5.4 Analysis of Complex Interactions
161
S
0
A T
0
B =
1
2
|ϕ 1 ϕ 2 |+|ϕ 2 ϕ 1 |
|ϕ 3 ϕ 4 |−|ϕ 4 ϕ 3 |
=
1
2
|ϕ 1 ϕ 2 ϕ 3 ϕ 4 |
−|ϕ 1 ϕ 2 ϕ 4 ϕ 3 |+|ϕ 2 ϕ 1 ϕ 3 ϕ 4 |−|ϕ 2 ϕ 1 ϕ 4 ϕ 3 |
= S
0 T
0
(5.35)
S
0
A S
0
B =
1
2
|ϕ 1 ϕ 2 |+|ϕ 2 ϕ 1 |
|ϕ 3 ϕ 4 |+|ϕ 4 ϕ 3 |
=
1
2
|ϕ 1 ϕ 2 ϕ 3 ϕ 4 |
+|ϕ 1 ϕ 2 ϕ 4 ϕ 3 |+|ϕ 2 ϕ 1 ϕ 3 ϕ 4 |+|ϕ 2 ϕ 1 ϕ 4 ϕ 3 |
= S
0 S
0
(5.36)
These six CSFs can be combined to form spin eigenstates; three states with local
triplet coupling on both magnetic centers and three more with at least one magnetic
center in a locally excited (non-Hund) state.
Q =
2
3
T
0 T
0 +
1
2
(T
+ T
− )
(5.37a)
T =
1
√
2
T
+ T
− − T
− T
+
(5.37b)
S =
1
√
3
T
0 T
0 − T
+ T
− − T
− T
+
(5.37c)
NH1 =
1
√
2
T
0 S
0 + S
0 T
0
(5.38a)
NH2 =
1
√
2
T
0 S
0 − S
0 T
0
(5.38b)
NH3 = S
0 S
0
(5.38c)
These six CSFs are the basis of the model space of the determinants of the fourelectron/four-orbital CAS calculation with the restriction of one electron per orbital.
The matrix representation of the model space is
|Q| T | S| NH1| NH2| NH3
Q|
E 0 − 2K − 2K ′
T |
0
E 0 − 2K
S|
00
E 0 − 2K + K ′
NH1|
000
E 0 + 2K ′
NH2|
0
K 24 − K 13
00
E 0 + 2K "
NH3|
00
√
3(K ′′ − K ′ )
00
E 0 + 2K − K ′
where E 0 contains all the one-electron terms and the Coulomb integrals. K is the
average of the on-site exchange integrals 2K = K 12 + K 34 . K ′ and K ′′ are sums
of two-center exchange integrals, and hence, much smaller. 2K ′ = K 13 + K 24 ;
2K ′′ = K 14 + K 23 . The different exchange integrals K ij == ϕ i ϕ j |1/r ij |ϕ j ϕ i are
defined in Fig. 5.13.