1.2 Generation of Many Electron Spin Functions
17
which are (anti-)symmetric under the permutation of electron 3 and 4. Furthermore,
these two linear combinations clearly reveal the triplet coupling of electron 1 and 2,
and the singlet (
Ψ ) or triplet coupling (
Ψ ′ ) for electron 3 and 4.
Remains to evaluate the function generated by the incorporation of the fourth
electron spin by subtraction from the three-electron doublet function. This gives
a singlet spin function characterized by the triplet coupling of electron 1 and 2
(following Hund’s rule) and of electron 3 and 4. To apply Eq. 1.43,theΨ(3,
1
2 , −
1
2 )
function has to be generated by acting with ˆ
S − on Ψ(3,
1
2 ,
1
2 ).
Ψ(4, 0, 0) =
− (0 − 0 + 1)
1
2
1
√
6
[−2β(1)β(2)α(3) + β(1)α(2)β(3)
+ α(1)β(2)β(3)]α(4) + (0 + 0 + 1)
1
2
1
√
6
[2α(1)α(2)β(3)
− α(1)β(2)α(3) − β(1)α(2)α(3)]β(4)
(2 · 0 + 2)
−
1
2
=
1
2
√
3
[2(ααββ + ββαα) − αβαβ − αββα − βααβ − βαβα]
=
1
2
√
3
[2(ααββ + ββαα) − (αβ + βα)(αβ + βα)]
(1.51)
1.9 (a) Construct a branching diagram and mark the path to generate the
N = 4 triplet and singlet spin states with singlet coupling for electron 1 and
2. (b) Construct Ψ(2, 0, 0) with the genealogical approach.
Two-by-two additions: The process of generating spin functions by the genealogical
approach can be made a little less tedious by considering the incorporation of two
electrons at the same time. An additional advantage of doing so is that one better
controls the spin coupling of electron pairs. The triplet functions with four electrons,
Ψ(4, 1, 1) and Ψ ′ (4, 1, 1) Eqs. 1.48 and 1.49 do have triplet coupling among electron
1 and 2, but turn out to be mixtures of singlet and triplet coupling for electrons 3 and 4.
Only after taking the correct linear combination, spin functions could be constructed
with clear-cut spin couplings of both electron pairs. This can be achieved directly
with the Serber variant of the genealogical approach [3, 4] illustrated in the branching
diagram of Fig. 1.5.
Starting with an N − 2-electron spin function of spin S ′ , singlet or triplet coupled
two-electron functions are added to obtain Ψ(N , S) with S = S ′ +1, S ′ or S ′ −1. The
branching diagram shows that four different cases can be distinguished, for which
the following formulas need to be considered:
• case 1: Singlet incorporation (gray solid lines); S = S ′
Ψ(N , S, M S ) = Ψ(N − 2, S, M S )Φ a
(1.52)
17
which are (anti-)symmetric under the permutation of electron 3 and 4. Furthermore,
these two linear combinations clearly reveal the triplet coupling of electron 1 and 2,
and the singlet (
Ψ ) or triplet coupling (
Ψ ′ ) for electron 3 and 4.
Remains to evaluate the function generated by the incorporation of the fourth
electron spin by subtraction from the three-electron doublet function. This gives
a singlet spin function characterized by the triplet coupling of electron 1 and 2
(following Hund’s rule) and of electron 3 and 4. To apply Eq. 1.43,theΨ(3,
1
2 , −
1
2 )
function has to be generated by acting with ˆ
S − on Ψ(3,
1
2 ,
1
2 ).
Ψ(4, 0, 0) =
− (0 − 0 + 1)
1
2
1
√
6
[−2β(1)β(2)α(3) + β(1)α(2)β(3)
+ α(1)β(2)β(3)]α(4) + (0 + 0 + 1)
1
2
1
√
6
[2α(1)α(2)β(3)
− α(1)β(2)α(3) − β(1)α(2)α(3)]β(4)
(2 · 0 + 2)
−
1
2
=
1
2
√
3
[2(ααββ + ββαα) − αβαβ − αββα − βααβ − βαβα]
=
1
2
√
3
[2(ααββ + ββαα) − (αβ + βα)(αβ + βα)]
(1.51)
1.9 (a) Construct a branching diagram and mark the path to generate the
N = 4 triplet and singlet spin states with singlet coupling for electron 1 and
2. (b) Construct Ψ(2, 0, 0) with the genealogical approach.
Two-by-two additions: The process of generating spin functions by the genealogical
approach can be made a little less tedious by considering the incorporation of two
electrons at the same time. An additional advantage of doing so is that one better
controls the spin coupling of electron pairs. The triplet functions with four electrons,
Ψ(4, 1, 1) and Ψ ′ (4, 1, 1) Eqs. 1.48 and 1.49 do have triplet coupling among electron
1 and 2, but turn out to be mixtures of singlet and triplet coupling for electrons 3 and 4.
Only after taking the correct linear combination, spin functions could be constructed
with clear-cut spin couplings of both electron pairs. This can be achieved directly
with the Serber variant of the genealogical approach [3, 4] illustrated in the branching
diagram of Fig. 1.5.
Starting with an N − 2-electron spin function of spin S ′ , singlet or triplet coupled
two-electron functions are added to obtain Ψ(N , S) with S = S ′ +1, S ′ or S ′ −1. The
branching diagram shows that four different cases can be distinguished, for which
the following formulas need to be considered:
• case 1: Singlet incorporation (gray solid lines); S = S ′
Ψ(N , S, M S ) = Ψ(N − 2, S, M S )Φ a
(1.52)