6.3 A Quantum Chemical Approach to Magnetic Interactions in the Solid State
191
where K is the set of ions that belong to the cluster and L contains all other ions,
q corresponds to the formal ionic charge of each center. This interaction is easily
included in the calculation by placing an array of point charges around the cluster at
the lattice sites. Their value is either taken as the formal ionic charge (with fractional
charges on the edge of the array to ensure charge neutrality) or fitted in such a way
that a relatively small set of point charges reproduces the electrostatic effect of the
whole crystal.
The presence of point charges at lattice sites in the immediate neighbourhood of the
cluster often artificially polarizes the electron density of the cluster. This polarization
is especially large when the cluster has anions on the outside and the first shell
of charges contains positive charges. Actually, this is the common situation when
magnetic interactions in ionic transition metal compounds are studied. The usual
cluster has two (or sometimes more) transition metals and the anions (O 2− ,F − , etc.)
of the first coordination sphere. The first shell around this cluster is formed by either
the transition metal ions, ternary cations or a combination of these, depending on the
crystal structure. In any case, positive point charges are located directly around the
highly polarizable anions causing important distortions of the electronic structure
not only in the border regions of the cluster, but also in the central part. To improve
the description a border region is created between the point charges and the cluster.
In this intermediate region the lattice sites are occupied by potentials that model the
Coulomb and exchange interactions between the electron density of the cluster and
the ions in the intermediate region [11]. Figure 6.7 shows how the potentials separate
the cluster from the bare point charges and avoid the artificial polarization of the
cluster electron density.
Density-based embeddings: This approach starts with a calculation on the whole
system to construct an approximate yet accurate representation of the total density
ρ tot by performing a periodic DFT calculation. Then, a guess density of the cluster is
constructed from a calculation on the isolated unit or using some simple embedding
scheme as described above. The total density is now divided in two parts ρ tot =
ρ 1 + ρ 2 and the one-electron embedding potential is constructed from the functional
derivative of interaction energy with respect to the cluster density ρ 1 .
E
int = T
int
s + E
int
ne + E
int
xc + E
int
H + E
int
nn
ν emb (r) =
∂E int
∂ρ 1
(6.37)
where the interaction energy is written as a sum of the kinetic, electron-nuclear,
exchange correlation, Coulomb repulsion, and nuclear repulsion energy. This embedding potential is added to the standard Kohn-Sham equation for the cluster and a new
energy and density ρ 1 are calculated. Since the embedding potential depends on the
density of the cluster, ν emb is updated and the Kohn-Sham equations of the cluster are
solved again. This process is repeated until a self-consistent description is obtained.
In addition to the here sketched DFT in DFT (cluster in embedding) procedure, the
variants with wave function (WF) based methods have also been described. The
191
where K is the set of ions that belong to the cluster and L contains all other ions,
q corresponds to the formal ionic charge of each center. This interaction is easily
included in the calculation by placing an array of point charges around the cluster at
the lattice sites. Their value is either taken as the formal ionic charge (with fractional
charges on the edge of the array to ensure charge neutrality) or fitted in such a way
that a relatively small set of point charges reproduces the electrostatic effect of the
whole crystal.
The presence of point charges at lattice sites in the immediate neighbourhood of the
cluster often artificially polarizes the electron density of the cluster. This polarization
is especially large when the cluster has anions on the outside and the first shell
of charges contains positive charges. Actually, this is the common situation when
magnetic interactions in ionic transition metal compounds are studied. The usual
cluster has two (or sometimes more) transition metals and the anions (O 2− ,F − , etc.)
of the first coordination sphere. The first shell around this cluster is formed by either
the transition metal ions, ternary cations or a combination of these, depending on the
crystal structure. In any case, positive point charges are located directly around the
highly polarizable anions causing important distortions of the electronic structure
not only in the border regions of the cluster, but also in the central part. To improve
the description a border region is created between the point charges and the cluster.
In this intermediate region the lattice sites are occupied by potentials that model the
Coulomb and exchange interactions between the electron density of the cluster and
the ions in the intermediate region [11]. Figure 6.7 shows how the potentials separate
the cluster from the bare point charges and avoid the artificial polarization of the
cluster electron density.
Density-based embeddings: This approach starts with a calculation on the whole
system to construct an approximate yet accurate representation of the total density
ρ tot by performing a periodic DFT calculation. Then, a guess density of the cluster is
constructed from a calculation on the isolated unit or using some simple embedding
scheme as described above. The total density is now divided in two parts ρ tot =
ρ 1 + ρ 2 and the one-electron embedding potential is constructed from the functional
derivative of interaction energy with respect to the cluster density ρ 1 .
E
int = T
int
s + E
int
ne + E
int
xc + E
int
H + E
int
nn
ν emb (r) =
∂E int
∂ρ 1
(6.37)
where the interaction energy is written as a sum of the kinetic, electron-nuclear,
exchange correlation, Coulomb repulsion, and nuclear repulsion energy. This embedding potential is added to the standard Kohn-Sham equation for the cluster and a new
energy and density ρ 1 are calculated. Since the embedding potential depends on the
density of the cluster, ν emb is updated and the Kohn-Sham equations of the cluster are
solved again. This process is repeated until a self-consistent description is obtained.
In addition to the here sketched DFT in DFT (cluster in embedding) procedure, the
variants with wave function (WF) based methods have also been described. The