MO Diagrams for
More Complex Molecules
Chapter 5
Wednesday, October 14, 2015
Boron trifluoride
1. Point group D3h
2.
x y
z
3. Make reducible reps for outer atoms
z x
y
y z
x
4. Get group
orbital
symmetries by
reducing each Γ
Γ2s
3
0
1
3
0
1
Γ2pz
3
0
-1
-3
0
1
Γ2px
3
0
-1
3
0
-1
Γ2py
3
0
1
3
0
1
Γ2s = A1’ + E’
Γ2pz = A2’’ + E’’
Γ2px = A2’ + E’
Γ2py = A1’ + E’
Boron trifluoride
Γ2s = A1’ + E’
Γ2pz = A2’’ + E’’
Γ2px = A2’ + E’
Γ2py = A1’ + E’
What is the shape of the group orbitals?
2s:
A1’
?
?
E’(y)
E’(x)
Which combinations of the
three AOs are correct?
The projection operator method provides a systematic way to find
how the AOs should be combined to give the right group orbitals
(SALCs).
BF3 - Projection Operator Method
In the projection operator method, we pick one AO in each set of
identical AOs and determine how it transforms under each
symmetry operation of the point group.
Fa
Fb
Fc
σh
S3
S32
Fb
Fa
Fb
Fc
Fa
Fc
Fb
1
1
1
1
1
1
1
C32 C2(a) C2(b) C2(c)
AO
E
C3
Fa
Fa
Fb
Fc
Fa
Fc
A1’
1
1
1
1
1
σv(a) σv(b) σv(c)
A 1 ’ = Fa + F b + F c + F a + Fc + Fb + Fa + F b + F c + Fa + F c + F b
A1’ = 4Fa + 4Fb + 4Fc
A1’
The group orbital wavefunctions are determined by multiplying the
projection table values by the characters of each irreducible
representation and summing the results.
BF3 - Projection Operator Method
In the projection operator method, we pick one AO in each set of
identical AOs and determine how it transforms under each
symmetry operation of the point group.
Fa
Fb
Fc
σh
S3
S32
Fb
Fa
Fb
Fc
Fa
Fc
Fb
-1
1
1
1
-1
-1
-1
C32 C2(a) C2(b) C2(c)
AO
E
C3
Fa
Fa
Fb
Fc
Fa
Fc
A2’
1
1
1
-1
-1
σv(a) σv(b) σv(c)
A 2 ’ = Fa + F b + F c – Fa – Fc – Fb + Fa + F b + F c – Fa – Fc – Fb
A2 ’ = 0
There is no A2’ group orbital!
The group orbital wavefunctions are determined by multiplying the
projection table values by the characters of each irreducible
representation and summing the results.
BF3 - Projection Operator Method
In the projection operator method, we pick one AO in each set of
identical AOs and determine how it transforms under each
symmetry operation of the point group.
Fa
Fb
Fc
σh
S3
S32
Fb
Fa
Fb
Fc
Fa
Fc
Fb
0
2
-1
-1
0
0
0
C32 C2(a) C2(b) C2(c)
AO
E
C3
Fa
Fa
Fb
Fc
Fa
Fc
E’
2
-1
-1
0
0
σv(a) σv(b) σv(c)
E’ = 2Fa – Fb – Fc + 0 + 0 + 0 + 2Fa – Fb – Fc + 0 + 0 + 0
E’ = 4Fa – 2Fb – 2Fc
E’(y)
The group orbital wavefunctions are determined by multiplying the
projection table values by the characters of each irreducible
representation and summing the results.
BF3 - Projection Operator Method
Γ2s = A1’ + E’
Γ2pz = A2’’ + E’’
Γ2px = A2’ + E’
Γ2py = A1’ + E’
What is the shape of the group orbitals?
2s:
A1’
?
?
E’(y)
E’(x)
We can get the third group orbital, E’(x), by using normalization.
𝜓 2 𝑑𝜏 = 1
Normalization condition
BF3 - Projection Operator Method
Let’s normalize the A1’ group orbital:
𝜓𝐴′1 = 𝑐a [𝜙(2𝑠Fa ) + 𝜙(2𝑠Fb ) + 𝜙(2𝑠Fc )]
𝜓 2 𝑑𝜏 = 1
𝑐a2
𝑐a2
𝑐a2
A1’ wavefunction
Normalization condition for group orbitals
[𝜙(2𝑠Fa ) + 𝜙(2𝑠Fb ) + 𝜙(2𝑠Fc )]2 𝑑𝜏 = 1
𝜙 2 (2𝑠Fa )𝑑𝜏 +
1+1+1 =1
𝜙 2 (2𝑠Fb )𝑑𝜏 +
𝑐a =
So the normalized A1’ GO is: 𝜓𝐴′1 =
nine terms, but the six
overlap (S) terms are zero.
𝜙 2 (2𝑠Fc )𝑑𝜏 = 1
1
3
1
3
[𝜙(2𝑠Fa ) + 𝜙(2𝑠Fb ) + 𝜙(2𝑠Fc )]
BF3 - Projection Operator Method
Now let’s normalize the E’(y) group orbital:
𝜓𝐸′ (𝑦) = 𝑐a [2𝜙(2𝑠Fa ) − 𝜙(2𝑠Fb ) − 𝜙(2𝑠Fc )]
𝑐a2
[2𝜙(2𝑠Fa ) − 𝜙(2𝑠Fb ) − 𝜙(2𝑠Fc )]2 𝑑𝜏 = 1
𝑐a2 4
𝑐a2
𝜙 2 (2𝑠Fa )𝑑𝜏 +
4+1+1 =1
𝜙 2 (2𝑠Fb )𝑑𝜏 +
𝑐a =
E’(y) wavefunction
nine terms, but the six
overlap (S) terms are zero.
𝜙 2 (2𝑠Fc )𝑑𝜏 = 1
1
6
So the normalized E’(y) GO is: 𝜓𝐸′ (𝑦) =
1
6
[2𝜙(2𝑠Fa ) − 𝜙(2𝑠Fb ) − 𝜙(2𝑠Fc )]
BF3 - Projection Operator Method
𝜓𝐴′1 =
1
3
𝜓𝐸′ (𝑦) =
[𝜙(2𝑠Fa ) + 𝜙(2𝑠Fb ) + 𝜙(2𝑠Fc )]
1
6
[2𝜙(2𝑠Fa ) − 𝜙(2𝑠Fb ) − 𝜙(2𝑠Fc )]
𝒄𝟐𝒊 𝐢𝐬 the probability of finding an electron in 𝝓𝒊 in a group orbital,
so 𝒄𝟐𝒊 = 𝟏 for a normalized group orbital.
So the normalized E’(x) GO is:
𝜓𝐸′ (𝑥) =
1
2
[𝜙(2𝑠Fb ) − 𝜙(2𝑠Fc )]
BF3 - Projection Operator Method
Γ2s = A1’ + E’
Γ2pz = A2’’ + E’’
Γ2px = A2’ + E’
Γ2py = A1’ + E’
What is the shape of the group orbitals?
notice the GOs are
orthogonal (S = 0).
?
2s:
A1’
E’(y)
E’(x)
Now we have the symmetries and wavefunctions of the 2s GOs.
We could do the same analysis to get the GOs for the px, py, and pz
orbitals (see next slide).