Semiempirical Pseudopotentials

What follows is not intended to be an encyclopaedia of band structures. It is offered as an introduction to some of the major classes of simple solids whose band structure has been studied, with illustrations taken from the recent literature. For more comprehensive bibliographies the reader is referred to Slater, Dimmock (for recent applications of the APW and related methods), and Cohen and Heine (for recent applications of semiempirical pseudopotential methods). 

The systems discussed in this chapter give some examples using different theoretical models for the interpretation of, primarily, UPS valence band data, both for pristine and doped systems as well as for the initial stages of interface formation between metals and conjugated systems. Among the various methods used in the examples are the following semiempirical Hartree-Fock methods such as the Modified Neglect of Diatomic Overlap (MNDO) [31, 32) and Austin Model 1 (AMI) [33] the non-empirical Valence Effective Hamiltonian (VEH) pseudopotential method [3, 34J and ab initio Hartree-Fock techniques. 

Other complications are associated with the partitioning of the core and valence space, which is a fundamental assumption of effective potential approximations. For instance, for the transition elements, in addition to the outermost s and d subshells, the next inner s and p subshells must also be included in the valence space in order to accurately compute certain properties (54). A related problem occurs in the alkali and alkaline earth elements, involving the outer s and next inner s and p subshells. In this case, however, the difficulties are related to core-valence correlation. Muller et al. (55) have developed semiempirical core polarization treatments for dealing with intershell correlation. Similar techniques have been used in pseudopotential calculations (56). These approaches assume that intershell correlation can be represented by a simple polarization of one shell (core) relative to the electrons in another (valence) and, therefore, the correlation energy adjustment will be... 

Due to the shallow wells involved in some off-centre instabilities, the three parameter hybrid semiempirical B3LYP functional [246] implemented in Gaussian 98 [247] has also been used in shallow well cases. These calculations use the double zeta LANL2DZ basis, which employ Gaussian type orbitals and pseudopotentials to simulate the core electrons. Some semiempirical Self Consistent Charge Extended Hiickel (SCCEH) calculations have also been performed in order to study the effect of the removal of some orbitals from the basis set on the off-center instabilities. More details about this method can be found in [248]. 

Les, A., and Ortega-Blake, I. (1985). Inti. J. Quantum Chem. 27, 567—semiempirical (PCILO.MNDO) and ab initio pseudopotential calculations for relative stabilities of tautomers of 2- and 4-oxopyrimidine. 

In the simulations reported above the interactions between the various components were mostly based on semiempir-ical potentials. In contrast, Price, Halley, and their collaborators attempt to model the whole interface in the spirit of the Car-Parinello method [16]. One of the first systems investigated was the interface between a copper electrode and water [67]. For this purpose these authors set up a simulation cell with approximate dimensions of 42 A X 15 A X 15 A. Each cell contained a slab of copper atoms that were five-layers thick, the two surfaces having (100) structure. The remaining space was filled with water molecules. CycKc boundary conditions were applied in all directions. Obviously, an ab initio, all electrons calculation is quite impossible for such a system, and may not even be desirable. Instead, Halley and collaborators used a mixture of pseudopotentials... 

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],... 

Fig. 10. A typical pseudopotential (for Sn), as calculated by Heine and Animalu/" shown together with points which have been fitted in semiempirical studies of (O) (semiconducting) gray and ( ) (metallic) white...

The methods of the preceding sections involve, at the outset, few necessary approximations. Even the muffin tin approximation can be dispensed with in the APW method. In practice, of course, all kinds of approximations are involved in the practical application of such schemes, especially when they are used in a semiempirical manner. This is not simply for reasons of convenience but also because one may find that a theory which has all the elements of a first principles calculation has too many free parameters in semiempirical work. For instance, if one chooses to use the parameters v(g) in pseudopotential theory as fitting parameters, one is forced to neglect (or fix the values of) the higher components g > 2kp entirely (Ref. 51, pp. 83-86), otherwise the number of fitting parameters would be too great and they would be undetermined by the available data. ... 

Any of the existing band structure methods can be adapted for use as a semiempirical scheme, or an interpolative scheme to facilitate the calculation of quantities which depend on interband integrals and the like. Tight binding theory, reduced to its bare essentials, with the overlap parameters used to fit experimental data or as an interpolation scheme in band structure calculation is generally referred to as Slater-Koster theory. Pseudopotential theory used in this way has been dubbed the empirical pseudopotential method (EPM) and has been the subject of a recent comprehensive review. Some comparisons of parameters t>(g), which have been fitted to experiment, with theoretical calculations have already been shown in Figure 12. 

Some further remarks are in order on the subject of truncation of the pseudopotential. Most current semiempirical studies involve quite large secular determinants (say 50 x 50) but set f(g) equal to zero for g 2kp. However, a somewhat cruder procedure, that of truncating the basis set at g = 2kp, resulting in a smaller secular determinant, has also been widely used. This procedure may be put on a formal basis by the use of Lowdin perturbation theory, by which a larger secular determinant is reexpressed as a smaller one, with correction terms. For a local pseudopotential the correction terms are given by (Ref 51, pp. 78-83)... 

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