7.5 Double Groups
175
as internal spin states. In spin-orbit coupling the internal spin degrees of the electron
are coupled to its external momentum, and this can be based on the embedding of
SO(3) in SU(2).
7.5 Double Groups
The spinor basis enables us to obtain a two-dimensional matrix representation of the
point-group operations. Let us first limit ourselves to the relationships for the proper
rotations. The counterclockwise rotation of the vector over an angle α with the pole
at the positive z-axis is given by the matrix:
ˆ
R
|x| y| z
=
|x| y| z
⎛
⎝
cos α − sin α 0
sin α cos α 0
00
1
⎞
⎠
(7.40)
According to Eqs. (7.34) and (7.35), the corresponding rotation matrix in the spinor
basis is determined up to a sign by
ˆ
R
|α| β
=±
|α| β
exp(
−iα
2 )
0
exp(
iα
2 )
(7.41)
How does one deal with this ambiguity of sign? A plausible way is to use a continuity argument [8]. If we approach the neighborhood of the unit element for both
spinor and vector by letting α decrease to zero, we should converge to the unit matrix and, hence, take the + sign in Eq. (7.41) with α = 0. Now let the rotation angle
increase continuously from 0 to 2π . While the matrix O(R) is periodic in α and
passes again to the unit matrix, the spinor matrix becomes minus the unit matrix.
Continuing the path in parameter space and increasing the angle to 4π drive the
vector rotation once again over the same interval, while the spinor rotation finally
completes its path and reaches the unit element again. So, the difference between
U and −U can be interpreted as a rotation over a full angle of 2π . From the topological point of view, the path that we have described corresponds to a full circle in
the 4-parameter space of SU(2), and the rotation over 2π connects a point in this
space to its antipode. In this space the SO(3) operations may be identified as the set
of straight lines connecting antipodal points.
Our real interest at present is molecular Hamiltonians that are characterized by a
point group G. However, as compared with the Hamiltonian considered in Chap. 5,
we should also include spin-orbit coupling operators. These will be invariant only
under concerted transformations of the orbital and spin parts. The homomorphism
between SO(3) and SU(2) provides a straightforward algebraic way to construct the
spinor group associated with the point group. For each operation, ˆ
R ∈ G, a matrix
O(R) is defined, which offers a faithful representation of G and, in turn, gives rise
to two spinor matrices, ±U(R), which describe the spinor transformation. The set of
these matrices forms a group, which contains twice as many elements as G and thus
175
as internal spin states. In spin-orbit coupling the internal spin degrees of the electron
are coupled to its external momentum, and this can be based on the embedding of
SO(3) in SU(2).
7.5 Double Groups
The spinor basis enables us to obtain a two-dimensional matrix representation of the
point-group operations. Let us first limit ourselves to the relationships for the proper
rotations. The counterclockwise rotation of the vector over an angle α with the pole
at the positive z-axis is given by the matrix:
ˆ
R
|x| y| z
=
|x| y| z
⎛
⎝
cos α − sin α 0
sin α cos α 0
00
1
⎞
⎠
(7.40)
According to Eqs. (7.34) and (7.35), the corresponding rotation matrix in the spinor
basis is determined up to a sign by
ˆ
R
|α| β
=±
|α| β
exp(
−iα
2 )
0
exp(
iα
2 )
(7.41)
How does one deal with this ambiguity of sign? A plausible way is to use a continuity argument [8]. If we approach the neighborhood of the unit element for both
spinor and vector by letting α decrease to zero, we should converge to the unit matrix and, hence, take the + sign in Eq. (7.41) with α = 0. Now let the rotation angle
increase continuously from 0 to 2π . While the matrix O(R) is periodic in α and
passes again to the unit matrix, the spinor matrix becomes minus the unit matrix.
Continuing the path in parameter space and increasing the angle to 4π drive the
vector rotation once again over the same interval, while the spinor rotation finally
completes its path and reaches the unit element again. So, the difference between
U and −U can be interpreted as a rotation over a full angle of 2π . From the topological point of view, the path that we have described corresponds to a full circle in
the 4-parameter space of SU(2), and the rotation over 2π connects a point in this
space to its antipode. In this space the SO(3) operations may be identified as the set
of straight lines connecting antipodal points.
Our real interest at present is molecular Hamiltonians that are characterized by a
point group G. However, as compared with the Hamiltonian considered in Chap. 5,
we should also include spin-orbit coupling operators. These will be invariant only
under concerted transformations of the orbital and spin parts. The homomorphism
between SO(3) and SU(2) provides a straightforward algebraic way to construct the
spinor group associated with the point group. For each operation, ˆ
R ∈ G, a matrix
O(R) is defined, which offers a faithful representation of G and, in turn, gives rise
to two spinor matrices, ±U(R), which describe the spinor transformation. The set of
these matrices forms a group, which contains twice as many elements as G and thus