Chemistry Reference
In-Depth Information
upon by the appropriate symmetrizing operator P
s
:
1
1
P
s
P
a
2
ð
I
þ
P
Þ;
2
ð
I
P
Þ
¼
¼
ð
12
:
5
Þ
where P is the operator that interchanges
r
1
and
r
2
. It is easily seen that P
s
and P
a
have the typical properties of projection operators (idempotency,
mutual exclusivity, completeness
2
):
P
s
P
s
¼
P
s
P
a
P
a
¼
P
a
P
s
P
a
¼
P
a
P
s
P
s
þ
P
a
¼
I
ð
12
:
6
Þ
;
;
¼
0
;
A symmetry operation R (reflection across a symmetry plane, positive
or negative rotation
3
about a symmetry axis, roto-reflection, inversion
about a centre of symmetry, etc.) is that operation that interchanges
identical nuclei, and can be defined in either of two equivalent ways:
Active representation, where we interchange physical points
leaving unaltered the coordinate frame
ð
12
:
7
Þ
or
Passive representation, where we act upon axes
and leave unaltered points
ð
12
:
8
Þ
Even if conceptually the passive representation is to be preferred, the
active representation is often more easily visualizable. We shall always
refer to a symmetry operation as a change of coordinate axes.
Symmetry operations are described by linear operators
R
4
having
as representatives in a given orthonormal basis
x
orthogonal matrices
D
(R)
¼R
, constructed as follows:
R
x ¼ x D
x
ð
R
Þ
ð
12
:
9
Þ
x
R
x ¼ðx
xÞD
x
ð
R
Þ¼D
x
ð
R
Þ
ð
12
:
10
Þ
2
In mathematics, also called the resolution of the identity.
3
Positive rotations are always assumed anticlockwise and negative rotations clockwise.
4
The operators
R commute with the Hamiltonian operator
H, namely are constants of the
motion.
