1. CHEM 515 Spectroscopy Vibrational Spectroscopy III

2. 2 Normal Modes in Water Molecule Q3Q3 Q2Q2 Q1Q1 z x

3. 3 Internal Coordinates Mathematically, the mass- weighted Cartesian coordinates (Cartesian force constant model) are very convenient. Ab initio calculation utilizes such a model in molecule optimization. A more recognizable way to define the atomic displacement in molecular modeling program is by utilizing internal coordinates.

4. 4 Internal Coordinate Definitions It is much convenient to transform from Cartesian coordinates to internal coordinates. These are the main four internal coordinate out of which other internal coordinated can be defined.

5. 5 Internal Coordinate for Water Molecule The internal displacement of atoms can be defined as change in the three coordinates. Δr 1 Δr 2 Δθ r1r1 r2r2 θ

6. 6 Symmetry Coordinates More preferably than internal coordinates is the use of symmetry coordinates. The concept of molecule symmetry “group theory” is applied with help of projection operators to generate the required 3N – 6 symmetry coordinates. Projection operator Symmetry specie Normalization factor Operation Character for symmetry specie γ New coordinate obtained from Int i upon operation R

7. 7 Symmetry Coordinates This totally symmetric projection operator is used to get a set of symmetric coordinates by linear combinations of internal coordinates. This method is also known as “Symmetry- Adapted Linear Combinations; SALCs)” as proposed by A. F. Cotton. Symmetry specie Normalization factor Operation Character for symmetry specie γ New coordinate obtained from Int i upon operation R

8. 8 Symmetry Coordinates of Ethylene Using SALC Method D 2d EC 2 (z)C 2 (y)C 2 (x)iσ (xy)σ (xz)σ (yz) AgAg 11111111 B 1g 11 11 B 2g 11 1 1 B 3g 1 11 1 AuAu 1111 B 1u 11 11 B 2u 11 1 1 B 3u 1 1 11

9. 9 Symmetry Coordinates of Ethylene Using SALC Method D 2d EC 2 (z)C 2 (y)C 2 (x)iσ (xy)σ (xz)σ (yz) O R (r 1 )r1r1 r4r4 r2r2 r3r3 r4r4 r1r1 r3r3 r2r2 AgAg 11111111 B 1g 11 11 B 2g 11 1 1 B 3g 1 11 1 AuAu 1111 B 1u 11 11 B 2u 11 1 1 B 3u 1 1 11

10. 10 Determining the Symmetry Species for the Vibrations in a Molecule We are very concerned with the symmetry of each normal mode of vibration in a molecule. Each normal mode of vibration will form a basis for an irreducible representation (Γ) of the point group of the molecule. The objective is to determine what the character (trace) is for the transformation matrix corresponding to a particular operation in a specific molecule.

11. 11 Symmetry of Normal Modes of Vibrations in H 2 O H 2 O has C 2v symmetry. Operation E results in the following transformations:

12. 12 Symmetry of Normal Modes of Vibrations in H 2 O The transformations in the x, y and z modes can be represented with the following matrix transformation: Trace of E matrix is equal to 9.

13. 13 Symmetry of Normal Modes of Vibrations in H 2 O The operation C 2 is more interesting! Operation C 2 results in the following transformations:

14. 14 Symmetry of Normal Modes of Vibrations in H 2 O The transformations in the x, y and z modes can be represented with the following matrix transformation: Trace of C 2 matrix is equal to –1.

15. 15 Determining the Symmetry Species for the Vibrations in a Molecule: a Shorter Way The matrix transformation method is very cumbersome. However, it can be streamlined tremendously another procedure. Alternative Method: 1.Count unshifted atoms per each operation. 2.Multiply by contribution per unshifted atom to get the reducible representation (Γ). 3.Determine (Γ) for each symmetry operation. 4.Subtract Γ trans and Γ rot from Γ tot. Γ vib = Γ tot – Γ trans – Γ rot.

16. 16 Determining the Irreducible Representation for the H 2 O Molecule 1. Count unshifted atoms per each operation. C 2v EC2C2 σ (xz)σ (yz) Unshifted atoms 3113

17. 17 Determining the Irreducible Representation for the H 2 O Molecule 2. Multiply by contribution per unshifted atom to get the reducible representation (Γ). C 2v EC2C2 σ (xz)σ (yz) Unshifted atoms 3113 Contribution per atom (Γ xyz ) 3–1+1

18. 18 Determining the Irreducible Representation for the H 2 O Molecule 2. Multiply by contribution per unshifted atom to get the reducible representation (Γ). C 2v EC2C2 σ (xz)σ (yz) Unshifted atoms 3113 Contribution per atom (Γ xyz ) 3–1+1 Γ 9–1+1+3

19. 19 Determining the Irreducible Representation for the H 2 O Molecule 3. Determine (Γ) for each symmetry operation. η i : number of times the irreducible representation (Γ) appears for the symmetry operation i. h : order of the point group. R : an operation of the group. χ R : character of the operation R in the reducible represent. χ i R : character of the operation R in the irreducible represent. C R : number of members of class to which R belongs.

20. 20 Determining the Irreducible Representation for the H 2 O Molecule C 2v EC2C2 σ (xz)σ (yz) Γ 9–1+1+3

21. 21 Determining the Irreducible Representation for the H 2 O Molecule C 2v EC2C2 σ (xz)σ (yz) Γ 9–1+1+3

22. 22 Determining the Irreducible Representation for the H 2 O Molecule 3. Determine (Γ) for each symmetry operation. Γ tot = 3A 1 + A 2 + 2B 1 + 3B 2 Number of irreducible representations Γ tot must equal to 3N for the molecule.

23. 23 Determining the Irreducible Representation for the H 2 O Molecule Subtract Γ trans and Γ rot from Γ tot. Γ tot = 3A 1 + A 2 + 2B 1 + 3B 2

24. 24 Determining the Irreducible Representation for the H 2 O Molecule Γ vib = 2A 1 + B 2 The difference between A and B species is that the character under the principal rotational operation, which is in this case C 2, is always +1 for A and –1 for B representations. The subscripts 1 and 2 are considered arbitrary labels. A1A1 A1A1 B2B2

25. 25 Determining the Irreducible Representation for the H 2 O Molecule Γ vib = 2A 1 + B 2 None of these motions are degenerate. One can spot the degeneracy associated with a special normal mode of vibration when the irreducible representation has a value of 2 at least, such as E operation in C 3v and C 4v point groups. A1A1 A1A1 B2B2

26. 26 Determining the Irreducible Representation for Ethene D 2d EC 2 (z)C 2 (y)C 2 (x)iσ (xy)σ (xz)σ (yz) AgAg 11111111 B 1g 11 11 RzRz B 2g 11 1 1 RyRy B 3g 1 11 1RxRx AuAu 1111 B 1u 11 11z B 2u 11 1 1y B 3u 1 1 11 x

27. 27 Determining the Irreducible Representation for Ethene

28. 28 Determining the Irreducible Representation for Ethene

29. 29 Normal Modes in Ethene Physical Chemistry By Robert G. Mortimer

30. 30 Mutual Exclusion Principle For molecules having a center of symmetry (i), the vibration that is symmetric w.r.t the center of symmetry is Raman active but not IR active, whereas those that are antisymmetric w.r.t the center of symmetry are IR active but not Raman active.

31. 31 Vibrations in Methyl and Methylene Groups Ranges in cm -1 : C-H stretch 2980 – 2850 CH 2 wag 1470 – 1450 CH 2 rock 740 – 720 CH 2 wag 1390 – 1370 CH 2 twist 1470 - 1440