The crystal orbital approach

2 The electronic structure of solids 6.2.1 The crystal orbital approach 

So far, no specification of the actual form of Bloch functions in terms of atomic orbitals has been given. Their definition, equation 6.7, forces them to be composed of a product of a plane wave and a function, S7(r, k), which has the same translational periodicity of the lattice  

It can easily be verified [4] that the above equation satisfies the Bloch condition 6.7. With the Schrodinger problem written in k-dependent form, the solutions are sought in the form of a linear combination of Bloch functions 6.10  

R being any lattice translation, and xjR a translated AO, the above equation is nothing else but the replacement of a single AO in the Ref-cell with a sum over all translation-ally equivalent AOs in the extended crystal. Equations 6.12 and 6.13 define the crystal orbitals. Substitution of 6.13 into 6.10 yields the final form of the Bloch function in the crystal orbital approach  

The solution of equation 6.11 with the hamiltonian 3.39 and under the conditions 6.12-6.14 is conceptually similar to the solution of equation 3.4 under the LCAO assumption 3.46, although (not unexpectedly) a number of computational complications arise [5]. Averaging over k-space and transformations between real and reciprocal space have to be painstakingly carried out. A major difference is that while for the molecular problem one solution of the Fock equations is sufficient, for the crystal problem the periodicized Fock equations are a function of k and therefore the Abs X Abs variational problem must be solved a number of times equal to the number of sampling points within the first Brillouin zone. At the time of writing, these computational difificulties limit the applicability of the crystal orbital method to rather small molecules and unit cells [6]. 

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The crystal orbital approach (see ref. 94 for a review of the recent computational developments in this field) has dominated the electronic structure calculations on polymers for several years. However, the recently published reports on the finite-cluster calculations reveal that the latter methodology has several definite advantages over the traditional approach. Let P(N) be an extensive property of a finite cluster X-(-A-)j -Y, where N is the number of repeating units denoted by A, while X and Y stand for terminal groups. The corresponding intensive properties, p(N) = P(N)/N, are known only for integer values of N. However, provided the polymer in question is not metallic, P(v) can be approximated by a smooth function p(v) of v = 1/N, which in turn can be extrapolated to v = 0 yielding the property of the bulk polymer. 

In Chapter 9, we considered a simple picture of metallic bonding, the electron-sea model The molecular orbital approach leads to a refinement of this model known as band theory. Here, a crystal of a metal is considered to be one huge molecule. Valence electrons of the metal are fed into delocalized molecular orbitals, formed in the usual way from atomic... 

The formalism to incorporate translational symmetry into the usual Hartree-Fock approach, the crystal orbital technique, is not new at all 74,75). Reviews of recent devel-opements and applications of the Hartree-Fock crystal orbital method may be found in refs. 76 79). However, only few investigations on the evaluation of equilibrium geometries and other properties derived from computed potential surfaces of one-dimensional infinite crystals or polymers have been reported. 

Karpfen" was able to consider an infinitely extended linear chain of HCN molecules using an ab initio crystal-orbital approach, with basis sets ranging from STO-3G to [641/41], albeit without any consideration of electron correlation. Polynomials of second and third order were fit to the results, computed for a grid of values of r(CH), r(CN), and... 

The chief drawbacks to the crystal field approach are in its concept of the repulsion of orbitals by the ligands and its lack of any explanation for bonding in coordination complexes. As we have seen in all our discussions of molecular orbitals, any interaction between orbitals leads to both higher and lower energy molecular orbitals. The purely electrostatic approach does not allow for the lower (bonding) molecular orbitals, and thus fails to provide a complete picture of the electronic structure. 

In tune with the above introductory remarks, we have arranged this review in the following way Section II deals with the oriented gas model that employs simple local field factors to relate the microscopic to the macroscopic nonlinear optical responses. The supermolecule and cluster methods are presented in Section III as a means of incorporating the various types of specific interactions between the entities forming the crystals. The field-induced and permanent mutual (hyper)polarization of the different entities then account for the differences between the macroscopic and local fields as well as for part of the effects of the surroundings. Other methods for their inclusion into the nonlinear susceptibility calculations are reviewed in Section IV. In Section V, the specifics of successive generations of crystal orbital approaches for determining the nonlinear responses of periodic infinite systems are presented. Finally,... 

The basic theoretical treatment devolves largely on the crystal-field approach, although as will be seen shortly this can also approximate to a molecular-orbital method. The effect of the environment on the energies of the 3[Pg.100]

We will first describe the common structures of crystals and then consider then-bonding according to the molecular orbital approach. Finally, we will describe some of the thermodynamic and electronic properties of these materials and their uses. 

Some early applications of the method were made by Japanese workers [15], but the 1950s were the heyday of crystal- and ligand-field theory [16]. It was not until the early 1960s that the molecular orbital approach for transition-metal complexes gained momentum. This interest was driven by a combination of spectroscopists ... 

TABLE 10.2. Theoretical Values of the Longitudinal Elastic Modulus of Polyethylene Obtained by Various Methods Using the Molecular Orbital (MO) Approach to Clusters or the Crystal Orbital (CO) Approach to Infinite Periodic Polymers ... 

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