Polymers, periodic band structures

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

While this well defined band structure was originally interpreted as indicating a very narrow molecular-weight distribution, it now seems more probable to the present authors that the chains are folded as they are in other crystalline polymers. The long fold period may well be related to the great stiffness of the fluorocarbon chains. 

For the discussion of the physicochemical properties of conductive conjugated polymers, it is most important to set up the appropriate model and to employ the proper method of calculations. In this article most of the analyses have been based on the one-dimensional tight-binding CO methodology for polymers with periodical unit cells. This approach is useful because it gives not only the band structure but also information reflecting the nature of each atomic orbital in the unit cell. 

Correlation Corrected Energy Band Structures of Different Periodic Polymers... 

This result was based partly on earlier arguments by Blount [30] who discussed how to represent r for infinite, periodic crystals. Since this discussion is of fundamental importance to the present study, we shall reproduce it here with some modifications that take into account the procedures of performing band-structure calculations for a polymer. 

Usually, a band-structure calculation for a polymer that is considered periodic in the z direction is performed by considering a discrete, equidistant set of k points. 

A completely different viewpoint is adopted in calculations of infinite periodic structures (molecular crystals, semiconductors, large polymers). Band structure approaches that focus on the dynamics of electron-hole pairs are then used. " " Band theories may not describe molecular systems with significant disorder and deviations from periodicity, and because they are formulated in momentum (k) space they do not lend themselves very easily to real-space chemical intuition. The connection between the molecular and the band structure pictures is an important theoretical challenge. ... 

The correlated band structure of the one-periodic systems (polymers) was efficiently studied using the AO Laplace-transformed MP2 theory discussed in next section. 

The increase in the number of fluorine atoms is correlated with a shift of about 1 eV per F atom to lower energies for the peak D, while the other peaks do not change appreciably. The spectrum predicted from the calculated band structure for PTFE is in good agreement with experiment, underlining once more that calculations of periodic polymers are quite successful in interpreting ESCA spectra. This agreement will be more comprehensive when allowance is made for the dependence of the ionization cross section of the crystal orbitals on their atomic constituents (intensity calculations), and correlation effects are also taken into account. Furthermore, better-resolved experimental spectra are probably also needed to check the detailed theoretical information obtained for relatively more complex polymers. 

The variation dJ = 0 yields the so-called time-dependent coupled Hartree-Fock equations. When the field does not break the periodic symmetry of the polymer, these equations are already formulated for a polymer. Since, however, the different cases of the effects of stationary magnetic fields on the band structure of a polymer have not yet been programmed, and therefore no calculations are available for this problem, we shall not give here the rather complicated equations obtained for the case of nonstationary magnetic fields. 

CORRELATION CORRECTIONS FOR THE BAND STRUCTURE OF PERIODIC POLYMERS 591... 

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