Pseudopotential

The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

Figure Al.3.10. Pseudopotential model. The outer electrons (valence electrons) move in a fixed arrangement of chemically inert ion cores. The ion cores are composed of the nucleus and core electrons.

One can quantify the pseudopotential by writing the total crystalline potential for an elemental solid as... 

With the density fiinctional theory, the first step in the constmction of a pseudopotential is to consider the solution for an isolated atom [27]. If the atomic wavefiinctions are known, tire pseudo-wavefiinction can be constmcted by removing the nodal stmcture of the wavefiinction. For example, if one considers a valence... 

Figure Al.3.13. All-electron and pseudopotential wavefiinction for the 3s state in silicon. The all-electron 3s state has nodes which arise because of an orthogonality requirement to tlie Is and 2s core states.

Since and depend only on die valence charge densities, they can be detennined once the valence pseudo- wavefiinctions are known. Because the pseudo-wavefiinctions are nodeless, the resulting pseudopotential is well defined despite the last temi in equation Al.3.78. Once the pseudopotential has been constructed from the atom, it can be transferred to the condensed matter system of interest. For example, the ionic pseudopotential defined by equation Al.3.78 from an atomistic calculation can be transferred to condensed matter phases without any significant loss of accuracy. 

There are complicating issues in defmmg pseudopotentials, e.g. the pseudopotential in equation Al.3.78 is state dependent, orbitally dependent and the energy and spatial separations between valence and core electrons are sometimes not transparent. These are not insunnoimtable issues. The state dependence is usually weak and can be ignored. The orbital dependence requires different potentials for different angular momentum components. This can be incorporated via non-local operators. The distinction between valence and core states can be addressed by incorporating the core level in question as part of the valence shell. For... 

There are a variety of other approaches to understanding the electronic structure of crystals. Most of them rely on a density functional approach, with or without the pseudopotential, and use different bases. For example, instead of a plane wave basis, one might write a basis composed of atomic-like orbitals ... 

An approach closely related to the pseudopotential is the orthogonalizedplane wave method [29]. In this method, the basis is taken to be as follows ... 

Figure Al.3.14. Band structure for silicon as calculated from empirical pseudopotentials [25],...

Figure Al.3.15. Density of states for silieon (bottom panel) as ealeulated from empirieal pseudopotential [25], The top panel represents the photoemission speetra as measured by x-ray photoemission speetroseopy [30], The density of states is a measure of the photoemission speetra.

Figure Al.3.16. Reflectivity of silicon. The theoretical curve is from an empirical pseudopotential method calculation [25], The experimental curve is from [31],...

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Figure Al.3.22. Spatial distributions or charge densities for carbon and silicon crystals in the diamond structure. The density is only for the valence electrons the core electrons are omitted. This charge density is from an ab initio pseudopotential calculation [27].

Figure Al.3.23. Phase diagram of silicon in various polymorphs from an ab initio pseudopotential calculation [34], The volume is nonnalized to the experimental volume. The binding energy is the total electronic energy of the valence electrons. The slope of the dashed curve gives the pressure to transfomi silicon in the diamond structure to the p-Sn structure. Otlier polymorphs listed include face-centred cubic (fee), body-centred cubic (bee), simple hexagonal (sh), simple cubic (sc) and hexagonal close-packed (licp) structures.

Figure Al.3.24. Band structure of LiF from ab initio pseudopotentials [39],...

Figure Al.3.27. Energy bands of copper from ab initio pseudopotential calculations [40].

Figure B3.2.1. The band structure of hexagonal GaN, calculated using EHT-TB parameters detemiined by a genetic algorithm [23]. The target energies are indicated by crosses. The target band structure has been calculated with an ab initio pseudopotential method using a quasiparticle approach to include many-particle corrections [194].

The pseudopotential is derived from an all-electron SIC-LDA atomic potential. The relaxation correction takes into account the relaxation of the electronic system upon the excitation of an electron [44]- The authors speculate that ... the ability of the SIRC potential to produce considerably better band structures than DFT-LDA may reflect an extra nonlocality in the SIRC pseudopotential, related to the nonlocality or orbital dependence in the SIC all-electron potential. In addition, it may mimic some of the energy and the non-local space dependence of the self-energy operator occurring in the GW approximation of the electronic many body problem [45]. 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

One current limitation of orbital-free DFT is that since only the total density is calculated, there is no way to identify contributions from electronic states of a certain angular momentum character /. This identification is exploited in non-local pseudopotentials so that electrons of different / character see different potentials, considerably improving the quality of these pseudopotentials. The orbital-free metliods thus are limited to local pseudopotentials, connecting the quality of their results to the quality of tlie available local potentials. Good local pseudopotentials are available for the alkali metals, the alkaline earth metals and aluminium [100. 101] and methods exist for obtaining them for other atoms (see section VI.2 of [97]). 

Pulci O, Onida G, Shkrebtii A I, Del Sole R and Adolph B 1997 Plane-wave pseudopotential calculation of the optical properties of GaAs Phys. Rev. B 55 6685... 

Stampfl C, van de Walle C G, Vogel D, Kruger P and Pollmann J 2000 Native defects and impurities in InN First-principles studies using the local-density approximation and self-interaction and relaxation-corrected pseudopotentials Phys. Rev. B 61 R7846-9... 

Singh D J 1994 Planewaves, Pseudopotentials and the LAPW Method (Norweii, MA Kiuwer)... 

Watson S, Jesson B J, Carter E A and Madden P A 1998 Ab initio pseudopotentials for orbital-free density functional Europhys. Lett. 41 37-42... 

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