How Many Vibrational Modes Belong To Each Irrep?

From the sketch of the molecular geometry and the character table, we can fairly easily determine how many vibrational modes there will be of each symmetry type (i.e., each irreducible representation).

The process is as follows. Apply each of the symmetry operations of the point group ($E$, $C_2(z)$, etc.) to the molecule, and determine how many atoms are not moved by the operation. Then, multiply this number by the so-called character contribution of that symmetry operation. This will yeild a series of $h$ numbers, where $h$ is the number of distinct symmetry operations in the point group (8 for $D_{2h}$).

What is this mysterious character contribution? Technically speaking, it is the trace of the matrix representation in $xyz$ Cartesian coordinates of that operation. However, it is usually easier just to memorize the character contributions of the most commonly used symmetry operations. A partial table of character contributions is given in Table 3.

Table 3: Character Contributions of Some Common Symmetry Operations

$E$	3
$\sigma$	1
$C_2$	-1
$i$	-3
$C_3$	0

Using these rules, we can obtain an 8-member array of integers usually denoted $\Gamma_{red}$, a reducible array. This is done in Table 3. The next step is to decompose the reducible array into a unique linear combination of irreducible representations (irreps). This is easily accomplished using dot products. For example, to get the number of $a_g$ modes, we take the dot product of $\Gamma_{red}$ with the row of the character table for $a_g$, and divide by the number of operations in the group (8 for $D_{2h}$). So, $\Gamma_{red} \cdot \Gamma_{a_g} / h$ = (12 + 4)/8 = 2. In a similar manner, we can determine the contributions from the other irreps, to obtain a decomposition of $\Gamma_{red}$ as 2 $a_g$ + 2 $b_{1g}$ + $b_{2g}$ + $b_{3g}$ + $a_u$ + $b_{1u}$ + 2 $b_{2u}$ + 2 $b_{3u}$.

Table 4: Symmetry Decomposition of Atomic Motions

	$E$	$C_2(z)$	$C_2(y)$	$C_2(x)$	$i$	$\sigma(xy)$	$\sigma(xz)$	$\sigma(yz)$
Stationary Atoms	4	0	0	0	0	4	0	0
Char contrib	3	-1	-1	-1	-3	1	1	1
$\Gamma_{red}$	12	0	0	0	0	4	0	0

Next, we need to subtract out the translations and rotations. The irreps of the translations can be found in most character tables by looking for which row contains $x$, $y$, and $z$ on the right-hand side of the table. Here, this gives $b_{1u}$, $b_{2u}$, and $b_{3u}$. Likewise, rotations are denoted in the table by $R_x,R_y,R_z$, which correspond to $b_{1g}$, $b_{2g}$, and $b_{3g}$. So, subtracting these out from $\Gamma_{red}$, we find that the vibrations are described by: 2 $a_g$, $b_{1g}$, $a_u$, $b_{2u}$, and $b_{3u}$. There are a total of six vibrations, which is correct according to the $3N-6$ rule.

Below is the output from a Q-Chem calculation. These normal modes are sketched in Figure 2, along with the irreducible representations of each.

 **********************************************************************
 **                                                                  **
 **                       VIBRATIONAL ANALYSIS                       **
 **                       --------------------                       **
 **                                                                  **
 **        VIBRATIONAL FREQUENCIES (CM**-1) AND NORMAL MODES         **
 **                  INFRARED INTENSITIES (KM/MOL)                   **
 **                                                                  **
 **********************************************************************


 Frequency:      -119.42                 126.71                 131.78
 IR Active:          YES                    YES                    YES
 IR Intens:        0.466                  0.000                  0.000
 Raman Active:       YES                    YES                    YES
               X      Y      Z        X      Y      Z        X      Y      Z
 O         -0.500  0.000  0.000    0.500 -0.001  0.000    0.000  0.000 -0.500
 O          0.500  0.000  0.000    0.500  0.001  0.000    0.000  0.000  0.500
 O         -0.500  0.000  0.000   -0.500  0.001  0.000    0.000  0.000 -0.500
 O          0.500  0.000  0.000   -0.500 -0.001  0.000    0.000  0.000  0.500

 Frequency:       281.48                1689.70                1849.60
 IR Active:          YES                    YES                    YES
 IR Intens:        0.000                *******                  0.000
 Raman Active:       YES                    YES                    YES
               X      Y      Z        X      Y      Z        X      Y      Z
 O          0.457 -0.202  0.000    0.000  0.500  0.000    0.001  0.500  0.000
 O         -0.457 -0.202  0.000    0.000 -0.500  0.000    0.001 -0.500  0.000
 O         -0.457  0.202  0.000    0.000  0.500  0.000   -0.001 -0.500  0.000
 O          0.457  0.202  0.000    0.000 -0.500  0.000   -0.001  0.500  0.000

Figure 2: Sketches of Normal Modes of O$_4^+$