120
4 From Orbital Models to Accurate Predictions
According to the HTH model this gives a reduction of the antiferromagnetic contribution to the magnetic coupling (see Eq. 4.21) and illustrates the anti-synergistic
effect or counter-complementarity of the two ligands.
4.3 Accurate Computational Models
Although the qualitative models discussed so far are very useful for a basic understanding of the magnetic interactions between two spin moments, more quantitative
predictions can only be obtained by going beyond the valence-only description considered so far. As shown in the previous chapter, the magnetic interaction parameter J
of the Heisenberg Hamiltonian can in many cases be related to the energy difference
of electronic states with different spin couplings. Hence, precise theoretical estimates
of the magnetic coupling strengths are intimately related to the correct application
of high-level computational schemes.
As shown in Sect. 3.1, the basic description of the magnetic coupling problem is
intrinsically multideterminantal and in most cases one needs a multiconfigurational
description for minimally accurate results. Before discussing the different computational schemes that can be used for quantitative estimates, we want to stress that
a multideterminantal wave function is not necessarily a multiconfigurational wave
function. This is best illustrated for the 2-electrons/2-orbitals case discussed before.
The simplest representation of the triplet state is obtained with a single Slater determinant
Φ T =|φ a φ b |
(4.28)
where all the doubly occupied orbitals have been omitted. On the other hand, the
most basic description of the open-shell singlet requires a wave function with two
Slater determinants to fulfill the requirements of the spin symmetry.
Φ S =
|φ a φ b |−|φ a φ b |
√ 2
(4.29)
This multideterminantal wave function only describes one electronic configuration
since the occupation of the orbitals is the same in both determinants and is in general
known as a configuration state function. In this simple monoconfigurational description, the energy of the triplet is lower than the singlet by twice the exchange integral
K ab . A more satisfactory description is obtained with a multiconfigurational singlet
wave function by adding the Slater determinants with two electrons in the same
orbital
Φ
′
S = c 1
|φ a φ b |−|φ a φ b |
+ c 2
|φ a φ a |+|φ b φ b |
(4.30)
where c 1 is much larger than c 2 for biradicalar systems, and their precise value has
to be determined in a configuration interaction calculation. Wave functions of this
4 From Orbital Models to Accurate Predictions
According to the HTH model this gives a reduction of the antiferromagnetic contribution to the magnetic coupling (see Eq. 4.21) and illustrates the anti-synergistic
effect or counter-complementarity of the two ligands.
4.3 Accurate Computational Models
Although the qualitative models discussed so far are very useful for a basic understanding of the magnetic interactions between two spin moments, more quantitative
predictions can only be obtained by going beyond the valence-only description considered so far. As shown in the previous chapter, the magnetic interaction parameter J
of the Heisenberg Hamiltonian can in many cases be related to the energy difference
of electronic states with different spin couplings. Hence, precise theoretical estimates
of the magnetic coupling strengths are intimately related to the correct application
of high-level computational schemes.
As shown in Sect. 3.1, the basic description of the magnetic coupling problem is
intrinsically multideterminantal and in most cases one needs a multiconfigurational
description for minimally accurate results. Before discussing the different computational schemes that can be used for quantitative estimates, we want to stress that
a multideterminantal wave function is not necessarily a multiconfigurational wave
function. This is best illustrated for the 2-electrons/2-orbitals case discussed before.
The simplest representation of the triplet state is obtained with a single Slater determinant
Φ T =|φ a φ b |
(4.28)
where all the doubly occupied orbitals have been omitted. On the other hand, the
most basic description of the open-shell singlet requires a wave function with two
Slater determinants to fulfill the requirements of the spin symmetry.
Φ S =
|φ a φ b |−|φ a φ b |
√ 2
(4.29)
This multideterminantal wave function only describes one electronic configuration
since the occupation of the orbitals is the same in both determinants and is in general
known as a configuration state function. In this simple monoconfigurational description, the energy of the triplet is lower than the singlet by twice the exchange integral
K ab . A more satisfactory description is obtained with a multiconfigurational singlet
wave function by adding the Slater determinants with two electrons in the same
orbital
Φ
′
S = c 1
|φ a φ b |−|φ a φ b |
+ c 2
|φ a φ a |+|φ b φ b |
(4.30)
where c 1 is much larger than c 2 for biradicalar systems, and their precise value has
to be determined in a configuration interaction calculation. Wave functions of this