The linearized augmented-plane-wave method (LAPW) is an implementation of Kohn-Sham density functional theory (DFT) adapted to periodic materials.
These features make it one of the most precise implementations of DFT, applicable to all crystalline materials, regardless of their chemical composition.
A method implementing Kohn-Sham DFT has to realize these different steps of the sketched iterative algorithm.
The physical description and the representation of the Kohn-Sham orbitals, the charge density, and the potential is adapted to this partitioning.
The central aspect of practical DFT implementations is the question how to solve the Kohn-Sham equations with the single-electron kinetic energy operator
The LAPW basis is designed to enable a precise representation of the orbitals and an accurate modelling of the physics in each region of the unit cell.
is hereby the solution of the Kohn-Sham Hamiltonian for the spherically averaged potential with regular behavior at the nucleus for the given energy parameter
these augmentations of the plane wave in each MT sphere enable a representation of the Kohn-Sham orbitals at arbitrary eigenenergies linearized around the energy parameters.
In each MT sphere, the expansion into spherical harmonics is limited to a maximum number of angular momenta
While the LAPW basis functions are used to represent the valence states, core electron states, which are completely confined within a MT sphere, are calculated for the spherically averaged potential on radial grids, for each atom separately applying atomic boundary conditions.
For the latter choice the linearized representation is not sufficient because the related eigenenergy is typically far away from the energy parameters.
In the MT spheres this setup is also simple and computationally inexpensive for the kinetic energy and the spherically averaged potential, e.g., in the muffin-tin approximation.
is set up, the Kohn-Sham orbitals are obtained as eigenfunctions from the algebraic generalized dense Hermitian eigenvalue problem where
The strong localization of core electrons due to the singularity of the effective potential at the atomic nucleus is connected to large kinetic energy contributions and thus a fully relativistic treatment is desirable and common.
Spin-orbit coupling can additionally be included, though this leads to a more complex Hamiltonian matrix setup or a second variation scheme,[14][15] connected to increased computational demands.
In the interstitial region it is reasonable and common to describe the valence electrons without considering relativistic effects.
The Fermi level itself is determined in this process by keeping charge neutrality in the unit cell.
In its construction a common approach is to employ Weinert's method for solving the Poisson equation.
[16] It efficiently and accurately provides a solution of the Poisson equation without shape approximation for an arbitrary periodic charge density based on the concept of multipole potentials and the boundary value problem for a sphere.
Because they are based on the same theoretical framework, different DFT implementations offer access to very similar sets of material properties.
The most basic quantity provided by DFT is the ground-state total energy of an investigated system.
The force exerted on an atom, which is given by the change of the total energy due to an infinitesimal displacement, has two major contributions.
The other, computationally more elaborate contribution, is due to the related change in the atom-position-dependent basis functions.
[19][20] Beyond forces, similar method-specific implementations are also needed for further quantities derived from the total energy functional.
Deviating interpretations of such quantities from experiments or other DFT implementations may lead to differences when comparing results.
On a side note also some atom-specific LAPW inputs relate directly to the respective MT region.
[23] A strength of the LAPW approach is the inclusion of all electrons in the DFT calculation, which is crucial for the evaluation of certain quantities.
One of which are hyperfine interaction parameters like electric field gradients whose calculation involves the evaluation of the curvature of the all-electron Coulomb potential near the nuclei.
For example, most energy eigenvalues of the Kohn-Sham states are not directly related to the real interacting many-electron system.
For the prediction of optical properties one therefore often uses DFT codes in combination with software implementing the GW approximation (GWA) to many-body perturbation theory and optionally the Bethe-Salpeter equation (BSE) to describe excitons.