92
3 Two (or More) Magnetic Centers
ˆ
P 1234 (αβαβ − βαβα) = βαβα − αβαβ
(3.78)
Note that the wave function with only one of the terms is not an eigenfunction of the
permutation operator ˆ
P 1234 . To determine the result of the sum of four-spin operators,
we will develop step-by-step the action of ( ˆ
S A · ˆ
S D )( ˆ
S B · ˆ
S C ). The other two terms
can be done by the reader as an exercise. In the first place, we need to establish the
result of acting with ˆ
S i · ˆ
S j on the different two-electron determinants. By writing ˆ
S
as ˆ
S x + ˆ
S y + ˆ
S z and using Eq. 1.20a, the following relations are easily derived:
ˆ
S 1 · ˆ
S 2 αα =
1
4
αα
ˆ
S 1 · ˆ
S 2 αβ =
1
2
βα −
1
4
αβ
ˆ
S 1 · ˆ
S 2 ββ =
1
4
ββ
ˆ
S 1 · ˆ
S 2 βα =
1
2
αβ −
1
4
βα
(3.79)
Next, we use these results to determine how ˆ
S B · ˆ
S C and ˆ
S A · ˆ
S D act on αβαβ
ˆ
S B · ˆ
S C α(1)β(2)α(3)β(4) =
ˆ
S B · ˆ
S C β(2)α(3)
α(1)β(4)
=
1
2
α(2)β(3) −
1
4
β(2)α(3)
α(1)β(4) =
1
2
ααββ −
1
4
αβαβ
(3.80)
with ˆ
S A · ˆ
S D αβαβ =
1
2 ββαα −
1
4 αβαβ and ˆ
S A · ˆ
S D ααββ =
1
2 βαβα −
1
4 ααββ the
product ( ˆ
S A · ˆ
S D )( ˆ
S B · ˆ
S C ) acting on αβαβ gives
( ˆ
S A · ˆ
S D )( ˆ
S B · ˆ
S C )αβαβ = ( ˆ
S A · ˆ
S D )
1
2
ααββ −
1
4
αβαβ
=
1
4
βαβα −
1
8
ααββ −
1
8
ββαα +
1
16
αβαβ
(3.81)
Repeating this for the other two products of bilinear operators and summing the
results of acting on βαβα as well, we obtain
1
4
βαβα −
1
8
αββα −
1
8
βααβ +
1
16
αβαβ −
1
4
αβαβ +
1
8
βααβ
+
1
8
αββα −
1
16
βαβα +
1
4
βαβα −
1
8
ααββ −
1
8
ββαα +
1
16
αβαβ
−
1
4
αβαβ +
1
8
ββαα +
1
8
ααββ −
1
16
βαβα −
1
16
αβαβ +
1
16
βαβα
=
7
16
(βαβα − αβαβ)
(3.82)
This shows that, except for a constant that can be absorbed in the interaction constant
J r , the action of the cyclic permutation operator is identical to the linear combina-