Application of a new projection technique to the 2D Heisenberg Hamiltonian

A recently developed new form of projection technique is applied to the computation of the ground-state energyE ₀ of the antiferromagnetic Heisenberg Hamiltonian on a square lattice, demonstrating its usufulness. The calculations are done analytically and do neither require the use of time-dependent operators nor Wick's theorem. A value ofE ₀=–2NJ0.3337 is found whereN is the number of sites in good agreement with series expansions.     

Reducing Ga?H and Ga?C Bonds in Close Proximity to Oxidizing Peroxo Groups: Conflicting Properties in Single Molecules

Treatment of [Li(H₂Ga{CH(SiMe₃)₂}₂)] ? 2 OEt₂ ( 1? 2 OEt₂) with two equivalents of tert‐butyl hydrogen peroxide, H‐O‐O‐CMe₃, afforded the organogallium peroxide [({(Me₃Si)₂HC}₂Ga(OH)(OOCMe₃)Li)₂] ( 3 ), which possesses oxidizing peroxo groups in close proximity to reducing Ga? C bonds. The lithium atoms of the dimeric formula units are coordinated by both oxygen atoms of the peroxides and by two hydroxo groups. The cleavage of the Ga? C bond was not observed, even when an excess of H‐O‐O‐CMe₃ was applied. Instead, the adduct [{(Me₃Si)₂HC}₂Ga(OH)(OOCMe₃)₂Li₂(HOOCMe₃)] ( 4 ) was isolated, which has an intact H‐O‐O‐CMe₃ molecule terminally attached to lithium. A similar structural motif was found for the compound [(LiOOCMe₃)₂(HOOCMe₃)₂] ( 5 ). The trihydrido gallanate [Li(H₃Ga? {CH(SiMe₃)₂})] ? OEt₂ ( 2 ) yielded the unique peroxide [({(Me₃Si)₂HC}? Ga(H)(OOCMe₃)₂Li)₂] ( 6 ) under similar conditions that possesses Ga? C and even more reactive Ga? H bonds beside peroxo groups. It decomposed at room temperature by the insertion of oxygen atoms into the Ga? H bonds and the formation of [({(Me₃Si)₂HC}? Ga(OH)(OCMe₃)(OOCMe₃)Li)₂] ( 7 ), which was isolated in a low yield. Further decomposition gave the complete degradation of all peroxo groups with the formation of a relatively complicated Li₄Ga₄O₈ cage ( 8 ).     

The influence of nucleus density on optical properties in nucleated isotactic polypropylene

The effect and efficiency of three nucleating agents, a sorbitol based clarifier, a traditional heterogeneous nucleating agent and poly(vinylcyclohexane) (PVCH) was studied in polypropylene (iPP) homopolymer. The nucleating agents were added to iPP in different amounts; PVCH in 0–200 ppm, while the other two in 0–2000 ppm. Optical and mechanical properties were determined on injection molded plates or bars, respectively. Nucleation efficiency was studied by thermal analysis, while structure was characterized by polarized light (PLM), scanning electron (SEM) and atomic force microscopy (AFM). Nucleus density was calculated using the method of Lamberti, which is based on the kinetic theory of the crystallization developed by Lauritzen and Hoffmann. The results proved that the nucleating agents modify properties in different ways and extent. PVCH is very efficient already at small concentrations and increases the stiffness of iPP considerably more than the other two compounds. On the other hand, the clarifier and the traditional nucleating agent induce better optical properties even at smaller efficiency. The structure developing in the presence of the three nucleating agents is also different. The clarifier forms a network in iPP and induces the formation of a microcrystalline structure according to the former literature data. Microspherulitic structure develops in the presence of the heterogeneous nucleating agent studied, while relatively large supermolecular units form in iPP nucleated by PVCH even under the conditions of injection molding. The calculation of nucleus density by existing models and the comparison of the results to optical properties proved that haze is determined by the size of the supermolecular units of the polymer and this latter depends on nucleus density.