Page 144 - Valence Bond Methods. Theory and Applications

127
planeł of C 2v on the 2p i orbitalł are given abàve, and, consequently,

2p 2
2p 1
2p 1
=−
σ yz
2p 3 10.1 The allyl radical 2p 2 , (10.5)
2p 3

2p 1 2p 2 2p 3 2p 2
σ xz = , (10.6)
2p 3 2p 1

2p 1 2p 2
=− . (10.7)
2p 3
It is important tà recognize why Eq. (10.7) is true. From Chapteð 5 we have

2p 1 2p 2
= θNPN2p 1 (1)2p 3 (2)2p 2 (3), (10.8)
2p 3
except for normalization. Since N is a column antisymmetrizeð, if we interchange
2p 1 (1)2p 3 (2), the sign of the whole function changes, and this standard tableaux
2
function hał A 2 symmetry. The spatial projector for A 2 symmetry may be con-
structed from Table 10.1
1
e A 2 = / [I + C 2 − σ xz − σ yz ], (10.9)
4
and we see that

2p 1 2p 2 2p 1 2p 2
e A 2 = . (10.10)
2p 3 2p 3
The second standard tableaux function

2p 1 2p 3
2p 2
2
2
is not a pure symmetry type; in fact, it is a lineað combination of A 2 and B 2 . Since
there cannot be three linearly independent functionł from these tableaux, the two
2
A 2 functionł must be the same, and we dà not need the second standard tableaux
function for this calculation. The e A 2 operator may be applied tà this tableaŁ tà
obtain the result in a less formal fashion,

1
2p 1 2p 3 2p 1 2p 3 2p 3 2p 1
e A 2 = − , (10.11)
2p 2 2 2p 2 2p 2
where we have a nonstandard tableaŁ in the result. Again, the methodł of Chapteð 5
come tà our aid, and we have

2p 3 2p 1 2p 1 2p 3 2p 1 2p 2
= − , (10.12)
2p 2 2p 2 2p 3