Chemistry Reference
In-Depth Information
the bond energy becomes:
8
<
2S
2
jba
S
j
D
E
ðu ¼
0
Þ¼
S
2
ðba
S
Þ
1
1
S
2
2S
2
ðba
S
Þ
¼
S
2
ðba
S
Þþ
ð
:
Þ
2
103
1
1
S
2
:
2
b
a
S
1
¼
þ
S
as it must be for homonuclear bonding. Hence, the correction term in
Equation (2.100) is essential in order to avoid overestimation of the bond
energy. This is in agreement with the well- known asymmetric splitting
of the MO levels occurring in H
0, where nonortho-
gonality of the basic AOs yields a bonding level less bonding, and an
antibonding level more antibonding, than those of the symmetric splitting
occurring for S
€
uckel theory for S
=
0.
As far as the bonding term in Equation (2.100)
¼
is concerned,
Equation (2.81) shows that
D
, in turn, depends: (i) on the atomic energy
difference
ða
B
a
A
Þ
; and (ii) on the product of bond energy integrals,
cos
2
u
, arising from the exchange-overlap densities
ðba
A
S
Þðba
B
S
Þ
Sa
2
(
r
)] on A and [b(
r
)a(
r
)
Sb
2
(
r
)] on B, respectively, and
[a(
r
)b(
r
)
which contains all dependence of
on the orientation
u
. So, it is apparent
that theMOdescription of bonding and of its directional properties in the
general case B
D
A involves a rather complicated dependence (through
the square root defining
=
) on such exchange-overlap densities. On the
other hand, both factors above contribute to the determination of the
polarity parameter
D
l
of the bonding MO
f
(Magnasco, 2003):
þlx
B
1
b
A
p
f ¼
ð
2
:
104
Þ
2
þl
þ
2
l
Scos
u
Rather than from the homogeneous system corresponding to the
pseudosecular equation (2.77), it is convenient to obtain
for the lowest
eigenvalue
«
as the appropriate solution of the quadratic equation
14
:
ðba
B
S
l
2
Þ
cos
u l
ða
B
a
A
Þlðba
A
S
Þ
cos
u ¼
0
ð
2
:
105
Þ
s
s
Dða
B
jba
A
S
ða
B
a
A
ÞD
a
A
Þ
j
l ¼
cos
u
¼
ð
2
:
106
Þ
ðba
B
S
Þ
Dþða
B
a
A
Þ
jba
B
S
j
2
14
Arising from the matrix formulation of the full 2
2 non-orthogonal eigenvalue problem
(Magnasco, 2007).
