Chem 355 10 Lecture 17

1. Chem 355 10 Lecture 17 Symmetry and IR- and Raman- . active-Vibrations.

2. Symmetry operations on the C2v point group vectors representing translation motions of the molecules.

3. The C2v Character Table

4. The C2v Character Table

5. The C2v Character Table

6. The C2v Character Table

7. The C2v Character Table

8. The influence of C2 on the rotation about z: Rz

9. +1 +1 -1 -1 The characters associated each of the symmetry operations in C2v on Rz

10. The C2v Character Table

11. The C2v Character Table

12. The total representation Gtotal specific for H2O can be found from the general Gx,y,z for the C2v group given at the bottom of the table times the number of unmoved atoms during each operation: While Gx,y,z would be the same for pyridine the # of unmoved atom during each operation, and therefore Gx,y,z would also differ.

13. The differences in Gtotal for the 2 molecules lies in the # of times a particular symmetry species contributes to the characters for the reducible representation. The Gtotal for the 2 molecules can then be decomposed to produce the # of times each of the irreducible symmetry species for a C2v molecule contribute to the total..

14. The Vectorlike Properties of Character Tables The set of characters along any row in a character table behaves like the components of a vector in space with dimensions equal to the number of operations of the group.

15. The Vectorlike Properties of Character Tables The set of characters along any row in a character table behaves like the components of a vector in space with dimensions equal to the number of operations of the group. a) The sum of the squares of the entries for each . symmetry species ie equal to 4.

16. The Vectorlike Properties of Character Tables The set of characters along any row in a character table behaves like the components of a vector in space with dimensions equal to the number of operations of the group. a) The sum of the squares of the entries for each . symmetry species ie equal to 4 (C2V). b) The, the sum of the term-by- term products over all . the operations for any 2 different symmetry species is . zero.

17. These 2 factors can be expressed mathematically. Let i refer to any one row of the character table and j to another row. Let R represent any column of a character table. Thus R is a symmetry operation of any of the classes of symmetry operations. Let nR be the number of operations in the class (a number given in the first row of the character table.). It follows that:

18. These 2 factors can be expressed mathematically. Let i refer to any one row of the character table and j to another row. Let R represent any column of a character table. Thus R is a symmetry operation of any of the classes of symmetry operations. Let nR be the number of operations in the class (a number given in the first row of the character table.). It follows that: Eq.1 where h is the number of symmetry operations in the group, or the order.

19. These vectorlike properties allow us to derive an expression that gives the number of times each of the irreducible representations occurs in any reducible representations.*

20. These vectorlike properties allow us to derive an expression that gives the number of times each of the irreducible representations occurs in any reducible representations.* The idea that the characters of any reducible representation can be made up of the characters of some of the irreducible representations can be expressed by: This is where the difference between H2O and pyridine is found. They both share the same set of irreducible representations, but due to the larger size (number of nuclei) of pyridine the number of time any particular irreducible representation (symmetry species) appears is larger. *

21. For: represents the character for the class containing the Rth symmetry operation for a reducible representation and . represents the character for the jth irreproducible operation. The aj’s are the number of times that jth irreducible representation, i.e. in the jth row of the character table, occurs in the reducible representation.

22. For: represents the character for the class containing the Rth symmetry operation for a reducible representation and . represents the character for the jth irreproducible operation. The aj’s are the number of times that jth irreducible representation, i.e. in the jth row of the character table, occurs in the reducible representation. The number of times each irreproducible representation (each row of the character table), occurs in a reproducible representation can be found. Focusing on the ith row, both sides of the equation above are multiplied by nRci(R) and sum over all classes of symmetry operations.

23. According to Eq.1 the right side will yield zero contributions except when j = i. Then the value on the right side is ai times h, where h is the order of the group

24. According to Eq.1 the right side will yield zero contributions except when j = i. Then the value on the right side is ai times g, where h is the order of the group or For C2V there is only 1 member of each class, nR = 1

25. The number of times each of the symmetry species in H2O contributes can be found from the relation, e.g.: where h = the order (# of operations), ci = # members of each class, ci = the character associated with the ith operation and ctotal = the total character associated with each operation, that is: e.g.

26. The number of times a member of each of the symmetry species characteristic for C2v can be found from the relation, e.g.: where h = the order (# of operations), ci = # members of each class, ci = the character associated with the ith operation and ctotal = the total character associated with each operation, that is: e.g.

27. The number of times a member of each of the symmetry species characteristic for C2v can be found from the relation, e.g.: where h = the order (# of operations), ci = # members of each class, ci = the character associated with the ith operation and ctotal = the total character associated with each operation, that is: e.g.

28. and therefore: Gtotal = 3A1 + A2 + 2B1 + 3B2 Note that the total number of symmetries species = 3N which it must be, i.e. in this case 9 symmetry species corresponding to the total of 9 coordinates required for the H2O molecule. It should also be noted that while Gx,y,z characterizes the sum of the characters for x,y and z for each operation. It is specific for the particular group, in this case C2v. The pyridine molecule: Also has C2v symmetry and therefore would possess the same symmetry species (irreducible representations) as H2O. However, Gtotal would differ and be specific for pyridine.

29. and therefore: Gtotal = 3A1 + A2 + 2B1 + 3B2 Note that the total number of symmetries species = 3N which it must be, i.e. in this case 9 symmetry species corresponding to the total of 9 coordinates required for the H2O molecule. It should also be noted that while Gx,y,z characterizes the sum of the characters for x,y and z for each operation. It is specific for the particular group, in this case C2v. The pyridine molecule: Also has C2v symmetry and therefore would possess the same symmetry species (irreducible representations) as H2O. However, Gtotal would differ and be specific for pyridine.

30. and therefore: Gtotal = 3A1 + A2 + 2B1 + 3B2 Note that the total number of symmetries species = 3N which it must be, i.e. in this case 9 symmetry species corresponding to the total of 9 coordinates required for the H2O molecule. It should also be noted that while Gx,y,z characterizes the sum of the characters for x,y and z for each operation. It is specific for the particular group, in this case C2v. The pyridine molecule: also has C2v symmetry and therefore would possess the same symmetry species (irreducible representations) as H2O. However, Gtotal would differ and be specific for pyridine.

31. The number of unmoved atoms under each of the operations would clearly be larger for the latter molecule.

32. Returning to the H2O example, The irreducible representations associated specifically with the vibrations in the molecule can be found by subtracting the symmetry species associated with translations and rotations of the entire molecule. The irreducible representations associated with rotations can be found from the Rz, Rx and Ry designations in the right hand column. • Gvib = 3A1 + A2 + 2B1 + 3B2 • – (A1 + B1 + B2) – (A2 + B2 + B1) • = 2A1 + B2

33. Returning to the H2O example, The irreducible representations associated specifically with the vibrations in the molecule can be found by subtracting the symmetry species associated with translations and rotations of the entire molecule. The irreducible representations associated with rotations can be found from the Rz, Rx and Ry designations in the right hand column. • Gvib = 3A1 + A2 + 2B1 + 3B2 • – (A1 + B1 + B2) – (A2 + B2 + B1) • = 2A1 + B2

34. Returning to the H2O example, The irreducible representations associated specifically with the vibrations in the molecule can be found by subtracting the symmetry species associated with translations and rotations of the entire molecule. The irreducible representations associated with rotations can be found from the Rz, Rx and Ry designations in the right hand column. • Gvib = 3A1 + A2 + 2B1 + 3B2 • – (A1 + B1 + B2) – (A2 + B2 + B1) • = 2A1 + B2

35. Since A1 transforms as z, the transition dipole lies along z. The displacement vectors for the symmetric stretch and the bending modes do not change signs under the symmetry operations of the group. The displacement vectors for the asymmetric stretching mode change signs under C2 and s(x,z), i.e. transform as B2: the transition dipole lies along y. The B1 irreducible representation (transforms as x) and does not represent a normal vibrational mode.

36. Since A1 transforms as z, the transition dipole lies along z. The displacement vectors for the symmetric stretch and the bending modes do not change signs under the symmetry operations of the group. The displacement vectors for the asymmetric stretching mode change signs under C2 and s(x,z), i.e. transform as B2: the transition dipole lies along y. The B1 irreducible representation (transforms as x) and does not represent a normal vibrational mode.

37. Since A1 transforms as z, the transition dipole lies along z. The displacement vectors for the symmetric stretch and the bending modes do not change signs under the symmetry operations of the group. The displacement vectors for the asymmetric stretching mode change signs under C2 and s(x,z), i.e. transform as B2: the transition dipole lies along y. The B1 irreducible representation (transforms as x) and does not represent a normal vibrational mode.

39. In order for a vibrational mode to appear in the Raman spectrum of a molecule or be “Raman active, requires that the related integral for the transition: where a is the polarizability tensor rather than the transition dipole. It is sufficient that if the overall integrand in the integral above is a symmetric function, then mtran 0. There are 6 components of a which transform as x2, y2, z2, xy, xz and yz. These and certain linear combinations appear at the right of character tables. Raman transitions are allowed then if any of the vibratonal mode transform as any one of the components above or: G(yv)xG(a) xG(yo) = A1

40. The C2v Character Table The far right hand column added to the character table reveals that all 4 symmetry species (irreducible representations) are Raman active.

41. Conversely if the integral is antisymmetric, then the transition dipole vanishes and the transition is forbidden. Since in the v = 0 state of a harmonic oscillator the wavefunction is symmetric with respect to the normal coordinate q, the product of the excited state wavefunction and the normal coordinate must be symmetric, or the excited-state wavefunction must be of the same symmetry of at least one of the Cartesian coordinates of that of the overall changing dipole. Using H2O as a simple example it can be noted that in the column on the right of the C2v character table which applies to H2O:

42. A1, B1 and B2 irreducible representations (symmetry species) transform under the operations of the group in the same manner as z, x and y, respectively. Irreducible representations which transform as x, y or z “could” represent IR-active normal modes of vibration in that they represent components of a dipole moment: or a dipole moment operator. In the C2v example z, and then x and y are seen from the character table to transform as the symmetry species A1 and B2, respectively. In the C2v example z, and then x and y are seen from the character table to transform as the symmetry species A1 and B2, respectively.

43. A1, B1 and B2 irreducible representations (symmetry species) transform under the operations of the group in the same manner as z, x and y, respectively. Irreducible representations which transform as x, y or z “could” represent IR-active normal modes of vibration in that they represent components of a dipole moment: or a dipole moment operator. In the C2v example z, and then x and y are seen from the character table to transform as the symmetry species A1 and B2, respectively. In the C2v example z, and then x and y are seen from the character table to transform as the symmetry species A1 and B2, respectively.

44. Since B2xB2 = A1, which can readily be shown from the multiplication of their characters in the Table for all 4 operations, then the fundamental transitions involving any of the 3 normal vibrational modes of H2O could be allowed, if the overall representation for the vibrations in the molecule contain symmetry species which transform as A1 or B2. The total representation Gtotal specific for H2O can be found from the general Gx,y,z for the C2v group given at the bottom of the table times the number of unmoved atoms during each operation:

45. Since B2xB2 = A1, which can readily be shown from the multiplication of their characters in the Table for all 4 operations, then the fundamental transitions involving any of the 3 normal vibrational modes of H2O could be allowed, if the overall representation for the vibrations in the molecule contain symmetry species which transform as A1 or B2. The total representation Gtotal specific for H2O can be found from the general Gx,y,z for the C2v group given at the bottom of the table times the number of unmoved atoms during each operation: