that are either symmetric (G) or antisymmetric (U) with respect to the center of inversion. Due to its center of inversion acetylene has no dipole moment.

At this point it is helpful to remember that only normal modes accompanied by a change in dipole moment will be IR active. Also, only those modes involving a change in polarizability will be Raman active. For the current case we note that all modes of G-type (symmetric with respect to the center of inversion) do NOT lead to a change in dipole moment and consequently describe IR-inactive vibrational frequencies. This characteristic is quite helpful in assigning the last two vibrational frequencies located in the region of C-H stretching vibrations. The higher of these at 3719.1787 cm^{-1} is of SGG type and therefore must correspond to the **symmetric** combination of the two C-H stretching motions. The lower value at 3606.9174 cm^{-1} is of SGU type and must consequently correspond to the **antisymmetric** combination of the two C-H stretching motions.

In a similar manner the symmetry properties can be used to assign the central C-C stretching mode at 2246.9882 cm^{-1} and the out-of-plane H-C-C bending modes at lower wavenumbers.

Methanol is a much less symmetric system whose most stable conformation is of C_{s} symmetry. This property can be represented in the following Z-Matrix:

With this definition, all atoms but H5 and H6 are located in the principal symmetry plane of the system. The computed vibrational normal modes are given in the output file in a format that includes, among others, 1) the irreducible representation, 2) the vibrational frequency (in cm^{-1}), 3) the force constant of the normal mode, 4) the IR intensity, and 5) the X-, Y-, and Z-components of each of the atoms in the eigenvector. The entries for the highest three vibrational frequencies are:

In the last of these three entries it can be seen that the eigenvector is described by the movement of essentially only one atom (hydrogen atom H4), moving in negative direction on the X- and in positive direction on the Y-coordinate axes. These components are given with respect to the molecular orientation listed as ** Standard orientation **in the upper part of the output file. In the current example, the standard orientation assumes the principal plane of the molecule to coincide with the XY-plane of the cartesian coordinate system. In combination with the wavenumber of 4118 cm^{-1}, this mode can easily be assigned as the O-H stretching vibration. The experimentally measured (anharmonic value) for this vibration is 3681 cm^{-1}. For the other two vibrations at 3231 and 3305 cm^{-1}, respectively, a straight forward interpretation is much more difficult due to the more complex composition of the eigenvectors. A third vibration with similar composition (contributions by hydrogens H5 and H6) is located close by at 3185 cm^{-1} and it may be speculated that all three vibrations describe some sort of C-H stretching vibration. In this situation, the Mulliken representations of these three vibrations can help to differentiate the modes in terms of their symmetry properties. The mode at 3231 cm^{-1} is of A'' symmetry describing vibrations which are antisymmetric with respect to the principal mirror plane, while the modes at 3185 and 3305 cm^{-1} are of A' symmetry describing vibrations which are symmetric with respect to the principal mirror plane. These two representations (A' and A'') are the only two irreducible representations of the C_{s} point group.

### 2) Assignment using animation with MOLDEN

The normal modes described in the output file can be animated with **MOLDEN**. After loading the output file and activating the **Norm. Mode** button in the upper right corner of the **MOLDEN** control panel, a window appears containing a list of all normal modes. Selection of one of the modes with the mouse pointer starts the animation of the corresponding normal mode, the displacements used in the animation being controlled by the **scale facto**r field in the lower left corner of the window. The animation can be terminated through de-selection of the **Norm. Mode** button.

### 3) Assignment using animation with GaussView

The normal modes described in the output file can be animated with** GaussView**. After loading the output file ("Open . . " option in the "File" menu) visualization of the normal modes can be activated selecting the "Vibrations . . " option in the "Results" menu. Similar to **MOLDEN** a window appears containing a list of all normal modes. Selection of one of the modes with the mouse pointer and activation of the "Start" button initiates animation of the vibrational motion. Details of the animation can be influenced with the "Frames/Cycle" and "Displacement" controls. An overall vibrational frequency spectrum can be generated with the "Spectrum . . . " button.

Using these visual aids it is also possible to identify the lowest vibrational mode at 348 cm^{-1} as the torsional vibration of the CH_{3} group (A'' symmetry). "Vibrations" of 300 cm^{-1} and below are frequently hindered rotations which should, ideally, not be treated as true vibrations.