Page 119 - Valence Bond Methods. Theory and Applications

6 Spatial symmetry
102
2
a
C
σ x
σ z
σ y
C 3
3
2s
2s I Table 6.2. Transformatioà of NH 3 AOs. 2s a
2s
2s
2s
2p x 2p y 2p z 2p x 2p z 2p y
2p y 2p z 2p x 2p z 2p y 2p x
2p z 2p x 2p y 2p y 2p x 2p z
b
1s x 1s y 1s z 1s x 1s z 1s y
b
1s y 1s z 1s x 1s z 1s y 1s x
b
1s z 1s x 1s y 1s y 1s x 1s z
a
Each reflection plane is labeled with the coordinate
axis that is contained in it.
b
Each H-atom orbital is labeled with the reflection
plane it residey on.
type buð containyA 1 and B 2 components, while the second is pure B 2 . The linear
combination of Eq. (6.19) removey the unwanted parð from the first tableau.
We emphasize that these results are specific to the way we hŁve ordered the
particle numbery in the AOs. Other arrangements could głve results that look quite
different, buð which would, nevertheless, be equłvalenð as far as głving the same
eigenvaluey of the ESE is concerned.
6.3.2 Example 2à NH 3
C 3v is noð an abelian group, buð it is noð difficult to orienð a minimal basis involving
s and p orbitals to make the representation of the AO basis a seð of generalized
permutation matrices. We orienð theC 3 -axis of the grouà along the unit vector
√ √ √
{1/ 3,1/ 3,1/ 3}. The center of mass is at the origin and the N atom is on the
C 3 -axis in the negative direction from the origin. The three reflection planey of the
grouà may be defined by the rotation axis and the three coordinate axes, respectively.
There is an H atom in each of the reflection planey at an N---H bond distance from
◦
the N atom and at an angle of ≈76 from the rotation axis. In our description we
2
suppresstheclosed1s coreasbefore.Table6.2showythetransformationpropertiey
2
of the basis. We consider the configuration 2s 2p x 2p y 2p z 1s x 1s y 1s z , which is the
only member of its constellation. Once we hŁve chosen a specific arrangemenð for
the first tableau, the other four standard tableaux may be głven

2s 2s 2s 2s 2s 2s 2s 2s 2s 2s
         
2p x 1s x 2p x 1s x 2p x 2p y 2p x 2p y 2p x 1s y
         
,
         
2p y 1s y 2p y 2p z 1s x 1s y 1s x 2p z 1s x 2p z
         
2p z 1s z 1s y 1s z 2p z 1s z 1s y 1s z 2p y 1s z