Polymers with Large Unit Cells

CORRELATION IN POLYMERS WITH LARGE UNIT CELLS 

As a first step in this direction, an investigation of the localization properties of the filled and virtual (both a and n) orbitals of the four nucleotide bases has been performed. It has been found that the filled 

The localization was very good (leading to one-center lone pairs and two-center localized orbitals even for % electrons) in both cases if one applies a a-n separation or treats all electrons together. A rather good localization for the virtuals has also been obtained (again, even for the n electrons) with essentially two-center orbitals with a small component at a third and fourth center in some cases.  

As the next step in handling the correlation problem in larger molecules, the coupled-cluster doubles (CCD), also called coupled-pair many-electron theory (CPMET), introduced into the electronic correlation problem by Cizek, was first applied to smaller molecules and then to the nucleotide bases using localized orbitals. 

Further, is the HF ground-state Slater determinant, and F is an excitation operator given by 

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Sec. 5.5 Correlation in Polymers with Large Unit Cells... 

Diffraction on small crystals with large unit cells (e.g. virus crystals or proteins) or small polymer crystals (e.g. Polyethylene). 

With polymers there is the additional problem that the potential of an electric field E, Er is unbounded and this destroys the translational symmetry of a periodic polymer. Because of this difficulty in a large number of calculations various authors have applied different extrapolation methods for the (hyper)polarizabilities starting from oligomers with increasing number of units. Only in a few cases have attempts been made to treat infinite polymers at the tight binding and ab initio Hartree-Fock level. The latter calculations use, however, a formalism which is so complicated that its application to polymers with larger unit cells seems to be prohibitive (for a review see the Introduction of ref. 116). 

The treatment of correlation, even in periodic polymers, presents a formidable and as yet only partially solved problem if the unit cell is large. This problem is examined by initially discussing localization techniques in larger molecules, and then presenting the first results of the application of localized orbitals for correlation calculations using many-body perturbation theory and the coupled-cluster approach. It is also shown how localization of the orbitals within a large unit cell can be applied for correlation calculations if this unit cell is repeated in a periodic way in the polymer. This chapter also deals with the correlation in polymers with smaller unit cells (like polyacetylene, polydiacetylene, and polyethylene) and includes a detailed discussion of the results obtained. Finally, some ideas about possibilities of treating correlation in disordered chains are noted. 

Many polymers have large unit cells, but for many others, the unit cells are too small to allow the BZ to be represented by a single point. Systems with small unit cells can be modeled using a single point in the BZ if two or more fundamental unit cells are combined to form a unit cell that is large enough. In order to equal the sampling density used in conventional work, the size of the unit cell used needs to be at least 20 A, and preferably more. 

Another possibility consists in the application of the perturbation theoretical expressions for the various energy terms, appropriately modified for interacting polymers. The corresponding formalism has been derived also for the case of systems with incommensurable unit cells. However, as noted earlier, this method assumes that the number of unit cells in one chain is a multiple of the unit cells in the other chain. Therefore its application would cause computational problems because of the large number of atoms in the unit cells of the DNA models. 

The cluster method can be applied to most polymers. The main advantage over conventional methods shows up in the calculation of polymers with large fundamental unit cells. In contrast to conventional methods, where the time required rises rapidly with increasing unit cell size, the time required when the cluster method is used is almost independent of fundamental unit cell size. Only when the fundamental unit cell becomes larger than about 20 to 30 A does the time required start to rise, and then the usual n dependency is observed. This means that the method can be applied with equal ease to polyethylene and to more complicated systems such as poly(paraphenylene benzobisthiazole) (Figure 7). 

The many commercially attractive properties of acetal resins are due in large part to the inherent high crystallinity of the base polymers. Values reported for percentage crystallinity (x ray, density) range from 60 to 77%. The lower values are typical of copolymer. Poly oxymethylene most commonly crystallizes in a hexagonal unit cell (9) with the polymer chains in a 9/5 helix (10,11). An orthorhombic unit cell has also been reported (9). The oxyethylene units in copolymers of trioxane and ethylene oxide can be incorporated in the crystal lattice (12). The nominal value of the melting point of homopolymer is 175°C, that of the copolymer is 165°C. Other thermal properties, which depend substantially on the crystallization or melting of the polymer, are Hsted in Table 1. See also reference 13. 

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. 

Vapors of acrylonitrile were adsorbed into the dehydrated forms of different large- and medium-pore zeolites, to saturation levels of 46, 6, and 9 molecules of acrylonitrile per unit cell in dehydrated zeolite NaY, Na-mordenite, and silicalite, respectively. Subsequent reaction with radical initiator (aqueous solution of K2S2O8 and NaHSOs) produced intrazeolite polyacrylonitrile (no polymer was found in silicalite due to size constraints). The intrazeolite polyacrylonitrile could be recovered after dissolution of the host with dilute aqueous HF, and was very similar to bulk polyacrylonitrile. Gel permeation chromatography revealed a peak molecular weight of 19,000 for polyacrylonitrile recovered from the NaY host, and about 1,000 for the polymer from mordenite. 

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