Chemistry Reference
In-Depth Information
calculations one must usually introduce approximate Hamiltonians and eigenspaces.
However, there is at least one simple way to endow these approximate eigenfunc-
tions with an exact property, i.e., by making sure that they are eigenfunctions of
the symmetry group of the exact Hamiltonian. The great success of semiempirical
theories, such as ligand-field theory for transition-metal and lanthanide complexes,
is in fact due to this consistent use of atomic and molecular symmetry. But also in
computational chemistry the effort of symmetrizing the orbital basis pays off in a
considerable gain of computation time, and it facilitates the assignment of spectro-
scopic data.
5.3 How to Structure a Degenerate Space
The eigenfunctions of an
n
-fold degenerate state are defined up to unitary equiva-
lence, which means that any complex linear combination of eigenfunctions is again
an eigenfunction. It is convenient to define a standard basis, which brings some
structure in this degenerate space. The tool that can be used for this is furnished by
the following lemma due to Schur.
Theorem 11
If a matrix commutes with all the matrices of an irreducible represen-
tation
,
the matrix must be a multiple of the unit matrix
:
∀
R
∈
:UD
=D
U→U∼I
G
(R)
(R)
(5.9)
This theorem implies that, if we combine the basis functions of a space in such a
way that their representation matrices of the group generators coincide with a canon-
ical choice, then the entire basis set will be completely fixed, up to a global phase
factor. This remaining phase freedom is external to the symmetry group. A basis set
that complies with a specified set of representation matrices is called a
canonical
basis. A convenient strategy for defining a canonical basis is founded on a
split-
ting field
. In this case the basis is chosen in such a way that it is diagonal with
respect to a particular generator, usually a principal axis of rotation. The compo-
nents then appear as eigenfunctions of the splitting field. In order for the splitting
to be unequivocal, all eigenvalues should be different. The splitting field then “rec-
ognizes” each component by its individual eigenvalue. As an example, consider the
twofold degenerate
E
-state in an octahedron. As a splitting field, we take the
C
4
axis along the
z
-direction. The standard components, which are recognized individ-
ually by this field, are denoted as
. They are respectively symmetric
and anti-symmetric with respect to the rotation axis:
|
Eθ
and
|
E
C
4
|
Eθ
=|
Eθ
(5.10)
C
4
|
E
=−|
E
In octahedral transition-metal complexes, the
d
-orbitals, which transform as the
E
irrep, are
d
z
2
and
d
x
2
y
2
. They are seen to match
|
Eθ
and
|
E
, respectively. The
−
