Chemistry Reference
In-Depth Information
where 1 and 6 are now adjacent atoms. By expanding the determinant we
obtain a sixth degree equation in x that can be easily factorized into the
three quadratic equations:
2
x
6
6x
4
9x
2
x
2
x
2
D
6
¼
þ
4
¼ð
4
Þð
1
Þ
¼
0
ð
2
:
283
Þ
with the roots, written in ascending order:
x
¼
2
;
1
;
1
;
1
;
1
;
2
ð
2
:
284
Þ
Because of the high symmetry of the molecule, two levels are now
doubly degenerate. The calculation of the MO coefficients can be done
using elementary algebraic methods in solving the linear homogeneous
system corresponding to Equation (2.282). With reference to Fig-
ure 2.29, a rather lengthy calculation (Section 2.9.3) shows that the
real MOs are:
<
1
p ðx
1
þx
2
þx
3
þx
4
þx
5
þx
6
Þ
f
1
¼
1
2
ðx
1
x
3
x
4
þx
6
Þ/
f
2
¼
x
1
f
3
¼
1
p ðx
1
þ
2
x
2
þx
3
x
4
2
x
5
x
6
Þ/
y
ð
2
:
285
Þ
:
1
x
2
y
2
1
p
ðx
1
f
4
¼
2
x
2
þx
3
þx
4
2
x
5
þx
6
Þ/
1
2
ðx
1
x
3
þx
4
x
6
Þ/
f
5
¼
xy
1
p ðx
1
x
2
þx
3
x
4
þx
5
x
6
Þ:
f
6
¼
The first degenerate MOs
35
f
3
transform like (x, y) and are
bonding MOs (HOMOs), the second degenerate MOs
f
2
and
f
4
and
f
5
trans-
form like (x
2
y
2
, xy) and are antibonding MOs (LUMOs).
35
Loosely speaking, we attribute to MOs a property (degeneracy) of energy levels.
