190
6 Magnetism and Conduction
long-range magnetic ordering, Kondo effect, domain formation, superconductivity,
metal-insulator transitions, etc. belongs typically to the field of condensed matter
physics and several excellent books have been published on this topic, see for example Refs. [7–10]. This does however not mean that quantum chemistry cannot contribute to the understanding of magnetic phenomena in solid state compounds. We
have already seen in Sect. 3.3 how the calculation of the magnetic interaction parameters can serve as the basis for the calculation of the magnetic susceptibility and
the determination of the magnetic structure, or more precise the magnetic unit cell.
In fact, a large part of the parameters that typically appear in the model Hamiltonians
of condensed matter physics can be calculated accurately through quantum chemical
calculations provided that one can establish an accurate finite representation of the
relevant part of the crystal. In the case of molecular crystals, this issue is nearly trivially answered: taking one or several discrete units as model often suffices to calculate
the desired microscopic electronic structure parameters. The situation becomes more
complicated when dealing with ionic lattices (oxides, pnictides among others) and is
even worse for crystals with only covalently bonded atoms (e.g. silicon or graphene
doped with holes). There are however several well-established approaches to extract
reliable information at least for the ionic crystals. Also in the more difficult case of
(partly) covalent lattices quantum chemical strategies can offer interesting insights
in the electronic structure related to magnetic interactions.
6.3.1 Embedded Cluster Approach
The intrinsic local nature of the interaction between two localized spin moments suggests the possibility to study the magnetic interactions in solids with a cluster model.
In this approach, a small yet relevant piece is cut from the crystal and treated like a
molecule. These bare clusters are only a reasonable choice in the case of molecular
crystals, but otherwise nearly always too crude a representation. Therefore it is necessary to account for the effect of the rest of the crystal especially when dealing with
ionic or covalent lattices. Here, we will shortly review a few representative examples
of the different approaches for improving the bare cluster model that find their basis
in the theory of electron separability of McWeeny, the subsystem formulation of DFT
of Cartona or the incremental scheme of Fulde and Stoll.
Electrostatic embedding: In the case of ionic compounds, the largest contribution
to the potential exerted by the rest of the crystal on the (central region of the) cluster
is due to long-range electrostatic interactions. These are accurately represented by
the point charge approximation, that is, the Madelung potential:
V M (κ ∈ K) =
λ∈L
q κ q λ
r κλ
(6.36)
6 Magnetism and Conduction
long-range magnetic ordering, Kondo effect, domain formation, superconductivity,
metal-insulator transitions, etc. belongs typically to the field of condensed matter
physics and several excellent books have been published on this topic, see for example Refs. [7–10]. This does however not mean that quantum chemistry cannot contribute to the understanding of magnetic phenomena in solid state compounds. We
have already seen in Sect. 3.3 how the calculation of the magnetic interaction parameters can serve as the basis for the calculation of the magnetic susceptibility and
the determination of the magnetic structure, or more precise the magnetic unit cell.
In fact, a large part of the parameters that typically appear in the model Hamiltonians
of condensed matter physics can be calculated accurately through quantum chemical
calculations provided that one can establish an accurate finite representation of the
relevant part of the crystal. In the case of molecular crystals, this issue is nearly trivially answered: taking one or several discrete units as model often suffices to calculate
the desired microscopic electronic structure parameters. The situation becomes more
complicated when dealing with ionic lattices (oxides, pnictides among others) and is
even worse for crystals with only covalently bonded atoms (e.g. silicon or graphene
doped with holes). There are however several well-established approaches to extract
reliable information at least for the ionic crystals. Also in the more difficult case of
(partly) covalent lattices quantum chemical strategies can offer interesting insights
in the electronic structure related to magnetic interactions.
6.3.1 Embedded Cluster Approach
The intrinsic local nature of the interaction between two localized spin moments suggests the possibility to study the magnetic interactions in solids with a cluster model.
In this approach, a small yet relevant piece is cut from the crystal and treated like a
molecule. These bare clusters are only a reasonable choice in the case of molecular
crystals, but otherwise nearly always too crude a representation. Therefore it is necessary to account for the effect of the rest of the crystal especially when dealing with
ionic or covalent lattices. Here, we will shortly review a few representative examples
of the different approaches for improving the bare cluster model that find their basis
in the theory of electron separability of McWeeny, the subsystem formulation of DFT
of Cartona or the incremental scheme of Fulde and Stoll.
Electrostatic embedding: In the case of ionic compounds, the largest contribution
to the potential exerted by the rest of the crystal on the (central region of the) cluster
is due to long-range electrostatic interactions. These are accurately represented by
the point charge approximation, that is, the Madelung potential:
V M (κ ∈ K) =
λ∈L
q κ q λ
r κλ
(6.36)