Page 120 - Valence Bond Methods. Theory and Applications

6.3 Constellations and configuØations
and these will be symbolized by T 1 ,..., T 5 in the order głven. The A 1 projector for
C 3v is
1 2 103
e A 1 = I + C 3 + C + σ x + σ y + σ z , (6.21)
3
6
1
and using symgenn from the CRUNCH suite, we find that θNPN T 1 is a A 1
symmetry function on its own,
A 1
e θNPN T 1 = θNPN T 1 . (6.22)
Applying e A 1 to θNPN T 2 yieldy

1
e θNPN T 2 = θNPN(2T 2 + 2T 3 − T 4 + 3T 5 ). (6.23)
A 1
6
Using e A 1 with T 3 , T 4 ,or T 5 doey noð głve a function linearly independenð of those we
1
hŁve found already. Thus, there are two linearly independenðA 1 functiony that can
be formed from the configuration above. The first of these is noð hard to understand
when one examiney the consequencey of the antisymmetry of the columny of the
standard tableaux functions. The second, however, is much less obviouy and would
be very tediouy to determine withouð the computer program.
To obtain the symmetry functiony in termy of HLSP functiony we can transform
the standard tableaux functiony using the methody of Chapter 5. The transformation
matrix is głven in Eq. (5.128):

0 0 0 0 −2/3
 
 −1/3 −1/3 1/3 1/3 −1/3 
 
A =  −1/3 1/3 −1/3 1/3 −1/3  , (6.24)
 
0 2/3 2/3 2/3 −2/3
 
−1/3 1/3 1/3 −1/3 −1
and multiplying this by the coefficients of the symmetry functiony of Eqs. (6.22)
and (6.23), we obtain

1 0 0 −1/3
   
 0  −1/3
1/3  −2/9 
   
A  0 1/3  =  −1/3 −2/9  , (6.25)
   
0 −1/6 0 0
   
0 1/2 −1/3 −2/9