One-electron Approximation for Crystals

One-electron Approximation for Crystals 113 The two-electron part (4.35) contains one-electron density matrix elements... 

A quantitative consideration on the origin of the EFG should be based on reliable results from molecular orbital or DPT calculations, as pointed out in detail in Chap. 5. For a qualitative discussion, however, it will suffice to use the easy-to-handle one-electron approximation of the crystal field model. In this framework, it is easy to realize that in nickel(II) complexes of Oh and symmetry and in tetragonally distorted octahedral nickel(II) complexes, no valence electron contribution to the EFG should be expected (cf. Fig. 7.7 and Table 4.2). A temperature-dependent valence electron contribution is to be expected in distorted tetrahedral nickel(n) complexes for tetragonal distortion, e.g., Fzz = (4/7)e(r )3 for com-... 

Currently the problems involved in calculating the electronic band structures of molecular crystals and other crystalline solids centre around the various ways of solving the Schrodinger equation so as to yield acceptable one-electron solutions for a many-body situation. Fundamentally, one is faced with an appropriate choice of potential and of coping with exchange interactions and electron correlation. The various computational approaches and the many approximations and assumptions that necessarily have to be made are described in detail in the references cited earlier. 

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. 

Although in the dielectrics the mobilities of the carriers are too small and their effective masses too large for the formation of shallow donor and acceptor states, the initial Hamiltonian describing the system of the host and Ln"" is the same as that for donor and acceptor impurities in semiconductors. In there is a one-electron approximation after the IT, the system consists of an unperturbed crystal with periodic potential—ionized lanthanide ion —and a free electron (in the... 

We consider a three-dimensional periodic polymer or molecular crystal containing m orbitals in the elementary cell of one or more atoms. For the sake of simplicity the number of elementary cells in the direction of each crystal axis is taken equal to an odd number Ni = N2 = Ni = 27V 4-1. We assume further that there is an interaction between orbitals belonging to different elementary cells. In that case we can describe, in the one-electron approximation, the delocalized crystal orbitals of the polymer with the aid of the linear-combination-of-atomic-orbitals (LCAO) approximation in the form... 

For atoms, it has been possible during the last few years to program the calculations in the ordinary HF scheme for some electronic computers, and, as soon as one has obtained enough experience in this connection, there will probably be no difficulties in doing the same also for the extended HF scheme sketched in this section. For molecules and crystals, on the other hand, one has probably to be satisfied with comparatively rough approximations for a long time. 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

An interesting and useful method of theoretical treatment of certain properties of complexes and crystals, called the ligand field theory, has been applied with considerable success to octahedral complexes, especially in the discussion of their absorption spectra involving electronic transitions.66 The theory consists in the approximate solution of the Schrddinger wave equation for one electron in the electric field of an atom plus a perturbing electric field, due to the ligands, with the symmetry of the complex or of the position in the crystal of the atom under consideration. 

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