Polymer aperiodic

Aoki, Y., S. Suhai, and A. Imamura. 1994. A Density Functional Elengation Method for Theoretical Synthesis of Aperiodic Polymers. Inti. J. Quant. Chem. 52, 367. 

Application of Dean s Negative Eigenvalue Theorem to Aperiodic Polymers.—The... 

In many cases of alloys the density-of-state curves of the constituents are really just shifted by a constant value and therefore many calculations in this field employ equation (4.65). Unfortunately, this is not the case for aperiodic polymers, where the DOS curves usually have a quite different shape. This will be shown in the case of the chains of... 

Therefore, for aperiodic polymers equation (4.65) generally cannot be used and one must work with the k- and energy-dependent 2 (k, E) self-energies. 

To determine the level distribution (density of states, DOS) in an aperiodic polymer chain one can apply the negative factor counting (NFC) method /2/. According to this method if we write for the disordered chain a Huckel determinant which is tridiagonal due to the fact that only first neighbors interactions are taken into account... 

Aoki, Y., Imamura, A. Local density of states of aperiodic polymers using the localized orbitals from an ab initio elongation method. J. Chem. Phys. 97, 8432-8440 (1992)... 

The representation of a polymer unit by one orbital is, of course, a rather crude approximation but calculations of this type49-50 may still throw light on the qualitative effects of aperiodicity. There seems to be, however, no difficulty in applying Dean s method in a more sophisticated way, namely substituting the matrix elements a and bt by Fock matrix blocks which could be taken from appropriate cluster calculations.51... 

Impurity and Aperiodicity Effects in Polymers.—The presence of various impurity centres (cations and water in DNA, halogens in polyacetylenes, etc.) contributes basically to the physics of polymeric materials. Many polymers (like proteins or DNA) are, however, by their very nature aperiodic. The inclusion of these effects considerably complicates the electronic structure investigations both from the conceptual and computational points of view. We briefly mentioned earlier the theoretical possibilities of accounting for such effects. Apart from the simplest ones, periodic cluster calculations, virtual crystal approximation, and Dean s method in its simplest form, the application of these theoretical methods [the coherent potential approximation (CPA),103 Dean s method in its SCF form,51 the Hartree-Fock Green s matrix (resolvent) method, etc.] is a tedious work, usually necessitating more computational effort than the periodic calculations... 

According to the results obtained already at 3 percent H spikes and dips occur in the density of states curve of the mixed system. At larger concentrations of impurities due to clustering effect of the impurity, a part of the gaps disappear (16). Finally, it should be mentioned that with the increase of the impurity concentration, as the calculations show, the density of states at the Fermi level increases and therefore we would expect an increase of the transition temperature between the superconducting and normal states. (This experiment, as far as we know, has not been performed yet). We can conclude from this CPA calculation that aperiodicity (disorder) has a rather serious effect on the band structure of polymers. 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

In the final part of the lecture methods to treat the electronic correlation and the aperiodicity of polymers will be outlined. For the latter case CPA results for a (SN) -(SN)... 

In describing the normal modes of a protein, it is instructive to compare them conceptually with those of a simple model of a polymer, such as a chain of atoms, both periodic and aperiodic. In a harmonic periodic chain, the normal modes carry energy without resistance from one end of the ID crystal to the other. On the other hand, the vast majority of normal modes of an aperiodic chain are spatially localized [138]. Protein molecules, which are of course not periodic, can be better characterized as an aperiodic chain of atoms, and most normal modes of proteins are likewise localized in space [111,112,126-128]. If a normal mode a is exponentially localized, then the vibrational amplitude of atoms in mode a decays from the center of excitation, Ro, as... 

In addition to this SCLCP host, MCLCPs have also been used. Li et al. [ 168] dissolved molecules of an oligomeric fragment of po-ly(p-phenylenebenzobisthiazole) 10, which was terminated by opposing nitro and di-methylamino groups, in the polymer itself. The resultant material has a value of 6x10 esu. Stupp et al. [169, 170] carried out a careful analysis of the system of Disperse Red 1 (11) dissolved in a chemically aperiodic nematic polymer containing the following three structural units (12). 

The most striking feature of the X-ray data is that there are several meridional maxima, and that these are not orders of a single repeat distance but are aperiodic and also vary in position depending on the monomer ratio. At present there is no information on the monomer sequence distribution from NMR analysis. (The polymer has... 

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