Crystal orbital approach

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... 

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 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. 

The electronic structure of solids 6.2.1 The crystal orbital approach... 

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 ... 

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. 

Fig. 10. Molecular orbital rationalization of the rates of reaction of hexaquo ions (a) experiment (b) theory and (c) crystal field approach.

Through a short description of the semi-empirical MNDO crystal orbital method we demonstrate in Section 4 how the computational difficulties of the a priori approach can be reduced. While the procedures described in the above Sections all refer to strictly periodic polymers, we present in Section 5 three methods which also can be efficiently applied for the treatment of deviations... 

The introduction of the concept of one-electron crystal orbitals (CO s) considerably reduces difficulties associated with the many-electron nature of the crystal electronic structure problem. The Hartree-Fock (HF) solution represents the best possible description of a many-electron system with a one-determinantal wavefunction built from symmetry-adapted one-electron CO s (Bloch functions). The HF approach is, of course, only a first approximation to the many-particle problem, but it has many advantages both from practical and theoretical points of view ... 

Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs ate called COs. Crystal orbitals must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. 

Finally we should note a valuable approach developed by Pyper and coworkers " for ionic sohds in which HF methods are used first to obtain a set of crystal orbitals, the interactions between which are calculated as a function of distance including a full explicit evaluation of the exchange term (unlike the local density approximation used in the electron gas method). Estimates of the dispersive energy are then added to the resulting interaction energy. This approach is particularly successful for strongly ionic hahdes and oxides. 

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. 

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