Chemistry Reference
In-Depth Information
Optimization of the linear coefficients in this simple H
€
uckel scheme
including overlap gives the 2
2 pseudosecular equation:
¼
a
1
«b
«
S
0
ð
2
:
5
Þ
b«
S
a
2
«
where
a
1
<
0
; a
2
<
0 are atomic integrals specifying the energy levels of
AOs
x
1
and
x
2
;
b<
0 the bond integral describing formation of a bond
between
x
1
and
x
2
;
ð
d
r x
1
ðrÞx
2
ðrÞ¼hx
1
jx
2
i
S
¼
ð
2
:
6
Þ
the overlap integral giving the superposition between the normalized AOs
x
1
and
x
2
. S depends in an exponentially decreasing way on the internu-
clear distance R between atoms A and B. It is important to note that
b
depends on S and that no bond can be formed between AOs for which
S
¼
0 by symmetry.
According to Equations (1.26) and (1.27) with:
A
11
¼ a
1
;
A
22
¼ a
2
;
A
12
¼
A
21
¼ b; l
¼ «
1
; l
¼ «
2
ð
2
:
7
Þ
1
2
solution of Equation (2.5) gives the real roots:
a
1
þa
2
2
b
S
D
«
1
¼
ð
2
:
8
Þ
2
ð
1
S
2
Þ
a
1
þa
2
2
b
S
þD
«
2
¼
ð
2
:
9
Þ
2
ð
1
S
2
Þ
with:
2
1
=
2
D ¼½ða
2
a
1
Þ
þ
4
ðba
1
S
Þðba
2
S
Þ
>
0
ð
2
:
10
Þ
The roots
«
i
of the pseudosecular equation are called molecular orbital
energies, while the differences
a
i
are assumed to give the
contribution of the ithMO to the bond energy. The energy of the chemical
bond will, in general, depend on
b; a
1
D«
i
¼ «
i
; a
2
, and S. The solutions become
particularly simple in the two cases schematically shown in Figure 2.1.
If
a
1
¼ a
2
¼ a
, we have degeneracy of the atomic levels, and we obtain
for orbital energies and MOs the following results:
a
þ
b
1
b
a
S
1
«
1
¼
S
¼ aþ
ð
:
Þ
2
11
þ
þ
S
