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Molecular Workbench Practical Group Theory and , 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 . 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 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- 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 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 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 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 ν Stretching species, respectively. A list of Mul- δ Deformation (bending) liken symbols and the meaning of

ρw Wagging each is given in Table II. Here are a few examples that you may have ρ Rocking r encountered in the past. A Raman ρt 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. spectrum. A list of vibrational mode symbols and the description of each is given in Table III. These symbols supplement those of the character table symmetry species and the mo- tions they describe are often useful in understanding changes to band position and width because of ef- fects of the molecular environment such as solvation, H bonding, and molecular aggregation on the mo- ν3 (B2) νas ν3 (B1) νas lecular vibrations.

Normal Vibrational Z Z Y Modes of Y Let’s continue working with the C2v X point group and analyze the nor- X mal vibrational modes of the water molecule. It is not the purpose of Figure 2: Different coordinate systems for the molecule lead to different species assignments for this installment to take the reader the antisymmetric stretch of H2O. Both coordinate systems obey the right-hand rule. step-by-step through the process of determining all the normal vi- Table IV: Character table for the C3v point group. brational modes of a molecule. Raman C E 2C 3σ IR Activity Excellent guidance for that activity 3V 3 v Activity can be found in the books listed in 2 2 2 A1 1 1 1 z x + y , z the references. Rather, we want to A 1 1 -1 R xy highlight those aspects of molecular 2 z 2 2 E 2 -1 0 (x,y) (R , R ) (x – y ,xy) symmetry and group theory that x y (xz, yz) will allow the reader to beneficially apply group theory and polariza- tion selection rules in both data acquisition and interpretation of the spectra. Choosing the x, y, and z axes of each in the molecule as a basis, the reducible representation for the displacement of the water molecule is

Γ3N = 3A1 + A2 + 2B1 + 3B2. [1]

The translation and rotation species from the second-to-last column of ν1 (A1) νs ν2 (A1) δs the C2v character table are

Γ(T + R) = Figure 3: A1 symmetric stretching and bending modes of NH3. (A1 + B1 + B2) + (A2 + B1 + B2), [2] with the species in the first and second parentheses corresponding to the translational and rotational degrees of freedom, respectively. Consequently, the complete set of vibrational modes is given by

Γvib = Γ3N - Γ(T + R) = 2A1 + B2. [3]

The three (two A1 and one B2) nor- mal modes of vibration for water are shown in Figure 1. The ν1 and ν bands are of A symmetry, a 2 1 ν (E) νν (E) ν symmetric stretch and symmetric 3a 3b bending mode, respectively. The ν3 band is an antisymmetric stretch of Figure 4: N-H doubly degenerate stretching modes of NH3. species B2. Now let’s confirm our normal main unchanged, thereby confirm- vibrational modes of water are IR vibrational mode symmetry assign- ing the modes as A1 species. If we and Raman active. ments. The A1 species are totally apply those same symmetry opera- Remember that group theory only symmetric. That means that if we tions to the ν3 mode we see that the predicts whether or not a vibra- apply any of the C2v symmetry op- characters for C2 and σν(xz) are -1, tional mode will be Raman or IR ac- erations to a basis of the molecule, and so the antisymmetric stretch is tive. It does not predict the strength the basis will remain unchanged. consistent with the B2 assignment of the corresponding bands in the Let’s designate the directional ar- because the characters of B2 under spectra. The strength of an indi- rows in Figure 1 as bases for the the symmetry operations C2 and vidual IR or Raman band depends molecular point group. If we apply σν(xz) are both -1. Inspection of the on the corresponding vibrational the E, C2, σν(xz), and σ′ν(yz) sym- last two columns of the C2v charac- transition dipole moment or Raman metry operations to both vibrational ter table indicates that both A1 and polarizability, respectively. There- modes the directional arrows re- B2 species, and therefore all three fore, group theory can account for F N = N F

Molecular center of symmetry

Figure 6: Molecular structure of trans-N2F2.

ν4a (E) δν4b (E) δ tional axes and reflection planes) in the application of group theory to Figure 5: H-N-H doubly degenerate bending modes of NH . 3 molecular vibrations. Having stated that, please note that the species Table V: Character table for the C2h point group subscripts exceed 2 in some point Raman C E C i σ IR Activity groups, in which case the subscript 2h 2 h Activity numbering can be arbitrary. 2 2 2 Ag 1 1 1 1 Rz x , y , z , xy

Bg 1 -1 1 -1 Rx ,Ry xz, yz Degeneracy in

Au 1 1 -1 -1 z Vibrational Modes Degeneracy in vibrational spec- Bu 1 -1 -1 1 x, y troscopy pertains to multiple vibra- tional modes of a molecule all at the the number of bands in a vibrational rule whereby the fingers originally same energy state and same sym- spectrum, but not how strong they aligned with the positive x-axis and metry. Here we use the ammonia will be. then folded into the positive y-axis molecule NH3 of point group C3v to In writing this installment, I points with the thumb to the posi- understand the role of degeneracy in uncovered a somewhat confusing tive z-axis. The difference is that spectra. The character table for the but interesting situation. Many of these coordinate systems are rotated C3v point group is shown in Table the group theory and vibrational 90° with respect to each other. Both IV. spectroscopy books list the vibra- species assignments are technically Choosing the x, y, and z axes of tional modes for water as 2A1 + correct because in each case the each atom in the molecule as a basis, B2 (2,5,6,7,10). However, using the C2 operation yields a character of the reducible representation for the exact same character table for C2v -1 and the vertical reflection plane displacement of the NH3 molecule is other book authors list the modes perpendicular to the plane of the as 2A1 + B1 (1,9,11). This really had molecule yields a character of -1. Γ3N = 3A1 + A2 + 4E. [4] me puzzled. I wondered how there Subsequently, I searched for a could be such a difference among nonarbitrary basis for the assign- The translation and rotation species fine textbooks on group theory and ment of molecular coordinates and from the second-to-last column of vibrational spectroscopy, whether found that J. Michael Hollas had the C3v character table are there was an error in one of the encountered the same situation (16). assignments, and whether anyone He states, “For a planar C2v mol- Γ(T+R) = (A1 + E) + (A2 + E), [5] else had noticed this difference. I ecule the convention is that the C2 solved the puzzle by scrupulously axis is taken to be the z axis and the with the species in the first and working through the written de- x axis is perpendicular to the plane. second parentheses corresponding scriptions by which the authors This is the convention used here for to the translational and rotational arrived at these different Mulliken H2O.” Following this convention degrees of freedom, respectively. symbols for the antisymmetric would lead to the B2 assignment Consequently, the complete set of stretch of water. shown in Figure 2. vibrational modes is given by The answer lies in the molecular The take-home message of this coordinate systems chosen by the cautionary tale is that great care and Γvib = Γ3N - Γ(T+R) = 2A1 + 2E. [6] authors. The coordinate systems consistency need to be exercised in The two A1 totally symmetric nor- leading to the different species as- the designation of molecular coor- mal modes of vibration for NH3 are signments are shown in Figure 2. dinate systems and all of the sym- shown in Figure 3. The ν1 and ν2 Each system follows the right-hand metry elements (for example, rota- bands are of A1 symmetry, a sym- metric stretch and symmetric bend- Table VI: Spectral characteristics of the trans form of N2F2 belonging to the C2h ing mode, respectively. Now let’s point group confirm our normal vibrational Vibrational Mode Symmetry Species Band Position (cm-1) Activity mode symmetry assignments. The ν (NF) A species are totally symmetric. s Ag 1010 Raman 1 ν (NN) Therefore, if we apply any of the Ag 1636 Raman δ (NNF) Ag 592 Raman C3v symmetry operations to a basis s of the molecule that basis should ρt Au 360 IR remain unchanged. Let’s designate νas (NF) Bu 989 IR the directional arrows in Figure δas (NNF) Bu 421 IR 3 as bases for the molecular point group. If we apply the E, C3, and conform to the character of the of i is either 1 for the symmetric g σν symmetry operations to both E species even though their mo- species or -1 for the antisymmetric vibrational modes, the directional tions are different. And, like the E u species. Furthermore, you can arrows remain unchanged, thereby stretching modes, these bending see from the last two columns that confirming the modes as A1 spe- modes are both IR and Raman ac- no vibrational mode of a molecule cies. tive in the xy-plane. In summary, belonging to the C2h point group The two ν3 doubly degenerate vi- both symmetry species and all six can be both IR and Raman active, brational modes are N-H stretching vibrational modes of NH3 are both thereby obeying the rule of mutual modes of species E and are shown IR and Raman active. exclusion. in Figure 4. The ν3a and ν3b stretch- Let’s continue working with the ing modes are doubly degenerate Centrosymmetric Molecules C2h point group and analyze the insofar as their excited states are of and the Rule of Mutual normal vibrational modes of trans- equal energy and their directional Exclusion N2F2, the structure that is shown in arrow bases conform to the char- Our last topic addresses the well- Figure 6. Please note that the mo- acter of the E species even though known . lecular center of symmetry need not their motions are different. The in- This rule states that for molecules be an atom in the molecule. As we terpretation of the bases and sym- with a center of symmetry all vibra- see here, the inversion symmetry el- metries of degenerate vibrational tional modes that are Raman active ement i is the midpoint between the modes is beyond the scope of this will be IR inactive and all IR-active N=N double bond. It is worth not- installment. To understand how to modes will be Raman inactive. Cen- ing that the cis form of N2F2 belongs determine the character of a de- trosymmetric molecules possess the to the C2v point group, and not only generate symmetry species, I would symmetry element i such that the are the symmetry species different highly recommend the treatment of process of inversion (the symmetry but so are the energies these same degenerate vibrational operation of i) changes all molecu- and number of spectroscopically modes of NH3 described on pages lar positions from (x,y,z) to (-x,-y,- active vibrational modes. The sym- 69–74 of the book by Hollas (3). z). Point groups of molecules with a metry species, band positions, and Inspection of the last two columns center of symmetry will contain the spectroscopic activities for trans- of the C3v character table indicates symmetry element i in the top row N2F2 are shown in Table VI (17). that the vibrational transition of the character table, and the sym- Note that both the symmetries dipole moment and Raman polar- metry species will have either a g or and energies of the Raman- and izabilities occur in the xy-plane u subscript in the Mulliken symbols IR-active vibrational modes are dif- perpendicular to the principal mo- indicating symmetric or antisym- ferent. It can be helpful to think of lecular axis; therefore, ν3a and ν3b metric, respectively. The symbols g the differences between the energies stretching modes are both IR and and u have their origin in the Ger- of these modes and their correlation Raman active. man words gerade and ungerade, to the entirely different symmetric The H-N-H doubly degenerate which here mean even and odd, and antisymmetric motions of the bending modes of NH3 are shown respectively. An example of a point molecule. The vibrational modes in Figure 5. Unlike the ν3a and ν3b group for a centrosymmetric mol- here do not share IR and Raman modes, which consist primarily of ecule is C2h, and its character table activity, and so we would expect stretching motions, the ν4a and ν4b is shown in Table V. the IR and Raman spectra of trans- modes are primarily bending mo- We see that i is one of the symme- N2F2 to be different with respect to tions in which the bond angles vary. try operations and all of the sym- the appearance of bands at specific

The ν4a and ν4b bending modes are metry species have a designation of energies. And of course, because the doubly degenerate insofar as their either g or u in their Mulliken sym- only Raman bands allowed are of Ag excited states are of equal energy bols. Note that consistent with the symmetry, the entire Raman spec- and their directional arrow bases inversion operation, the character trum consists of polarized bands; that is, they will be seen with the mathematical descriptions of group (9) A. Vincent, Molecular Symmetry and Raman analyzer oriented parallel theory. In part II of this series, we Group Theory, 2nd Ed. (John Wiley & to the incident laser polarization will see how to beneficially apply Sons, Chichester, UK, 2001). but will be absent or weak with the the Raman polarization selection (10) P.H. Walton, Beginning Group Theory analyzer in a perpendicular con- rules in the acquisition of the Raman for Chemistry (Oxford University figuration. spectra along with the application of Press, Oxford, UK, 1998). In summary, using Raman polar- group theory to the interpretation of (11) M. Diem, Introduction to Modern Vi- ization selection rules in the acqui- the Raman spectra for structure de- brational Spectroscopy (Wiley-Inter- sition of the Raman spectra along termination and characterization. science, New York, New York, 1993). with the application of the rule of (12) D.W. Ball, Spectroscopy 24(12), mutual exclusion can be a powerful References 63–66 (2009). and effective method of structure (1) J.S. Ogden, Introduction to Molecular (13) D.W. Ball, Spectroscopy 25(1), 18–23 determination and characterization. Symmetry (Oxford University Press, (2010). Having said that, let me close with a Oxford, UK, 2001). (14) D.W. Ball, Spectroscopy 25(4), 16–21 word of caution: The rule of mutual (2) R.L. Carter, Molecular Symmetry and (2010). exclusion does not always present or Group Theory (John Wiley & Sons, (15) D.W. Ball, Spectroscopy 25(9), 18–21 manifest itself in the most obvious New York, New York, 1998). (2010). ways for the easy determination of (3) J.M. Hollas, Symmetry in Molecules (16) J.M. Hollas, Modern Spectroscopy, 4th molecular structure. Weak bands or (Chapman and Hall, London, UK, Ed. (John Wiley & Sons, Chichester, those only partially resolved from 1972). UK, 2004), p. 89. an adjacent, more intense band (4) S.F.A. Kettle, Symmetry and Structure: (17) A.F. Fadini and F.-M. Schnepel, Vibra- can easily be missed or overlooked. Readable Group Theory for Chemists tional Spectroscopy: Methods and Ap- There is no substitute for attention (John Wiley & Sons, New York, New plications (Ellis Horwood, Chichester, to detail and careful analysis and York, 1995). UK, 1989), p. 98. interpretation by the spectroscopist. (5) D.C. Harris and M.D. Bertolucci, Sym- metry and Spectroscopy: An Intro- Conclusions duction to Vibrational and Electronic In part I of this two-part series we Spectroscopy (Dover, New York, David Tuschel is a have discussed and highlighted those 1989). Raman applications manager aspects of molecular symmetry and (6) B.E. Douglas and C.A. Hollingsworth, at Horiba Scientific, in Edison, group theory that are of relevance Symmetry in Bonding and Spectra: New Jersey, where he works to spectroscopists. An explanation An Introduction (Academic Press, Or- with Fran Adar. David is of the nomenclature of group theory lando, Florida, 1985). sharing authorship of this and vibrational spectroscopy has (7) M. Orchin and H.H. Jaffe, Symmetry, column with Fran. He can be been provided to better equip read- Orbitals and Spectra (S.O.S.) (Wiley- reached at: [email protected]. ers with the scientific literature. Interscience, New York, New York, Small- molecule examples have been 1971). For more information on presented on how to understand the (8) F.A. Cotton, Chemical Applications of this topic, please visit: correlation between depictions of Group Theory, 3rd Ed. (John Wiley & www.spectroscopyonline.com normal vibrational modes and the Sons, New York, New York, 1990).

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