## 3 Approximation for the exchange-correlation energy

In the literature one can find several approximations of the , for instance,the local density approximation (LDA) [144,145], the general gradient approximation (GGA) [146], etc. LDA is a rather simple approximation which has yielded accurate results in many cases, and may be expressed in the form:

 (82)

where 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) [147]. 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:

 (83)

represents the Pauli matrices and is the unit matrix.

In this context the Kohn-Sham equation (KSE) may be expressed by:

 (84)

where 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:

 (85) (86)

Another condition that has to be satisfied is the charge conservation. This requirement allows us to determine the Fermi level , from the next relation:

 (87)

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).

Rocio Yanes