5.4 Analysis of Complex Interactions
173
there is only one sizeable four-center interaction, namely the one with the ˆ
P 1234
operator associated to it.
The expectation values of the M S = 0 determinants can be found in Eq. 3.84,but
for the M S = 1 and M S = 2 determinants we will derive them here. Among the four
degenerate M S = 1 determinants we will focus on |abcd|,or|αααβ| in a spin-only
notation. With the following ingredients for the two-center interactions:
−J 1 ( ˆ
S A ˆ
S B + ˆ
S C ˆ
S D )αααβ =−J 1
1
4
αααβ +
1
2
ααβα −
1
4
αααβ
−J 2 ( ˆ
S A ˆ
S D + ˆ
S B ˆ
S C )αααβ =−J 2
1
2
βααα −
1
4
αααβ +
1
4
αααβ
(5.62)
−J 3 ( ˆ
S A ˆ
S D + ˆ
S B ˆ
S C )αααβ =−J 3
1
4
αααβ +
1
2
αβαβ −
1
4
αααβ
and for the four-center interaction:
J r ( ˆ
S A ˆ
S B )( ˆ
S C ˆ
S D )αααβ = J r ( ˆ
S A ˆ
S B )
1
2
ααβα −
1
4
αααβ
= J r
1
8
ααβα −
1
16
αααβ
J r ( ˆ
S A ˆ
S D )( ˆ
S B ˆ
S C )αααβ = J r ( ˆ
S A ˆ
S D )
1
4
αααβ
(5.63)
= J r
1
8
βααα −
1
16
αααβ
−J r ( ˆ
S A ˆ
S C )( ˆ
S B ˆ
S D )αααβ =−J r ( ˆ
S A ˆ
S C )
1
2
αβαα −
1
4
αααβ
=−J r
1
8
αβαα −
1
16
αααβ
the matrix element becomes
αααβ| ˆ
H |αααβ=−
1
16
J r
(5.64)
Applying the same procedure to |αααα| leads to
αααα| ˆ
H |αααα=−
1
2
(J 1 + J 2 + J 3 ) +
1
16
J r
(5.65)
5.14 Check the matrix element of the M S = 2 determinant of the spin Hamiltonian given in Eq. 3.83.