The Hartree-Fock method for crystals

In many quantum-mechanical calculations, use is made of the wave functions obtained by the Dirac—Slater and the Hartree—Fock methods for the approximate solution of the Schrddinger equation for free atoms. It woiild be very interesting to determine whether these functions could be refined specifically for crystals and whether the problem could be solved using relatively simple analytic approximations to the calculated functions. In particular, the approximation by Gaussian functions demands attention. 

What we expect to obtain finally in the Hartree-Fock method for an infinite crystal are the molecular orbitals, which in this context will be called the crystal orbitals (COs). As usual, we will plan to expand the CO as linear combinations of atomic orbitals (cf. 427). Which atomic orbitals Well, those that we consider appropriate for a satisfactory description of the crystal e.g., the atomic orbitals of all the atoms of the crystal. We feel, however, that we are going to have a big problem trying to perform this task. 

Specific Features of the Hartree-Fock Method for a Cyclic Model of a Crystal... 

Fig. 15. (a) Intramolecular hydrogen bonds in urea crystal with displacement ellipsoids at 50% probability, (b) Static deformation density obtained from the multipolar analysis of the experimental data corrected for the thermal diffuse scattering. Theoretical deformation density obtained using (c) the Hartree-Fock method (d) the DFT method by generalized gradient approximation (contours at 0.0675 eA-3) (reproduced with permission from Zavodnik et al. [69]). 

When the Hartree-Fock method is used for the ionic calculation, the error due to the neglect of electron correlation is significant ifthe number of electrons in the crystal ion is different from that in the reference gas phase ion (this occurs for oxides, where the crystal ion is O2- and the reference gas phase ion is 0-). Earlier electron gas calculations estimated the selfenergy due to electron correlation [24], but in more recent calculations an... 

In practical calculations of the band structure of crystals by the Hartree-Fock method, the one-electron density matrix Prr (k) in the reciprocal space can be calculated only for a rather small finite set of special points (see Sect. 4.2). Let us consider such a set of special points kj, j = 1,2,..., Nq. Then, in the calculation of the density matrix of the infinite crystal the integration over the BriUouin-zone volume Vg is changed by the sum over the special points chosen... 

As was the case with lanthanide crystal spectra (25), we found that a systematic analysis could be developed by examining differences, AP, between experimentally-established actinide parameter values and those computed using Hartree-Fock methods with the inclusion of relativistic corrections (24), as illustrated in Table IV for An3+. Crystal-field effects were approximated based on selected published results. By forming tabulations similar to Table IV for 2+, 4+, 5+ and 6+ spectra, to the extent that any experimental data were available to test the predictions, we found that the AP-values for Pu3+ provided a good starting point for approximating the structure of plutonium spectra in other valence states. However,... 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

The theoretical prediction of the optical absorption profile of a solid using first-principles methods has produced results in reasonable agreement with experiment for a variety of systems [2-4], For example, several ionic crystals were studied extensively, generally using the Hartree-Fock one-electron approximation [5], through the extreme-ultraviolet. Lithium fluoride was the focus of a particularly detailed comparison [6-8], providing excellent confirmation of the applicability of the band theory of solids for optical absorption. 

Impurity and Aperiodicity Effects in Polymers.—The presence of various impurity centres (cations and water in DNA, halogens in polyacetylenes, etc.) contributes basically to the physics of polymeric materials. Many polymers (like proteins or DNA) are, however, by their very nature aperiodic. The inclusion of these effects considerably complicates the electronic structure investigations both from the conceptual and computational points of view. We briefly mentioned earlier the theoretical possibilities of accounting for such effects. Apart from the simplest ones, periodic cluster calculations, virtual crystal approximation, and Dean s method in its simplest form, the application of these theoretical methods [the coherent potential approximation (CPA),103 Dean s method in its SCF form,51 the Hartree-Fock Green s matrix (resolvent) method, etc.] is a tedious work, usually necessitating more computational effort than the periodic calculations... 

---