
Adding a new dimension to DFT calculations of solids ...
The Full Potential APW methods
Recently, the development of the Augmented Plane Wave (APW) methods from
Slater's APW, to LAPW and the new APW+lo was described by Schwarz et al. 2001.
The 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 , but recent improvements using the generalized
gradient approximation (GGA) are available too.
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 spinorbit
coupling (Macdonald 80, Novak 97).
Core states are treated fully relativistically
(Desclaux 69).
A description of this method to linearize Slater's old APW method
(i.e. the LAPW formalism) and further programming hints 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). An excellent book by
D. Singh (Singh 94) describes all the details of the LAPW method
and is highly recommended to the
interested reader. Here only the basic ideas are summarized; details
are left to those references.
Like most ``energyband methods``, the LAPW method is a procedure for
solving the KohnSham equations for the ground state density, total
energy, and (KohnSham) eigenvalues (energy bands) of a manyelectron
system (here a crystal) by introducing a basis set which is especially
adapted to the problem.
Figure 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)
nonoverlapping atomic spheres (centered at the atomic sites)
and (II) an interstitial region. In the two types of regions different
basis sets are used:
 (I) inside atomic sphere t of radius R_{t} a
linear combination of radial functions times spherical harmonics
Y_{lm}(r) is used (we omit the index t when it is
clear from the context)

(5) 
where u_{l}(r,E_{l})
is the (at the origin) regular solution of the
radial Schroedinger equation for energy
E_{l}
(chosen normally at
the center of the corresponding band with llike character) and the
spherical part of the potential inside sphere t
is the energy derivative of u_{l}
taken at the same energy E_{l}.
A linear combination of these two functions
constitute the linearization of the radial function; the
coefficients
A_{lm}
and
B_{lm}
are functions of k_{n}
(see
below) determined by requiring that this basis function matches (in
value and slope) the corresponding basis function of the
interstitial region; and are obtained by numerical
integration of the radial Schroedinger equation on a radial mesh
inside the sphere.
 (II) in the interstitial region a plane wave expansion is used

(6) 
where k_{n}=k+K_{n};
K_{n}
are the reciprocal lattice vectors and
k
is
the wave vector inside the first Brillouin zone. Each plane
wave is augmented by an atomiclike function in every atomic
sphere.
The solutions to the KohnSham equations are expanded in this
combined basis set of LAPW's according to the linear variation method

(7) 
and the coefficients c_{n}
are determined by the RayleighRitz variational
principle. The convergence of this basis set is controlled by
a cutoff parameter
R_{mt}K_{max}= 6  9, where
R_{mt}
is the smallest
atomic sphere radius in the unit cell and
K_{max}
is the magnitude
of the largest K
vector.
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 (k_{n}
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):

(8) 
The coefficients
A_{lm},
B_{lm}, and
C_{lm},
are determined by
the requirements that should be normalized and has zero value
and slope at the sphere boundary.
The APW+lo method
Sjöstedt, Nordström and Singh (2000) have shown
that the standard LAPW method with the additional constraint on the PWs
of matching in value AND slope to the solution inside the sphere is not
the most efficient way to linearize Slater's APW method. It can be
made much more efficient when one uses the standard APW basis, but of
course with u_{l}(r,E_{l}) at a fixed energy
E_{l} in order to keep the
linear eigenvalue problem. One then adds a new local
orbital (lo) to have enough variational flexibility in the radial
basisfunctions:

(8) 

(9) 
This new lo (denoted with lower case to distinguish it from the
LO given above) looks almost like the old ``LAPW''basis set, but here the
A_{lm} and B_{lm} do not depend on
k_{n} and are determined by the
requirement that the lo is zero at the sphere boundary and
normalized.
Thus we construct basis functions that have ``kinks'' at the sphere boundary,
which makes it
necessary to include surface terms in the kinetic energy part of the
Hamiltonian. Note, however, that the total wavefunction is of course smooth
and differentiable.
As shown by Madsen et al. (2001) this new scheme converges practically to
identical
results as the LAPW method, but allows to reduce ``RKmax'' by about one,
leading to significantly smaller basis sets (up to 50 %) and thus the
corresponding
the computational time is drastically reduced (up to an order of magnitude).
Within one calculation a mixed ``LAPW and APW+lo''
basis can be used for different atoms and even different lvalues
for the same atom (Madsen et al. 2001). In general one describes
by APW+lo those orbitals which
converge most slowly with the number of PWs (such as TM 3d states) or the
atoms with a small sphere size, but the rest with ordinary LAPWs.
One can also add a second ``lo'' at a different energy
so that both, semicore and valence states, can be described simultaneously.
General considerations
In its general form the LAPW method expands the potential in
the following form



(9) 
and the charge densities analogously. Thus no shape approximations
are made, a procedure frequently called the ``fullpotential`` method.
The ``muffintin`` approximation used in early band calculations
corresponds to retaining only the L=0 and M=0
component in the
first expression of equ. (9) and only the K=0
component 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 atomiclike
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) and Madsen et al. 2001. 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.
Spinorbit interactions can be considered via a second variational step
using the scalarrelativistic eigenfunctions as basis. (See MacDonald 80,
Singh 94 and Novak 97). In order to overcome the problems due to the missing
p_{1/2} radial basis function in the scalarrelativistic
basis (which corresponds to p_{3/2}, we have
recently extended the standard LAPW basis by an additional
p_{1/2}local orbital'', i.e. a LO with
a p_{1/2} basis function, which is added in the
secondvariational SO calculation (Kunes et al. 2001).
It is well known that for localized electrons (like the 4f states in lanthanides
or 3d states in some TMoxides) the LDA (GGA) method is not accurate enough
for a proper description. Thus we have implemented various forms of the
LDA+U method as well as the ``Orbital polarization method'' (OP)
(see Novak 2001 and references therein).
One can also consider interactions with an external magnetic (see Novak 2001)
or electric field (via a supercell approach, see Stahn et al. 2000).
PROPERTIES:
The density of states (DOS) can be calculated using the modified tetrahedron
method of Blöchl et al. 94.
Xray 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)
Xray 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 KramersKronig transformation is also possible.
An analysis of the electron density according to Bader's ``atoms in molecules''
theory can be made using a program by J. Sofo and J. Fuhr (2001)
References
 Abt R., AmbroschDraxl C. and Knoll P. 1994 Physica B 194196
 Abt R. 1997 PhD Theses, Univ.Graz
 Andersen O.K. 1973 Solid State Commun. 13, 133
  1975 Phys. Rev. B 12, 3060
 AmbroschDraxl C., Majewski J. A., Vogl P., and Leising G. 1995, PRB 51 9668
 Blaha P. and Schwarz K. 1983 Int. J. Quantum Chem. XXIII, 1535
 Blaha P., Schwarz K., and Herzig P 1985 Phys. Rev. Lett. 54, 1192
 Blaha P., Schwarz K., Sorantin P.I. and Trickey S.B. 1990 Comp. Phys. Commun. 59, 399
 Blöchl P.E., Jepsen O. and Andersen O.K. 1994, Phys. Rev B 49, 16223
 Desclaux J.P. 1969 Comp. Phys. Commun. 1, 216; note that the
actual code we use is an apparently unpublished relativistic
version of the nonrelativistic code described in this paper.
Though this code is widely circulated, we have been unable to
find a formal reference for it.
  1975 Comp. Phys. Commun. 9, 31; this paper contains much
of the DiracFock treatment used in Desclaux's relativistic LSDA
code.
 Jansen H.J.F. and Freeman A.J. 1984 Phys. Rev. B 30, 561
  1986 Phys. Rev. B 33, 8629
 Koelling D.D. 1972 J. Phys. Chem. Solids 33, 1335
 Koelling D.D. and Arbman G.O. 1975 J.Phys. F: Met. Phys. 5, 2041
 Koelling D.D. and Harmon B.N. 1977 J. Phys. C: Sol. St. Phys. 10, 3107
 Kohler B., Wilke S., Scheffler M., Kouba R. and AmbroschDraxl C. 1996
Comp.Phys.Commun. 94, 31
 Krimmel H.G., Ehmann J., Elsässer C., Fähnle M. and
Soler J.M. 1994, Phys.Rev. B50, 8846
 Kuneš J, Novák P., Schmid R., Blaha P. and Schwarz K. 2001, Phys. Rev. B64, 153102
 MacDonald A. H., Pickett, W. E. and Koelling, D. D. 1980 J. Phys. C 13, 2675
 Madsen G. K. H., Blaha P, Schwarz K, Sjöstedt E and Nordström L 2001, Phys. Rev. B
 Mattheiss L.F. and Hamann D.R. 1986 Phys. Rev. B 33, 823
 Neckel A., Schwarz K., Eibler R. and Rastl P. 1975 Microchim.Acta, Suppl.6, 257
 Novak P. 1997 to be published, see also $WIENROOT/SRC/novak_lecture_on_spinorbit.ps
 Novák P. , Boucher F., Gressier P., Blaha P. and
Schwarz K. 2001 Phys. Rev. B 63, 235114
 Schwarz K. and Blaha P.: Lecture Notes in Chemistry 67, 139 (1996)
 Schwarz K., P.Blaha and Madsen, G. K. H. 2001 Comp.Phys.Commun.
 Singh D. 1991, Phys.Rev. B43, 6388
 Singh D. 1994, Plane waves, pseudopotentials and the LAPW method,
Kluwer Academic
 Sjöstedt E, Nordström L and Singh D. J. 2000 Solid State Commun. 114, 15
 Sofo J and Fuhr J 2001: $WIENROOT/SRC/aim_sofo_notes.ps
 Soler J.M. and Williams A.R. 1989, Phys.Rev. B40, 1560
 Stahn J, Pietsch U, Blaha P and Schwarz K. 2001, Phys.Rev. B63, 165205
 Wei S.H., Krakauer H., and Weinert M. 1985 Phys. Rev. B 32, 7792
 Weinert M. 1981 J. Math. Phys. 22, 2433
 Weinert M., Wimmer E., and Freeman A.J. 1982 Phys. Rev. B26, 4571
 Wimmer E., Krakauer H., Weinert M., and Freeman A.J. 1981
Phys. Rev. B24, 864
 Yu R., Singh D. and Krakauer H. 1991, Phys.Rev. B43, 6411
©2001 by P. Blaha and K. Schwarz
