DENSITY FUNCTIONAL CALCULATIONS AS A NEW TOOL TO STUDY PHASE TRANSITIONS
Karlheinz Schwarz
Technische Universität
Wien, A-1060 Vienna, Getreidemarkt 9/156, Austria
E-mail: kschwarz@email.tuwien.ac.at
www-page: http://www.tuwien.ac.at/theochem/
Calculation of solids can be performed with a variety of methods from classical to quantum mechanical (QM) approaches. The former are semi-empirical schemes, in which the forces that determine the interactions between the atoms are parameterized such that they allow to reproduce a series of experimental data. These schemes have reached a high level of sophistication and are often useful within a given class of materials provided good parameters are known from closely related systems. If, however, such parameters are not available, or if a system shows unusual phenomena that are not yet understood, one often must rely on ab initio calculations. They are more demanding in terms of computer requirements but have the advantage that they do not require any experimental knowledge as input for such calculations.
The following aspects characterize the commonly used ab initio methods:
i) the treatment of exchange and correlation effects:
The Hartree-Fock (HF) method is based on a wave-function description (with one Slater determinant) and treats exchange exactly but contains - by definition- no correlation effects. Density Functional Theory (DFT) is an alternative approach in which both effects, exchange and correlation, are treated in a combined but approximate scheme (see below).
ii) The choice of basis sets and wave functions:
Essentially all methods use an LCAO (linear combination of atomic orbitals) scheme but they differ in the basis sets. Some use Gaussian or Slater type orbitals (GTOs or STOs) others use plane wave (PW) basis sets with or without augmentations. Closely related to the basis set is the explicit form of the wave-functions, node-less pseudo-wave-functions or all-electron wave-functions with a proper description of the radial nodal structure and accurate close to the nucleus.
iii) The form of the potential:
Schemes that make no shape approximation in the form of the potential are termed full-potential schemes. In the muffin-tin or the atomic sphere approximation (MTA or ASA) the potential is assumed to be spherically symmetric within each atom in the crystal.
iv) Relativistic effects:
If a solid contains heavier elements, relativistic effects can no longer be neglected. Scalar relativistic schemes are often used, which describe the main contraction or expansion of various orbitals (due to the Darwin s-shift or the mass-velocity term) but they omit spin-orbit splitting that can be included in a second-variational treatment. For very heavy elements it may be necessary to solve Dirac's equation, which has all these terms included.
v) Software packages:
Let me mention three representative examples for different types of program packages:
Density functional theory
For solids the most calculations are based on
density functional theory (DFT), which is a universal approach to
the quantum mechanical many-body problem, where the interacting
system is mapped in a unique manner onto an effective
non-interacting system with the same total density. The electron
density plays the key role in this formalism. The non-interacting
particles of this auxiliary system move in an effective local
one-particle potential, which consists of a mean-field (Hartree)
part and an exchange-correlation part that, in principle,
incorporates all correlation effects exactly. The exact
functional form of this potential is not known and thus one needs
to make approximations. The most commonly used one is the local
density approximation (LDA) but it has some shortcomings mostly
due to the tendency of overbinding, which cause for example too
small lattice constants. In recent years improvements beyond the
LDA have been developed and especially the version of DFT based
on the generalized gradient approximation (GGA) has reached
(almost) chemical accuracy. Such DFT calculations can be applied
to problems which require a very good description of the
electronic structure, such as electron density distribution,
disproportionation, phonons, phase transitions, electric field
gradients, or typical questions of chemical interest, such as
catalysis.
The full-potential linearized augmented plane wave (LAPW) method
One among the most accurate schemes for solving
the Kohn Sham equations is the full-potential
linearized-augmented-plane-wave (FP-LAPW) method on which our
WIEN code is based [1,2] that has been developed in my group and
is frequently used worldwide. Its newest version (WIEN97) is available
on request. Interested users should contact the author,
preferably by email. For continuously updated information see our
World Wide Web - homepage (Error! Bookmark not defined.)
Results of LAPW calculations and properties
The results of LAPW calculations will be given which provide the basis for interpretation and comparison with several experimental data:
Acknowledgments: I want to thank all my
coworkers who contributed to the present paper: P.Blaha, the main
author of WIEN97, P.Dufek (local orbitals), J.Luitz (graphical
user interface) and D.Kvasnicka (matrix diagonalization).