We propose a new method for decomposing electron density of a crystal into contributions associated with pair-wise chemical bonds. To this end, an ion-covalent crystal is represented using a neutral, closed shell cluster assembled from identical structural elements (SE) and embedded into the lattice electrostatic potential. The wave function of this cluster is calculated using the one determinant approximation. Then, a set of orthonormal, noncanonical, multicenter orbitals of the cluster valence states is generated, so as each orbital is localized on one structural element. The projection operators technique is used here, the valence molecular orbitals of the cluster being taken for the orthonormal basis set. In this construction, the first-order reduced density matrix of the cluster valence electrons is exactly the sum of the first-order reduced density matrices of the SE, and the latter is the exact sum of localized on this cluster orbitals densities. The localized orbitals are then transformed into directed orbitals corresponding to the ion-covalent bonds in each structural element. The first-order reduced density matrix of each structural element is exactly the sum of densities of all such corresponding directed orbitals. This method is demonstrated on the examples of MgO, cubic ZrO 2 , and rutile TiO 2 crystals.

The embedded cluster method for ion-covalent crystal band structure calculations is proposed. This method uses the results of embedded cluster electronic structure calculations within one-determinant Hartree-Fock approximations. The band structure of high-temperature cubic phase ZrO 2 crystal is calculated and found to be in good agreement with calculations in the literature, which applied periodic boundary conditions at the same theory level. Figure 8. The band structure of ZrO 2 cubic phase.

ABSTRACT:In calculation, the electronic structure of crystals, especially those containing point defects, the embedding approach is proved to be useful and convenient. In this approach, a finite part of the crystal, referred to as cluster, is considered instead of infinite crystal and the influence of the rest of the crystal is simulated by the embedding potential. The key problems of this approach are the cluster selection and the embedding potential generation. To select a cluster, the Wigner-Seitz unit cell is used in the present approach and every border atom, situated at the unit cell face, edge, or vertex is symmetrically "divided" among adjacent unit cells sharing this atom. The atomic hybrid orbitals are used for the border atoms partition between neighboring clusters. It is shown that contrary to the conventional hybridization scheme, the nonorthogonal and even linearly dependent atomic hybrid orbitals can be used to construct the border atom density matrix. This density matrix can be made to satisfy the proper point symmetry and to match the number of equivalent hybrid orbitals and the number of nearest neighbors. Two different types (one-center and multicenters) of the embedding potential corresponding to the border atom are considered in the article. As a particular example of the border atom the oxygen ion in ZrO 2 , MgO, and TiO 2 rutile crystals is considered.

Many important properties of crystals are the result of the local defects. However, when one address directly the problem of a crystal with a local defect one must consider a very large system despite the fact that only a small part of it is really essential. This part is responsible for the properties one is interested in. By extracting this part from the crystal one obtains a so-called cluster. At the same time, properties of a single cluster can deviate significantly from properties of the same cluster embedded in crystal. In many cases, a single cluster can even be unstable. To bring the state of the extracted cluster to that of the cluster in the crystal one must apply a so-called embedding potential to the cluster. This article discusses a case study of embedding for ion-covalent crystals. In the case considered, the embedding potential has two qualitatively different components, a long-range (Coulomb), and a short-range. Different methods should be used to generate different components. A number of approximations are used in the method of generating an embedding potential. Most of these approximations are imposed to make the equations and their derivation simple and these approximations can be easily lifted. Besides, the one-determinant approximation for the wave function is used. This is a reasonably good approximation for ion-covalent systems with closed shells, which simplifies the problem considerably and makes it tractable. All employed approximations are explicitly stated and discussed. Every component of generation methods is described in details. The proofs of used statements are provided in a relevant appendix. V C 2015 Wiley Periodicals, Inc. DOI: 10.1002/qua.25041Introduction An immediate solution of the Schroedinger equation for a real crystal is impossible due to an enormous number of particles in the system and a complicated system structure. However, a real crystal consists of similar, almost identical, repeating parts. Therefore, one can substitute a real crystal with its model as an infinite system with translational symmetry. This model is usually referred to as a perfect crystal. The first methods which were applied to a perfect crystal problem are: the WignerSeitz cell method, [1,2] the tight-binding method, [3,4] the orthogonalized plane wave method, [5] the augmented plane wave method, [6] the Korringa-Kohn-Rostoker method [7,8] and their modifications. There are, at present, a wide variety of methods that can be used to calculate the perfect crystals properties. However, many technologically important crystal properties are the result of crystal defects and impurities. These defects and impurities break down the perfect crystal translational symmetry. Many of these defects are local and their properties are determined by a comparatively small region of the crystal. Although the actual region is small, it is still part of the large system. There are two general approaches to this problem. One of them uses the well-developed methods of band structure calculations. The system is accord...