Chemistry Reference
In-Depth Information
j
(R) for each
operation R of the group. This projector is needed in the case of multi-
dimensional irreps, but for one-dimensional irreps it is sufficient to use the
simpler projector involving the characters
which requires the complete knowledge of matrices
D
/
X
P
j
ð
R
Þ
R
j
R
x
ð
12
:
30
Þ
12.3 ILLUSTRATIVE EXAMPLES
Use of symmetry-adapted functions is particularly useful for predicting
which matrix elements of an operator are zero (selection rules, mostly in
spectroscopy), even without doing any effective calculation, and in the
factorization of the secular equations arising either in MO, CI, or VB
calculations. We shall illustrate below these factorizations for the
minimal basis set H
€
uckel calculations for the H
2
O(C
2v
)andNH
3
(C
3v
) molecules.
12.3.1 Use of Symmetry in Ground-state H
2
O(
1
A
1
)
The H
2
Omolecule is chosen to lie in the yz-plane, with the O atom at the
origin of a right-handed coordinate system having z as binary symmetry
axis (Figure 12.2). The nuclear symmetry implies two symmetry planes (yz
σ
v
′
O
.
y
σ
v
.
.
H
1
H
2
C
2
z
x
Figure 12.2
Elements of C
2v
symmetry in H
2
O
