The Electronic Structure of Polymers

During the past decade there has been very intense experimental and theoretical research on the physics and chemistry of polymeric materials. This work has led to the discovery of a great number of organic and inorganic crystals built from polymers or molecular stacks which have a quasi-one-dimensional electronic system and exhibit fascinating electric, magnetic, etc. properties. Other polymers have proved to play a fundamental role in the biosciences. 

Owing to recent developments in theoretical and computational methods, the quantum mechanical approach to the polymer electronic structure problem has begun to associate very fruitfully with experimental research in this field. Combination of the methods of molecular quantum theory with the ideas of theoretical solid-state physics has provided a really efficient tool, not only for the interpretation of experimental results, but also for investigation of fine details in the electronic structure which would be only barely accessible in experiments. 

Calculation of the infinite lattice sums remains the most difficult step in ab initio polymer calculations. They can be evaluated in configuration space as well as in momentum space (or the two procedures can be also combined). There is not enough experience accumulated in the literature to decide which approach 

In conclusion we can state that careful handling of lattice sums and efficient use of symmetry makes it now possible to perform a priori studies even for polymers with larger elementary cells. These computations are not horrendous as claimed earlier and this approach is thus able economically to supplement experimental work on these systems. 

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. 

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Sometimes the estimation of the electronic structures of polymer chains necessitates the inclusion of long-range interactions and intermolecular interactions in the chemical shift calculations. To do so, it is necessary to use a sophisticated theoretical method which can take account of the characteristics of polymers. In this context, the tight-binding molecular orbital(TB MO) theory from the field of solid state physics is used, in the same sense in which it is employed in the LCAO approximation in molecular quantum chemistry to describe the electronic structures of infinite polymers with a periodical structure -11,36). In a polymer chain with linearly bonded monomer units, the potential energy if an electron varies periodically along the chain. In such a system, the wave function vj/ (k) for electrons at a position r can be obtained from Bloch s theorem as follows(36,37) ... 

Wilson reviews in detail many-body perturbation theory of molecules, which is one very useful technique for the inclusion of electron correlation in molecular calculations for small molecules. Ladik and Suhai at the other extreme describe the important advances which have recently been made in the study of the electronic structure of polymers, with emphasis on the use of ab initio methods, which have become practicable in recent years following the development of new computational schemes. Finally, March surveys the current status of the density functional approach, which gives an alternative approach to the description of atoms and molecules. 

X-ray photoelectron spectroscopy used to study the electronic structure of polymers through the analysis of their valence band spectra was found to present very interesting potentialities, but also some limits. 

D. R. Armstrong et al. have a paper on polymers The electronic structure of polymers by FSGO . ... 

Such a situation applies to many polymers and, in order to establish the relationship between the NMR chemical shift and the electronic structure of polymers, it is necessary to use a sophisticated theoretical method which takes account of the characteristics of polymers. 

Some methodologies for obtaining structures and the electronic structures of polymers, both in the solution and solid state, involve a combination of the observation and calculation of NMR chemical shifts. This approach has been applied to various polymer systems. Theoretical calculations of NMR chemical shifts for polymer systems have been achieved using two main approaches. One approach is that model compounds, such as the dimer,... 

Some aspects of computational quantum chemistry applied to the analysis of the electronic structure of polymers are reviewed in connection with the timely trends observed in their electrical and optical properties. The paper is organized as follows after an introduction (Section 36.1), the basic theory of the quantum chemical methodologies as applied to periodic chains is summarized (Section 36.2). Several fields of applications are then presented photoelectron spectra (Section 36.3), conducting and semiconducting conjugated polymers (Section 36.4), hnear and non-linear optical properties (Section 36.5) and the role of charge transfer in organic chains (Section 36.6). Possible developments for the near future are also sketched. 

Janos Ladik contributed a chapter entitled The Electronic Structure of Polymers to the 1981 Volume 4 of the Theoretical Chemistry SPR. This again... 

At the age of 23, Felix Bloch published an article called t/ter die Quantenmechanik derElektronen in Kristcdlgittern" in Zeitschriftflir Physik, 52,555 (1928) (only two years after Schrbdinger s historic publication) on the translational symmetry of the wave function. This was also the first application of LCAO expansion. In 1931. Leon Brillouin published a book entitled Quantenstatistik (Springer Verlag, Berlin), in which the author introduced some of the fundamental notions of band theory. The first ab initio calculations for a poljmer were made by Jean-Marie Andre in a paper Self-consistentfield theory for the electronic structure of polymers f publidied in the Journal cf Chemical Physics, 50, 1536 (1969). 

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... 

Duke, C. D., and Ford, W. K., The electronic structure of polymers relation to molecular architecture, Int. J. Quant. Chem. Quant. Chem. Symp., 17, 597-608 (1983). 

In the past few years, the work of Ladik, Andre, Karpfen, Suhai, Kerstesz and others (27) has led the way to a greater under standing of the electronic structure of polymers. Using ab-initio methods mainly, these authors have shown that high quality electronic structure calculations can be done on polymers in the same way as on semiconductors and metals. At the same time, semi-empirial methods have been applied to this problem with success. Duke al. (28) have used CNDO methods, while Bredas et (29) have useT"valence effective Hamiltonian (VEH) methods. 

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