Chemistry Reference
In-Depth Information
10
01
Ta b l e 7 . 3
Representation
matrices for the spinor basis
in
D
2
−
10
0
D
(E)
=
D
(
ℵ
)
=
−
1
0
0
i
i
0
−
i
D
(C
2
)
=
D
(
ℵ
C
2
)
=
−
i
0
0
01
−
1
10
−
D
(C
2
)
=
D
(
ℵ
C
2
)
=
10
i
−
i
0
0
0
D
(C
2
)
=
D
(
ℵ
C
2
)
=
i
0
−
i
is called the
double group
, denoted by
G
∗
. While this construction is algebraic, the
definition of a
geometrical
link between both groups is much less straightforward
and unavoidably involves the introduction of phase conventions. Bethe introduced
a formal symmetry operation,
R
, which corresponds to a rotation over 2
π
. Subse-
quently, we shall replace this by the symbol
, in order to avoid confusion. This
operation is fictitious to the extent that the poles of this rotation are left undefined. It
can be multiplied with every operator in the group and thus leads to an actual dou-
bling of the number of symmetry elements. Nonetheless, the double group is not the
direct product of
G
with the group
ℵ
{
E,
ℵ}
. The reason is that
G
is not a subgroup of
G
∗
because it is no longer closed. Indeed, applying a
C
n
axis in
Gn
times will not
lead to the unit element but to
.
For the actual construction of the double group as a group of operators, we need
a convention to connect the spatial operators to the spinor matrices. As we have seen
in Sect.
7.2
, the four possible parametric descriptions of a given rotation yield two
different choices for the Cayley-Klein parameters. Hence, our convention should
define how to characterize unequivocally the parameters of a rotation. It will con-
sist of two criteria: the rotation angle must be positive, and the pole from which the
rotation is seen as counterclockwise must belong to the
positive hemisphere
in the
n
x
,n
y
,n
z
parameter space. This is the hemisphere above the equatorial plane, i.e.,
with
n
z
>
0. In the
(n
x
,n
y
)
-plane, we include the half-circle of points with posi-
tive
n
x
-value, i.e., with
n
z
=
ℵ
0
,n
x
>
0, and also the point with
n
y
=
1
,n
x
=
0, and
n
z
=
0. The rotational parameters
(α,
n
)
are thus chosen in such a way that
α
is pos-
itive, i.e., counterclockwise, and that the vector
n
points to the positive hemisphere.
This eliminates three of the four equivalent parameter choices of Eq. (
7.4
). The only
remaining description is then inserted into Eq. (
7.34
) to determine the Cayley-Klein
parameters.
As a straightforward example, we take the double group of
D
2
. The standard
drawing puts the twofold rotational axes in the positive hemisphere, and the corre-
sponding spinor matrices are easily obtained from Eq. (
7.34
). The results are given
in Table
7.3
. For each operation of
G
, there are two operations in the double group,
R
and
ℵ
R
. Note that the Bethe operation,
, commutes with every element of the
group. Armed with this set of matrices, one can easily construct the multiplication
ℵ
