62
3 Two (or More) Magnetic Centers
tions corresponding to ...φ 1
1 φ 1
2 belong also to a different representation compared to
those of ...φ 2
1 and ...φ 2
2 . As a result, in those cases there is no Hamiltonian matrix
element of Φ 11 (0, 0) and Φ 22 (0, 0) with Φ 12 (0, 0):
Φ
11 (0, 0)| ˆ
H |Φ
12 (0, 0)==Φ
22 (0, 0)| ˆ
H |Φ
12 (0, 0)=0
(3.6)
Then, improved singlet variational wave functions can be formed by making linear
combinations of only Φ 11 (0, 0) and Φ 22 (0, 0), whereas the singlet wave function corresponding to ...φ 1
1 φ 1
2 cannot be improved within this two electrons in two orbitals
scheme. In other magnetic systems there is no strict but only approximate symmetry,
giving rise to non-zero but still quite small matrix elements Φ 11 (0, 0)| ˆ
H |Φ 12 (0, 0)
and Φ 11 (0, 0)| ˆ
H |Φ 12 (0, 0). In the following we will assume symmetry, hence
assume that these matrix elements are zero and turn our attention to the Hamiltonian matrix element Φ 11 (0, 0)| ˆ
H |Φ 22 (0, 0) that leads to an improved singlet
wave function
Ψ(0, 0) = c 1 Φ
11 (0, 0) + c 2 Φ
22 (0, 0)
(3.7)
where c 1 and c 2 are chosen to minimize the energy of Ψ(0, 0). Using the Slater–
Condon rules we find for the Hamiltonian matrix element between the two closed
shell determinants
Φ
11 (0, 0)| ˆ
H |Φ
22 (0, 0)==φ 1 (1)φ 1 (2)|
1
r 12
|φ 2 (1)φ 2 (2)
(3.8)
By rearranging the (real) functions in this integral, it becomes clear that the matrix
element is equal to the exchange integral K 12
φ 1 (1)φ 1 (2)|
1
r 12
|φ 2 (1)φ 2 (2)==φ 1 (1)φ 2 (1)|
1
r 12
|φ 2 (2)φ 1 (2)=K 12
(3.9)
3.1 Demonstrate that Φ 11 (0, 0)| ˆ
H |Φ 22 (0, 0)=K 12 .
The exchange integral K 12 does not vanish, not even in case of weakly interacting
A and B centers. This becomes clear if we introduce the localized orthogonal orbitals
ψ a and ψ b that can be constructed from the delocalized molecular orbitals φ 1 and φ 2 :
ψ a =
1
√
2
(φ 1 + φ 2 )ψ b =
1
√
2
(φ 1 − φ 2 )
(3.10a)
If A and B are strongly coupled, ψ a will be mainly localized on A, but with important
orthogonalization tails on B and vice versa.OnlyifA and B are weakly coupled,
ψ a and ψ b are nearly completely localized on A and B, respectively.
3 Two (or More) Magnetic Centers
tions corresponding to ...φ 1
1 φ 1
2 belong also to a different representation compared to
those of ...φ 2
1 and ...φ 2
2 . As a result, in those cases there is no Hamiltonian matrix
element of Φ 11 (0, 0) and Φ 22 (0, 0) with Φ 12 (0, 0):
Φ
11 (0, 0)| ˆ
H |Φ
12 (0, 0)==Φ
22 (0, 0)| ˆ
H |Φ
12 (0, 0)=0
(3.6)
Then, improved singlet variational wave functions can be formed by making linear
combinations of only Φ 11 (0, 0) and Φ 22 (0, 0), whereas the singlet wave function corresponding to ...φ 1
1 φ 1
2 cannot be improved within this two electrons in two orbitals
scheme. In other magnetic systems there is no strict but only approximate symmetry,
giving rise to non-zero but still quite small matrix elements Φ 11 (0, 0)| ˆ
H |Φ 12 (0, 0)
and Φ 11 (0, 0)| ˆ
H |Φ 12 (0, 0). In the following we will assume symmetry, hence
assume that these matrix elements are zero and turn our attention to the Hamiltonian matrix element Φ 11 (0, 0)| ˆ
H |Φ 22 (0, 0) that leads to an improved singlet
wave function
Ψ(0, 0) = c 1 Φ
11 (0, 0) + c 2 Φ
22 (0, 0)
(3.7)
where c 1 and c 2 are chosen to minimize the energy of Ψ(0, 0). Using the Slater–
Condon rules we find for the Hamiltonian matrix element between the two closed
shell determinants
Φ
11 (0, 0)| ˆ
H |Φ
22 (0, 0)==φ 1 (1)φ 1 (2)|
1
r 12
|φ 2 (1)φ 2 (2)
(3.8)
By rearranging the (real) functions in this integral, it becomes clear that the matrix
element is equal to the exchange integral K 12
φ 1 (1)φ 1 (2)|
1
r 12
|φ 2 (1)φ 2 (2)==φ 1 (1)φ 2 (1)|
1
r 12
|φ 2 (2)φ 1 (2)=K 12
(3.9)
3.1 Demonstrate that Φ 11 (0, 0)| ˆ
H |Φ 22 (0, 0)=K 12 .
The exchange integral K 12 does not vanish, not even in case of weakly interacting
A and B centers. This becomes clear if we introduce the localized orthogonal orbitals
ψ a and ψ b that can be constructed from the delocalized molecular orbitals φ 1 and φ 2 :
ψ a =
1
√
2
(φ 1 + φ 2 )ψ b =
1
√
2
(φ 1 − φ 2 )
(3.10a)
If A and B are strongly coupled, ψ a will be mainly localized on A, but with important
orthogonalization tails on B and vice versa.OnlyifA and B are weakly coupled,
ψ a and ψ b are nearly completely localized on A and B, respectively.