Chapter 3
Two (or More) Magnetic Centers
Abstract The description of the magnetic interactions is now extended to more
than one magnetic center. First it is shown that the two-electron/two-orbital system
can be approached from different viewpoints using (de-)localized, (non-)orthogonal
orbitals. After this quantum chemical description of the magnetic interaction we discuss the more phenomenological approach based on spin operators. Starting with
the standard Heisenberg Hamiltonian for isotropic bilinear interactions, the chapter
discusses how biquadratic, anisotropic and four-center interactions can be accounted
for within this spin formalism. Furthermore, it is shown how the microscopic electronic interaction parameters can be used to describe macroscopic properties by
diagonalization of model Hamiltonians, Monte Carlo simulations and some other
techniques.
3.1 Localized Versus Delocalized Description
of the Two-Electron/Two-Orbital Problem
The most simple magnetic systems have only two magnetic sites, each with spin
1
2 . Examples are doubly bridged binuclear Cu II complexes. The energy splitting
between the lowest singlet and triplet spin states in such complexes turns out to
depend strongly on the geometry of the bridging Cu–L 2 –Cu units (R 1 , R 2 , α, β, etc.)
depicted in Fig. 3.1 and this magneto-structural correlation can be well explained by
using simple quantum theoretical models that can be developed “on the back of an
envelope”. The basis for such models is provided in this section, they are further
elaborated in Chap. 4. We consider a many-electron system, which, in addition to
closed shells of electrons, has two magnetic electrons which are mainly localized
on two magnetic sites A and B. We assume that the orbitals of the complex, i.e. its
molecular orbitals (MOs), have been determined by a self-consistent field procedure.
Configuration Interaction using delocalized orbitals: We first consider the case
where the two magnetic electrons are coupled to a spin triplet, S = 1. The simplest
description of the M S = 1 component of the lowest lying triplet state is one single
Slater determinant
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5_3
59
Two (or More) Magnetic Centers
Abstract The description of the magnetic interactions is now extended to more
than one magnetic center. First it is shown that the two-electron/two-orbital system
can be approached from different viewpoints using (de-)localized, (non-)orthogonal
orbitals. After this quantum chemical description of the magnetic interaction we discuss the more phenomenological approach based on spin operators. Starting with
the standard Heisenberg Hamiltonian for isotropic bilinear interactions, the chapter
discusses how biquadratic, anisotropic and four-center interactions can be accounted
for within this spin formalism. Furthermore, it is shown how the microscopic electronic interaction parameters can be used to describe macroscopic properties by
diagonalization of model Hamiltonians, Monte Carlo simulations and some other
techniques.
3.1 Localized Versus Delocalized Description
of the Two-Electron/Two-Orbital Problem
The most simple magnetic systems have only two magnetic sites, each with spin
1
2 . Examples are doubly bridged binuclear Cu II complexes. The energy splitting
between the lowest singlet and triplet spin states in such complexes turns out to
depend strongly on the geometry of the bridging Cu–L 2 –Cu units (R 1 , R 2 , α, β, etc.)
depicted in Fig. 3.1 and this magneto-structural correlation can be well explained by
using simple quantum theoretical models that can be developed “on the back of an
envelope”. The basis for such models is provided in this section, they are further
elaborated in Chap. 4. We consider a many-electron system, which, in addition to
closed shells of electrons, has two magnetic electrons which are mainly localized
on two magnetic sites A and B. We assume that the orbitals of the complex, i.e. its
molecular orbitals (MOs), have been determined by a self-consistent field procedure.
Configuration Interaction using delocalized orbitals: We first consider the case
where the two magnetic electrons are coupled to a spin triplet, S = 1. The simplest
description of the M S = 1 component of the lowest lying triplet state is one single
Slater determinant
© Springer International Publishing Switzerland 2016
C. Graaf and R. Broer, Magnetic Interactions in Molecules and Solids,
Theoretical Chemistry and Computational Modelling,
DOI 10.1007/978-3-319-22951-5_3
59