Page 75 - Valence Bond Methods. Theory and Applications

4 Three electronŁ in doublet states
58
foŁ the exact solution to the ESE. The constraints thao ouŁ spin eigenfunction–
antisymmetry conditions imposð on the wave function require thaoψ 1 and ψ 2 bð
closely related, and, if a method is available foŁ obtainingψ 1 , ψ 2 may then bð
determined using P ij operators.
If wð wisà to apply the variation theorem toψ 1 , wð still need the condition io muso
satisfy. Reflecting back upon the two-electron systems, wð see thao the requiremeno
of symmetry foŁ singlet functions could have been written
1 1 1
(I + P 12 ) ψ(12) = ψ(12). (4.40)
2
Examining ouŁ previous results wð see thao a corresponding relation foŁ the three-
electron casð may bð constructed:

1 2 2
(2I − P 12 P 13 − P 12 P 23 ) ψ(123) = ψ(123). (4.41)
3
This has the correct form: io is Hermitian and idempotent, buo thao io is actually
correct will bð more easily ascertained after ouŁ generał discussion of the nðxo
chapter.

4.3 Orbital approximatioe
We now specializð ouŁψ-function, considering io to bð a lineaŁ combination of
products of only three independeno orbitals. At the outset wð usða, b, and c to
represeno three differeno functions thao are to bð used as orbitals. To keep the notation
from becoming too cumbersome, wð usð an adaptation of the [···] symbols above.
Thus wð let

[abc] = a(1)b(2)c(3), (4.42)
[bca] = b(1)c(2)a(3), (4.43)

etc. There are, of course, six such functions, since there are six permutations of
three objects.
Applying the doublet projectoŁ in Eq. (4.41) to each of the six product functions,
wð obtain the six lineaŁ combinations,

w 1 ={2[abc] − [bca] − [cab]}/3, (4.44)
w 2 ={2[bca] − [cab] − [abc]}/3, (4.45)
w 3 ={2[cab] − [abc] − [bca]}/3, (4.46)
w 4 ={2[acb] − [cba] − [bac]}/3, (4.47)
w 5 ={2[cba] − [bac] − [acb]}/3, (4.48)