Biomedical Engineering Reference
In-Depth Information
15.1 Hydrogen Molecule Ion, H
2
+
The simplest molecule is the hydrogen molecule ion, H
2
+
, shown in Figure 15.1. It is
formed by passing an electric discharge through dihydrogen, and it has been well studied
experimentally. Its dissociation energy is known to be
D
e
=
269.6 kJ mol
−
1
and it has an
equilibrium bond length of 106 pm. There is only one well-established electronic state,
namely the ground state.
electron
r
A
B
R
A
R
B
Origin
Figure 15.1
Hydrogen molecule ion
First of all, in quantum mechanics just as in classical mechanics, we can rigorously
separate off the translational motion of the molecule. That leaves us to concentrate on
the two nuclei (each of mass
m
p
=
1.673
×
10
−
27
kg) and the electron (of mass
m
e
=
9.109
10
−
31
kg) about the centre of mass. The wavefunction therefore depends on the
coordinates of the electron (
r
) and the two nuclei (
R
A
and
R
B
):
×
Ψ
tot
=
Ψ
tot
(
R
A
,
R
B
,
r
)
If we regard such a system from the viewpoint of classical mechanics, we would be
tempted to try and separate the motions of the nuclei and the electron, because of the great
difference in their masses (a factor of 1:1836). This does not mean that the motions are
truly separable like the molecular translational motion of the centre of mass and the relative
motion of the nuclei and electrons within the molecule, more that the separation can be
done with only a small error.
Born and Oppenheimer (1927) first investigated this possibility and wrote
Ψ
tot
(
R
A
,
R
B
,
r
)
=
Ψ
nuc
(
R
A
,
R
B
) ψ
e
(
R
A
,
R
B
,
r
)
I am using a convention that lower case ψ refers to one electron, whilst upper case Ψ refers
to many electrons. In this case there is only one electron and so the electronic wavefunction
is technically an
orbital
; in this case a
molecular orbital
(MO). Born and Oppenheimer
showed that the approximation was good to almost (but not exactly) the ratio of the particle
masses and so we normally glue the nuclei to fixed positions in space and concentrate on
the electron(s) when investigating many problems in molecular electronic structure theory.
