Core-pseudopotential

An approach based on orbital radii of atoms effectively rationalizes the structures of 565 AB solids (Zunger, 1981). The orbital radii derived from hard-core pseudopotentials provide a measure of the effective size of atomic cores as felt by the valence electrons. Linear combinations of orbital radii, which correspond to the Phillips structural indices and have been used as coordinates in constructing structure maps for AB solids. 

We can understand the behaviour of the binding energy curves of monovalent sodium and other polyvalent metals by considering the metallic bond as arising from the immersion of an ionic lattice of empty core pseudopotentials into a free-electron gas as illustrated schematically in Fig. 5.15. We have seen that the pseudopotentials will only perturb the free-electron gas weakly so that, as a first approximation, we may assume that the free-electron gas remains uniformly distributed throughout the metal. Thus, the total binding energy per atom may be written as... 

Fig. 5.15 An ionic lattice of Ashcroft empty core pseudopotentials immersed in a free-electron gas.

Table 6.3 Contributions to the binding energy (in Ry per atom) of sodium, magnesium, and aluminium within the second order real-space representation, eqn (6.73), using Ashcroft empty-core pseudopotentials. L/gf is defined by eqn (6.75). The numbers in brackets correspond to the simple expression, eqn (6.77), for = 0) and to the experimental values of the binding energy and negative cohesive energy respectively.

Defining the normalized ion-core pseudopotential matrix element by... 

The wave vector, k , and the screening length, 1/ , depend only on the density of the free-electron gas through the poles of the approximated inverse dielectric response function, whereas the amplitude, A , and the phase shift, a , depend also on the nature of the ion-core pseudopotential through eqs (6.96) and (6.97). For the particular case of the Ashcroft empty-core pseudopotential, where tfj fa) = cos qRc, the modulus and phase are given explicitly by... 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

Explain the concept of a pseudopotential. Aluminium is fee with a lattice constant of a = 7.7 au. It is well described by an Ashcroft empty core pseudopotential of core radius 1.1 au. Show that the lattice must be expanded by 14% for the 2n/a(200) Fourier component of the pseudopotential to vanish. 

Analytic derivatives have been reported for both the LSCF and GHO models, making them attractive options for MD simulations (Amara et al. 2000). Their generalization to ab initio levels of theory through the use of core pseudopotentials (along the lines of the pseudohalogen capping atoms described above) ensures that they will see continued development. 

Bouteiller, Y., Mijoule, C., Nizam, M., Barthelat, J. C., Daudey, J. P., Pelissier, M. and Silvi, B. (1988) Extended gaussian-type valence basis sets for calculations involving non-empirical core pseudopotentials Mol. Phys., 65, 295. 

Except for occasional discussions of the basis set dependence of the results, the numerical implementation issues such as grid integration techniques, electron-density fitting, frozen-cores, pseudopotentials, and linear-scaling techniques, are omitted. 

Usually the atoms on the local region include all electrons or a small core pseudopotential whereas the atoms in the outer region include a larger core and most often only the outermost ns electrons necessary to represent the metal conduction band are explicitly included in the quantum mechanical calculation. This is nowadays a standard procedure to model Ni, Cu, Ag and Pt surfaces. " " In addition, a mixed basis scheme is usually employed to further reduce the computational cost while attempting to preserve the quality of the cluster model ab initio calculations. In this scheme, the atoms in the outer region are treated with a rather limited, even minimal, basis set whereas atoms in the local region are treated with a more extended basis set. 

The free-clectron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pscudopotential, not the true potential, which is treated as the perturbation. If we were to start with a frcc-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these arc drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pscudopotential, the core states arc already there, and full, and the effects of the pseudopotential arc small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

It should be noticed that curves like those shown in Fig. 16-1 arcdclined only for q 0. The q = 0 value is best obtained by combining all of the terms in the energy, as we did in the discussion in Chapter 15. Notice, however, that in the discussion of cohesion in Chapter 15 we used the same that we use here for the discussion of w, for q 0. This remarkable feature appears to be special to the empty-core pseudopotential. 

The lowest-lying s state for the empty-core pseudopotential is illustrated in F ig. 16-13,a. Its energy f can be computed by numerical integration and obtained as a function of Z and c. We shall make an approximate solution. 

There the relation was made in terms of the splitting at F rather than X, since the corresponding formulae are simpler.) The first comparison we make is between the LCAO values and the empty-core pseudopotential. We shall find only qualitative correspondence between the values because of errors in the empty-core model, which become large here. We shall then go on to consider other properties, using pseudopotential matrix elements obtained without resort to the empty-core model. 

Making first a comparison of the covalent energy, notice that in homopolar semiconductors, Wy, becomes simply w, The various geometrical factors in the empty-core pseudopotential may be directly evaluated. Then, the pseudopotential matrix element becomes... 

Polarities predicted from the empty-core pseudopotential and the relations given at Eq, 18-5. Values from LCAO theory (Table 4-1) are given in parentheses. 

Serving so broad an audience has dictated a simplified analysis that depends on three approximations a one-electron framework,simple approximate interatomic matrix elements, and empty-core pseudopotentials. Refinement of these methods is not difficult, and is in fact carried out in a series of appendixes. The text begins with an introduction to the quantum mechanics needed in the text. An introductory course in quantum mechanics can be considered a prerequisite. What is reviewed here will not be adequate for a reader with no background in quantum theory, but should aid readers with limited background. 

As the atom becomes larger, the number of basis functions needed to describe it increases as well. However, since one is most interested in the valence shell where most of the action occurs, the increasingly larger number of inactive or core functions become more and more of a nuisance. One cannot simply omit them as the valence orbitals would then collapse into smaller core orbitals (which are of much lower energy). One solution is development of core pseudopotentials or effective core potentials (ECP) which eliminate the need to include core functions explicitly, yet keep the valence functions from optimizing themselves into core orbitals ° . Such pseudopotentials are commonly used in elements of the lower rows of the periodic table, like Br or I. 

Table 2.22 Properties of binary complexes, calculated at the SCF level, using core pseudopotentials .

---