1. Review: Applications of Symmetry, cont.
II. Molecular Vibrations We will learn a few different methods of determining symmetry and IR/Raman activity of molecular vibrations. A. Determining All Vibrational Motions in a Molecule 1. Use group theory to classify symmetries of all motions of a molecule
(translations, rotations, vibrations). 2. Identify and exclude the translations and rotations, leaving only the vibrations Use the character table to determine IR/Raman activity of each vibration.
Degrees of freedom: Total: 3N (N = # of atoms) Translation: 3 (x, y, and z directions) Rotation 3* (Rx, Ry, Rz) *only 2 for linear molecules Vibration N-6 normal modes (*Linear: 3N-5)

2. Generalized Procedure for Applying Group Theory
(For Spectroscopy, MO Theory, etc.)
Find the point group of the molecule. All questions you seek to answer depend on the group to which the molecule belongs.
Determine a basis set of objects you care about. It may include: a point (x, y, z); all 3N molecular motions; specific vibrations; a set of atomic orbitals. . . Use a vector on the molecule to represent each item in the basis set (n items = n vectors) .
Find the reducible representation identifying the symmetry of all items in the basis set.
Reduce the representation to find the irreducible representations it contains (i.e., which rows in the character table are represented).
The irreducible representations provide the symmetries of the individual elements in your basis set. Use these to answer your desired question(s) [identifying IR-active modes, creating MO correlation diagrams, etc.].
This will begin to make sense once you’ve seen some examples

3. Example: Characterizing Vibrational Modes in H2Oy z y x z
C2v E C2(z) v(xz) v(yz)
Objective: Identify the symmetries of all vibrations in H2O; predict the number of IR- and Raman-active stretches and the number of bands (or peaks) observed in each spectrum.
Strategy:
Use all 3N molecular motions as the basis set; represent with vectors [(x,y,z) on each atom]
Determine a (matrix) reducible representation describing the symmetry of all 3N motions, taken together
Reduce that representation to find symmetries of individual motions
Use character table to exclude rotations and translations; then, determine activity of vibrations

4. Example: Vibrational modes in H2O, cont.y x z x y z
C2v E C2(z) v(xz) v(yz)
Generate a transformation matrix for each class in C2v, describing how it transforms the nine original vectors xO yO zO xA yA zA xB yB zB xO yO zO xA yA zA xB yB zB O
3 axes x
3 atoms
9 x 9
matrix = HA HB E
(E) = 9

5. = Consider each atom: What happens to the axes when you do a C2?
O (stays in place) x → -x; y → -y; z → z
HA → HB: xA → -xB; yA → -yB; zA → zB
HB → HA: xB → -xA; yB → -yA; zB → zA
C2: xO yO zO xA yA zA xB yB zB =
-xO
-yO
-xB
-yB
-xA
-yA
(C2) = -1
Note: HA & HB have all 0’s on the diagonal.

6. A Shortcut to Transformation Matrices
Only atoms that stay in place contribute non-zeros to  --
Find those atoms; consider x, y, z axes. Ignore atoms that move.
E: O, HA and HB stay in place
For each, x → x, y → y, z → z All 1’s on diagonal;  =
v(xz):
v(yz): 9
O stays in place; HA  HB
 = 1 + (-1) (60) = 1
For O, xx, y -y, zz x y z
O, HA, HB stay in place
For each atom, x -x, yy, zz
 = 3(-1) + 3(1) + 3(1) = 3 y x z
We’ve now developed a reducible rep. for the 3N motions of H2O:
C2v E C2 v(xz) v(yz) R

7. Reducing R
Characters of transformation matrices form a reducible representation:
Represents the symmetry of all molecular motions (together)
[Notice: 3N = 9 = (E). This tells you that 9 items are represented.]
Break down (reduce) into irreducible reps. to find symmetries of individual translations, rotations, vibrations
C2v E C2 v(xz) v(yz) R
C2v E C2
v(xz)
v(yz) R 9 -1 3 1 A1 A2 B1 B2
Consider the irreducible representations in C2v (A1, B1, etc.) to see how many times each one appears in R.

8. Reducing R, cont. R = 3 A1 + 1 A2 + 2 B1 + 3 B2
C2v E C2
v(xz)
v(yz) R 9 -1 1 3 A1 A2 B1 B2
R =
3 A1 + 1 A2 + 2 B1 + 3 B2
Check: Add up ’s under E – should get 9 (should be able to add all ’s to retrieve R)
[Arithmetic version of “block diagonalizing”]
# i = (1/h)   [(# ops)  (R)  (i)]
class
# A1 = (1/4) [(191) + (1-11) + (111) + (131)] = (1/4)(12) = 3
# A2 =
# B1 =
# B2 =
(1/4)[9 + (-1) + (-1) + (-3)] = (1/4)[4] = 1
(1/4)[ (-3)] = (1/4)[8] = 2
(1/4)[ (-1) + 3] = (1/4)[12] = 3

9. Identifying vibrations from the 3N molecular motions
We have identified symmetries of the 9 molecular motions of H2O:
R = 3 A1 + A2 + 2 B1 + 3 B2
Step 2: Use character table to find translations and rotations:
C2v E C2
v(xz)
v(yz) R 9 -1 3 1 A1 z
x2, y2, z2 A2 Rz xy B1
x, Ry xz B2
y, Rx yz
Translations:
Rotations:
The remaining irreducible reps. in R must be vibrations:
along x, y, z: A1, B1, B2
Rx, Ry, Rz: A2, B1, B2
2 A1, B2
(note: 3N-6 = 3 vibrations, as expected)

10. Discussion Question for Thurs., 2/26/15
Review your notes from Tuesday; rework the example to predict the total number of peaks in the IR spectrum of H2O and make note of your questions.
If you have learned vibrational spectroscopy in PChem II, review the selection rules for IR and Raman spectroscopy. What is required for a molecule to be IR-active? Raman-active?
You may have covered the IR portion of this in Organic, too. What must a vibration do to the molecule in order for it to be detected in IR spectroscopy?

11. Determining IR Activity
2/26/15 L12
Determining IR Activity
Selection Rule: To be IR-active, a vibration must result in . . .
H2O example:
Vibrational modes: 2 A1 + B2
How many peaks in the IR spectrum?
a change of dipole moment
Change in dipole moment results from net translational motion
IR-active modes have the same symmetry as x, y, or z axes.
3 – each vibration will have its own frequency
C2v E C2
v(xz)
v(yz) R 9 -1 3 1 A1 z
x2, y2, z2 A2 Rz xy B1
x, Ry xz B2
y, Rx yz

12. Determining Raman Activity
Selection Rule: To be Raman-active, a vibration must result in . . .
H2O example:
Which vibrational modes are Raman active?
IR and Raman are complementary methods, often used for structural determination. A couple of factors help make this possible:
If there is an inversion center (i), a particular vibration cannot be both IR and Raman active (won’t see same frequency in both spectra)
When radio-labeling is used (heavy isotopes), the measured frequency for a vibration will only change if the heavy atom moves during the vibration (used to assign frequencies to specific stretches/bends)
a change of molecular polarizability (tensor)
related to size/shape of e- cloud
Raman-active modes have the same symmetry as quadratic terms: x2, y2, z2, xy, xz, yz
all – 2 A1 and B2
3 peaks at same freq. as IR

13. Activity of a Specific Vibration (of the entire molecule)
It is also possible to determine the IR (or Raman) activity of a particular vibrational mode, if you can diagram it accurately.
Consider the symmetric stretch of water (represented by red arrows below): y x z
Use char. table to find the symmetry of the vibration:
1. “Do” one operation in each class (pictorially) and determine the character:
+1 if it is unchanged; -1 if it “changes sign”
Symm. stretch
2. Compare  to irreducible reps to find symmetry; then check for IR and/or Raman activity.
a1 – IR active
Antisymm. stretch b2
C2v E C2
v(xz)
v(yz)  A1 1 z
x2, y2, z2 A2 -1 Rz xy B1
x, Ry xz B2
y, Rx yz 1 1 1 1
Bend a1

14. IIB. Predicting Vibrational Peaks for Functional Groups
It is often convenient to determine IR activity of vibrations of specific functional groups – e.g., C-O stretches
Example: How many C-O peaks are there in the IR spectrum of fac-[ML3 (CO)3 ]?

15. Review: General Procedure for Applying Group Theory
(for Spectroscopy, MO Theory, etc.)
Find the point group of the molecule. All questions you seek to answer depend on the group to which the molecule belongs.
Determine a basis set of objects you care about. It may include: a point (x, y, z); all 3N molecular motions; specific vibrations; a set of atomic orbitals. . . Use a vector (or orbital) on the molecule to represent each item (n items = n vectors).
Find the reducible representation identifying the symmetry of all items in the basis set.
Reduce the representation to find the irreducible representations it contains (i.e., which rows in the character table are represented).
The irreducible representations provide the symmetries of the individual elements in your basis set. Use these to answer your desired question(s) [identifying IR-active modes, creating MO correlation diagrams, etc.].

16. IIB. Predicting Vibrational Peaks for Functional Groups
It is often convenient to determine IR activity of vibrations of specific functional groups – e.g., C-O stretches
Example: How many C-O peaks are there in the IR spectrum of fac-[ML3 (CO)3 ]? 1 2 3
1. Point group?
C3v
2. Basis set?
We don’t care about all vibrations of the molecule, only the 3 C-O vibrations. Use three vectors to represent what we care about.
3. Reducible representation?
Find the symmetry of the 3 vectors in the basis set (the 3 C-O vibrations) taken as a group.

17. IR-active C-O stretches in fac-[ML3(CO)3], cont.
3. Reducible representation: 1
Perform one operation from each class. What happens to the 3 basis set vectors?
Each operation is represented by a 3  3 transformation matrix (for 3 vectors). We need the characters of these matrices. 2 3
If a vector moves, it counts 0 toward the character of R (0 on the diagonal)
If a vector stays the same, it counts 1 toward the character of R (1 on the diagonal) E
2C3
3v A1 1 z A2 -1 Rz 2
(x,y), (Rx,Ry) R
{1, 2, 3}
C3v E:
C3:
v:
all unchanged -- (E) = 3
all move -- (C3) = 0
one stays -- (v) = 1 3 1

18. IR-active C-O stretches in fac-[ML3(CO)3], cont.
C3v E
2C3
3v A1 1 z
x2+y2, z2 A2 -1 Rz 2
(x,y), (Rx,Ry)
(x2-y2, xy) (xz, yz) R 3
{1, 2, 3}
4. Reduce R to find symmetries of individual C-O stretches:
# A1 =
# A2 =
# E =
5. Answer the question: Number of C-O peaks?
IR active: Peaks in spectrum:
Raman active: Peaks in spectrum:
(1/6)[(131) + (201) + (311)] = (1/6)[3 + 3] = 1
R = A1 + E
(1/6)[ (-3)] = 0
Only 2 symmetries? But there are 3 C-O vibrations . . .
(1/6)[ ] = 1
E is 2-dimensional; represents the symmetry of two degenerate vibrations
a1 + e (3 vib.)
2 – e vibrations degenerate
a1 + e (3 vib.)
2 – e vibrations degenerate

19. Distinguishing Isomers via IR Spectroscopy?
Can we differentiate between the fac- and mer- isomers of ML3(CO)3 based on the number of C-O bands (or peaks) in their IR spectra? x z y 1 1 2 3 2 3
fac
mer
2 IR bands ?? IR bands
Your turn

20. Review: General Procedure for Applying Group Theory
(for Spectroscopy, MO Theory, etc.)
Find the point group of the molecule. All questions you seek to answer depend on the group to which the molecule belongs.
Determine a basis set of objects you care about. It may include: a point (x, y, z); all 3N molecular motions; specific vibrations; a set of atomic orbitals. . . Use a vector (or orbital) on the molecule to represent each item (n items = n vectors).
Find the reducible representation identifying the symmetry of all items in the basis set.
Reduce the representation to find the irreducible representations it contains (i.e., which rows in the character table are represented).
The irreducible representations provide the symmetries of the individual elements in your basis set. Use these to answer your desired question(s) [identifying IR-active modes, creating MO correlation diagrams, etc.].

21. How many C-O bands in the IR of mer-ML3(CO)3?x z y
1. Find the point group:
C2v 1 2 3
2. Identify a basis set.
stretching vectors 1, 2, and 3
3. Determine a reducible representation describing the symmetry of your basis set.
4. Decompose R into irreducible reps to find symmetries of stretches.
C2v E C2
v(xz)
v(yz) R A1 1 z
x2, y2, z2 A2 -1 Rz xy B1
x, Ry xz B2
y, Rx yz 3 1 1 3

22. C-O bands in IR of mer-ML3(CO)3, cont.1
4. Decompose R into irreducible reps to find symmetries of individual C-O stretches. 2
C2v E C2
v(xz)
v(yz) R A1 1 z
x2, y2, z2 A2 -1 Rz xy B1
x, Ry xz B2
y, Rx yz 3 3 1 1 3 x z y
# A1 = # B1 =
# A2 = # B2 =
(1/4)[ ] = 2
(1/4)[ ] = 0
(1/4)[ ] = 0
(1/4)[ ] = 1
R = 2 A1 + B2
How many IR-active C-O stretches?
How many C-O bands in IR spectrum?
3; two a1, b2
3; two a1, b2

23. Distinguishing Isomers via IR Spectroscopy?
Can we differentiate between the fac- and mer- isomers of ML3(CO)3 based on the number of C-O stretches in their IR spectra? x z y 1 1 2 3 2 3
fac
mer
2 IR bands
3 IR bands – Yes!
Try the mer-isomer for additional practice!