6.3 A Quantum Chemical Approach to Magnetic Interactions in the Solid State
197
Examples of strongly correlated systems are transition metal and rare-earth metal
compounds. In these materials on-site Coulomb repulsion between the metal valence
electrons dominates the width of the corresponding one-electron energy bands.
Widely used independent electron methods such as DFT in the local density approximation (LDA) are not suited to study the magnetic properties of such systems. Therefore, a correction term U to the LD functional has been introduced that accounts for
the strong on-site Coulomb interactions between d (or f ) electrons on the metal
ions, giving rise to the LDA+U method [14]. Although LDA+U was introduced as
a method without adjustable parameters, the values used for U vary significantly in
different studies on the same compound.
Within Green function theory, many-electron effects can be introduced through a
non-local and energy-dependent self-energy operator [15]. Since the self-energy is
hard to calculate, various approximations are introduced and among the simplest ones
is the so-called GW approximation, which is derived from many-body perturbation
theory. Although the GW approximation offers in principle a sophisticated account
of the electron correlation effects, practical realizations are commonly also based on
the LDA method.
Finally, algorithms have been developed which incorporate electron correlation
effects explicitly in wave function based band theory for crystalline solids [16, 17].
These algorithms construct the many-electron Hamiltonian matrix for a periodic system by extracting the matrix elements from calculations on finite embedded clusters.
In this way the incorporation of correlation effects leads to many-electron energy
bands, not only associated with hole states and added-electron states but also with
excited states. More recently, Pisani and co-workers [18] introduced a post-HartreeFock program based on periodic local second order Møller-Plesset perturbation
theory.
A word of warning is in place when these techniques are employed for the study
of magnetic interactions. The tiny energy differences associated with these interactions demand that the procedure is capable to deliver not only an accurate but also
a balanced treatment of the various states involved. This means that approximate
computation and cut-offs of integrals etc. have to be exactly the same for all states.
6.4 Goodenough–Kanamori Rules
The Goodenough–Kanamori (GK) rules have evolved from the studies to explain the
magnetism in manganese oxides in the 1950s and have been applied ever since mostly
in the field of ionic insulators; often oxides of one or several third-row transition metal
ions. Studies of the magnetic interactions in these compounds commonly reduce to
a three center problem with two metals that carry a spin moment and a non-magnetic
anion in between. Before explaining the rules, which are sometimes (incorrectly)
referred to as the Goodenough–Kanamori–Anderson rules, we need to introduce
some concepts related with the electron hopping involving the magnetic sites and the
ligand that connects them. Goodenough defined superexchange as the virtual electron
197
Examples of strongly correlated systems are transition metal and rare-earth metal
compounds. In these materials on-site Coulomb repulsion between the metal valence
electrons dominates the width of the corresponding one-electron energy bands.
Widely used independent electron methods such as DFT in the local density approximation (LDA) are not suited to study the magnetic properties of such systems. Therefore, a correction term U to the LD functional has been introduced that accounts for
the strong on-site Coulomb interactions between d (or f ) electrons on the metal
ions, giving rise to the LDA+U method [14]. Although LDA+U was introduced as
a method without adjustable parameters, the values used for U vary significantly in
different studies on the same compound.
Within Green function theory, many-electron effects can be introduced through a
non-local and energy-dependent self-energy operator [15]. Since the self-energy is
hard to calculate, various approximations are introduced and among the simplest ones
is the so-called GW approximation, which is derived from many-body perturbation
theory. Although the GW approximation offers in principle a sophisticated account
of the electron correlation effects, practical realizations are commonly also based on
the LDA method.
Finally, algorithms have been developed which incorporate electron correlation
effects explicitly in wave function based band theory for crystalline solids [16, 17].
These algorithms construct the many-electron Hamiltonian matrix for a periodic system by extracting the matrix elements from calculations on finite embedded clusters.
In this way the incorporation of correlation effects leads to many-electron energy
bands, not only associated with hole states and added-electron states but also with
excited states. More recently, Pisani and co-workers [18] introduced a post-HartreeFock program based on periodic local second order Møller-Plesset perturbation
theory.
A word of warning is in place when these techniques are employed for the study
of magnetic interactions. The tiny energy differences associated with these interactions demand that the procedure is capable to deliver not only an accurate but also
a balanced treatment of the various states involved. This means that approximate
computation and cut-offs of integrals etc. have to be exactly the same for all states.
6.4 Goodenough–Kanamori Rules
The Goodenough–Kanamori (GK) rules have evolved from the studies to explain the
magnetism in manganese oxides in the 1950s and have been applied ever since mostly
in the field of ionic insulators; often oxides of one or several third-row transition metal
ions. Studies of the magnetic interactions in these compounds commonly reduce to
a three center problem with two metals that carry a spin moment and a non-magnetic
anion in between. Before explaining the rules, which are sometimes (incorrectly)
referred to as the Goodenough–Kanamori–Anderson rules, we need to introduce
some concepts related with the electron hopping involving the magnetic sites and the
ligand that connects them. Goodenough defined superexchange as the virtual electron