Treatment of core electrons

The FLAPW method allows to incorporate all electrons into the DFT calculations. While the valence electrons are represented by the LAPW basis the core electron wave functions are calculated on a radial grid within the MT spheres and an extension of this grid beyond the sphere boundaries. As an approximation for the calculation of the core electron states Fleur only takes the spherical part of the effective potential $V_{l=0}^\alpha(r_\alpha)$ into account. Beyond the MT sphere boundary this part of the potential is extended by a cone-like confining potential which has the same value at the MT boundary as $V_{l=0}^\alpha(r_\alpha)$.

Due to the singularity of the effective potential at the atomic nucleus together with the concentration of the core electron states near the nucleus core electrons feature a very high kinetic energy. Therefore the determination of the core electron wave functions and eigenenergies are treated by a fully-relativistic solver on the basis of the Kohn-Sham-Dirac equations

$$\begin{equation*} \renewcommand{\vec}[1]{\mathbf{#1}} \left[c\vec{\alpha}\cdot\vec{p} + (\beta-1) m_\text{e}c^2 + V_{l=0}(r)\right] \psi_\nu(r) = E_\nu \psi_\nu(r), \end{equation*}$$

where $\psi_\nu$ the 4-component wavefunction, $E_\nu$ the $\nu$-th eigenenergy, $c$ is the speed of light, and $\vec{p}$ the momentum operator. $\vec{\alpha}$ is the vector of the $4 \times 4$ matrices

$$\begin{equation*} \renewcommand{\vec}[1]{\mathbf{#1}} \alpha_i = \left(\begin{matrix}0 & \sigma_i \\ \sigma_i & 0 \end{matrix}\right) \end{equation*}$$

in which $\sigma_i$ are the Pauli spin matrices. Finally, $\beta$ is the $4 \times 4$ matrix

$$\begin{equation*} \beta = \left(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix}\right) \end{equation*}$$

and $m_\text{e}$ is the rest mass of the electron. In the here-used Hartree atomic units this is actually 1. It is shown in the equation explicitely to make the user aware of the substraction of the rest mass energy. This is done to establish a consistent treatment between the core electrons and the valence electrons. In the determination of the energy eigenvalues for the valence electrons the electron rest mass was also not considered.

There are several aspects of the treatment of the core electrons that can be controlled by the user. First the user can decide whether the tail of the core electron density reaching out of the MT sphere is accurately added to the total density or the charge within this tail is just evenly distributed over the interstitial region. While the latter option is less accurate it may significantly speed up the calculation because the re-expansion of the core-tail density in the other MT spheres is computationally expensive. For details on the re-expansion the user may consult the respective section on the density.

The second aspect that can be controlled by the user is the choice of whether the core density is actually updated in every iteration of the SCF calculation or a frozen core approximation is used and a previously calculated core electron density is held fixed.

The last aspect to be controlled by the user is related to spin-polarized potentials. Spin is not a good quantum number in fully-relativistic calculations. This implies that the correct handling of spin-polarized potentials is more complex. It is possible to use an exact treatment of spin-polarized systems with the approach described in H. Ebert, Fully relativistic tretment of core states for a spin-dependent potential, J. Phys: Condens. Matter 1, 9111 (1989). But as an approximation Fleur also implements a relativistic solver that treats each spin channel separately. If only the core electron density is required this already provides highly accurate results. If the core level spectrum is of interest, the exact solver should be used.