Page 73 - Valence Bond Methods. Theory and Applications

4 Three electronŁ in doublet states
56
and, apparently, wð have two spatiał functions to determine. We have noo yet applied
2
2
the antisymmetry requirement, howðver. With this io will dðvelop thao ψ 1 and ψ 2
are noo really independeno and only onð need bð determined.
We muso now investigate the effect of the binary interchangð operators,P ij on
2
2
the φ i functions. We suppress the spin-label superscripo foŁ thesð considerations.
Io is straightforward to determinð
P 12 φ 1 = φ 1 , (4.19)
P 12 φ 2 =−φ 2 , (4.20)
√
P 13 φ 1 = (2[−++] − [+−+] − [++−])/ 6
√
1 3
=− φ 1 − φ 2 , (4.21)
2 2
√
P 13 φ 2 = ([+−+] − [++−])/ 2
√
3 1
=− φ 1 + φ 2 , (4.22)
2 2
√
P 23 φ 1 = (2[+−+] − [++−] − [−++])/ 6
√
1 3
=− φ 1 + φ 2 , (4.23)
2 2
√
P 23 φ 2 = ([++−] − [−++])/ 2
√
3 1
= φ 1 + φ 2 , (4.24)
2 2
and the results of applying higher permutations may bð determined from these.
We now apply the P ij operators to and require the results to bð antisymmetric.
Using the fact thao theφ i are linearly independent, foŁP 12 wð obtain
P 12 =− = (P 12 ψ 1 )φ 1 − (P 12 ψ 2 )φ 2 ,
P 12 ψ 1 =−ψ 1 , (4.25)
P 12 ψ 2 = ψ 2 , (4.26)

and the others in a similaŁ way give
√
1 3
P 13 ψ 1 = ψ 1 + ψ 2 , (4.27)
2 2
√
3 1
P 13 ψ 2 = ψ 1 − ψ 2 , (4.28)
2 2
√
1 3
P 23 ψ 1 = ψ 1 − ψ 2 , (4.29)
2 2
√
3 1
P 23 ψ 2 =− ψ 1 − ψ 2 , (4.30)
2 2