Plane-wave methods

Jansen H J F and Freeman A J 1984 Total-energy full-potential linearized augmented plane-wave method for bulk solids electronic and structural properties of tungsten Phys. Rev. B 30 561-9... 

Theileis, V. and Bross, H. (2000) Relativistic modified augmented plane wave method and its application to the electronic structure of gold and platinum. Physical Review B - Condensed Matter, 62, 13338-13346. 

Wdowik, U.D., Ruebenbauer, K. Calibration of the isomer shift for the 77.34 keV transition in 197-Au using the full-potential linearized augmented plane-wave method. J. Chem. Phys. 129 (10), 104504 (2008)... 

Hamada, N. and Ohnishi, S. (1986) Self-interaction correction to the local-density approximation in the calculation of the energy band gaps of semiconductors based on the full-potential linearized augmented-plane-wave method, Phys. Rev., B34,9042-9044. 

For the conduction electrons, it is reasonable to consider that the inner-shell electrons are all localized on individual nuclei, in wave functions very much like those they occupy in the free atoms. The potential V should then include the potential due to the positively charged ions, each consisting of a nucleus plus filled inner shells of electrons, and the self-consistent potential (coulomb plus exchange) of the conduction electrons. However, the potential of an ion core must include the effect of exchange or antisymmetry with the inner-shell or core electrons, which means that the conduction-band wave functions must be orthogonal to the core-electron wave functions. This is the basis of the orthogonalized-plane-wave method, which has been successfully used to calculate band structures for many metals.41... 

Augmented-plane-wave method, 34 246 Austemite, decarburization of, 21 332-334 Autocatalysis, 25 275, 34 15, 36 Automobile exhaust emission control, 34 275, 278... 

In the next two subsections, we describe collections of calculations that have been used to probe the physical accuracy of plane-wave DFT calculations. An important feature of plane-wave calculations is that they can be applied to bulk materials and other situations where the localized basis set approaches of molecular quantum chemistry are computationally impractical. To develop benchmarks for the performance of plane-wave methods for these properties, they must be compared with accurate experimental data. One of the reasons that benchmarking efforts for molecular quantum chemistry have been so successful is that very large collections of high-precision experimental data are available for small molecules. Data sets of similar size are not always available for the properties of interest in plane-wave DFT calculations, and this has limited the number of studies that have been performed with the aim of comparing predictions from plane-wave DFT with quantitative experimental information from a large number of materials. There are, of course, many hundreds of comparisons that have been made with individual experimental measurements. If you follow our advice and become familiar with the state-of-the-art literature in your particular area of interest, you will find examples of this kind. Below, we collect a number of examples where efforts have been made to compare the accuracy of plane-wave DFT calculations against systematic collections of experimental data. 

Wang, D. S., Freeman, A. J., Krakauer, H., and Postemak, M. (1981). Self-consistent linearized-argumented-plane-wave-method determination of electronic structure and surface states on Al(lll). Phys. Rev. B 23, 1685-1692. 

Extensions of this model in which the atomic nuclei and core electrons are included by representing them by a potential function, V, in Equation (4.1) (plane wave methods) can account for the density of states in Figure 4.3 and can be used for semiconductors and insulators as well. We shall however use a different model to describe these solids, one based on the molecular orbital theory of molecules. We describe this in the next section. We end this section by using our simple model to explain the electrical conductivity of metals. 

This method can be applied to individual atomic layers or to a thick layer representing the entire surface, but remains relatively time-consuming compared to the plane-wave methods, except when the separation between individual layers becomes small (5 0.5 A). 

As mentioned earlier, the existence of surface shifted core levels has been questioned.6 Calculated results for TiC(lOO) using the full potential linearized augmented plane wave method (FLAPW) predicted6 no surface core level shift in the C Is level but a surface shift of about +0.05 eV for the Tis levels. The absence of a shift in the C Is level was attributed to a similar electrostatic potential for the surface and bulk atoms in TiC. The same result was predicted for TiN because its ionicity is close to that of TiC. This cast doubts on earlier interpretations of the surface states observed on the (100) surface of TiN and ZrN which were thought to be Tamm states (see references given in Reference 4), i.e. states pulled out of the bulk band by a shift in the surface layer potential. High resolution core level studies could possibly resolve this issue, since the presence of surface shifted C Is and N Is levels could imply an overall electrostatic shift in the surface potential, as suggested for the formation of the surface states. 

Loucks, T.L. (1967). Augmented Plane Wave Method (Benjamin, New York), pp. 98-103. 

Note A/B implies A grown or strained to B and vice versa. A B implies no growth direction or explicit strain dependence, i.e. natural. ) T = theoretical E = experimental AVL = average lattice XPS = X-ray photoelectron spectroscopy PL = photoluminescence CL = cathodoluminescence UPS = ultraviolet photoelectron spectroscopy LMTO = linear muffin tin orbital method LAPW = linearised augmented plane wave method PWP = plane wave pseudopotential method VCA = virtual crystal approximation. 

Parameters originating from the plane-wave methods kinetic energy... 

APW, self-consistent energy-band calculation by the augmented plane-wave method KKR, Korringa-Kohn-Rostoker method for electronic band calculations in solids. 

Perhaps the most successful representation of the wave functions for band calculations for semiconductors has been the OPW method (orthogonalized plane-wave method), developed by Herring (1940). The success of the method has been due to the ease of obtaining and using realistic potentials in the calculation, in contrast to methods that utilize the muffin-tin approximation to the potential (discussed in Chapter 20). Only recently have difficulties with the application of muffin-tin potentials to semiconductors been overcome. (P or discussion and references see Johnson, Norman, and Connolly, 1973.) For any given potential, any of the accurate methods should give the same bands if the necessary effort is applied. 

We have seen, particularly in the discussion of covalent crystals in terms of pseudopotentials, the importance of recognizing which matrix elements or effects are dominant and which should be treated as corrections afterward. Tliis is also true in transition-metal systems, and different effects arc dominant in different transition-metal systems thus the correct ordering of terms is of foremost importance. For many transition-metal systems, we find that band calculations, particularly those by L. F. Mattheiss, provide an invaluable guide to electronic structure. Mattheiss uses the Augmented Plane Wave method (APW method), which is analogous to the OPW method discussed in Appendix D. 

If I is replaced by a plane wave on the right side of Eq. (D-1), this gives exactly what is called an orthogonalizedplane ivave, or OPW, The orthogonalized plane wave method of band calculation consists of expanding the true wave function in OPW s. It was invented by Herring (1940) and provides the conceptual basis of pseudopotential theory. 

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