Chemistry Reference
In-Depth Information
great electron transitions from the low-energy levels to the higher-energy ones. This
diene synthesis can also take place photochemically, but the process is energetically
disadvantaged (Simon
1973
, pp. 29-53; Avram
1994
, pp. 349-353).
In the case of propene with
σ
π
conjugation (hyperconjugation) the following
parametric scheme is chosen (Nenitzescu
1966
, p. 70; Streitwieser
1961
):
−
α
C
=
β
α
−
0
.
1
β
C
α
−
0
.
2
β
C
α
−
0
.
5
β
H
3
0
.
7
β
≡
(11.42)
The algorithm related to quantum-mechanics calculations leads to the system of
homogeneous equations beside the normalization condition of the molecular orbitals:
⎧
⎨
c
1
(
α
−
ε
)
+
c
2
β
=
0
c
1
β
+
c
2
(
α
−
0
.
1
β
−
ε
)
+
c
3
0
.
7
β
=
0
(11.43)
c
2
0
.
7
β
+
c
3
(
α
−
0
.
2
β
−
ε
)
+
c
4
2
β
=
0
⎩
c
3
2
β
+
c
4
(
α
−
0
.
5
β
−
ε
)
=
0
c
1
+
c
2
+
c
3
+
c
4
=
1
Considering abbreviation as in (
11.43
) we obtain the Hückel secular determinant:
K
1
0
0
1(K
−
0.1)
0.7
0
=
0
(11.44)
0
0.7
(K
−
0.2)
2
0
0
2
(K
−
0.5)
which is successively rewritten till obtaining the simple equation:
(K
2
−
0.1K)[(K
−
0.2)(K
−
0.5)
−
4]
−
0.7K[0.7(K
−
0.5)]
−
−
−
−
=
[(K
0.2)(K
0.5)
4]
0
K
4
0.8K
3
5.32K
2
−
−
+
1.335K
+
3.9
=
0
(11.45)
Using the standard algebraic computational analysis, we obtain the solution values of
the secular equation and of the coefficient sets of the molecular orbitals (Table
11.4
).
Based on Table
11.4
data, the structural indices are calculated as follows:
charge densities
2[0
.
1878
2
0
.
7073
2
]
ρ
1
=
+
=
1
.
0709
2[0
.
3475
2
0
.
5938
2
]
ρ
2
=
+
=
0
.
9465
2[0
.
6996
2
0
.
2135)
2
]
ρ
3
=
+
(
−
=
1
.
0699
2[0
.
5954
2
0
.
3188)
2
]
ρ
4
=
+
−
=
(
0
.
9122
(11.46)
