Pseudopotential perturbation

The remarkable conclusion of this argument is that though pseudopotentials can be used to describe semiconductors as well as metals, the pseudopotential perturbation theory which is the essence of the theory of metals is completely inappropriate in semiconductors. Pseudopotenlial perturbation theory is an expansion in which the ratio W/Ey of the pseudopotential to the kinetic energy is treated as small, whereas for covalent solids just the reverse quantity, Ey/W, should be treated as small. The distinction becomes /wimportant if we diagonalize the Hamiltonian matrix to obtain the bands since, for that, we do not need to know which terms are large. Thus the distinction was not essential to the first use of pseudopotentials in solids by Phillips and KIcinman (1959) nor in the more recent application of the Empirical Pseudopotenlial Method u.scd by M. L. Cohen and co-workers. Only in approximate theories, which are the principal subject of this text, must one put terms in the proper order. 

The expansion in F/( , - tj.) would suggest the possibility of incorporating the effect of the resonance in perturbation theory and thus extending the pseudopotential perturbation theory of simple metals to transition metals. This has in fact been done (Harrison, 1969) and the pseudopotential approach has been extensively developed by Moriarty (1970, I972a,b,c), but the application.s have been largely restricted to the ends of the transition scries where the expansion is clearest. 

The model is shown to give results that are very often in quantitative agreement with experimental data and serve in most other cases as a good starting point for the calculation of the effects of the ionic structure, e.g. via pseudopotential perturbation theory. 

Main experimental findings both for the ground state (magic numbers for the stability of clusters [3] and the existence of supershells [4]) and for excited states (the dominance of collective states in the photoabsorption of metal clusters MeA with N > S) were predicted [5] before their experimental confirmation. Recently we were able to explain the temperature dependence of the absorption of small metal clusters as observed by Haber-land s group [6]. If the model is complemented by pseudopotential perturbation theory [7] the results obtained match qualitatively those obtained by demanding quantum-chemical methods (e.g. the photoabsorption spectra of Na6). Further improvement of the model includes the removal of self-interaction effects, the so-called SIC [8-10] (a consequence of using the local density approximation (LDA) to general density functional theory (DFT)). 

Figure 1.15 Comparison of the electron affinity within the spheroidal jellium model plus SIC and various experimental results (circles [44], triangles [45], squares [46]). For N = 30 theory predicts two isomers (prolate and oblate), which are nearly degenerate. But both do have different affinities and in the beam the signal will come from those clusters having the lower affinities (connected by dashed lines). Clearly, shell effects are very pronounced. Qualitatively theory and experiment agree rather well. In order to achieve quantitative agreement one has to introduce pseudopotential perturbation theory as sketched above. Reproduced by permission of Springer Verlag...

Figure 1.20 Optical absorption of Nag in its ground-state structure D2d. This result from pseudopotential perturbation theory is explained in the main text. The spherical plasmon line at about 2.5 eV (see Figure 1.21) is split into two components which can be understood as follows. The moments of inertia of the structure D2d point to a prolate spheroid within the jellium approximation to the distribution of ions. In such a system there are two collective excitations one at higher frequencies (perpendicular to the axis of symmetry) and one for the motion along the axis of symmetry. Because the motion perpendicular is twofold degenerate its intensity is twice that of the low-frequency motion (with the cluster being statistically oriented in the beam (see [30])...

Now we have a picture about how demanding temperature-dependent calculations are we need at each T and colO Monte Carlo points and 10 frequencies in order to cover the experimental range. Furthermore we need the three different directions jc, y and z. So we had to solve three million times the TDLDA integral equation with full inclusion of the ionic structure (via pseudopotential perturbation theory). Needless to say it seems almost impossible to perform calculations of this type for transition metals with the additional complication of the d-electrons ... 

We conclude with two more examples which demonstrate the power of pseudopotential perturbation theory. Figure 1.23 shows the geometry and absorption spectra of Na6 which... 

Figure 1.25 Geometry and optical absorption of NaQo. For N = 90 there are no determinations of the geometry at the ab initio level available. We have therefore simply calculated the total energy within pseudopotential perturbation theory of second order for various model clusters built as small crystal fractions of fee, bcc, hep and icosahedral type. For the case in question the hep structure (see upper part of the figure) has the lowest total energy...

The peak positions of the hard-wall jellium model are slightly blue-shifted compared to the experimental data [8, 27, 36-39]. Agreement can be obtained by using a soft-wall jellium model [16, 36], or, more correctly, by pseudopotential perturbation theory [38, 39]. 

The phonon calculation from the pseudopotential perturbation theory, with the Heine-Animalu and the BHS pseudopotential, raises the question whether the perturbation approach is sufficiently adequate. 

The first two terms are precisely the electronic contribution to the dynamical matrix, as obtained from the pseudopotential perturbation theory. 

In the first and second order contributions to the dielectric matrix, only pseudopotential matrix elements appear at reciprocal lattice vectors. If the pseudopotential perturbation for the energy makes sense, these matrix elements can be considered as a perturbation. Consequently, the contributions required are these, where the potential u is needed at wave vectors smaller than a reciprocal lattice vector. This is obviously only the case for... 

For illustrative purposes, this exchange treatment was introduced in the pseudopotential perturbation theory, and revealed substantial exchange effects in the phonon dispersion of simple metals (in contrast to Harrison s... 

The simple metals, whose conduction bands correspond to s and p shells in isolated atoms, include the alkali metals, the divalent metals Be, Mg, Zn, Cd, and Hg, the trivalent metals Al, Ga, In, and Tl, and the tetravalent metals (white) Sn and Pb. Almost all of their properties which are related to electronic band structure are explicable by nearly-free-electron theory using pseudopotentials (Sections 3.5 and 3.6). The extent to which they conform in detail to this generalization varies from one case to another. For all the metals cited simple pseudopotential theory is fairly successful in predicting or fitting Fermi surface properties. This will be evident from a consideration of the comparisons of theoretical and fitted pseudopotential parameters already shown in Figure 12. However, the use of perturbation theory is not very critical in this context [i.e., the contribution of screening to the values of v q) which are of interest is not large]. In other contexts the validity of perturbation theory is more critical, and indeed the use of pseudopotential-perturbation theory is then not always so successful. An example is the calculation of phonon dispersion relations by such methods, which has enjoyed remarkable success for Na, Mg, and j(i2i,i22) jjjjQ difficulties for the heavier metals and those... 

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