In the literature one can find several approximations of the , for instance,the local density approximation (LDA) [144,145], the general gradient approximation (GGA) , etc. LDA is a rather simple approximation which has yielded accurate results in many cases, and may be expressed in the form:
is the exchange-correlation energy per particle for a homogeneous electron gas of density
Within this approximation it is assumed that the energy of a system with an inhomogeneous electron distribution
at each point () has the same value as in the case of homogeneous electron gas with the same density.
We are working with a magnetic system, therefore we have to use an approximation that takes into account the spin-polarization of the system, for this reason we used a generalization of the LDA approximation for a spin polarized system: local spin density approximation (LSDA) .
In the case of spin-polarized calculations, the spin density
has to be included and the non-spin polarized density is replaced by the density matrix
. The density matrix can be expressed as:
represents the Pauli matrices and
is the unit matrix.
In this context the Kohn-Sham equation (KSE) may be expressed by:
are the resulting eigenfunctions of the auxiliary non-interacting system. These eigenstates are dependent on the eigenenergy and on the spin up or down (
) configuration, with respect to the local magnetization direction
along the effective magnetic field
is the effective potential which collects the contributions of the external potential, the Hartree and the exchange-correlation potentials, the last one in practice is unknown and must be approximated.
We have to complete the set of Kohn-Sahm equations with the charge and spin density ones:
Another condition that has to be satisfied is the charge conservation. This requirement allows us to determine the Fermi level , from the next relation:
The equations (6.7-6.10) are the set of equations that must be solved self-consistently in the DFT calculations to obtains the ground state energy of the system.
Note that in principle the Kohn-Sham equation corresponds to a non-relativistic formulation of the DFT. If we work within the relativistic DFT for instance, in the case where the spin-orbit coupling (SOC) can not be treated as a perturbation term, we have to replace the Kohn-Sham equation by the corresponding Kohn-Sham-Dirac equation (KSDE).