5.4 Analysis of Complex Interactions
169
|T | NH2| I 2| I 4| I 6| I 8
T |
0
NH2|
02 K
I 2|
t 13
−t 13
U
I 4|
t 13
−t 13
0
U
I 6|
−t 24
−t 24
00
U ′
I 8|
−t 24
−t 24
000
U ′
|S| NH3| I 1| I 3| I 5| I 7
S|
0
NH3|
04 K
I 1|
−3t 13 /
√
6
t 13 /
√
2
U
I 3|
−3t 13 /
√
6
t 13 /
√
20U
I 5|
3t 24 /
√
6
−t 24 /
√
20
0
U ′
I 7|
3t 24 /
√
6
−t 24 /
√
20
0
0
U ′
The second-order correction to the energy is obtained from the expression
E
(2) =
Φ I | ˆ
H |Φ α Φ α | ˆ
H |Φ I
E I − E α
(5.47)
which becomes
E
(2)
T =
i=2,4,6,8
T | ˆ
H |IiIi| ˆ
H |T
−U i
=−
2t 2
13
U
−
2t 2
24
U ′
(5.48)
for the triplet and
E
(2)
S =
i=1,3,5,7
S| ˆ
H |IiIi| ˆ
H |S
−U i
=−
3t 2
13
U
−
3t 2
24
U ′
(5.49)
for the singlet. From these equations we can define J (2) , the second-order estimate
of J ,as−
t 2
13
U −
t 2
24
U ′ . The fourth-order contribution is determined with Eq. 5.12 and
counts with much more terms, which are summarized below and illustrated for one
of the cases in Fig. 5.16.