An expression for the ’’correlation energy’’ of a multiconfiguration wave function is developed using perturbation theory. The asymptotic form of this expression for an N-configuration pair natural orbital expansion is Error(N×N)?(Σμ = 1NCμ)2 (−225/4608)N−1. The asymptotic form attributes the dominant variation in multiconfiguration pair correlation errors to an interference effect between low-lying natural orbitals. Three levels of extrapolation based on the asymptotic convergence of pair natural orbital expansions are examined. The first requires separate calculations with 5 and 14 natural orbitals. When applied to the helium atom, for which E(5) = −2.897 484 and E(14) = −2.901 697, the extrapolated value, E = −2.903 724, is accurate to within 0.05% of the error from the 14 natural orbital wave function (i.e., the absolute accuracy is ≲0.000 001 hartree). The second extrapolation requires separate calculations with 5 and 14 pair MCSCF configurations and is accurate to within 2% of the MCSCF (14) error (i.e., the absolute accuracy is ≲0.000 05 hartree) for the helium isoelectronic series. The third extrapolation requires only the 5-configuration MCSCF calculation. This extrapolation is accurate to ∼10% of the MCSCF (5) error (i.e., the absolute accuracy is ∼0.0005 hartree) for the cases examined, including CH2, Ne, He, and H2. This is comparable to the accuracy of an MCSCF calculation including ten times as many natural orbitals (which would require a factor of ∼104 more computing time).
A series of four propylene/ethylene, metallocene-catalyzed random copolymer samples, with ethylene mole fractions ranging from 0.8% to 7.5% and melt crystallization histories of cooling at 1 °C/min, were studied by 13C solid-state NMR techniques. The principal objective of the study was to determine the partitioning of the ethylene “defect” residues within the semicrystalline morphology of these isotactic poly(propylene/ethylene) copolymers. Signals from the crystalline (CR) and the noncrystalline (NC) regions were separated on the basis of contrasting T 1 ρ H behaviors. Four new resonances, three distinct and one strongly overlapping, were identified in the spectrum of the CR regions. The assignment of these new defect resonances to specific carbons at or near the ethylene defect site was made principally on the basis of quantum mechanical chemical shift calculations. These calculations were performed on two methyl-terminated oligomers of about 6.5 monomers in length with a 31 helical backbone conformation, characteristic of the iPP backbone conformation in the CR state. One oligomer was the pure iPP chain, and the other contained one centrally located ethylene repeat unit. Good agreement between the experimental shifts associated with the ethylene defect and the computed shifts supported the assumption that the chain conformation in the CR regions in the vicinity of the ethylene defect remained a 31 helix. This good agreement between shifts was obtained when the computed shifts were not used directly, but used in a difference mode. This mode was based on the computed shift differences for corresponding carbons on the two oligomers where these differences were applied to the experimental shifts of the main iPP peaks with the same chemical identity. The assignment of the defect resonances, along with the loss of chemical shift equivalences seen in solution-state spectra, was also rationalized in the context of γ-gauche and vicinal−gauche interactions as applied to the 31 helical structure. Defect line width differences that parallel the line width differences of the main iPP resonances also aid in assigning the defect resonances to particular types of carbons. Over the range of ethylene concentrations studied herein, the partitioning coefficient, P CR(eth), given by the ratio of the concentration of ethylene residues in the CR region to the sample-average concentration of ethylene residues, is found to be constant, taking a value of 0.42 with a standard uncertainty of 0.03. On the basis of measurements of the NMR crystallinities, this partitioning translates to a fraction of the total ethylene residues in CR regions ranging from 0.24 to 0.30 and an average concentration of ethylenes in the NC region about twice the overall concentration. We also looked for evidence that the ethylene residues become highly concentrated at the CR/NC interface. While we cannot say whether this is happening on the NC side of the interface, since we cannot identify any NC defect resonances, we can claim that a high concentration of et...
We report defect-resonance patterns associated with two kinds of low-concentration defects typically found in metallocene-synthesized isotactic polypropylenes (iPP's). These defects are the simple mrrm stereo defect and the regio 2,1-erythro defect. This work is a critical part of our effort to determine the extent to which various defects, typically found in isotactic polypropylene (iPP) samples, are incorporated into the crystalline regions of this semicrystalline polymer. The relationship between defect concentrations and mechanical (as well as thermal) properties is quite dependent on the extent of incorporation of defects into the crystalline regions. Several melt-crystallized (at a cooling rate of 1 °C/ min) iPP samples, whose concentrations of various stereo and regio defects are known from high-resolution NMR, have been examined in the solid state by 13 C NMR. Using a method based on differences in the rotating-frame proton relaxation times of the crystalline (CR) and the noncrystalline (NC) regions, signals from the CR and the NC regions are separated. The resulting "CR" spectra, pertaining to the CR regions of the iPP, are examined for distinct resonances associated with such defects; relative integrals associated with these resonances are also determined. Definite defect-resonance patterns associated with both the simple mrrm stereo defect and the regio 2,1-erythro defect have been identified. One of our samples, having a rather low molecular weight, contained a substantial amount of the regio 1,3 defect. The corresponding CR spectrum had no sharper resonances that would indicate the presence of 1,3 defects in the CR lattice. Associated with each type of defect, "i", we define a partitioning coefficient, P CR(i), as the ratio of the ith-defect concentration in the CR region to the overall ith-defect concentration. While we cannot, at this point, be absolutely sure about assignments which ultimately dictate the crucial correspondence between defect populations and defect intensities, we can make arguments or assumptions about this correspondence and then suggest P CR values for the stereo and regio defects. On the basis of the arguments and assumptions made herein, the following values are obtained: PCR(stereo: mrrm) ) 0.48 ( 0.06 and PCR(regio: 2,1-erythro) ) 0.28 ( 0.08. In principle, partitioning coefficients might depend on both on the crystallization kinetics and the crystal habit. Many of our samples possessed mixed amounts of R-and γ-crystallites. The few indications we have suggest that there is only a weak dependence, if any, on kinetics or crystal habit. The hypothesis is considered that those defects seen in the CR spectrum are highly concentrated at the CR/NC interface. On the basis of a modeling of the experimental proton polarization, including spin diffusion, it is concluded that the defects are not highly concentrated on the CR side of the interface; at the same time, we have no information about the possibility of defect concentration on the NC side of the interface. Finally, one of our...
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