Crystal orbital method

BAC-MP4 method, description, 344-346 Band theory, crystal orbital method, 149 Be clusters, structures, 24-25 Bifurcation... 

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. 

In Section 2 we briefly summarize the basic mathematical expressions of the LCAO Hartree-Fock crystal orbital method both in its closed-shell and DODS (different orbitals for different spin) forms and describe the difficulties encountered in evaluating lattice sums in configuration space. Various possibilities for calculating optimally localised Wannier functions are also presented. They can be efficiently used in the calculation of excited states and correlation effects discussed in Section 3. 

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... 

Hartree-Fock LCAO Crystal Orbital Method... 

B. Ionic and Covalent Crystals Interaction Schemes Crystal Orbital Methods... 

Selected References Employing Crystal Orbital Methods to Evaluate the... 

Crystal orbital methods have also been used and developed in order to obtain the nonlinear optical responses of stereoregular polymers (Table VI). The rationale is related to the large nonlinear susceptibilities that Ji-conjugated polymers can have. The initial study, carried out by Agrawal et al. [173], employed the Genkin and Mednis [120] formalism within the tight-binding (Hiickel) model and... 

For the analysis of the spectra of trans-(CH)x, valence-force-field calculations have been performed, the force constant being empirically fitted so as to reproduce the observed vibrational frequencies (Inagaki et al., 1975 Schiigerl and Kuzmany, 1981). The energy gradient scheme based on the ab initio crystal orbital method has also been attempted for the study of the vibrational structure of trans- and c -(CH)x (Teramae et al., 1984). The calculated result of the vibrational frequencies has been found to depend upon the quality of the basis set. It has been demonstrated that, e.g., for the 4-31G basis, a uniform scaling of the force constants, multiplying by 0.8, is required to adjust the vibrational frequencies of the trans-(CH), (Table III). 

The calculated band gap values by the crystal orbital methods (Ker-tesz et al., 1978 Yamabe et al., 1979a Suhai, 1980) are rather too large... 

The solution of Eq. (1.1.20) with the hamiltonian of Eq. (1.1.7) and under the Bloch conditions (1.1.22) is conceptually similar to the solution under the LCAO-MO assumption (1.1.12). A major difference is that while for the molecular problem one solution is sufficient, the crystal problem is a function of A and therefore the variational problem must be solved a number of times equal to the number of sampling points in the independent part of A-space (the first Brillouin zone). These computational difficulties limit the applicability of the crystal orbital method to rather small molecules and unit cells, the urea crystal being probably the limit. Moreover, the crystal orbital method cannot be applied with a proper consideration of correlation energy because of computational limitations. 

The electronic structures of poiy(fluoroacetylene) and poly(difluoroacetylene) have been investigated previously using the ab initio Hartree-Fock crystal orbital method with a minimum basis set (42). Only the cis and trans isomers with assumed, planar geometries were studied. The trans isomer was calculated to be more stable in both cases, and the trans compounds were predicted to be better intrinsic semiconductors and more conductive upon reductive doping than trans polyacetylene. However, our results show that head-to-tail poly(fluoroacetylene) prefers the cis structure and that the trans structure for poly(difluoroacetylene) will not be stable. Thus the conclusions reached previously need to be re-evaluated based on our new structural information. Furthermore, as noted above, addition of electrons to these polymers may lead to structural deformations that could significantly change the conductive nature of the materials. 

The self-consistent-field (SCF) ab initio Hartree-Fock crystal orbital method is applied with success to polysulfur nitride (SN)X, chains using non-local exchange and evaluating all integrals over atomic orbitals within 5 atomic neighbours accurately. 

We have applied the a. i. Hartree-Fock crystal orbital method to three nuclear configurations I was an equidistant chain with rg =1.62 K and all bond angles taken as 115° II was a chain taken from Boudelle s structure (4) chain III corresponded to the "Penn" structure (5). 

Bakhshi and coworkers [19-22] then reported studies of the electronic structures of polymers 8-21 in which the ah initio Hartree-Fock crystal orbital method was applied to optimized unit cell geometries obtained from the MNDO—AMI method. It was pointed out that this method largely overestimated all... 

These results were in agreement with a previous calculation done using an ab initio Hartree-Fock crystal orbital method with a 7s/3p minimal basis set with the zW-trans geometry taken from the Extended Huckel calculation [12]. This method allowed complete... 

Kertesz, M., J.Koller and A.Azman. On the Electronic Structure of Polydiacetylenes as Studied by the Ab Initio Crystal Orbital Method, Chem.Phys. 27 (1978)... 

Trans to Gauche, Helical angle dependence of the band structures of parent polysilanes was calculated by the ab initio crystal orbital method . When the screw axis is taken to be coincident with the Cartesian z axis, the pxi and pyJ basis functimis belonging to the j i cell from the reference cell can be obtained by the relationships of the Cartesian Px-) and pyj orbitals... 

This chapter presents some examples of the application of the ab initio crystal-orbital method described in Chapter 1. Though these applications range from the field of plastics (polyethylene and its fluoro derivatives) through highly conducting polymers [polyacetylenes and polydiacetylenes, (SN) c, TCNQ and TTF stacks] to biopolymers (homopolynucleotides and homopolypeptides), they are only illustrative. No attempt has been made to review the numerous other applications performed by the Namur group and by other researchers, as this would increase unduly the size of this book. 

Owing to the central role of DNA in biochemistry and biophysics, the computation of the electronic structure of periodic polymers constructed from nucleotide bases, base pairs, and nucleotides by applying the ab initio Hartree-Fock crystal-orbital method has attracted much... 

It can be seen from Table 2.6 that the physically most important valence and conduction bands in the later ab initio calculations are much broader (0.435-0.789 and 0.245-0.820 eV, respectively) than those obtained by application of different semiempirical crystal-orbital methods. With the simple PPP-CO approximation, the corresponding values for the highest filled bands are 0.218-0.299 eV and those for the conduction band are about 0.109 eV. Energy-band-structure calculations for the base stacks taking into account the effect of the other valence electrons with the aid of the CNDO/2 CO method give again broader bands (valence bandwidths of 0.136-0.490 eV and conduction band-widths of 0.109-0.245 eV), while the MINDO/2 CO results indicate somewhat less-broad bands (valence bandwidths of 0.027-0.299 eV and conduction bandwidths of 0.027-0.163 eV). For futher details on the semiempirical crystal-orbital calculations see also Chapter 3. 

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