Lecture 5

See: Chemical Applications of Group Theory by F. A. Cotton Symmetry point groups The set of all possible symmetry operations on a molecule is called the point group (there are 28 point groups) The mathematical treatment of the properties of groups is Group Theory In chemistry, group theory allows the assignment of structures, the definition of orbitals, analysis of vibrations, ... See: Chemical Applications of Group Theory by F. A. Cotton 2

To determine the point group of a molecule 3

4

Groups of low symmetry 5

6

Principal rotation axis?

Any C2 axes perpendicular to principal axis? D Groups C or S Groups

How many C2 axis perpendicular to principal axis, Cn? There are n C2 axes !!

sh ? If yes then done!

D3h D? D? C? or S4 D? C2h C? or S6 C? or S4 C? or S4 Is there a horizontal plane, sh , perpendicular to principal axis, Cn? If so, then can assign D  Dnh and C  Cnh. D3h D? D? C? or S4 D? Recall classified as D vs C/S on basis of perpendicular C2 C2h C? or S6 C? or S4 C? or S4

Recapitulation C3, 3C2 , sh C2, no C2 , sh D3h C2h

s containing the principal axis?

We have no sh. Next question: any plane containing Cn? D  Dnh if present; D  Dn if not. (C or S)  Cnv. if present Dn or Dnd Dnd Cn, Cnv, or S2n Dn or Dnd D3 Cn, Cnv, or S2n C3v Cn, Cnv, or S2n

D3 Point group? C3, C2 , no sh, no sv C3, no C2 , no sh, 3sv C3v C3, C2 , no sh, 3sv D3d

S2n ?

Cn or S2n? If there is an S2n axis colinear with Cn then S2n otherwise Cn. C2, no C2 , no sh, no sv, S4 colinear with C2 S4 C2, no C2 , no sh, no sv, no S2n colinear with C2 C2

D classifications C classifications General Case: Look for nC2 axes perpendicular to the highest order Cn axis Dnh or Dnd or Dn Cnh or Cnv or S2n or Cn Sub categories horizontal plane of symmetry perpendicular to Cn if present. n vertical planes, containing principal axis, if present S2n axis, if present. If above criteria not met Dnh Dnd NA Dn Cnh Cnv S2n Cn Notes Vertical planes contain the highest order Cn axis. In the Dnd group the planes are designated dihedral as they lie between the C2 axes. Simply having a Cn axis does not guarantee that a molecule will be D or C. Other possigilities include Td, Oh, Ih, and related groups. If doubts persist consult the character tables.

The point groups of common molecular shapes D∞h C2v D3h C2v C3v

The point groups of common molecular shapes Td D4h C2v

The point groups of common molecular shapes C4v D3h

The point groups of common molecular shapes Oh D4h D3d

Character Tables Group theory makes use of the properties of matrices Each symmetry 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 Leaves z unchanged = Results of transformation. Transformation matrix Original coordinates

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 + 1 * y + 0 * z = y

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):

Operations Applied to Functions - 1 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.

Plots of Functions 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

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. X3 is an eigenfunction of s with eigenvalue -1

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.

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.

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

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

Different ways that objects can behave for a group consisting of E and the reflection plane. Basis of Irreducible Representations. Cs E sh A’ A” 1 1 -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’ pz orbital has a multiplicative factor of z times a spherical factor. Behaves as A”.

sp Hybrids Reflection Do not get the same hybrid back. hybrid The two hybrids form the basis of a reducible representation of the operation z 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.

Reducible Representations Use the two spz 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 + 0 = 0 1 + 1 = 2

Representations Cs E sh A’ A” 1 1 -1 x, y,Rz z, Rx,Ry x2,y2,z2,xy 1 -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 2 0 (h1, h2) Note that G = A’ + A”

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 irreducible representation since they show what the s operator does. Both x2 and 4 are unchanged by the s operator. sx2 = x2 and s4 = 4. Eigenfunctions with eigenvalue +1 4x becomes multiplied by -1 as a result of the reflection sx = -x eigenvalue =1 We can reduce the reducible representation to a combination of irreducible reps. f(x) = x2 + 4 – 4x g(x) = x2 + 4 + 4x F and g are reducible, x2,4 are irreducible of sym A, x is irreducible of sym A’

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

Each row is an irreducible representation Character Table Symmetry operations Point group Characters +1 symmetric behavior -1 antisymmetric Mülliken symbols Each row is an irreducible representation

Symmetry of translations (p orbitals) 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