3 Approximation for the exchange-correlation energy $ E_{xc}[n]$

In the literature one can find several approximations of the $ E_{xc}[n]$, 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:

$\displaystyle E_{xc}[n]\approx E_{xc}^{LDA}[n]=\int n(\vec{r})\epsilon_{xc}^{LDA}(n(\vec{r}))d^{3}r$ (82)

where $ \epsilon_{xc}^{LDA}$ is the exchange-correlation energy per particle for a homogeneous electron gas of density $ n(\vec{r})$. Within this approximation it is assumed that the energy $ E_{xc}$ of a system with an inhomogeneous electron distribution $ n(\vec{r})$ at each point ($ \vec{r}$) 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 $ \vec{m}(\vec{r})$ has to be included and the non-spin polarized density $ n$ is replaced by the density matrix $ \mathbf{n}(\vec{r})$. The density matrix can be expressed as:

$\displaystyle \mathbf{n}(\vec{r})=n(\vec{r})\cdot \mathbf{1} + \mathbf{\vec{\sigma}}\cdot\vec{m}(\vec{r});$ (83)

$ \mathbf{\vec{\sigma}}$ represents the Pauli matrices and $ \mathbf{1}$ is the unit matrix.

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

$\displaystyle \bigl( -\frac{\hbar^{2}}{2m}\nabla^{2} +V_{eff}(\vec{r}) +\mathbf...
...\psi_{i\uparrow }(\vec{r}) \ \psi_{i\downarrow} (\vec{r}) \end{array} \Bigr)=0$ (84)

where $ \psi_{i\uparrow;\downarrow }(\vec{r})$ are the resulting eigenfunctions of the auxiliary non-interacting system. These eigenstates are dependent on the eigenenergy $ E_{i}$ and on the spin up or down ( $ \uparrow; \downarrow$) configuration, with respect to the local magnetization direction $ \mu (\vec{r})$ along the effective magnetic field $ \vec{B}_{eff}(\vec{r})$. $ V_{eff}(\vec{r})$ 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:

$\displaystyle n(\vec{r})$ $\displaystyle =$ $\displaystyle \sum_{E_{i}\leq E_{F}} \bigl(\mid\psi_{i\uparrow }(\vec{r}) \mid^{2} + \mid\psi_{i\downarrow }(\vec{r}) \mid^{2}\bigr)$ (85)
$\displaystyle \vec{m}(\vec{r})$ $\displaystyle =$ $\displaystyle \mu(\vec{r})\sum_{E_{i}\leq E_{F}}\bigl(\mid\psi_{i\uparrow }(\vec{r}) \mid^{2} - \mid\psi_{i\downarrow }(\vec{r}) \mid^{2}\bigr)$ (86)

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

$\displaystyle N=\int n(\vec{r}) d^{3}r =\int d^{3}r\sum_{E_{i}\leq E_{F}} \bigl...
...i\uparrow }(\vec{r}) \mid^{2} + \mid\psi_{i\downarrow }(\vec{r}) \mid^{2}\bigr)$ (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