Theoretical Analysis of Chemical Bonding in Crystals

The local properties of the electronic structure of a periodic system are defined by the density matrix p R,R ),see (4.91) the electron position vectors R and R vary within the basic domain of a crystal consisting of N primitive unit cells. For a one-determinant wavefunction the density matrix can be expressed through Bloch-type spin orbitals = ,/ )  

The expansion coefficients are calculated by solving the matrix equation of the CO LCAO method for crystals, see (4.67)  

6) F k) is the matrix of the Hartree-Fock (HP) or Kohn-Sham (KS) operator. The former operator includes a nonlocal exchange part, depending on the density matrix p R,R ), whereas the latter operator involves the electron density p R) = p R, R), that is, it depends only on the diagonal elements of the density matrix, see Chapters 4 and 7. 

In view of the translational symmetry of a crystal, one can introduce density normalization per primitive rmit cell containing n electrons  

The Mulhken population analysis may be extended to crystalline soUds [573,574,576] giving the following definitions for an electronic population Nao on an atom, atomic charge Qao bond order WAo,Bnt covalency Cao, and the total valency Vao (AO, Bn mean atom A in the reference rmit cell and atom B in the unit cell with a translation 

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Theoretical Analysis of Chemical Bonding in Crystals 349 Simple algebraic manipulations lead to the conclusion that... 

The first solution to this problem was produced phenomenologically by Mooser and Pearson. The solution for A B compounds is reproduced in Figure 9. Similar solutions apply not only to A"B semiconductors and insulators, but also to many intermetallic compounds including transition metals. This work provides the first step toward explaining structural and phase transitions in chemically homologous families of binary crystals. It has made the question of the proper treatment of chemical bonding in crystals susceptible to theoretical analysis, whereas formerly work based on mechanical models (ionic compounds) or quantum mechanical perturbation theory (nearly-free-electron metals) made the same problem appear insoluble. 

The problem of transference from the hydrogen bonds in the crystal to those in a biological process is not different, in principle, from the transference of molecular structural information determined by crystal structure analysis to the interpretation of the mechanism of a chemical reaction. In Chapter 4, we discuss the differences between the geometry of hydrogen bonds in crystals and in the free molecule models that are necessarily used by the theoretical methods. 

The titaniuin oxides are of considerable technological interest, so different theoretical studies of electronic and atomic structure and properties have been performed both for Ti02 [100,323,596,597] and Ti20s [598,599] crystals. We are not aware of the existence of the electronic-structure calculations of TiO. The available publications focus attention primarily on description of the band structure and phase stabihty of titanium oxides and restrict the discussion of the nature of chemical bonding in these compounds to an analysis of Mulhken atomic charges and overlap populations. 

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