Glossary

Here we will describe a few terms often used in the context of FLEUR calculations

atomic units

Almost all input and output in the FLEUR code is given in atomic units, with the exception of the U and J parameters for the LDA+U method in the input-file and the bandstructure and the DOS output-files where the energy unit is eV.

energy units: 1 Hartree (htr) = 2 Rydberg (Ry) = 27.21 electron volt (eV)
length units: 1 bohr (a.u.) = 0.529177 Ångström = 0.0529177 nm
electron mass, charge and Planks constant h / 2 π (ℏ) are unity
speed of light = e'^2^'/ℏ 1/ α ; fine-structure constant α: 1/α = 137.036

band gap

The band-gap printed in the output ([[out]] file) of the FLEUR code is the energy separation between the highest occupied Kohn-Sham eigenvalue and the lowest unoccupied one. Generally this value differs from the physical band-gap, or the optical band-gap, due to the fact that Kohn-Sham eigenvalues are in a strict sense Lagrange multipliers and not quasiparticle energies (see e.g. Perdew & Levy, PRL 51, 1884 (1983)).

core levels

States, which are localized near the nucleus and show no or negligible dispersion can be treated in an atomic-like fashion. These core levels are excluded from the valence electrons and not described by the FLAPW basisfunctions. Nevertheless, their charge is determined at every iteration by solving a Dirac equation for the actual potential. Either a radially symmetric Dirac equation is solved (one for spin-up, one for spin-down) or, if @@kcrel=1@@ in the input file, even a magnetic version (cylindrical symmetry) is solved.

distance (charge density)

In an iteration of the self consistency cycle, from a given input charge density, ρ'^in^', a output density, ρ'^out^', is calculated. As a measure, how different these two densities are, the distance of charge densities (short: distance, d) is calculated. It is defined as the integral over the unit cell: {$ d = \int || \rho^{in} - \rho^{out} || d \vec r $}\ and gives an estimate, whether self-consistency is approached or not. Typically, values of 0.001 milli-electron per unit volume (a.u.'^3^') are small enough to ensure that most properties have converged. You can find this value in the out-file, e.g. by @@grep dist out@@. In spin-polarized calculations, distances for the charge- and spin-density are provided, for non-Collinear magnetism calculations even three components exists. Likewise, in an LDA+U calculation a distance of the density matrices is given.

energy parameters

To construct the FLAPW basisfunctions such, that only the relevant (valence) electrons are included (and not, e.g. 1s, 2s, 2p for a 3d-metal) we need to specify the energy range of interest. Depending slightly on the shape of the potential and the muffin-tin radius, each energy corresponds to a certain principal quantum number "n" for a given "l". E.g. if for a 3d transition metal all energy parameters are set to the Fermi-level, the basis functions should describe the valence electrons 4s, 4p, and 3d. Also for the vacuum region we define energy parameters, if more than one principal quantum number per "l" is needed, local orbitals can be specified.

Fermi level

In a calculation, this is the energy of the highest occupied eigenvalue (or, sometimes it can also be the lowest unoccupied eigenvalue, depending on the "thermal broadening", i.e. numerical issues). In a bulk calculation, this energy is given relative to the average value of the interstitial potential; in a film or wire calculation, it is relative to the vacuum zero.

interstitial region

Every part of the unit cell that does not belong to the
muffin-tin spheres and not to the vacuum region. Here, the basis (charge density, potential) is described as 3D planewaves.

lattice harmonics

Symmetrized spherical harmonics. According to the point group of the atom, only certain linear combinations of spherical harmonics are possible. A list of these combinations can be found at the initial section of the out-file.

local orbitals

To describe states outside the valence energy window, it is recommended to use local orbitals. This can be useful for lower-lying semicore-states, as well as unoccupied states (note, however, that this just enlarges the basis-set and does not cure DFT problems with unoccupied states).

magnetic moment

The magnetic (spin) moment can be defined as difference between "spin-up" and "spin-down" charge, either in the entire unit cell, or in the muffin-tin spheres. Both quantities can be found in the out-file, the latter one explicitly marked by " --> mm", the former has to be calculated from the charge analysis (at the end of this file). \ The orbital moments are found next to the spin-moments, when SOC is included in the calculation. They are only well defined in the muffin-tin spheres as {$ m_{orb} = \mu_B \sum_i < \phi_i | r \times v | \phi_i > $}.\ The in a collinear calculation, the spin-direction without SOC is arbitrary, but assumed to be in z-direction. With SOC, it is in the direction of the specified spin-quantization axis. The orbital moment is projected on this axis. In a non-collinear calculation, the spin-directions are given explicitely in the input-file.

muffin-tin sphere

Spherical region around an atom. The muffin-tin radius is an important input parameter. The basis inside the muffin-tin sphere is described in spherical harmonics times a radial function. This radial function is given numerically on a logarithmic grid. The charge density and potential here are also described by a radial function times a the lattice harmonics.