MIT OpenCourseWare
http://ocw.mit.edu
5.04 Principles of Inorganic Chemistry II ��
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
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
+ φ 2 + φ 3 + φ 4 + φ5 + φ 6 )
E ⎛⎜ψ A1g ⎞⎟ =
∫
ψ A1g H ψ A1gdτ =
ψ A1g | H | ψ A1g
⎠
⎝
=
=
1
6
(φ
+ φ2 + φ3 + φ4 + φ5 + φ6
1
)
(φ
1
H
+ φ2 + φ3 + φ4 + φ5 + φ6
1
6
)
(
1
H11 + H12 + H13 + H14 + H15 + H16 + H21 + H22 + H23 + H24 + H25 + H26
6
↓
↓
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
E ⎛⎜ψ A1g ⎞⎟ = (6)(α + 2β ) = α + 2β
⎝
⎠ 6
The energy of the LCAO, ψB2g
E ⎛⎜ψ B2g ⎞⎟ = ψ B2g | H | ψ B2g
⎝
⎠
=
=
1
6
(φ
1
)
− φ2 + φ3 − φ4 + φ5 − φ6 | H |
1
6
(φ
1
− φ2 + φ3 − φ4 + φ5 − φ6
)
(
1
H11 − H12 + H13 − H14 + H15 − H16 + H2i (i = 1 − 6) + H3i + H4i + H5i + H6i
6
↓
α
↓
β
↓
β
↓
α–β
↓
↓
↓
α–2β
α–2β
α–2β
α–2β
1
E ⎛⎜ψ B2g ⎞⎟ = (6)(α − 2β ) = α − 2β
⎠ 6
⎝
5.04, Principles of Inorganic Chemistry II
Prof. Daniel G. Nocera
↓
)
Lecture 7
Page 1 of 4
The energies of the remaining LCAO’s are:
⎛
⎞
⎛
⎞
E
⎜⎜ψ a ⎟⎟ =E
⎜⎜ψ b ⎟⎟ =
α +
β
⎝ E1g ⎠
⎝ E1g ⎠
⎞
⎛
⎞
⎛
E
⎜⎜ψ a ⎟⎟ = E
⎜⎜ψ b ⎟⎟ =
α − β
⎝ E2u ⎠
⎝ E2u ⎠
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),
–2
B2g
–1
E2u
E/
0
1
E1g
2
A2u
6 π bonding electrons
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,
ψ 1(A) =
ψ 2 (B) =
1
2
1
2
(φ1 + φ2 )
(φ1 − φ2 )
5.04, Principles of Inorganic Chemistry II
Prof. Daniel G. Nocera
Lecture 7
Page 2 of 4
E(ψ 1 ) =
E(ψ 2 ) =
1
2
1
2
(φ
1
(φ
1
)
1
)
1
+ φ2 | H |
− φ2 | H |
(φ
+ φ2
)
=
1
(2α + 2β ) = β
2
(φ
− φ2
)
=
1
(2α − 2β ) = − β
2
1
2
2
1
E/
The 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:
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
cyclohexatriene
Etotal
benzene
The bond order is given by,
B.O. = ∑ necic j
coefficients of electron i and
electron j in a given bond
i, j
orbital e– occupancy 5.04, Principles of Inorganic Chemistry II
Prof. Daniel G. Nocera
Lecture 7
Page 3 of 4
Consider the B.O. between the C1 and C2 carbons of benzene
⎛
⎞⎛
⎞
[ ψ (A )] =
2⎜⎜
1
6
⎟⎟⎜⎜ 1
6
⎟⎟
1
2u
=
1
3
⎝
⎠⎝
⎠
⎛
⎞⎛
⎞
⎡ ψ (E a )⎤ =
2⎜ 2
⎟⎜ 1
⎟ =
1
⎢⎣ 3 1g ⎥⎦
⎜
12 ⎟⎜ 12 ⎟
3
⎝
⎠⎝
⎠
⎡ ψ (E b )⎤ =
1
0 ⎛⎜ 1
⎞⎟
=
0
⎢⎣ 4 1g ⎥⎦
2
⎜⎝ 2
⎟
⎠
2
3
()
5.04, Principles of Inorganic Chemistry II
Prof. Daniel G. Nocera
Lecture 7
Page 4 of 4