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4 From Orbital Models to Accurate Predictions
orbitals [1]. Let φ a and φ b be the optimal local orbitals for the unpaired electrons on
site A and B. These orbitals are normalized but not orthogonal
φ a |φ b =S
φ a |φ a ==φ b |φ b =1
(4.1)
Multiplying the spatial part of the wave function |φ a φ b |=| ab| with the singlet
and triplet (M S = 0) spin functions, the following normalized wave functions are
obtained
Ψ S =
|ab|+|ba|
√ 2 + 2S 2
Ψ T =
|ab|−|ba|
√
2 − 2S 2
(4.2)
4.1 Confirm that the norms of Ψ S and Ψ T are equal to 1.
As shown in the previous chapter, the energy difference between singlet and triplet
is proportional to the magnetic coupling strength. The energy expectation values of
Ψ S and Ψ T are
E S,T =
ab ± ba| ˆ
H|ab ± ba
ab ± ba|ab ± ba
=
ab ± ba| ˆ
H|ab ± ba
2 ± 2S
(4.3)
with
ˆ
H = ˆ
h 1 (1) + ˆ
h 1 (2) +
1 − ˆ
P 12
r 12
(4.4)
where ˆ
P 12 is the permutation operator. To avoid lengthy equations, some parameters
will be introduced to facilitate the derivation.
ε ==a| ˆ
h 1 |a==b| ˆ
h 1 |b
(4.5a)
β ==a| ˆ
h 1 |b==b| ˆ
h 1 |a
(4.5b)
J
C ==ab|
1
r 12
|ab
(4.5c)
K ==ab|
1
r 12
|ba
(4.5d)
This results in the following expressions for the energy of the singlet and triplet
states.