170
5 Towards a Quantitative Understanding
Fig. 5.16 Schematic
representation of one of the
fourth-order interactions
contributing to the energy of
the triplet state. The total
contribution of this path
equals −t 2
13 t 2
24 /UU ′ 2K
Φ I Φ α Φ β Φ γ Φ I
Φ I Φ α Φ β Φ γ Φ I
SI 1 NH3 I 1 S −
3
4 t 4
13 /U 2 4KT I 2 NH2 I 2 T −t 4
13 /U 2 2K
I 3
−
3
4 t 4
13 /U 2 4KI 4
−t 4
13 /U 2 2K
I 5
−
3
4 t 2
13 t 2
24 /UU ′ 4KI 6
t 2
13 t 2
24 /UU ′ 2K
I 7
−
3
4 t 2
13 t 2
24 /UU ′ 4KI 8
t 2
13 t 2
24 /UU ′ 2K
I 3
I 3
−
3
4 t 4
13 /U 2 4KI 4
I 4
−t 4
13 /U 2 2K
I 5
−
3
4 t 2
13 t 2
24 /UU ′ 4KI 6
t 2
13 t 2
24 /UU ′ 2K
I 7
−
3
4 t 2
13 t 2
24 /UU ′ 4KI 8
t 2
13 t 2
24 /UU ′ 2K
I 5
I 5
−
3
4 t 4
24 /U ′2 4KI 6
I 6
−t 4
24 /U ′2 2K
I 7
−
3
4 t 4
24 /U ′2 4KI 8
t 2
24 /U ′2 2K
I 7
I 7
−
3
4 t 4
24 /U ′2 4KI 6
I 6
−t 4
24 /U ′2 2K
The terms with the ionic states interchanged (Φ α = I 3, Φ γ = I 1, etc.) should also
be added to the perturbational estimate. This is easily done by multiplying all the
terms by two, except the ones with Φ α = Φ γ . Now the following corrections arise
E
(4)
T =
2t 4
13
U 2 K
+
2t 4
24
U ′2 K
−
4t 2
13 t 2
24
UU ′ K
=
2B
K
(5.50)
E
(4)
S =−
3
4
t 4
13
U 2 K
−
3
4
t 4
24
U ′2 K
−
3
2
t 2
13 t 2
24
UU ′ K
=−
3J (2)2
4K
(5.51)
with B = t 2
13 /U − t 2
24 /U ′ . Then the total perturbative estimate for the singlet and
triplet energies are