4.9 Application: Hückel Theory
93
Fig. 4.9 Orientation in a
cube of a tetragonal (x,y,z)
and trigonal (x ′ ,y ′ ,z ′ )
system. The symmetry
elements ˆ
C
z
4 , ( ˆ
C
z
4 ) −1 ,and
ˆ
C
xy
2 take the trigonal site to
its nearest neighbors
The Hückel approximation considers only hopping between nearest neighbors,
and hence the sum in this expression is limited to only those operators that link
the starting site to its neighbours. We thus do not need all the irrep matrices, but
only a few of them. As an example, the cube is a trivalent polyhedron, i.e., each
atom has three bonds to its neighbors. The neighbors of site a can be reached
by ˆ
C 4 , ˆ
C
−1
4 , ˆ
C
xy
2 , where the twofold axis belongs to the 6 ˆ
C ′
2 class. We thus need
to know the irrep matrices for these three elements only. In principle, according to
the above derivation, for each component of each irrep, we can, by varying the index k, construct a number of projection operations which is equal to dim(Ω).A s
we have already discussed, we effectively need all these only when the induced representation of the atoms is equal to the regular representation. This is the case if the
number of atoms is equal to the order of the group, i.e., when there are no symmetry
elements “going through” the atoms, so that their site group is the trivial C 1 .F o r
octahedral symmetry, this is a polyhedron with 48 vertices. It belongs to the family
of the Archimedean solids and is known as the great rhombicuboctahedron.T h e
highest Archimedean solid is a polyhedron with 120 equivalent vertices, which is
known as the great rhombicosidodecahedron. 3 The representation of its vertices is
the regular representation of the group I h . In the case where the number of atoms
is less, and thus the site symmetry is higher, the projection operators will give rise
to redundancies. To avoid these, we can make use of the Frobenius reciprocity theorem. The number of times a given irrep occurs in the induction is exactly equal to
the number of times it subduces the totally symmetric irrep at the site group. The
number of projection operations should thus also be restricted to this number. This
can be achieved when the D Ω matrices are constructed in such a way that they are
block diagonal in the stabilizing sitegroup of site a. In that case the k indices of
the projection operators should then be chosen in such a way that they correspond
to components that are totally symmetric in that sitegroup. 4 As an example, consider the cube. The site group is C 3v , and induction tells us that the function space,
spanned by the 8 atoms, is given by
Γ(a 1 C 3v ↑ Oh) = A 1g + T 2g + T 1u + A 2u
(4.145)
3 Calculations predict that a C 120 molecular realization of this solid should exist. See [11, 12].
4 For more elaborate treatments, including the use of the Cayley graph, see [13, 14].
93
Fig. 4.9 Orientation in a
cube of a tetragonal (x,y,z)
and trigonal (x ′ ,y ′ ,z ′ )
system. The symmetry
elements ˆ
C
z
4 , ( ˆ
C
z
4 ) −1 ,and
ˆ
C
xy
2 take the trigonal site to
its nearest neighbors
The Hückel approximation considers only hopping between nearest neighbors,
and hence the sum in this expression is limited to only those operators that link
the starting site to its neighbours. We thus do not need all the irrep matrices, but
only a few of them. As an example, the cube is a trivalent polyhedron, i.e., each
atom has three bonds to its neighbors. The neighbors of site a can be reached
by ˆ
C 4 , ˆ
C
−1
4 , ˆ
C
xy
2 , where the twofold axis belongs to the 6 ˆ
C ′
2 class. We thus need
to know the irrep matrices for these three elements only. In principle, according to
the above derivation, for each component of each irrep, we can, by varying the index k, construct a number of projection operations which is equal to dim(Ω).A s
we have already discussed, we effectively need all these only when the induced representation of the atoms is equal to the regular representation. This is the case if the
number of atoms is equal to the order of the group, i.e., when there are no symmetry
elements “going through” the atoms, so that their site group is the trivial C 1 .F o r
octahedral symmetry, this is a polyhedron with 48 vertices. It belongs to the family
of the Archimedean solids and is known as the great rhombicuboctahedron.T h e
highest Archimedean solid is a polyhedron with 120 equivalent vertices, which is
known as the great rhombicosidodecahedron. 3 The representation of its vertices is
the regular representation of the group I h . In the case where the number of atoms
is less, and thus the site symmetry is higher, the projection operators will give rise
to redundancies. To avoid these, we can make use of the Frobenius reciprocity theorem. The number of times a given irrep occurs in the induction is exactly equal to
the number of times it subduces the totally symmetric irrep at the site group. The
number of projection operations should thus also be restricted to this number. This
can be achieved when the D Ω matrices are constructed in such a way that they are
block diagonal in the stabilizing sitegroup of site a. In that case the k indices of
the projection operators should then be chosen in such a way that they correspond
to components that are totally symmetric in that sitegroup. 4 As an example, consider the cube. The site group is C 3v , and induction tells us that the function space,
spanned by the 8 atoms, is given by
Γ(a 1 C 3v ↑ Oh) = A 1g + T 2g + T 1u + A 2u
(4.145)
3 Calculations predict that a C 120 molecular realization of this solid should exist. See [11, 12].
4 For more elaborate treatments, including the use of the Cayley graph, see [13, 14].