ABSTRACT:An energy minimization method for seeking stable crystal structure in terms of the nonbonded repulsive energy alone was derived. The variable parameters are molecular positions and their internal coordinates; the unit cell dimensions and the space group are fixed to those obtained by X-ray analysis. By starting from the crystal structure models composed of uniform helical chains for polyisobutylene, poly(ethylene oxide), and isotactic polypropylene, the present method gave energetically stable structures which are very close to those determined by X-ray analyses. Furthermore, a constrained least-squares method was derived, in which the intra-and intermolecular repulsive energies were taken into account in addition to the differences of the observed and calculated structure factors. This constrained least-squares refinement was successfully applied to the crystal structure analysis of isotactic poly(4-methyl-1-pentene).KEY WORDS Energy Minimization / X-Ray Analysis / Crystal Structure / Polyisobutylene / Poly(ethylene oxide) / Isotactic Polypropylene / Isotactic Poly(4-methyl-1-pentene) / Repulsive Energy / In the previous papers of this series, several methods 1 • 2 for setting up molecular models were derived for X-ray structure analyses of polymers, based upon the intramolecular interaction energy calculations. These methods have been applied successfully to the structure analyses of several polymers. 3 • 4 However, the crystal structure models obtained by packing these molecular models in the unit cell are not always satisfactory as the models for carrying out the analysis, and some cases 5 have been found in which the influence of intermolecular interactions on the flexible molecular chains of polymers cannot be neglected. It is, therefore, quite important to seek more suitable crystal structure models by calculating the potential energy due to both the intra-and intermolecular interactions.parameters; the former uses line searches and quadratic interpolations and the latter the Newton-Raphson method.Although there have been papers concerning the energy minimization of three-dimensional crystals of high polymers by McGuire, et al.,6 and Hermans, et al.,7 their methods require a very large number of computations, because of the optimization problem of a large number ofIn the present work, the internal coordinates (bond lengths, bond angles, and internal rotation angles) were used as the parameters and the quadratic potential functions 8 were introduced as the potentials. Thereby we derived a new method, with fewer calculations, in which the energy minimization was carried out by the leastsquares method. Moreover, a constrained leastsquares refinement method was derived, in which the intra-and intermolecular repulsive energies were taken into account in addition to the differences of the observed and calculated structure factors. This constrained least-squares refinement was applied to the crystal structure analysis of isotactic poly(4-methyl-1-pentene).
METHOD AND ASSUMPTION
Potential FunctionsIn th...