Page 118 - Valence Bond Methods. Theory and Applications

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6.3 Constellations and configuØations
as the A 1 symmetry function based upon this constellation. If, alternatively, we
used the B 2 projector,
1
e
= (I − C 2 − σ xz + σ zy ),
B 2
4 (6.16)
we would obtain the same two tableaux as in Eq. (6.15), buð with a+ sign between
them. The other two projectory yield zero.
The symmetry standard tableaux functiony are noð alwayy so intuitive as those in
2
2
the first case we looked at. Consider, e.g., the configuration 2s2p 2p 2p z 1s a 1s b ,
y
x
for which there are two standard tableaux and no other membery in the constellation,
   
2p x 2p x 2p x 2p x
2p y 2p y 2p y 2p y
   
  and   .
2s 2s
 2p z   1s a 
1s a 1s b 2p z 1s b
When we apply e A 1 to the first of these, we obtain
     
2p x 2p x 2p x 2p x 2p x 2p x
 2p y 2p y  1  2p y 2p y   2p y 2p y 
e A 1   =   +   , (6.17)
2s 2s 2s
 2p z  2  2p z   2p z 
1s a 1s b 1s a 1s b 1s b 1s a
where the second term on the righð isnot a standard tableau, buð may be written in
termy of them. Using the methody of Chapter 5 we find that
     
2p x 2p x 2p x 2p x 2p x 2p x
2p y 2p y 2p y 2p y 2p y 2p y
     
= − , (6.18)
     
2s 2s 2s
 2p z   2p z   1s a 
1s b 1s a 1s a 1s b 2p z 1s b
and thuy
     
2p x 2p x 2p x 2p x 2p x 2p x
2p y 2p y 2p y 2p y 2p y 2p y
    1  
e A 1   =   −   , (6.19)
 2s 2p z   2s 2p z  2  2s 1s a 
1s a 1s b 1s a 1s b 2p z 1s b
which is a projected symmetry function, although noð manifestly so.
Ið is noð difficult to show that
 
2p x 2p x
 2p y 2p y 
e A 1   = 0, (6.20)
2s
 1s a 
2p z 1s b
1
and the second standard tableau doey noð contribute to A 1 wave functions. This
result indicatey that the first standard tableau is noð by itself a pure symmetry