Biomedical Engineering Reference
In-Depth Information
15
Simple Molecules
It is time to progress to the quantum theory of molecules, and it should come as no surprise
when I tell you that the orbital model is a good starting point for many of the calculations
we professionals do today.
Molecules are more complicated than atoms for the following reasons.
•
Whilst angular momentum considerations dominate the theory of atomic structure (it is
amazing what can be done with pencil and paper before any nasty integrals have to be
evaluated) the
L
2
and
L
z
operators do not commute with molecular Hamiltonians (except
for linear molecules; if the molecular axis is the
z
axis, then
L
z
does commute). To a
certain extent, molecular symmetry operators help us since they commute with molecular
Hamiltonians, but most molecules of any real chemical interest have no symmetry apart
from the identity operation. So we are stuck with the Hamiltonian, and nothing to help
us simplify the eigenvalue problem apart from electron spin. The Schrödinger equation
for a free molecule in the absence of an applied field does not contain spin and so both
S
2
and
S
z
inevitably commute with molecular Hamiltonians.
•
Molecules are not spherical, and the great simplifying feature of atomic HF theory
1
r
P
nl
(
r
)
Y
l
,
m
l
(θ , φ)
ψ (
r
)
=
is no longer appropriate. This statement follows from the one above.
•
The Hartree-Fock (HF) limit is attainable for atoms by direct numerical integration
of the HF equations. This limit is unattainable for molecules except in a few simple
cases of high symmetry, and numerical integration techniques that are fine for atoms are
inapplicable for molecules. In any case, we have to concern ourselves with calculations
that are beyond the HF limit if we want to study chemical reactions.
