Biomedical Engineering Reference
In-Depth Information
electron
r
A
r
B
φ
R
AB
A
B
Figure 15.2
Elliptic coordinates
where λ can take on positive and negative values, and j is the square root of
−
1. In the
limit as
R
AB
→
0, the quantum number λ becomes equivalent to the atomic quantum
number
m
l
.
Separation into a μ and a ν equation also proves possible, and the equations have
been solved by Teller (1930), by Burrau (1927) and by others leading to results in com-
plete agreement with experiment. Solution of the differential equations is far from easy
and the best references are Bates
et al.
(1953), Wind (1965) and of course Eyring
et al.
(1944).
15.2 LCAO Model
The hydrogenmolecule ion is unusual amongst molecules in that we can solve the electronic
problem exactly (by numerical methods). Once we consider polyelectron systems, we have
to seek approximate methods.Any chemist would argue that molecules are built from atoms
and so we should capitalize on this chemical knowledge by attempting to build molecular
wavefunctions from atomic ones.
Suppose we build the hydrogenmolecular ion starting from a hydrogen atom and a proton
initially separated by a large distance. The electronic wavefunction will be a hydrogen 1s
orbital until the proton is close enough to make any significant perturbation and so we
might guess that the molecular wavefunction should resemble an atomic 1s orbital, at least
near the appropriate nucleus.
We therefore guess that the low-energy molecular orbitals of H
2
+
might be represented as
=
c
A
1s
A
+
ψ
c
B
1s
B
where the coefficients
c
A
and
c
B
have to be determined. This technique is called the
linear
combination of atomic orbitals
(LCAO) and I have used the shorthand that 1s
A
is a hydrogen
1s orbital centred on nucleus A. In this particular case we can deduce the coefficients from
symmetry. According to the Born interpretation, ψ
2
dτ gives the chance that an electron
can be found in the volume element dτ . We have
=
c
A
1s
A
+
c
B
1s
B
dτ
ψ
2
dτ
2
c
A
c
B
1s
A
1s
B
+
