30
3 Groups
Fig. 3.4 Genealogical tree,
representing progressive
symmetry breaking of the
C 3v point group. The C s box
stands for the three equivalent
reflections groups
3.5 Cosets
A genuine partitioning of a group is achieved when the set of elements is divided
into separate subsets that do not exhibit any overlap and, together, constitute the
whole group. Subgroups clearly do not form a partitioning since, for instance, they
all share the same unit element. On the other hand, cosets do form a partitioning.
In molecules, the natural realizations of the cosets are the sets of equivalent sites.
These are atoms or groups of atoms that are permuted by the action of the molecular symmetry group. In the example of the ammonia molecule, each of the three
hydrogen atoms occupies an equivalent site with C s symmetry. The nitrogen atom,
however, occupies a unique site that has the full C 3v symmetry. Now consider the
site of one particular hydrogen atom, say A. The C s subgroup that leaves this site
invariant consists of only two symmetry elements: ˆ
E and ˆ
σ 1 . This subgroup is called
the stabilizer of the site. When we multiply each element of this subgroup (on the
left) with an element outside it, say ˆ
C 3 , we obtain two new elements, ˆ
C 3 and ˆ
σ 3 ,
which both share the property that they map A onto B. They form a (left) coset of
the original C s subgroup, and the element that we used to form this coset is the
coset-representative. There is still another coset, which may be generated by one
of the remaining elements, say ˆ
C 2
3 . In this way, one finds the coset, { ˆ
C 2
3 , ˆ
σ 2 },o f
elements which have the property that they both map A onto C. The sum of all the
cosets forms the total set, and hence,
C 3v ={ ˆ
E, ˆ
σ 1 }+{ ˆ
C 3 , ˆ
σ 3 }+
ˆ
C
2
3 , ˆ
σ 2
(3.19)
We can rewrite this in general as
G =
n
ˆ
R n H
(3.20)
where ˆ
R n denotes a coset representative, and the product ˆ
R n H denotes the nth coset,
obtained by multiplying every element of the subgroup on the left by the generator.
The choice of coset representatives is not unique since every element of a given
coset may act as representative. In the case of the present group, we can choose all
3 Groups
Fig. 3.4 Genealogical tree,
representing progressive
symmetry breaking of the
C 3v point group. The C s box
stands for the three equivalent
reflections groups
3.5 Cosets
A genuine partitioning of a group is achieved when the set of elements is divided
into separate subsets that do not exhibit any overlap and, together, constitute the
whole group. Subgroups clearly do not form a partitioning since, for instance, they
all share the same unit element. On the other hand, cosets do form a partitioning.
In molecules, the natural realizations of the cosets are the sets of equivalent sites.
These are atoms or groups of atoms that are permuted by the action of the molecular symmetry group. In the example of the ammonia molecule, each of the three
hydrogen atoms occupies an equivalent site with C s symmetry. The nitrogen atom,
however, occupies a unique site that has the full C 3v symmetry. Now consider the
site of one particular hydrogen atom, say A. The C s subgroup that leaves this site
invariant consists of only two symmetry elements: ˆ
E and ˆ
σ 1 . This subgroup is called
the stabilizer of the site. When we multiply each element of this subgroup (on the
left) with an element outside it, say ˆ
C 3 , we obtain two new elements, ˆ
C 3 and ˆ
σ 3 ,
which both share the property that they map A onto B. They form a (left) coset of
the original C s subgroup, and the element that we used to form this coset is the
coset-representative. There is still another coset, which may be generated by one
of the remaining elements, say ˆ
C 2
3 . In this way, one finds the coset, { ˆ
C 2
3 , ˆ
σ 2 },o f
elements which have the property that they both map A onto C. The sum of all the
cosets forms the total set, and hence,
C 3v ={ ˆ
E, ˆ
σ 1 }+{ ˆ
C 3 , ˆ
σ 3 }+
ˆ
C
2
3 , ˆ
σ 2
(3.19)
We can rewrite this in general as
G =
n
ˆ
R n H
(3.20)
where ˆ
R n denotes a coset representative, and the product ˆ
R n H denotes the nth coset,
obtained by multiplying every element of the subgroup on the left by the generator.
The choice of coset representatives is not unique since every element of a given
coset may act as representative. In the case of the present group, we can choose all