Incidence Category of the Young Lattice,Injections Between Finite Sets,and Koszulity

We study the quadratic quotients of the incidence category of the Young lattice defined by the zero relations corresponding to adding two boxes to the same row,or to the same column,or both.We show that the last quotient corresponds to the Koszul dual of the original incidence category,while the first two quotients are,in a natural way,Koszul duals of each other and hence they are in particular Koszul self-dual.Both of these two quotients are known to be basic representatives in the Morita equivalence class of the category of injections between finite sets.We also present a new,rather direct,argument establishing this Morita equivalence.     

trans‐Cyano(6‐methyl‐1,4,8,11‐tetraazacyclotetradecan‐6‐amine)cobalt(III) bis(perchlorate) hydrate and trans‐hydroxo(6‐methyl‐1,4,8,11‐tetraazacyclotetradecan‐6‐amine)cobalt(III) bis(perchlorate)

The crystal structures of a pair of closely related macrocyclic cyano‐ and hydroxopenta­amine­cobalt(III) complexes, as their perchlorate salts, are reported. Although the two complexes, [Co(CN)(C₁₁H₂₇N₅)](ClO₄)₂·H₂O and [Co(OH)(C₁₁H₂₇N₅)](ClO₄)₂, exhibit similar conformations, significant differences in the Co—N bond lengths arise from the influence of the sixth ligand (cyano as opposed to hydroxo). The ensuing hydrogen‐bonding patterns are also distinctly different. Disorder in the perchlorate anions was clearly resolved and this was rationalized on the basis of distinct hydrogen‐bonding motifs involving the anion O atoms and the N—H and O—H donors.     

Geometric isomers of chloro(6‐methyl‐1,4,8,11‐tetraazacyclotetradecane‐6‐amine)cobalt(III) tetrachlorozincate(II)

The crystal structures of a pair of cis and trans isomers of the macrocyclic chloro­penta­amine title complex, as their tetra­chloro­zincate(II) salts, [CoCl(C₁₁H₂₇N₅)][ZnCl₄], are re­ported. The two distinct isomeric forms lead to significant variations in the Co—N bond lengths and, furthermore, hydrogen bonding between the complex ions is influenced by the folded (cis) or planar (trans) conformations of the coordinated ligand.     

Coordination Polymers of Bipyridyldicarboxylates – a Cobalt Containing 12,3‐net with Potential Reactive Sites

The porous cobalt(II) coordination polymer [Co{2,2′‐bipy‐4,4′(CO₂)₂}(H₂O)₂] ( 1 ) was prepared via a hydrothermal synthesis approach starting from Co(NO₃)₂ · 6 H₂O and 2,2′‐bipyridine‐4,4′‐dicarboxylic acid in a one to one mixture of ethanol and water at 120 °C. The product was characterized by single crystal X‐ray diffraction, TGA and elemental analysis. Space group: P3₁21, unit cell dimensions at –73 °C: a = 11.980(2), c = 8.335(2) Å, R = 0.032, wR² (all data) = 0.048. The structure determination revealed 1 to be a network in which octahedrally coordinated cobalt atoms are arranged to form a (12,3) net.     

A Heterometallic Porphyrin Dimer as a Potential Quantum Gate: Magneto-Structural Correlations and Spin Coherence Properties

In the development of two-qubit quantum gates, precise control over the intramolecular spin-spin interaction between molecular spin units plays a pivotal role. A weak but measurable exchange coupling is especially important for achieving selective spin addressability that allows controlled manipulation of the computational basis states |00⟩ |01⟩ |10⟩ |11⟩ by microwave pulses. Here, we report the synthesis and Electron Paramagnetic Resonance (EPR) study of a heterometallic meso-meso (m-m) singly-linked V^IVO−Cu^II porphyrin dimer. X-band continuous wave EPR measurements in frozen solutions suggest a ferromagnetic exchange coupling of ca. 8 ⋅ 10⁻³ cm⁻¹. This estimation is supported by Density Functional Theory calculations, which also allow disentangling the ferro- and antiferromagnetic contributions to the exchange. Pulsed EPR experiments show that the dimer maintains relaxation times similar to the monometallic Cu^II porphyrins. The addressability of the two individual spins is made possible by the different g -tensors of V^IV and Cu^II-ions, in contrast to homometallic dimers where tilting of the porphyrin planes plays a key role. Therefore, single-spin addressability in the heterometallic dimer can be maintained even with small tilting angles, as expected when deposited on surface, unlocking the full potential of molecular quantum gates for practical applications.     

Phase Transition-Promoted Rapid Photomechanical Motions of Single Crystals of a Triene Coordination Polymer

Molecular crystals with the ability to transform light energy into macroscopic mechanical motions are a promising class of materials with potential applications in actuating and photonic devices. In regard to such materials, coordination polymers that exhibit dynamic photomechanical motion, associated with a phase transition, are unknown. Herein, we report an intriguing photoactive, one-dimensional Zn^II coordination polymer, 1 , derived from 1,3,5-tri-4-pyridyl-1,2-ethenylbenzene and 3,5-difluorobenzoate. Single crystals of 1 under UV light irradiation exhibit rapid shrinking and bending, violent bursting-jumping, splitting, and cracking behavior. Single-crystal X-ray diffraction analysis and ¹H NMR spectroscopy reveal an unusual photoinduced phase transition involving a single-crystal-to-single-crystal [2+2] cycloaddition reaction that results in photomechanical responses. Interestingly, crystals of 1 , which are triclinic with space group , are transformed into a higher symmetry, monoclinic cell with space group C2/c. This process represents a rare example of symmetry enhancement upon photoirradiation. The photomechanical activity is likely due to the sudden release of stress associated with strained molecular geometries and significant solid-state molecular movement arising from cleavage and formation of chemical bonds. A composite membrane fabricated from 1 and polyvinyl alcohol (PVA) also displays interesting photomechanical behavior under UV light illumination, indicating the material's potential as a photoactuator.