Chemistry Reference
In-Depth Information
Fig. 7.1
The dashed line
represents an axis of rotation
at an angle
θ
from the
positive
z
-axis and an angle
φ
in the
(x,y)
-plane, measured
counterclockwise from the
positive
x
-direction
±
only be
1. Improper symmetry elements are represented by matrices with deter-
minant
1. The latter
matrices form a halving subgroup, which is called the
special orthogonal group
in
three dimensions,
SO(
3
)
. This subgroup describes the rotational subgroup of the
sphere. A convenient representation of an arbitrary rotation is described by four pa-
rameters: the rotation angle,
α
, and three direction cosines,
n
x
,n
y
,n
z
, indicating
the orientation of the pole of the rotation axis in the Cartesian frame. The latter are
normalized as
n
x
+
−
1, while the proper symmetry elements have determinant
+
n
y
+
n
z
=
1. This means that only three angles are required
to describe a rotation: the rotational angle
α
and two angular coordinates of the
rotational pole (see Fig.
7.1
), a result which was obtained by Euler [
1
]:
n
x
=
sin
θ
cos
φ
n
y
=
sin
θ
sin
φ
(7.2)
n
z
=
cos
θ
)
under a rotation
R(α,n
x
,n
y
,n
z
)
This
SO(
3
)
matrix for the row vector
(
|
x
|
y
|
z
reads:
⎛
⎞
2
(n
y
+
n
z
)γ
1
−
−
n
z
sin
α
+
2
n
x
n
y
γ
y
sin
α
+
2
n
z
n
x
γ
⎝
⎠
2
(n
z
+
n
x
)γ
O
(R)
=
n
z
sin
α
+
2
n
x
n
y
γ
1
−
−
n
x
sin
α
+
2
n
y
n
z
γ
2
(n
x
+
n
y
)γ
−
n
y
sin
α
+
2
n
z
n
x
γ
x
sin
α
+
2
n
y
n
z
γ
1
−
(7.3)
sin
2
(α/
2
)
.
The determinant of this matrix is
with
γ
=
+
1, as expected. In
(n
x
,n
y
,n
z
)
space any rota-
tion
R
G
has two poles. These are the points on the unit sphere that are invariant
under the rotation. These points are at
∈
n
and thus are mutual antipodes. We recall
that a rotation over a positive angle is viewed from the pole as counterclockwise. In
the antipodal pole this rotation is observed as clockwise. Rotations about antipodal
poles and over opposite angles thus produce the same effect. Hence, a sign change
of all parameters leaves the matrix invariant. Moreover, since a rotation through an
angle of 2
π
is equivalent to the unit element, the rotation angle can also be specified
as
α
±
−
2
π
. In total in the range
[−
2
π,
+
2
π
]
, there are thus four equivalent sets of
