1. Lecture 6

2. Character Tables Group theory makes use of the properties of matrices
Idea: When an operation, O,
(proper rotation, improper rotation, reflection, inversion)
is done on the function, f(x,y,z)
O f(x,y,z)
the result is a value taken on by the original function at some other point (x’,y’,z’). Or
O f(x,y,z) = f (x’,y’,z’)
We know the function f(x,y,z) the problem is to determine the values (x’,y’,z’).
The nature of the operation tells us where to look. 2

3. Character Tables Group theory makes use of the properties of matrices
Each operation may be expressed as a transformation matrix:
[New coordinates] = [transformation matrix][old coordinates]
Example: in Cartesian coordinate system, reflection in x = 0 plane
Changes the value of x to –x (multiplies it by -1)
Leaves y unchanged (multiplies it by 1)
Leaves z unchanged (multiplies it by 1)
To see the result of the operation at (x, y, z) look at the original object at (x’, y’, z’). =
Results of transformation.
Transformation matrix
Original coordinates

4. Recall Technique of Matrix multiplication= V’ M V
To get an element of the product vector a row in the operation square matrix is multiplied by the original vector matrix.
For example
V’2 = y’ =
M2,1 * V1
+ M2,2 * V2
+ M2,3 * V3
y’ = 0 * x * y * z = y

5. Character Tables - 2
The matrix representation of the symmetry operations of a point group is
the set of matrices corresponding to all the symmetry operations in that group. The matrices record how the x,y,z coordinates are modified as a result of an operation.
For example, the C2v point group consists of the following operations
E: do nothing. Unchanged.
C2: rotate 180 degrees about the z axis: x becomes –x; y becomes –y and z unchanged.
sv (xz): y becomes –y
sv’ (yz): x becomes -x E C2
sv (xz):
sv’ (yz):

6. Operations Applied to Functions - 1
Transform the coordinates.
Consider
f(x) = x2
sv’ (f(x)) = sv(x2) = (-x)2 = x2 = f(x) or
sv’ (f(x)) = 1 * f(x)
f(x) is an eigenfunction of this reflection operator with an eigenvalue of +1. This is called a symmetric eigenfunction.
Similarly
f(x) = x3
sv’ (f(x)) = -1 * f(x)
f(x) is an eigenfunction of this reflection operator with an eigenvalue of -1. This is called a antisymmetric eigenfunction.

7. Plots of Functions, x2 Reflection yields.
Here f(x) is x2. It can be seen to be a symmetric function for reflection at x = 0 because of mirror plane.
The reflection carries out the mapping shown with the red arrows.
x2 is an eigenfunction of s with eigenvalue 1

8. Plots of Functions, x3 Reflection yields.
Here f(x) is x3. It can be seen to be a antisymmetric function for reflection at x = 0.
The reflection carries out the mapping shown with the red arrows.
x3 is an eigenfunction of s with eigenvalue -1

9. Plots of Functions - 2 Reflection yields.
Here f(x) is x3. It can be seen to be a antisymmetric function for reflection at x = 0.
The reflection carries out the mapping shown with the red arrows.

10. Operations Applied to Functions - 2
Now consider
f(x) = (x-2)2 = x2 – 4x + 4
sv’ (f(x)) = sv(x-2)2 = (-x-2)2 = x2 + 4x + 4
f(x) = (x-2)2 is not an eigenfunction of this reflection operator because it does not return a constant times f(x).
Reflection yields this function, not an eigenfunction.
Neither symmetric nor antisymmetric for reflection thru x = 0.

11. Let’s look at Atomic Orbitals
Reflection
Get the same orbital back, multiplied by +1, an eigenfunction of the reflection, symmetric with respect to the reflection. The s orbital forms the basis of an irreducible representation of the operation
s orbital z

12. Atomic Orbitals s Reflection
Get the same orbital back, multiplied by -1, an eigenfunction of the reflection, antisymmetric with respect to the reflection. The p orbital behaves differently from the s orbital and forms the basis of a different irreducible representation of the operation
p orbital z

13. Simplest ways that objects can behave for a group consisting of E and sh , the reflection plane.
Irreducible Representations.
Basis of the Irreducible Reps. Cs
E sh A’ A” 1
x, y,Rz
z, Rx,Ry
x2,y2,z2,xy
yz, xz
s orbital is spherical behaves as x2 + y2 + z2. s orbital is A’. The s orbital is an eigenfunction of both E and sh.
pz orbital has a multiplicative factor of z times a spherical factor. Behaves as A”. pz is an eigenfunction of both E and sh.

14. sp Hybrids
Reflection
Do not get the same hybrid back multiplied by +1 or -1 or some other constant. Not an eigenfunction.
hybrid z
The two hybrids form the basis of a reducible representation of the operation
Recall: the hybrid can be expressed as the sum of an s orbital and a p orbital. = +
Reduction: expressing a reducible representation as a combination of irreducible representations.

15. Reducible Representations
Use the two sp hybrids as the basis of a representation h1 h2
sh operation.
E operation.
h1 becomes h1; h2 becomes h2.
h1 becomes h2; h2 becomes h1. = =
The reflection operation interchanges the two hybrids.
The hybrids are unaffected by the E operation.
Proceed using the trace of the matrix representation.
= 0
= 2

16. Our Irreducible RepresentationsCs
E sh A’ A” 1
x, y,Rz
z, Rx,Ry
x2,y2,z2,xy
yz, xz
The reducible representation derived from the two hybrids can be attached to the table. G
(h1, h2)
Note that G = A’ + A”

17. Return to polynomials:
f(x) = (x-2)2 = x2 – 4x + 4
sv (f(x)) = sv(x-2)2 = (-x-2)2 = x2 + 4x + 4 =g(x)
Neither f nor g is an eigenfunction of s but, taken together, they do form an reducible representation since they show what the s operator does.
Approaching the problem in the same way as we did for hybrids we can carry out the reduction this way
u(x) = ½ (f(x) + g(x)) = ½ (f(x) + s f(x)) = x2 + 4 symmetric, unchanged by the s operator. Behaves as A’
v(x) = ½ (f(x) - g(x)) = ½ (f(x) - s f(x)) = -4x, antisymmetric, multiplied by -1 by the s operator. Behaves as A’’

18. Character Table Each row is an irreducible representation
Symmetry operations, Classes
Point group
x, y, z
Symmetry of translations (p orbitals)
Rx, Ry, Rz: rotations
Characters
+1 symmetric behavior
-1 antisymmetric
Mülliken symbols
Each row is an irreducible representation

19. Character Tables - 3
Irreducible representations are not linear combinations of other representation
(Reducible representations are)
# of irreducible representations = # of classes of symmetry operations
Instead of the matrices, the characters are used (traces of matrices)
A character Table is the complete set of irreducible representations of a point group

20. Operation E C2 sv sv’ Transformation 1 -1
Effect of the 4 operations in the point group C2v
on a translation in the x direction. The translation is simply
multiplied by 1 or -1. It forms a basis to show
what the operators do to an object.
Operation E C2 sv
sv’
Transformation 1 -1

21. Character Table
Verify this character. It represents how a function that behaves as x, Ry, or xz behaves for C2. 21

22. Symmetry of translations (p orbitals)
Another point group, C3v.
Classes of operations
x, y, z
Symmetry of translations (p orbitals)
Rx, Ry, Rz: rotations
dxy, dxz, dyz, as xy, xz, yz
dx2- y2 behaves as x2 – y2
dz2 behaves as 2z2 - (x2 + y2)
px, py, pz behave as x, y, z
s behaves as x2 + y2 + z2

23. Symmetry of Atomic Orbitals

24. Naming of Irreducible representations
One dimensional (non degenerate) representations are designated A or B.
Two-dimensional (doubly degenerate) are designated E.
Three-dimensional (triply degenerate) are designated T.
Any 1-D representation symmetric with respect to Cn is designated A; antisymmétric ones are designated B
Subscripts 1 or 2 (applied to A or B refer) to symmetric and antisymmetric representations with respect to C2  Cn or (if no C2) to  sv respectively
Superscripts ‘ and ‘’ indicate symmetric and antisymmetric operations with respect to sh, respectively
In groups having a center of inversion, subscripts g (gerade) and u (ungerade) indicate symmetric and antisymmetric representations with respect to i

25. Character Tables
Irreducible representations are the generalized analogues of s or p symmetry in diatomic molecules.
Characters in rows designated A, B,..., and in columns other than E indicate the behavior of an orbital or group of orbitals under the corresponding operations (+1 = orbital does not change; -1 = orbital changes sign; anything else = more complex change)
Characters in the column of operation E indicate the degeneracy of orbitals
Symmetry classes are represented by CAPITAL LETTERS (A, B, E, T,...) whereas orbitals are represented in lowercase (a, b, e, t,...)
The identity of orbitals which a row represents is found at the extreme right of the row
Pairs in brackets refer to groups of degenerate orbitals and, in those cases, the characters refer to the properties of the set

26. Definition of a Group
A group is a set, G, together with a binary operation : such that the product of any two members of the group is a member of the group, usually denoted by a*b, such that the following properties are satisfied :
(Associativity) (a*b)*c = a*(b*c) for all a, b, c belonging to G.
(Identity) There exists e belonging to G, such that e*g = g = g*e for all g belonging to G.
(Inverse) For each g belonging to G, there exists the inverse of g, g-1, such that g-1*g = g*g-1 = e.
If commutativity is satisfied, i.e. a*b = b*a for all a, b belonging to G, then G is called an abelian group.

27. Examples The set of integers Z, is an abelian group under addition.
What is the element e, identity, such that
a*e = a?
What is the inverse of the a element? -a

28. As applied to our symmetry operators.
For the C3v point group
What is the inverse of each operator? A * A-1 = E
E C3(120) C3(240) sv (1) sv (2) sv (3)
E C3(240) C3(120) sv (1) sv (2) sv (3)

29. Examine the matrix represetation of the C2v point group-
s’v(yz)
sv(xz)

30. Multiplying two matrices (a reminder)
Most of the transformation
matrices we use have the
form

31. C2 sv(xz) s’v(yz) E What is the inverse of C2? C2 =
What is the inverse of sv? sv =

32. C2 sv(xz) s’v(yz) E What of the products of operations? C2 E * C2 = ?
sv * C2 = ?
s’v =

33. Classes
Two members, c1 and c2, of a group belong to the same class if there is a member, g, of the group such that
g*c1*g-1 = c2

34. Properties of Characters of Irreducible Representations in Point Groups
Total number of symmetry operations in the group is called the order of the group (h). For C3v, for example, it is 6.
= 6
Symmetry operations are arranged in classes. Operations in a class are grouped together as they have identical characters. Elements in a class are related.
This column represents three symmetry operations having identical characters.

35. Properties of Characters of Irreducible Representations in Point Groups - 2
The number of irreducible reps equals the number of classes. The character table is square.
= 6
3 by 3 1 22 6
The sum of the squares of the dimensions of the each irreducible rep equals the order of the group, h.

36. Properties of Characters of Irreducible Representations in Point Groups - 3
For any irreducible rep the squares of the characters summed over the symmetry operations equals the order of the group, h.
A1: ( ) + = 6
A2: ( ) + ((-1)2 + (-1)2 + (-1)2 ) = 6
E: (-1)2 + (-1)2 = 6

37. Properties of Characters of Irreducible Representations in Point Groups - 4
Irreducible reps are orthogonal. The sum of the products of the characters for each symmetry operation is zero.
For A1 and E:
1 * (1 *(-1) + 1 *(-1)) + (1 * * * 0) = 0

38. Properties of Characters of Irreducible Representations in Point Groups - 5
Each group has a totally symmetric irreducible rep having all characters equal to 1

39. Reduction of a Reducible Representation
Irreducible reps may be regarded as orthogonal vectors. The magnitude of the vector is h-1/2
Any representation may be regarded as a vector which is a linear combination of the irreducible representations.
Reducible Rep = S (ai * IrreducibleRepi)
The Irreducible reps are orthogonal. Hence
S(character of Reducible Rep)(character of Irreducible Repi) = ai * h Or
ai = S(character of Reducible Rep)(character of Irreducible Repi) / h
Sym ops
Sym ops

40. These are block-diagonalized matrices
(x, y, z coordinates are independent of each other)
Irreducible
representations
Reducible Rep

41. C2v Character Table to be used for water
Symmetry operations
Point group
Characters
+1 symmetric behavior
-1 antisymmetric
Mülliken symbols
Each row is an irreducible representation

42. Let’s use character tables! Symmetry and molecular vibrations
# of atoms
degrees of freedom
Translational modes
Rotational modes
Vibrational modes
N (linear)
3 x 2 3 2
3N-5 = 1
Example
3 (HCN) 9 4
N (non- linear) 3N
3N-6
3 (H2O)

43. Symmetry and molecular vibrations
A molecular vibration is IR active
only if it results in a change in the dipole moment of the molecule
A molecular vibration is Raman active
only if it results in a change in the polarizability of the molecule
In group theory terms:
A vibrational mode is IR active if it corresponds to an irreducible representation
with the same symmetry of a x, y, z coordinate (or function)
and it is Raman active if the symmetry is the same as
A quadratic function x2, y2, z2, xy, xz, yz, x2-y2
If the molecule has a center of inversion, no vibration can be both IR & Raman active

44. You need the molecular geometry (point group) and the character table
How many vibrational modes belong to each irreducible representation?
You need the molecular geometry (point group) and the character table
Use the translation vectors of the atoms as the basis of a reducible representation.
Since you only need the trace recognize that only the vectors that are either unchanged or have become the negatives of themselves by a symmetry operation contribute to the character.

45. Apply each symmetry operation in that point group to the molecule
A shorter method can be devised. Recognize that a vector is unchanged or becomes the negative of itself if the atom does not move.
A reflection will leave two vectors unchanged and multiply the other by -1 contributing +1.
For a rotation leaving the position of an atom unchanged will invert the direction of two vectors, leaving the third unchanged.
Etc.
Apply each symmetry operation in that point group to the molecule
and determine how many atoms are not moved by the symmetry operation.
Multiply that number by the character contribution of that operation:
E = 3
s = 1
C2 = -1
i = -3
C3 = 0
That will give you the reducible representation

46. Finding the reducible representation
3x3 9
1x-1 -1
3x1 3
1x1 1
(# atoms not moving x char. contrib.) G

47. Now separate the reducible representation into irreducible ones
to see how many there are of each type S 9 -1 3 1 G
A1 = 1/4 (1x9x1 + 1x(-1)x1 + 1x3x1 + 1x1x1) = 3
A2 =
1/4 (1x9x1 + 1x(-1)x1 + 1x3x(-1) + 1x1x(-1)) = 1

48. Symmetry of molecular movements of water
Vibrational modes

49. Which of these vibrations having A1 and B1 symmetry are IR or Raman active?
IR active

50. Often you analyze selected vibrational modes
Example: C-O stretch in C2v complex.
n(CO)
2 x 1 2
0 x 1
2 x 1 2
0 x 1 G
Find: # vectors remaining unchanged after operation.

51. Both A1 and B1 are IR and Raman active 2 GG
= A1 + B1
A1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1
A2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x-1) = 0
B1 = 1/4 (1x2x1 + 1x0x1 + 1x2x1 + 1x0x1) = 1
B2 = 1/4 (1x2x1 + 1x0x1 + 1x2x-1 + 1x0x1) = 0

52. What about the trans isomer?
Only one IR active band and no Raman active bands
Remember cis isomer had two IR active bands and one Raman active

53. Symmetry and NMR spectroscopy
The # of signals in the spectrum
corresponds to the # of types of nuclei not related by symmetry
The symmetry of a molecule may be determined
From the # of signals, or vice-versa

54. Molecular Orbitals

55. Atomic orbitals interact to form molecular orbitals
Electrons are placed in molecular orbitals
following the same rules as for atomic orbitals
In terms of approximate solutions to the Scrödinger equation
Molecular Orbitals are linear combinations of atomic orbitals (LCAO)
Y = caya + cbyb (for diatomic molecules)
Interactions depend on the symmetry properties
and the relative energies of the atomic orbitals

56. If the total energy of the electrons in the molecular orbitals
As the distance between atoms decreases
Atomic orbitals overlap
Bonding takes place if:
the orbital symmetry must be such that regions of the same sign overlap
the energy of the orbitals must be similar
the interatomic distance must be short enough but not too short
If the total energy of the electrons in the molecular orbitals
is less than in the atomic orbitals, the molecule is stable compared with the atoms

57. Y = N[caY(1sa) ± cbY (1sb)]
Combinations of two s orbitals (e.g. H2)
Antibonding
Bonding
More generally:
Y = N[caY(1sa) ± cbY (1sb)]
n A.O.’s
n M.O.’s

58. (total energy is raised)
Electrons in antibonding orbitals cause mutual repulsion between the atoms
(total energy is raised)
Electrons in bonding orbitals concentrate between the nuclei and hold the nuclei together
(total energy is lowered)

59. Both s (and s*) notation means symmetric/antisymmetric with respect to rotation
Not s

60. Combinations of two p orbitals (e.g. H2)
s (and s*) notation means no
change of sign upon rotation
p (and p*) notation means
change of sign upon C2 rotation

61. Combinations of two p orbitals

62. Combinations of two sets of p orbitals

63. Combinations of s and p orbitals

64. Combinations of d orbitals
No interaction – different symmetry
d means change of sign upon C4

65. Is there a net interaction?NO NO
YES

66. Relative energies of interacting orbitals must be similar
Strong interaction
Weak interaction

67. for diatomic molecules From H2 to Ne2
Molecular orbitals
for diatomic molecules
From H2 to Ne2
Electrons are placed
in molecular orbitals
following the same rules
as for atomic orbitals:
Fill from lowest to highest
Maximum spin multiplicity
Electrons have different quantum numbers including spin (+ ½, - ½)

69. symmetric/antisymmetric
O2 (2 x 8e)
1/2 (10 - 6) = 2
A double bond
Or counting only
valence electrons:
1/2 (8 - 4) = 2
Note subscripts
g and u
symmetric/antisymmetric
upon i

70. Place labels g or u in this diagram
s*u
p*g pu sg

71. s*u sg
g or u?
p*g pu
d*u dg

72. Orbital mixing Same symmetry and similar energies !
shouldn’t they interact?

73. When two MO’s of the same symmetry mix
s orbital mixing
When two MO’s of the same symmetry mix
the one with higher energy moves higher and the one with lower energy moves lower

74. for diatomic molecules
Molecular orbitals
for diatomic molecules
From H2 to Ne2
H2 sg2 (single bond)
He2 sg2 s*u2 (no bond)

75. E (Z*) DE s > DE p C2 pu2 pu2 (double bond)
Paramagnetic
due to mixing
C2 pu2 pu2 (double bond)
C22- pu2 pu2 sg2(triple bond)
O2 pu2 pu2 p*g1 p*g1 (double bond)
paramagnetic
O22- pu2 pu2 p*g2 p*g2 (single bond)
diamagnetic

76. Bond lengths in diatomic molecules
Filling bonding orbitals
Filling antibonding orbitals

77. Photoelectron Spectroscopy

78. O2 N2 sg (2p) p*u (2p) pu (2p) sg (2p) pu (2p) s*u (2s) s*u (2s)
Very involved in bonding
(vibrational fine structure)
s*u (2s)
(Energy required to remove electron, lower energy for higher orbitals)