next up previous contents
Next: Quick Start Up: The basic concepts of Previous: The density functional theory

   
The Full Potential LAPW method

The linearized augmented plane wave (LAPW) method is among the most accurate methods for performing electronic structure calculations for crystals. It is based on the density functional theory for the treatment of exchange and correlation and uses e.g. the local spin density approximation (LSDA). Several forms of LSDA potentials exist in the literature , and also recent improvements using the generalized gradient approximation (GGA) are available too (see sec. 2.1). For valence states relativistic effects can be included either in a scalar relativistic treatment (Koelling and Harmon 77) or with the second variational method including spin-orbit coupling (Novak 97). Core states are treated fully relativistically (Desclaux 69).

Various parts of the LAPW formalism and programming methods are found in many references: Andersen 73, 75, Koelling 72, Koelling and Arbman 75, Wimmer et al. 81, Weinert 81, Weinert et al. 82, Blaha and Schwarz 83, Blaha et al. 85, Wei et al. 85, Mattheiss and Hamann 86, Jansen and Freeman 84, Schwarz and Blaha 96). Recently an excellent book by D. Singh (Singh 94) appeared and is highly recommended to the interested reader. Here only the basic ideas are summarized; details are left to those references.

Like most ``energy-band methods``, the LAPW method is a procedure for solving the Kohn-Sham equations for the ground state density, total energy, and (Kohn-Sham) eigenvalues (energy bands) of a many-electron system (here a crystal) by introducing a basis set which is especially adapted to the problem.


  
Figure 2.1: Partitioning of the unit cell into atomic spheres (I) and an interstitial region (II)

This adaptation is achieved by dividing the unit cell into (I) non-overlapping atomic spheres (centered at the atomic sites) and (II) an interstitial region. In the two types of regions different basis sets are used:

1.
(I) inside atomic sphere t, of radius Rt, a linear combination of radial functions times spherical harmonics Ylm(r) is used (we omit the index t when it is clear from the context)


 \begin{displaymath}
\phi_{k_n} = \sum_{lm} [ A_{lm} u_l(r,E_l) +
B_{lm} \dot u_l(r,E_l) ] Y_{lm}(\hat r)
\end{displaymath} (5)

where ul(r,El) is the (at the origin) regular solution of the radial Schroedinger equation for energy El (chosen normally at the center of the corresponding band with l-like character) and the spherical part of the potential inside sphere t; $\dot u_l(r,E_l)$ is the energy derivative of ul taken at the same energy El. A linear combination of these two functions constitute the linearization of the radial function; the coefficients Alm and Blm are functions of kn (see below) determined by requiring that this basis function matches (in value and slope) the corresponding basis function of the interstitial region; ul and $\dot u_l$ are obtained by numerical integration of the radial Schroedinger equation on a radial mesh inside the sphere.

2.
(II) in the interstitial region a plane wave expansion is used


 \begin{displaymath}
\phi_{k_n} = \frac{1}{\sqrt \omega} e^{i k_n r}
\end{displaymath} (6)

where kn=k+Kn; Kn are the reciprocal lattice vectors and k is the wave vector inside the first Brillouin zone. Each plane wave is augmented by an atomic-like function in every atomic sphere.

The solutions to the Kohn-Sham equations are expanded in this combined basis set of LAPW's according to the linear variation method


 \begin{displaymath}
\psi_k = \sum_n c_n \phi_{k_n}
\end{displaymath} (7)

and the coefficients cn are determined by the Rayleigh-Ritz variational principle. The convergence of this basis set is controlled by a cutoff parameter RmtKmax = 6 - 9, where Rmt is the smallest atomic sphere radius in the unit cell and Kmax is the magnitude of the largest K vector in equation (2.7).

In order to improve upon the linearization (i.e. to increase the flexibility of the basis) and to make possible a consistent treatment of semicore and valence states in one energy window (to ensure orthogonality) additional (kn independent) basis functions can be added. They are called ``local orbitals`` (Singh 91) and consist of a linear combination of 2 radial functions at 2 different energies (e.g. at the 3s and 4s energy) and one energy derivative (at one of these energies):


 \begin{displaymath}
\phi_{lm}^{LO} = [ A_{lm} u_l(r,E_{1,l}) + B_{lm} \dot u_l(r,E_{1,l}) +
C_{lm} u_l(r,E_{2,l}) ] Y_{lm} (\hat r)
\end{displaymath} (8)

The coefficients Alm, Blm and Clm are determined by the requirements that $\phi^{LO}$ should be normalized and has zero value and slope at the sphere boundary.

In its general form the LAPW method expands the potential in the following form


 
$\displaystyle V(r)=\left\{ {\begin{array}{ll} \sum
\limits_{lm}V_{lm}(r)Y_{lm}(...
...here}\\
\sum \limits_K V_K e^{iKr} & \mbox{outside sphere}
\end{array}}\right.$     (9)

and the charge densities analogously. Thus no shape approximations are made, a procedure frequently called the ``full-potential LAPW`` (FLAPW) method. WIEN97 is, in this nomenclature, a FLAPW package.

The ``muffin-tin`` approximation used in early band calculations corresponds to retaining only the L=0 and M=0 component in the first expression of equ. 2.9 and only the K=0component in the second. This (much older) procedure corresponds to taking the spherical average inside the spheres and the volume average in the interstitial region.

The total energy is computed according to Weinert et al. 82.

Rydberg atomic units are used except internally in the atomic-like programs (LSTART and LCORE) or in subroutine outwin (LAPW1, LAPW2), where Hartree units are used. The output is always given in Rydberg units.

The forces at the atoms are calculated according to Yu et al (91). For the implementation of this formalism in WIEN see Kohler et al (94). An alternative formulation by Soler and Williams (89) has also been tested and found to be equivalent, both in computationally efficiency and numerical accuracy and the respective code is available from M.Fähnle (Krimmel et al 94).

The Fermi energy and the weights of each band state can be calculated using a modified tetrahedron method (Bloechl et al. 94), a Gaussian or a temperature broadening scheme.

Spin-orbit interactions can be considered via a second variational step using the scalar-relativistic eigenfunctions as basis. (See Singh 94 and Novak 97)

PROPERTIES:

The density of states (DOS) can be calculated using the modified tetrahedron method of Bloechl et al. 94.

X-ray absorption and emission spectra are determined using Fermi's golden rule and dipole matrix elements (between a core and valence or conduction band state respectively). (Neckel et al. 75)

X-ray structure factors are obtained by Fourier Transformation of the charge density.

Optical properties are obtained using the ``Joint density of states'' modified with the respective dipole matrix elements according to Ambrosch et al. 95, Abt et al. 94, Abt 97. A Kramers-Kronig transformation is also possible.


next up previous contents
Next: Quick Start Up: The basic concepts of Previous: The density functional theory

2000-04-11