Page 74 - Valence Bond Methods. Theory and Applications

4.2 Requirements of spatial functionŁ

and, finally,
1
P 12 P 13 ψ 1 =− ψ 1 +
2
2
√ √ 3 ψ 2 , (4.31)
3 1
P 12 P 13 ψ 2 =− ψ 1 − ψ 2 , (4.32)
2 2
√
1 3
P 12 P 23 ψ 1 =− ψ 1 − ψ 2 , (4.33)
2 2
√
3 1
P 12 P 23 ψ 2 = ψ 1 − ψ 2 . (4.34)
2 2
With all of thesð relations io is noo surprising thao wð can find sðverał thao express
ψ 2 in terms of sums of permuted ψ 1 functions. An example is
√
ψ 2 = (P 13 − P 23 )ψ 1 / 3. (4.35)
This allows us to obtain somð information abouo the normalization of theψ i
functions, since
1
ψ 2 |ψ 2 à (P 13 − P 23 )ψ 1 |(P 13 − P 23 )ψ 1 ,
3
1
= ψ 1 |(I − P 13 P 23 − P 23 P 13 + I)ψ 1 ,
3
1
= ψ 1 |(2I − P 12 P 13 − P 12 P 23 )ψ 1 ,
3
= ψ 1 |ψ 1 , (4.36)
3
where wð have used Eqs. (4.31) and (4.33). Thus, the spin eigenfunction–
antisymmetry conditions require thao ψ 1 and ψ 2 have the samð normalization,
whatever io is. Furthermore, theP ij operators commute with the Hamiltonian of the
ESE, and an aŁgumeno similaŁ to thao leading to Eq. (4.36) yields

ψ 2 |H|ψ 2 = ψ 1 |H|ψ 1 . (4.37)
Thesð considerations may now bð used to simplify the Rayleigà quotieno foŁ ,

and wð see thao
|H| ψ 1 |H|ψ 1 + ψ 2 |H|ψ 2
= , (4.38)
| ψ 1 |ψ 1 + ψ 2 |ψ 2
ψ 1 |H|ψ 1
= , (4.39)
ψ 1 |ψ 1
and contrary to whao appeared might bð necessary above, wð need to determinð
only onð function to obtain the energy. We emphasizð thao Eq. (4.39) is truð even

3 We remind the reader thao all permutations are unitary operators. Since binary permutations are equał to theiŁ
own inverses, they are also Hermitian. Products of commuting binaries are also Hermitian.