Unbounded quasi-integrals

Let be a locally compact Hausdorff space. We define a quasi-measure in , a quasi-integral on , and a quasi-integral on . We show that all quasi-integrals on are bounded, continuity properties of the quasi-integral on , representation of quasi-integrals on in terms of quasi-measures, and unique extension of quasi-integrals on to .

     

Siamese‐Twin Porphyrin: A Pyrazole‐Based Expanded Porphyrin of Persistent Helical Conformation

The 3+3‐type synthesis of a pyrazole‐based expanded porphyrin 22 H₄ , a hexaphyrin analogue named Siamese‐twin porphyrin, and its homobimetallic diamagnetic nickel(II) and paramagnetic copper(II) complexes, 22 Ni₂ and 22 Cu₂ , are described. The structure of the macrocycle composed of four pyrroles and two pyrazoles all linked by single carbon atoms, can be interpreted as two conjoined porphyrin‐like subunits, with the two opposing pyrazoles acting as the fusion points. Variable‐temperature 1D and 2D NMR spectroscopic analyses suggested a conformationally flexible structure for 22 H₄ . NMR and UV/Vis spectroscopic evidence as well as structural parameters proved the macrocycle to be non‐aromatic, though each half of the molecule is fully conjugated. UV/Vis and NMR spectroscopic titrations of the free base macrocycle with acid showed it to be dibasic. In the complexes, each metal ion is coordinated in a square‐planar fashion by a dianionic, porphyrin‐like {N₄} binding pocket. The solid‐state structures of the dication and both metal complexes were elucidated by single‐crystal diffractometry. The conformations of the three structures are all similar to each other and strongly twisted, rendering the molecules chiral. The persistent helical twist in the protonated form of the free base and in both metal complexes permitted resolution of these enantiomeric helimers by HPLC on a chiral phase. The absolute stereostructures of 22 H₆ ²⁺, 22 Ni₂ , and 22 Cu₂ were assigned by a combination of experimental electronic circular dichroism (ECD) investigations and quantum‐chemical ECD calculations. The synthesis of the first member of this long‐sought class of expanded porphyrin‐like macrocycles lays the foundation for the study of the interactions of the metal centers within their bimetallic complexes.