Biomedical Engineering Reference
In-Depth Information
15.5 Dihydrogen MO Treatment
The MO treatment was given by Hellmann (1937) amongst others. He took the lowest
energy MO as a combination of hydrogen 1s orbitals, as for H
2
+
,
1
√
2 (1
ψ
+
=
S
)
(1
s
A
+
1
s
B
)
+
and wrote
1
2
ψ
+
(
r
1
) α (
s
1
)
ψ
+
(
r
1
) β (
s
1
)
D
7
=
ψ
+
(
r
2
) α (
s
2
)
ψ
+
(
r
2
) β (
s
2
)
!
The calculated bond length and dissociation energy are in poorer agreement with exper-
iment than those obtained from the simple VB treatment (Table 15.3), and this puzzled
many people at the time. It also led them to believe that the VB method was the correct
way forward for the description of molecular structure; in the event, advances in com-
puter technology and numerical linear algebra have proved them wrong, but that is for a
later chapter.
Table 15.3
Dihydrogen calculations
Comment
D
e
/eV
R
e
/
a
0
Experiment
4.72
1.40
Simple VB
3.14
1.64
Simple MO
2.65
1.60
James and Coolidge
4.698
1.40
But why is the simple MO treatment of dihydrogen so poor? If we expand
D
7
in terms
of the atomic orbitals 1s
A
and 1s
B
we find (apart from the normalizing constant)
D
7
=
+
+
+
(1
s
A
(
r
1
) 1
s
B
(
r
2
)
1
s
B
(
r
1
) 1
s
A
(
r
2
)
1
s
A
(
r
1
) 1
s
A
(
r
2
)
1
s
B
(
r
1
) 1
s
B
(
r
2
))
×
(
α
(
s
1
)
β
(
s
2
)
−
α
(
s
2
)
β
(
s
1
))
which is equal to the simple VB wavefunction but with ionic terms included and weighted
equally to the covalent terms. The solution to the problem is to include excited states in
the wavefunction, just as we did for helium, but with a variable weight. This process is
configuration interaction (CI). In the limit, once all the refinements aremade, the two refined
treatments (VB with CI and MO with CI) give exactly the same results and so there is no
particular reason to prefer one rather than the other. The starting points are different, but
not the refined end results.
To summarize, Figure 15.6 shows the potential energy curves for the VB, LCAO and CI
treatments. The full curve is the VB calculation, the dotted curve the MO and the dashed
curve is a CI treatment.
