LCAO theory

Where might these one-electron wavefunctions come from I explained the basic ideas of HF and HF-LCAO theory in Chapter 6 we could find the molecular orbitals as linear combinations of appropriate atomic orbitals by solving the HF eigenvalue problem... 

You probably noted that the original papers were couched in terms of HF-LCAO theory. From Chapter 6, the defining equation for a Hamiltonian matrix element (in the usual doubly occupied molecular orbital, closed-shell case) is... 

The calculation usually proceeds along the traditional lines of HF-LCAO theory. We make an initial estimate of the electron density matrix P, calculate a revised h and iterate until the electron density and the HP matrix are self-consistent. 

In HF-LCAO theory, the electronic ground state of pyridine is We... 

There is no systematic way in which the exchange correlation functional Vxc[F] can be systematically improved in standard HF-LCAO theory, we can improve on the model by increasing the accuracy of the basis set, doing configuration interaction or MPn calculations. What we have to do in density functional theory is to start from a model for which there is an exact solution, and this model is the uniform electron gas. Parr and Yang (1989) write... 

In the conventional MO-LCAO theory, the function u is approximated by a Is orbital, but better approximations may be obtained by including higher orbitals. The total wave function is such that, for separated atoms, there is a fifty per cent chance that the mole-... 

Before we close this MO/LCAO discussion of the generalized (8-N) rule, we note that a derivation of Eqn. II. 1 has been reported by Hulliger and Mooser (23) on a similar basis. However, a careful analysis of their treatment reveals that, in addition to features of general MO/LCAO theory (Thms. II. 1—II.3) and necessary assumptions (equivalents of Hyps. II.1—II.3), they also introduce some superfluous assumptions and specializations. This not only obscures the treatment, it also introduces new aspects which it may be instructive to dwell on in some detail. In order to keep the number of notational symbols to a minimum, the definitions already invoked in the preceding discussion will be utilized as far as possible. However, the disposition and layout of their paper differ significantly from ours since, moreover, many of Hulliger and Mooser s assumptions are to be classified as being only partly superfluous, some quotations are inescapable. 

Hartree-Fock atomic term values are used in LCAO theory [89]. 

In addition to explaining why hydrogen bonds with itself, LCAO theory also explains why some atoms do not. A beryllium atom, for instance, has four electrons, including two valence electrons in the 2s orbital. It would seem that the 2s orbitals of two beryllium atoms could join together to form a sigma bond. 

Linear combination of atomic orbitals (LCAO theory) A method for combining atomic orbitals to approximately compute molecular orbitals. 

Wc saw how formulae for the matrix elements can be obtained by equating band energies from LCAO theory and from frce-clcctron theory in Fig. 2-2. Froyen and Harrison (1979) made the corresponding treatment of the tetrahedral solids, again including only matrix elements between nearest-neighbor atoms. The form of their results is just as found for the simple cubic case... 

Let us make an elementary calculation of energy bands, using the notation of LCAO theory. For many readers the procedure will be familiar. Consider a ring of N atoms, each with an s orbital. We seek an electronic state in the form of an LCAO,... 

Comparison of LCAO bands and true bands indicates that indeed the valence bands arc very accurately represented by simple LCAO theory. The conduction bands are not nearly so accurately given, though they arc qualitatively correct. 

The response of covalent crystals to magnetic fields is very weak and of less interest than the dielectric response. It may nevertheless prove useful as a probe of the electronic structure. An early discussion of the problem, with references to still earlier work, was given by KrumhansI (1959). Two recent treatments in terms of LCAO theory have been given by Sukhatmc and Wolff (1975) and by Chadi, White, and Harrison (1975). We shall follow the latter treatment but give a more complete formulation than given there. 

In Chapter 3 we gave a preliminary discussion of the energy bands in terms of the simple LCAO theory, and illustrated, in Fig. 3-7, the form of more accurately determined energy bands. For most of the studies made in this text, that description will be sufficient. However, the bands are of some interest in their own right and are important to the understanding of the electronic properties of semiconductors, and a consideration of them increases one s understanding of the electronic structure of covalent solids. In this chapter, therefore, we shall look at a more extensive set of accurate bands and at their interpretation in terms of the con-... 

Wc saw the general form of the valence bands in Fig. 6-1 for a number of semiconductors, and discussed the general features there. In Fig. 6-6 we show another version of the bands for GaAs, which will be useful for reference as we construct the conduction bands from LCAO theory. 

A final interesting feature of the conduction bands, which we can discuss in the context of the F,-only bands, is the gap Eq between the nondegeneratc conduction-band level at F and the valence-band maximum at F. A formula for this gap was obtained from the full LCAO theory in Eq. (3-43), which is repeated... 

We note now that the momentum operator is = hli)d/d, so the matrix elements that enter the k p method are exactly the same matrix elements that entered the calculations of optical absorption. This remarkable fact enables us to obtain parameters from the LCAO theory given in Chapter 4. 

This does not mean that the LCAO approach of the type we have used is incorrect or not useful. Recent applications of LCAO theory, based only upon electron orbitals that are occupied in the free atom, have been made to the study of simple metals (Smith and Gay, 1975), noble-metal surfaces (Gay, Smith, and Arlinghaus, 1977), and transition metals (Rath and Callaway, 1973). In fact, the LCAO approach seems a particularly effective way to obtain self-consistent calculations. The difficulty from the point of view taken in this book is that, as with many other band-calculational techniques, LCAO theory has not provided a means for the elementary calculations of properties emphasized here, but pseudo-potentials have. 

Let us now complete the derivation of formulae for the interatomic matrix elemenfis, which was described in Section 2-D, by equating band energies obtained from LCAO theory and those obtained from nearly-free-electron bands. This analysis follows a study by Froyen and Harrison (1979). The band energies obtained from nearest-neighbor LCAO theory at symmetry points were given in... 

Froyen and Harrison (1979) also used this procedure to obtain for a nearest-neighbor fit to the bands for a face-centered cubic structure. The four values (in comparison to the tetrahedral values) were —0.62 (— 1.39), 2.33 (1.88), 2.47 (3.24), and 0 (—0.93). These are major differences, but the nearest-neighbor LCAO theory of close-packed structures is of little use in any case. The tetrahedral parameters seem to be adequate for covalent and ionic systems but not to be relevant to the simple metals. 

What has been accomplished is a very simple relation between the pseudopotential and the important gap in the band structure. What is more, we have provided such a simple representation of the band structure that we may use it to calculate other properties of the semiconductor, just as we did with the LCAO theory once we had made the Bond Orbital Approximation. 

The first such application is the completion of the identification of the parameters of the two theories that we mentioned earlier. The separation of the parallel band.s, which has been given here, for pseudopotential theory, by 2IF, was written for LCAO theory in Chapter 4 as 2 V + with and... 

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. 

Let us now turn to a few other informative applications of the pseudopotential model to properties (discussed extensively in Harrison, 1976a). For that purpose it will be most convenient to take the pseudopotential parameters from the LCAO values for V2 and Fj, using Eq. (18-5). We see from Table 18-1 that for V2 this is roughly equivalent to using Empirical Pseudopotential Model parameters. Let us look first at the dielectric susceptibility, which was so important in the development of the LCAO theory. For this, the most convenient form for the susceptibility is Eq. (4-5), which we rewrite in simple form,... 

This form does not resemble very closely the form of Eq. (4-26), which was obtained from LCAO theory. It can be brought into the same form, however, by multiplying by the square of V2 and then dividing by the square of the form 2A6h lmd, from (Eq. 4-16), for V. After a little algebra, this leads to... 

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