Chemistry Reference
In-Depth Information
transform as follows:
Γ(μ
z
)
=
a
1
Γ(μ
x
,μ
y
)
(5.2)
=
e
Hence, only the
z
-component is compatible with the existence of a dipole moment.
The principles of Neumann and Curie are of course based on classical physics.
They remain valid for quantum systems but do not include typical quantum phenom-
ena, such as the existence of electronic degeneracy or transitions between quantum
states. A proper quantum description of molecular symmetry is thus required.
5.2 The Schrödinger Equation
According to quantum mechanics, the stationary states of a molecule are described
by the eigenfunctions of the Hamiltonian,
H
, corresponding to a quantized set of
eigenvalues,
H
|
Ψ
j
=
E
k
|
Ψ
j
(5.3)
Ψ
j
Here,
E
k
is a fixed eigenvalue, and
is an associated eigenfunction. The index
j
takes into account the possibility that several eigenfunctions may be associated
with the same eigenvalue. In this case the set of these functions forms an eigenspace
{|
|
Ψ
j
}
j
=
1
,...,n
, where
n
denotes the dimension of this space, and the
E
k
eigenvalue
is said to be
n
-fold
degenerate
. As usual, the eigenspace will be taken to be or-
thonormal. The Hamiltonian expresses the kinematics of the electrons in the frame
of the nuclei subject to Coulomb forces. We shall study this in detail in Sect.
5.4
.
For the moment, all we need to know is that the operators of the molecular point
group leave
H
invariant:
∀
R
∈
G
→[
R,
H
]=
0
(5.4)
Now applying
R
to the Schrödinger equation yields
Ψ
j
=
H
R
Ψ
j
=
E
k
R
Ψ
j
R
H
(5.5)
Here, we have made use of the commutation relation in Eq. (
5.4
) and the property
that
R
as a linear operator does not affect the constant eigenvalue. The equation
signifies that if
is an eigenfunction, the transformed function,
R
|
Ψ
j
|
Ψ
j
,also
is an eigenfunction
with the same eigenvalue.
This is an important result, which
ties quantum mechanics and group theory together, and is essentially the reason
why group theory can be applied to chemistry! Now, there are two possibilities,
depending on the degeneracy.
1. The electronic state is nondegenerate
(n
1
)
. In this case the transformed eigen-
function must necessarily be proportional to the original one. Since the transfor-
mation does not change normalization, the proportionality constant must be a
=
