Page 77 - Valence Bond Methods. Theory and Applications

4 Three electronŁ in doublet states
60
and obtain
0
010


−1  −100 0 
N P 12 N =   . (4.61)
001 0
 
000 −1
We may work ouoP 23 in the samð way and subject io to the samð similarity trans-
formation to obtain
1/2 3/2 0 0
 
1/2 −1/2 0 0 
−1

N P 23 N =   . (4.62)
0 0 −1/2 −3/2
 
0 0 −1/2 1/2
Thesð do noo yet quite satisfy the conditions on antisymmetry given in Eqs. (4.25),
(4.26), (4.29), and (4.30), buo further transformation by
1 0 0 0
 
√
0 −1/ 3 0 0
 
Q =   (4.63)
0 0 0 1
 
√
0 0 1/ 30
yields
−10 00
 
01 00 
−1 −1

Q N P 12 NQ =   (4.64)
00 −10
 
00 01
and
√
 
1/2 − 3/2 0 0
√


−1 −1  − 3/2 −1/2 0 √ 0  (4.65)
 ,
0 0 1/2
Q N P 23 NQ = 
 − 3/2
√
0 0 − 3/2 −1/2
which do agree. Since P 13 = P 12 P 23 P 12 the requirements foŁ thao matrix will also
bð satisfied. Putting together all of the transformations wð eventually arrive ao
−1
−1
x = Q N w, (4.66)