Chemistry Reference
In-Depth Information
These results clearly illustrate the importance of the GOT. It not only provides a
selection rule at the level of the irreps, but also at the level of the components. Of
course, the latter selection rule will work only if we ensured that the symmetry adap-
tation of the basis set has been carried out at the component level, as was explained
in Sect.
5.3
.
A further consequence is that SALCs on peripheral atom sites can quite often
easily be derived from central symmetry-adapted orbitals. One simply has to make
sure that the SALCs have the same nodal characteristics as the central functions, so
as to guarantee maximal overlap. This is well illustrated in Fig.
4.4
.
6.2 The Coupling of Representations
Overlap integrals are scalar products of a bra and a ket function. A general matrix
element is an integral of the outer product of a bra, an operator, and a ket, giving
rise to a triad of irreps. The evaluation of such elements is based on the
coupling
of
irreps. This concept refers to the formation of a product space. The simplest example
is the formation of a two-electron wavefunction, obtained by multiplying two one-
electron functions. This section will be devoted entirely to the formation of such
product spaces.
Consider two sets of orbitals, transforming as the irreps
Γ
a
and
Γ
b
respectively,
each occupied by one electron. A two-electron wavefunction with electron 1 in the
γ
a
component of the first set, and electron 2 in the
γ
b
component of the second set
is written as a simple product function:
. Clearly, since the one-
electron function spaces are invariants of the group, their product space is invariant,
too. Now the question is to determine the symmetry of this new space. The recipe
to find this symmetry can safely be based on the character theorem: first determine
the character string for the product basis, and then carry out the reduction according
to the character theorem. Symmetry operators are all-electron operators affecting all
particles together; hence, the effect of a symmetry operation on a ket product is to
transform both kets simultaneously.
R
Γ
a
γ
a
(
1
)
Γ
b
γ
b
(
2
)
=
|
Γ
a
γ
a
(
1
)
|
Γ
b
γ
b
(
2
)
γ
b
γ
b
(R)
Γ
a
γ
a
(
1
)
Γ
b
γ
b
(
2
)
(6.5)
D
Γ
a
γ
a
γ
a
(R)D
Γ
b
γ
a
γ
b
The transformation of the product functions is thus expressed by a super matrix,
each element of which is a product of two matrix elements for the individual orbital
transformations. The trace of this super matrix is given by:
χ
Γ
a
×
Γ
b
(R)
D
Γ
a
γ
a
γ
a
(R)D
Γ
b
=
γ
b
γ
b
(R)
γ
a
γ
b
=
χ
Γ
a
(R)χ
Γ
b
(R)
(6.6)
This is a gratifying result. The character of a product space is simply the product of
the characters of the factor spaces. Accordingly, the symmetry of the product space
