Cluster models translational symmetry

The focus then shifts to the delocalized side of Fig. 1.1, first discussing Hartree-Fock band-structure studies, that is, calculations in which the full translational symmetry of a solid is exploited rather than the point-group symmetry of a molecule. A good general reference for such studies is Ashcroft and Mermin (1976). Density-functional theory is then discussed, based on a review by von Barth (1986), and including both the multiple-scattering self-consistent-field method (MS-SCF-ATa) and more accurate basis-function-density-functional approaches. We then describe the success of these methods in calculations on molecules and molecular clusters. Advances in density-functional band theory are then considered, with a presentation based on Srivastava and Weaire (1987). A discussion of the purely theoretical modified electron-gas ionic models is... 

The need to model this distribution means that it is difficult to theoretically study the electronic states of oxide glasses. There are several ways to theoretically study the electronic state of oxide materials, these include band calculations and molecular orbital methods. (9-13) The randomness is a problem for the band approach because it requires translational symmetry of the unit cell a large super-cell may be chosen, but this is at the cost of increased computer time and possible spurious interactions between cells. On the other hand, the molecular orbital (MO) approach is usually applied to isolated molecules, (14-16) and can not handle infinite numbers of atoms as in the solid state. The embedded potential method is one of the improvements in moleculeir orbited methods which have been introduced in order to study solid state materials. (17) Basically, the cluster Hamilto-... 

One of the most difficult problems for ab initio quantum chemistry is to determine the potential energy function for a chemical reaction on a metal surface. Why is this so First of all, the metal substrate is strongly delocalized. This means that the system cannot be modeled [1] by considering just a small or medium-sized cluster of metal atoms. On the other hand, the band structure techniques that would simplify calculations for a bare metal surface cannot be directly applied because the translational symmetry is broken by the presence of the reactants. As a result one has the difficulty of dealing with extended interactions without the benefit of simplifications due to symmetry. Many problems involving surfaces, interfaces, impurities, or defects in solid state materials fall under this broad rubric along with various solution phenomena as well. 

The cluster model approach can be viewed as the chemist s approach since it reduces a very large system to a supermolecule, yet it is currently the only way to study excited states and therefore to contribute to the interpretation of electronic spectra. On the other extreme, one finds the physicist point of view, which makes uses of translational symmetry and treats the system as a perfect periodic solid. Therefore, as in the cluster model approach, the periodic approach constitutes a severe approximation since the same structure is reproduced in two or three space directions. 

The prototype molecule (or cluster ) approach to the quantum chemical treatment of extended systems has proved to be a valuable tool, especially if the chemical phenomenon of interest is mainly of local character. The prototype molecule models were used e.g. in the quantum chemistry of silicates [58], zeolites [59] and enzymes [14]. In the solid state, quantum chemistry calculations on prototype molecules [60] represent an important alternative to crystal orbital type methods. Although these latter calculations may be very important as starting points even for the description of local phenomena which violate the exact translational symmetry, the cluster approaches have the advantage of providing a direct space representation of the wave function. 

In the theory of electronic structure two symmetric models of a boundless crystal are used or it is supposed that the crystal fills aU the space (model of an infinite crystal), or the fragment of a crystal of finite size (for example, in the form of a parallelepiped) with the identified opposite sides is considered. In the second case we say, that the crystal is modeled by a cyclic cluster which translations as a whole are equivalent to zero translation (Born-von Karman Periodic Boundary Conditions -PBC). Between these two models of a boundless crystal there exists a connection the infinite crystal can be considered as a limit of the sequence of cychc clusters with increasing volume. In a molecule, the number of electrons is fixed as the number of atoms is fixed. In the cyclic model of a crystal the number of atoms ( and thus the number of electrons) depends on the cyclic-cluster size and becomes infinite in the model of an infinite crystal. It makes changes, in comparison with molecules, to a one-electron density matrix of a crystal that now depends on the sizes of the cyclic cluster chosen (see Chap. 4). As a consequence, in calculations of the electronic structure of crystals it is necessary to investigate convergence of results with an increase of the cyclic cluster that models the crystal. For this purpose, the features of the symmetry of the crystal, connected with the presence of translations also are used. 

The overwhelming majority of band calculations performed assume perfect translational symmetry of surface layers. In cluster models of local... 

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