Group theory is an important component for understanding the fundamentals of vibrational spectroscopy. The molecular or solid
state symmetry of a material in conjunction with group theory form the basis of the selection rules for infrared absorption
and Raman scattering. Here we investigate, in a two-part series, the application of group theory for practical use in laboratory
In part I of this two-part series we present salient and beneficial aspects of group theory applied to vibrational spectroscopy
in general and Raman spectroscopy in particular. We highlight those aspects of molecular symmetry and group theory that will
allow readers to beneficially apply group theory and polarization selection rules in both data acquisition and interpretation
of Raman spectra. Small-molecule examples are presented that show the correlation between depictions of normal vibrational
modes and the mathematical descriptions of group theory.
Why Would I Want to Use Group Theory?
Many of us learned group theory in undergraduate or graduate school. For some, the last time that they used or understood
group theory was in preparation for a final exam. Furthermore, the application of group theory to anything other than the
small molecules covered in textbooks can seem like an ordeal in tedious mathematics and bookkeeping. But, it doesn't have
to be that way. So, let's brush up on those aspects of group theory that apply to vibrational spectroscopy, and you'll soon
find that there is a lot more information to be gained through the application of group theory and Raman polarization selection
rules in both data acquisition and interpretation of the spectra.
There are so many good instructional and reference materials in books (1–11) and articles (12–15) on group theory that there
did not seem to be any good reason to attempt to duplicate that work in this installment. Rather, I would like to summarize
and present the most salient and beneficial aspects of group theory when it is applied to vibrational spectroscopy in general
and Raman spectroscopy in particular.
The character table is at the heart of group theory and contains a great deal of information to assist vibrational spectroscopists.
Thus, let's examine and describe in detail one of the most simple character tables — the C
2v point group, which is shown in Table I. The top row consists of the type and number of symmetry operations that form a symmetry
class. The first column lists the symmetry species (represented by their Mulliken symbols) that comprise the C
2v point group. The symmetry species' irreducible representations of characters appear in the rows immediately to the right
of the Mulliken shorthand symbols. The individual characters indicate the result of the symmetry operation at the top of the
column on the molecular basis for that symmetry species. In vibrational spectroscopy, each normal mode of vibration consists
of stretches, bends, and other motions that form a basis for an irreducible representation in the character table of the point
group with the same symmetry as that of the molecule. The individual characters in the table indicate the effect of the symmetry
operation in the top row on the symmetry species in the first column. Each normal vibrational mode of the molecule will conform
to the irreducible representation of a symmetry species in the point group of the molecule. Consequently, the effect of the
symmetry operations on the vibrational mode must match the character value of that irreducible representation for the symmetry
species to be a valid or correct construction of the vibrational mode. What that means is that only those vibrational motions
with the symmetry properties described in the character table are allowed.
Table I: Character table for the C2v point group
And speaking of being allowed, it is important to note that not all vibrational modes of a molecule are spectroscopically
active. The complete set of normal vibrational modes of a molecule will belong to one of the following three categories: Raman
active, infrared active, or silent. Here is where the last two columns of the character table become particularly helpful
to the spectroscopist. The next to last column indicates the axes along which a change in the dipole moment will occur with
molecular vibration and thereby allow that vibration to be infrared (IR) active (that is, will absorb IR radiation at the
frequency of the changing dipole moment). Only the A
1, or B
2 species of normal vibrational modes of a molecule in this C
2v point group will be IR active. Likewise, all four symmetry species may be Raman active. The last column indicates the axes
along which a change in polarizability will occur with molecular vibration and thereby allow that vibration to be Raman active
(that is, will scatter radiation at the frequency of the vibrational modulation of the polarizability). One final point here
is that any Raman bands belonging to the totally symmetric A
1 (or A
g) symmetry species will be polarized; that is, allowed and observed with the Raman analyzer in a configuration parallel to
the incident polarization, but absent or weak when the analyzer is configured perpendicular to the incident polarization.
We will have more to say on the practical application of the Raman polarization selection rules in the second installment
of this two-part series.