1. Molecular Symmetry Symmetry Elements Group Theory
Photoelectron Spectra
Molecular Orbital (MO) Diagrams for Polyatomic Molecules

2. The Symmetry of Molecules
The shape of a molecule influences its physical properties, reactivity, and its spectroscopic behavior
Determining the symmetry of a molecule is fundamental to gaining insight into these characteristics of molecules
The chemist’s view of symmetry is contained in the study of group theory
This branch of mathematics classifies the properties of a molecule into groups, defined by the symmetry of the molecule
Each group is made up of symmetry elements or operations, which are essentially quantum operators disguised as matrices
Our goal is to use group theory to build more complex molecular orbital diagrams
CHEM 3722
Symmetry

3. Symmetry Elements You encounter symmetry every day
A ball is spherically symmetric
Your body has a mirror image (the left and right side of your body)
Hermite polynomials are either even (symmetric on both sides of the axis) or odd (symmetric with a twist).
Symmetry operations are movements of a molecule or object such that after the movement the object is indistinguishable from its original form
Symmetry elements are geometric representations of a point, line, or plane to which the operation is performed
Identity element (E)
Plane of reflection (s)
Proper rotation (Cn)
Improper rotation (Sn)
Inversion (i)
CHEM 3722
Symmetry

4. Symmetry Elements II Identity
If an object (O) has coordinates (x,y,z), then the operation E(x,y,z) (x,y,z)
The object is unchanged E
Plane of reflection
s(xz) (x,y,z) = (x,-y,z)
s(xy) (x,y,z) = (x,y,-z)
s(yz) (x,y,z) = (-x,y,z)
s(xz) z y
s(xy) x
CHEM 3722
Symmetry

5. How many C2 operations are there for benzene?
Symmetry Elements III
Proper Rotation
Cn where n represents angle of rotation out of 360 degrees
C2 = 180°, C3 = 120°, C4 = 90° …
C2(z) (x,y,z) = (-x,-y,z) C6 C6 C3
How many C2 operations are there for benzene? C6 C2
CHEM 3722
Symmetry

6. Symmetry Elements IV Inversion Improper Rotation
Takes each point through the center in a straight line to the exact distance on the other side of center
i (x,y,z) = (-x,-y,-z) 1 4 6 2 3 5 5 3 2 6 4 1
Improper Rotation
A two step operation that first does a proper rotation and then a reflection through a mirror plane perpendicular to the rotational axis.
S4(z) (x,y,z) = (y,-x,-z)
Same as (s)(C4) (x,y,z)
Note, symmetry operations are just quantum operators: work from right to left 1 6 6 2 5 1
sh C6 = S6 5 3 4 2 4 3
CHEM 3722
Symmetry

7. A Few To Try
Determine which symmetry elements are applicable for each of the following molecules
CHEM 3722
Symmetry

8. Point Groups
We can systematically classify molecules by their symmetry properties
Call these point groups
Use the flow diagram to the right
Special groups:
a) Cv,Dh (linear groups)
b) T, Th, Td, O, Oh, I, Ih
(1)
Start
(2)
No proper or improper axes
(3)
Only Sn (n = even) axis: S4, S6…
Cn axis
(4)
(5)
No C2’s  to Cn
n C2’s  to Cn sh
n sv’s
No s’s sh
n sd’s
No s’s
Cnh
Cnv Cn
Dnh
Dnv Dn
CHEM 3722
Symmetry

9. Some Common Groups D3h Point Group C3v Point Group C4h Point Group
AB3
D3h Point Group
C3, C32, 3C2, S3, S35, 3sv, sh
Trigonal planar
C3v Point Group
C3, 3sv
Trigonal pyramid
Trigonal bi-pyramid
C4h Point Group
C2, 2C2’, 2C2’’, C4, S4, S42, 2sv, 2sd, sh, i
Square planar
C4v Point Group
C2, C4, C42, 2sv, 2sd
Square pyramid
AB3
AB5
AB4
AB5
CHEM 3722
Symmetry

10. Character Tables
Character tables hold the combined symmetry and effects of operations
For example, consider water (C2v)
Point Group
Symmetry operations available
Effect of this operation on an orbital of this symmetry.
Coordinates and rotations of this symmetry
Symmetry of states
A, B = singly degenerate
E = doubly degenerate
T = triply degenerate
1 = symmetric to C2 rotation*
2 = antisymmetric to C2 rotation*
CHEM 3722
Symmetry

11. The Oxygen’s px orbital
For water, we can look at any orbital and see which symmetry it is by applying the operations and following the changes made to the orbital
If it stays the same, it gets a 1
If it stays in place but gets flipped, -1
If it moves somewhere else, 0 1 -1
The px orbital has B1 symmetry
CHEM 3722
Symmetry

12. The Projection Operator
Since we want to build MO diagrams, our symmetry needs are simple
Only orbitals of the same symmetry can overlap to form bonds
Each point group has many possible symmetries for an orbital, and thus we need a way to find which are actually present for the particular molecule of that point group
We’ll also look at collections of similar atoms and their collective orbitals as a group
The projection operator lets us find the symmetry of any orbital or collection of orbitals for use in MO diagrams
It will also be of use in determining the symmetry of vibrations, later
We just did this for the px orbital for water’s oxygen atom
It’s functional form is
But it’s easier to use than this appears
CHEM 3722
Symmetry

13. Ammonia Let’s apply this mess to ammonia
First, draw the structure and determine the number of s and p bonds
Then look at how each bond changes for the group of hydrogen atoms, building a set of “symmetry adapted linear combinations of atomic orbitals” (SALC) to represent the three hydrogen's by symmetry (not by their individual atomic orbitals)
Finally, we’ll compare these symmetries to those of the s and p atomic orbitals of the nitrogen to see which overlap, thus building our MO diagram from the SALC
3 sigma bonds and no pi bonds, thus we’ll build our SALC’s from the projection operator and these 3 MO’s
Note, ammonia is in C3v point group (AB3)
CHEM 3722
Symmetry

14. Ammonia II The Character Table for C3v is to the right
Take the s bonds through the operations & see how many stay put
We now have a representation (G) of this group of orbitals that has the symmetry
(3)
(0)
(1)
CHEM 3722
Symmetry

15. Ammonia III
This “reducible representation” of the hydrogen’s s-bonds must be a sum of the symmetries available
Only one possible sum will yield this reducible representation
By inspection, we see that Gs = A1 + E
From the character table, we now can get the symmetry of the orbitals in N
N(2s) = A1 -- x2 + y2 is same as an s-orbital
N(2pz) = A1
N(2px) = N(2py) = E
So, we can now set up the MO diagram and let the correct symmetries overlap
CHEM 3722
Symmetry

16. Ammonia, The MO Diagram s* 2p’s 2s 3A1 2Ex 2Ey A1, Ex, Ey 2A1 Ex, Ey
CHEM 3722
Symmetry

17. Methane
The usual view of methane is one where four equivalent sp3 orbitals are necessary for the tetrahedral geometry
sp3-s overlap for a s MO
sp3-s overlap for a s MO
Photoelectron spectrum (crude drawing) adapted from Roy. Soc. Chem., Potts, et al.
However, the photoelectron spectrum shows two different orbital energies with a 3:1 population ratio
Maybe the answer lies in symmetry
Let’s build the MO diagram using the SALC method we saw before
CHEM 3722
Symmetry

18. Methane: SALC Approach
Methane is a tetrahedral, so use Td point group
The character table is given below
To find the SALC’s of the 4H’s, count those that do not change position for each symmetry operation and create the reducible representation, GSALC.
GSALC = A1 + T2
The character table immediately gives us the symmetry of the s and p orbitals of the carbon:
C(2s) = A1
C(2px, 2py, 2pz) = T2
CHEM 3722
Symmetry

19. The MO Diagram of Methane
Using the symmetry of the SALC’s with those of the carbon orbitals, we can build the MO diagram by letting those with the same symmetry overlap.
s7*
s8*
s9* T2 A1
s6*
2px
2py
2pz T2 1s
A1 + T2 2s A1 s3 s4 s5 T2 s2 A1 s1 1s A1 A1
C CH H’s
CHEM 3722
Symmetry

20. An Example: BF3
BF3 affords our first look at a molecule where p-bonding is possible
The “intro” view is that F can only have a single bond due to the remaining p-orbitals being filled
We’ll include all orbitals
The point group for BF3 is D3h, with the following character table:
We’ll begin by defining our basis sets of orbitals that do a certain type of bonding
CHEM 3722
Symmetry

21. Worksheet for reducing Gss
F3 Residue Basis Sets
Looking at the 3 F’s as a whole, we can set up the s-orbitals as a single basis set:
Regular character table
Worksheet for reducing Gss
CHEM 3722
Symmetry

22. The Other Basis Sets
CHEM 3722
Symmetry

23. No s-orbital interaction from F’s is included in this MO diagram!
The MO Diagram for BF3
No s-orbital interaction from F’s is included in this MO diagram!
CHEM 3722
Symmetry

24. The MO Diagram for BF3 s-orbital interaction from F’s allowed
CHEM 3722
Symmetry

25. Using Hybrid Orbitals for BF3
If we use sp2 hybrids and the remaining p-orbital (pz) of the boron, we see how hybridization yields the same exact picture.
Build our sp2 hybrids and take them through the operations
Find the irreducible representations using the worksheet method and the reducible representation
pz is found in the character table to be A2”.
Result: identical symmetries for boron’s orbital’s in both cases
This is how it should be, since hybridization is an equivalent set of orbitals that are simply oriented in space differently.
CHEM 3722
Symmetry

26. p-Bonding in Aromatic Compounds
The -bonding in C3H3+1 (aromatic)
1 node
0 nodes
The -bonding in C4H4+2 (aromatic)
2 nodes
1 node
Aromatic compounds must have a completely filled set of bonding -MO’s.
This is the origin of the Hückel (4N+2) -electron definition of aromaticity.
0 nodes
CHEM 3722
Symmetry

27. Cyclopentadiene
As the other examples showed, the actual geometric structure of the aromatic yields the general shape of the p-MO region
The -bonding in C5H5-1 (aromatic)
2 nodes
1 node
0 nodes
CHEM 3722
Symmetry

28. Benzene The -bonding in C6H6 (aromatic) 3 nodes 2 nodes 1 node
CHEM 3722
Symmetry