The process is as follows. Apply each of the symmetry
operations of the point group (, , 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
numbers, where is the number of distinct symmetry
operations in the point group (8 for ).

What is this mysterious character contribution? Technically speaking, it is the trace of the matrix representation in 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.

Using these rules, we can obtain an 8-member array of
integers usually denoted , 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 modes, we take the dot product of with
the row of the character table for , and divide by the number
of operations in the group (8 for ). So,
= (12 + 4)/8 = 2. In a similar manner, we
can determine the contributions from the other irreps, to obtain
a decomposition of as 2 + 2 +
+ + + + 2 + 2 .

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 , , and on the right-hand side of the table. Here, this gives , , and . Likewise, rotations are denoted in the table by , which correspond to , , and . So, subtracting these out from , we find that the vibrations are described by: 2 , , , , and . There are a total of six vibrations, which is correct according to the 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