Pseudopotentials

In practice, the true crystal potential does not satisfy the criterion for the applicability of the nearly-free-electron approximation, but there are much weaker equivalent potentials, or pseudopotentials, which do. By equivalent we mean that they produce the same band structure for the valence and conduction bands (but not necessarily the same wave functions). The difference is that the potential must of necessity be strong enough to bind states at lower energies (core states, more or less) but the pseudopotential need not. The elimination of such bound states produces a potential which is much weaker in the region close to the ion cores. This cancellation of the strong inner part of the potential can be seen from many points of view but will here be accepted as a fact of life for s-p-bonded systems. 

It should be emphasized that while the utility of this approach is largely based on the NFE method, no such approximation is made at the outset. 

In practice a bare pseudopotential, equivalent to the potential produced by an ion core, is first calculated and then screened, i.e., the potential due to the outer electron is added. In this case the self-consistent incorporation of a screening correction is particularly simple because of the use of NFE theory. In lowest-order perturbation theory, performed self-consistently, the relation is just 

The function x derives from the use of perturbation theory, being simply 

The dielectric function e is often modified for exchange and correlation by the incorporation of an effective electron-electron interaction (Section 1.2) rather than the Coulomb interaction, which appears, in Fourier transform, as 4nlq in Eq. (23). 

Each pseudopotential is defined within a cut-off radius from the atom center. At the cut-off, the potential and wavefunctions of the core region must join smoothly to the all-electron-like valence states. Early functional forms for pseudopotentials also enforced the norm-conserving condition so that the integral of the charge density below the cut-off equals that of the aU-electron calculation [42, 43]. However, smoother, and so computationally cheaper, functions can be defined if this condition is relaxed. This idea leads to the so called soft and ultra-soft pseudopotentials defined by Vanderbilt [44] and others. The Unk between the pseudo and real potentials was formaUzed more clearly by Blochl [45] and the resulting 

In oxides, the division between core and valence states should be carried out with care. For example, in strongly ionic materials the cation may have lost all valence electrons to the anion and so the cation/anion interaction involves orbitals on the cation that are atomic core states. In this situation, expHcit inclusion of the outermost cation orbitals would be required for accurate results. Even so, the replacement of the core region with a smoother potential can reduce the calculation time even for H atoms, and so pseudopotentials are available even in this case. 

For consistency, pseudopotentials should be based on the same functionals as used for the valence states. This point has been illustrated by Gale and coworkers in their study of aluminum trihydroxides [47]. Table 8.3 compares their results of cell optimizations using various mixtures of psuedopotential and valence state functionals with the experimental structure of gibbsite [48]. LDA pseudopotentials in an LDA optimization of the cell gives underestimated cell parameters, consistent with the usual expectation that LDA gives over-binding in chemical bonds. 

Fluctuations in density lead to fluctuations of the potential Vq. In the Wigner-Seitz model the repulsive part of the potential is described by the hard core where V oo. A more realistic approach is the choice of a pseudopotential. A pseudopotential incorporates a dependence on r and it secures orthogonality of the wave functions. Several authors have described pseudopotentials which give a good description of the density dependence of Vq in argon, neon, and helium (Plenkiewicz et al., 1986 1991b) and in methane (Ishimaru and Fukui, 1975). 

The pseudopotential approximation was originally introduced by Hellmann already in 1935 for a semiempirical treatment of the valence electron of potassium [25], However, it took until 1959 for Phillips and Kleinman from the solid state community to provide a rigorous theoretical foundation of PPs for single valence electron systems [26]. Another decade later in 1968 Weeks and Rice extended this method to many valence electron systems [27,28], Although the modern PPs do not have much in common with the PPs developed in 1959 and 1968, respectively, these theories prove that one can get the same answer as from an AE calculation by using a suitable effective valence-only model Hamiltonian and pseudovalence orbitals with a simplified nodal structure [19], 

The analytical form applied for PPs nowadays is the semilocal ansatz (local in the coordinate r, but nonlocal in spherical angle coordinates 6 and (f) [6]), which goes back to Abarenkov and Heine working in the field of solid states [29,30] and was introduced a few years later to quantum chemistry by Schwarz [31] as well as Kahn and Goddard [32], In this ansatz besides the r-dependency of the PP also a /-dependency, i.e., a dependency on the angular momentum quantum number /, is taken into account. In the case of nonrelativis-tic and scalar-relativistic, i.e., one-component, PPs in equation 6.3 the following semilocal ansatz for the PP Vpp is used [32] 

This semilocal PP consists of a sum of local potentials V/(rii) acting separately on each angular momentum symmetry 0 / /max present in the core and a common local potential Vl(Ai) which acts on all angular momentum symmetries / /max not included in the core [19]. If /max is taken large enough, the leading local term Vl can be avoided [6]. 

The second term in equation 6.4 contains the angular momentum projection operator P[ based on spherical harmonics lm, I) 

Since there are no core functions in equation 6.4, the pseudovalence orbitals belonging to the lowest Hartree-Fock (HF) or Kohn-Sham solutions for each angular momentum / are thus nodeless [6]. 

The space of orthonormal orbitals of a system with a single valence electron outside a closed shell core may be partitioned into a subspace for the doubly occupied core orbitals (f c and a subspace for the singly occupied valence orbital (jDy. The space of the unoccupied virtual orbitals is not considered at this moment, however as described below in the section of shape-consistent PPs it cannot be neglected in accurate approaches. The Fock equation for the valence orbital (pv 

Reductions in the basis set used to represent the valence orbital can be only achieved if by admixture of core orbitals Pc the radial nodes are eliminated and the shape of the resulting pseudo ip) valence orbital Pp is as smooth as possible in the core region (pseudo-valence orbital transformation) 

Np denotes a normalization factor depending on the coefficients G)c- The origins valence orbital with the full nodal structure in terms of the pseudo valence orbital with the simplified nodal structure 

Using the so-called generalized Phillips-Kleinman pseudopotential [3] 

If one assumes the core orbitals (pc to be also eigenfunctions of the Fock operator i.e., [ v,Pc] = 0, and uses the idempotency of the projection operator = Pc ( 1) oil recovers a simplified pseudo eigenvalue problem 

the p-like state, X at the bottom of the energy gap must be associated with the NFE eigenfunction, px, which pushes charge away from the atomic centres, whereas the s-like state, Xv at the top of the energy gap must be associated with the NFE eigenfunction, pX) which pulls charge onto the atomic centres. From eqn (5.42) it follows that in order for the NFE approximation to fit the observed band structure, the (200) Fourier com- 

But this contradicts our picture of the crystalline potential, V(x), that is sketched in the middle of Fig. 5.10. It comprises to a good approximation the overlapping of attractive atomic potentials like those drawn for the diatomic molecule in Fig. 3.f. The (200) Fourier component of such a potential is large and negative, the ab initio crystalline potential for aluminium taking the value 

we have a paradox although the band structure of aluminium is nearly-free-electron-like, the actual Fourier components of the crystalline potential are large and negative, not small and positive as required by the NFE model. 

The resolution of this paradox is easily obtained once it is remembered that the NFE bands in aluminium are formed from the valence 3s and 3p electrons. These states must be orthogonal to the s and p core functions, so that they contain nodes in the core region as illustrated for the 2s wave function in Fig. 2.12. In order to reproduce these very short wavelength oscillations, plane waves of very high momentum must be included in the plane wave expansion of . Retaining only the two lowest energy plane waves in eqn (5.35) provides an extremely bad approximation. 

In 1940 Herring circumvented this problem by starting at the outset with a basis of plane waves that had already been orthogonalized to the core states, the so-called orthogonalized plane wave (OPW) basis. Retaining only the two lowest orthogonalized plane waves we can look for the OPW solution that is analogous to eqn (5.35), namely 

The discussion above points to the fact that large energy cutoffs must be used to include plane waves that oscillate on short length scales in real space. This is problematic because the tightly bound core electrons in atoms are associated 

The most important approach to reducing the computational burden due to core electrons is to use pseudopotentials. Conceptually, a pseudopotential replaces the electron density from a chosen set of core electrons with a smoothed density chosen to match various important physical and mathematical properties of the true ion core. The properties of the core electrons are then fixed in this approximate fashion in all subsequent calculations this is the frozen core approximation. Calculations that do not include a frozen core are called all-electron calculations, and they are used much less widely than frozen core methods. Ideally, a pseudopotential is developed by considering an isolated atom of one element, but the resulting pseudopotential can then be used reliably for calculations that place this atom in any chemical environment without further adjustment of the pseudopotential. This desirable property is referred to as the transferability of the pseudopotential. Current DFT codes typically provide a library of pseudopotentials that includes an entry for each (or at least most) elements in the periodic table. 

The analytical forms of the modern PPs used today have little in common with the formulas we obtain by a strict derivation of the theory (Dolg 2000). Formally, the pseudo-orbital transformation leads to nodeless pseudovalence orbitals for the lowest atomic valence orbitals of a given angular quantum number l (one-component) or Ij (two-component). The simplest and historically the first choice is the local ansatz for A VCy in Equation (3.4). However, this ansatz turned out to be too inaccurate and therefore was soon replaced by a so-called semilocal form, which in two-component form may be written as 

Pfi denotes a projection operator on spinor spherical harmonics centred at the core k 

For scalar-quasirelativistic calculations, i.e. when spin-orbit coupling is neglected, a one-component form may be obtained by averaging over the spin 

For some calculations (see below) it is advantageous to separate space and spin 

As has been said before, plane waves might serve as general and straightforward basis functions if there were not the rapid oscillations of the atomic wave functions close to the nuclei. If these oscillations are artificially suppressed, such as in the free-electron model, plane waves are the optimum choice. Since 

Given that the low-lying core orbitals core of an atom are known from an atomic calculation, however, it is possible to construct plane waves representing the valence levels by forcing these plane waves to be orthogonal to the core levels for a specified k, namely by writing 

In the late 1950s, pseudopotentials were re-invented for the solid state by replacing the OPW orthogonality recipe with an effective potential, the above pseudopotential [217,218], and the method has blossomed ever since. Indeed, pseudopotential theory is not a specialty of the solid state because a large part 

Within the molecular quantum-chemical regime, various strategies on how to generate sets of pseudopotentials have been followed, such as the shape-consistent pseudopotentials for which the pseudo-valence orbital is identical 

31) In Section 2.9, we have already illustrated the different effective potentials acting upon 2p (strong), 3p (weaker) and 4p orbitals (even weaker), and this is because a valence p orbital is most strongly pushed out by a filled core p function. Since there is no Ip function, only the 2p pseudopotential is dose to the real potential. 

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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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