Biomedical Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
0
1
2
3
4
5
6
Distance/atomic unit
Figure 15.3
Potential energy curves for lowest states of
H
2
+
Table 15.1
Hydrogen molecule ion results
D
e
/eV
R
e
/pm
Experiment
2.791
106
Simple LCAO with
ζ
=
1
1.76
132.3
Simple LCAO, best
ζ
=
1.238
2.25
106
James, elliptic
2.772
106
where the orbital exponent ζ
1 for a hydrogen atom, 2 for He
+
and so on. The next step
is to find the best value of ζ that is appropriate for a hydrogen atom within a molecule. The
best value turns out to be 1.238 showing that the 1s orbital contracts a little on molecule
formation. Table 15.1 shows better agreement with experiment.
=
15.3 Elliptic Orbitals
The best orbitals that have been obtained for H
2
+
were found by a different approach; the
natural coordinates to use are the elliptic coordinates μ, ν, φ and James (1935) found that
a good approximation to the lowest orbital is
δμ )
1
c
ν
2
ψ
=
exp (
−
+
(15.8)
His results are shown in Table 15.1.
