ELECTRONICALLY REPRINTED FROM FEBRUARY 2014 ® Molecular Spectroscopy Workbench Practical Group Theory and Raman Spectroscopy, Part I: Normal Vibrational Modes 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 vibrational spectroscopy. David Tuschel n part I of this two-part series we present salient polarization selection rules in both data acquisition and and beneficial aspects of group theory applied to interpretation of the spectra. Ivibrational spectroscopy in general and Raman spec- There are so many good instructional and reference troscopy in particular. We highlight those aspects of materials in books (1–11) and articles (12–15) on group molecular symmetry and group theory that will allow theory that there did not seem to be any good reason readers to beneficially apply group theory and polariza- to attempt to duplicate that work in this installment. tion selection rules in both data acquisition and inter- Rather, I would like to summarize and present the most pretation of Raman spectra. Small-molecule examples salient and beneficial aspects of group theory when it are presented that show the correlation between depic- is applied to vibrational spectroscopy in general and tions of normal vibrational modes and the mathemati- Raman spectroscopy in particular. cal descriptions of group theory. The character table is at the heart of group theory and contains a great deal of information to assist vibra- Why Would I Want to Use Group Theory? tional spectroscopists. Thus, let’s examine and describe Many of us learned group theory in undergraduate or in detail one of the most simple character tables — the graduate school. For some, the last time that they used C2v point group, which is shown in Table I. The top row or understood group theory was in preparation for a consists of the type and number of symmetry opera- final exam. Furthermore, the application of group the- tions that form a symmetry class. The first column lists ory to anything other than the small molecules covered the symmetry species (represented by their Mulliken in textbooks can seem like an ordeal in tedious math- symbols) that comprise the C2v point group. The sym- ematics and bookkeeping. But, it doesn’t have to be that metry species’ irreducible representations of characters way. So, let’s brush up on those aspects of group theory appear in the rows immediately to the right of the Mul- that apply to vibrational spectroscopy, and you’ll soon liken shorthand symbols. The individual characters find that there is a lot more information to be gained indicate the result of the symmetry operation at the top through the application of group theory and Raman of the column on the molecular basis for that symmetry vibrational modes of a molecule in Table I: Character table for the C2v point group Raman this C2v point group will be IR ac- C E C σ (xz) σ ‘(yz) IR Activity 2V 2 V V Activity tive. Likewise, all four symmetry 2 2 2 species may be Raman active. The A1 1 1 1 1 z x , y , z last column indicates the axes along A 1 1 -1 -1 R xy 2 z which a change in polarizability will B1 1 -1 1 -1 x, Ry xz occur with molecular vibration and B2 1 -1 -1 1 y, Rx yz thereby allow that vibration to be Raman active (that is, will scatter Table II: The Mulliken symbols used to describe the symmetry species of point groups radiation at the frequency of the vi- including their meaning with respect to molecular symmetry brational modulation of the polariz- Mulliken Symbols of ability). One final point here is that Symmetry Species (Column Meaning any Raman bands belonging to the 1 in Character Table) totally symmetric A1 (or Ag) sym- A Symmetric with respect to principal axis of symmetry metry species will be polarized; that B Antisymmetric with respect to principal axis of symmetry is, allowed and observed with the Doubly degenerate, two-dimensional irreducible Raman analyzer in a configuration E representation parallel to the incident polarization, Triply degenerate, three-dimensional irreducible but absent or weak when the ana- T representation lyzer is configured perpendicular to g Symmetric with respect to a center of symmetry the incident polarization. We will u Antisymmetric with respect to a center of symmetry have more to say on the practical application of the Raman polariza- Symmetric with respect to a C2 axis that is perpendicular to the principal axis. Where there is no such axis the tion selection rules in the second 1 (subscript) subscript indicates that reflection in a σv plane of installment of this two-part series. symmetry is symmetric. Antisymmetric with respect to a C2 axis that is Symbols in Group perpendicular to the principal axis. Where there is no 2 (subscript) such axis the subscript indicates that reflection in a Theory and Spectroscopy σv plane of symmetry is antisymmetric. The literature of atomic and mo- Symmetric with respect to reflection in a lecular spectroscopy is filled with ‘ (prime) horizontal plane of symmetry symbols that can sometimes seem to Antisymmetric with respect to reflection in a “ (double prime) be an unfathomable, cryptic code. horizontal plane of symmetry In reality, the symbolism of spec- troscopy is a logical and systematic species. In vibrational spectroscopy, vibrational motions with the sym- shorthand way of communicating a each normal mode of vibration con- metry properties described in the great deal of information about the sists of stretches, bends, and other character table are allowed. interaction of light with matter and motions that form a basis for an ir- And speaking of being allowed, the subsequent response by the ma- reducible representation in the char- it is important to note that not all terial. In vibrational spectroscopy, acter table of the point group with vibrational modes of a molecule each normal mode of vibration must the same symmetry as that of the are spectroscopically active. The derive from a basis (for example, molecule. The individual characters complete set of normal vibrational atomic positions, bond stretches, in the table indicate the effect of the modes of a molecule will belong to and bond angles) that conforms to symmetry operation in the top row one of the following three catego- an irreducible representation in the on the symmetry species in the first ries: Raman active, infrared active, character table of that point group column. Each normal vibrational or silent. Here is where the last with the same symmetry as that of mode of the molecule will conform two columns of the character table the molecule. When an author pro- to the irreducible representation of a become particularly helpful to the vides an accepted or newly assigned symmetry species in the point group spectroscopist. The next to last col- Mulliken symbol to a spectroscopic of the molecule. Consequently, the umn indicates the axes along which band, they are communicating in- effect of the symmetry operations a change in the dipole moment will formation to you about the particu- on the vibrational mode must match occur with molecular vibration and lar vibrational mode from which the the character value of that irreduc- thereby allow that vibration to be band has its origin and perhaps even ible representation for the symmetry infrared (IR) active (that is, will ab- the symmetry of the molecule. species to be a valid or correct con- sorb IR radiation at the frequency of The first column of the character struction of the vibrational mode. the changing dipole moment). Only table lists the Mulliken symbols What that means is that only those the A1, B1, or B2 species of normal frequently used in the designation Table III: The vibrational mode symbols used to describe the motion and symmetry of a of spectroscopic band symmetry normal vibrational mode associated with a specific spectroscopic band species. The last two columns for the band’s symmetry species pertain Symbol Description of Vibrational Mode to IR and Raman activity of that σ Stretching species, respectively. A list of Mul- σ Deformation (bending) liken symbols and the meaning of σw Wagging each is given in Table II. Here are a few examples that you may have σ Rocking r encountered in the past. A Raman σt Twisting band designated as A1 can be either π Out of plane a totally symmetric stretch or bend as Antisymmetric with respect to the principal axis of symmetry in a molecule without a s Symmetric center of symmetry. A band desig- d Degenerate nated Ag can also be either a totally symmetric stretch or bend, and here the subscript g informs us that the molecule is centrosymmetric. We will have more to say about centro- symmetric molecules and the rule of mutual exclusion later. A B2 desig- nation indicates that the stretching or bending mode is antisymmetric with respect to the principal axis of symmetry and antisymmetric with respect to either a vertical reflection ν1 (A1) νs ν2 (A1) δs plane or a C2 axis perpendicular to the principal axis of symmetry. In addition to the Mulliken symbols of character tables used to describe the symmetries of the vibrational modes, spectroscopists often use symbols to describe the type of motion (stretching, bending, ν (B ) ν 3 2 as twisting, and so on) associated with a particular band in the vibrational Figure 1: Normal vibrational modes of H2O.