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5.04 Principles of Inorganic Chemistry II ��
Fall 2008
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5.04, Principles of Inorganic Chemistry II Prof. Daniel G. Nocera
Lecture 7: Hückel Theory 2 (Eigenvalues)
The energies (eigenvalues) may be determined by using the Hückel approximation.
ψ A1g = 1
6
(
φ1 +φ2 +φ3 +φ4 +φ5 +φ6)
E ⎛⎜⎝ψ A1g ⎞⎟⎠ =∫ψ A1g H ψ A1gdτ = ψ A1g| H |ψ A1g
= 1
6
(
φ1 +φ2 +φ3 +φ4 +φ5 +φ6)
H 16(
φ1 +φ2 +φ3 +φ4 +φ5 +φ6)
= 1 6
(
H 11 + H 12 + H 13 + H 14 + H 15 + H 16 + H 21 + H 22 + H 23 + H 24 + H 25 + H 26↓ ↓ ↓ ↓ ↓ ↓
α β β β α β
⎞⎟
⎟⎠ + H3i(i = 1 − 6) + H4i(i = 1 − 6) + H5i(i = 1 − 6) + H6i(i = 1 − 6)
↓ ↓ ↓ ↓
α+2β α+2β α+2β α+2β
= 1 6 (6)(α + 2β) =α + 2β E ψ A1g ⎞⎟⎠
⎞⎟⎠
The energy of the LCAO, ψB2g
⎛⎜⎝
B2g = ψ 2g | H |ψ 2g
⎛⎜⎝
⎛⎜⎝ E ψ
E ψ
B B
1 1
= 6
(
φ1 −φ 2 +φ 3 −φ 4 +φ 5 −φ6)
| H | 6(
φ1−φ 2+φ 3 −φ 4 +φ 5 −φ6)
= 6
1
(
H11 − H12 + H13 − H14 + H15 − H16 + H2i(i = 1 − 6) + H3i + H4i + H5i + H6i)
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
α β β α–β α–2β α–2β
α–2β α–2β
= 1 6 (6)(α − 2β) =α − 2β
⎞⎟⎠
B2g
The energies of the remaining LCAO’s are:
⎞⎟
⎠
⎞⎟
⎠ =
ψ ⎟ = ⎜ψ ⎟
Note the energies of the E orbitals are degenerate. Constructing the energy level diagram, we set α = 0 and β as the energy parameter (a negative quantity, so an MO whose energy is positive in units of β has an absolute energy that is negative),
B2g –2
–1 E2u
⎛⎜
⎝ ψ ⎛⎜⎜ψ
⎝
⎛⎜⎜
⎝
⎛ ⎜⎜
⎝
α β
E E a E b +
1g E1g
⎞ ⎞
α −β
E ⎟⎟ = E =
⎠ ⎟⎟
b⎠
E2u a E2u
E/ 0
1 E1g
6 π bonding electrons
2 A2u
The energy of benzene based on the Hückel approximation is Etotal = 2(2β) + 4(β) = 8β
What is the delocalization energy (i.e. π resonance energy)?
To determine this, we consider cyclohexatriene, which is a six-membered cyclic ring with 3 localized π bonds; in other terms, cyclohexatriene is the product of three condensed ethylene molecules. For ethylene,
Following the procedures outlined above, we find,
ψ1(A) = 1 (φ1 +φ2) 2
E(ψ1) = 1 2
(
φ 1+φ2)
| H | 1 2(
φ1+φ2)
= 1 2 (2α + 2β) = βE(ψ2) = 1 2
(
φ 1−φ2)
| H | 1 2(
φ1 −φ2)
= 1 2 (2α − 2β) = −βThe above was determined in the C2 point group. Correlating to D2h point group gives A in C2 → B1u in D2h and B in C2 → B2g in D2h:
E/
The Hückel energy of ethylene is, Etotal = 2(β) = 2β
Therefore, the energy of cyclohexatriene is 3(2β) = 6β. The resonance energy is therefore,
Eres(C6H6) = 8β – 6β = 2β
↓ ↓
Etotal Etotal
benzene cyclohexatriene
The bond order is given by,
coefficients of electron i and electron j in a given bond B.O. = ∑necicj
orbital e– occupancy
j ,i
Consider the B.O. between the C1 and C2 carbons of benzene
⎤ ⎥⎦
⎡⎣
⎡⎣
⎛⎜
⎜
⎛⎜
⎜
⎞⎟
⎟
⎞⎟
⎟
1 1 1
[
)]
)
)⎤⎥⎦
ψ1(A 2
⎝
= =
2u 6⎠⎝ 6⎠ 3
⎛⎜
⎜
⎛⎜
⎜
⎞⎟
⎟⎝
⎠
⎞⎟ 2 ⎟
12
1 1
⎢ ψ3(E1ga 2
⎝
= =
12 ⎠ 3
( )
0⎛⎜⎜⎝⎞⎟⎟
1 2
⎢ ψ4(E1gb = 1 = 0
2⎠
2 3
