[40]π Fused and Nonfused Core‐Modified Nonaphyrins: Syntheses and Structural Diversity

The synthesis of fused and nonfused core‐modified 40π nonaphyrins are reported. Spectroscopic and X‐ray structural studies reveal a twisted figure‐eight conformation in the freebase form that is nonaromatic. Structural changes occur, from figure‐eight to open extended conformation, upon protonation, thereby adopting 4nπ Hückel antiaromatic character, which is reflected in spectroscopic and theoretical studies. Such a structural change also induces ring inversions of specific heterocyclic rings by 180°.     

Pd/CaCO3 in liquid poly(ethylene glycol) (PEG): an easy and efficient recycle system for partial reduction of alkynes to cis-olefins under a hydrogen atmosphere

Lindlar’s catalyst (Pd/CaCO₃) in PEG (400) has been found to be the most reusable reaction medium for selective reduction of alkynes to cis-olefins. The catalyst and PEG were recycled five times without loss of activity.     

Supramolecular Assemblies of Sulfur- and Selenium- Containing Expanded Porphyrins Mediated Through Noncovalent Interactions

Abstract

Various supramolecular assemblies based on expanded porphyrins building blocks containing sulfur and/or selenium in the core, formed through multiple non-covalent hydrogen bonding interactions are highlighted. Specifically, modified expanded porphyrins such as 22 π sapphyrins, 26 π rubyrins, and 34 π octaphyrins self assemble in solid state through C–H…O, C–H…N, C–H…S, C–H…Se, C–H…π, and C–H…Cl interactions to form dimeric, oligomeric, and three dimensional networks. Furthermore, the supramolecular networks promoted by trapped solvent molecules such as nitrobenzene and bound anions such as chloride or trifluoroacetate through noncovalent interactions will be discussed.     

On 3-Regular Tripartitions

Abstract In this article,we investigate the arithmetic behavior of the function D_3(n)which counts the number of 3-regular tripartitions of n.For example,we show that forα≥1 and n≥0,■and ■     

Integrability on lightlike lines in six dimensional superspace

The on-shell constraint equations of extended supersymmetric Yang-Mills theories in six dimensions are shown to be integrability conditions for the superspace gauge potential on super-null-lines.     

Fractional recurrence in discrete-time quantum walk

Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum Pólya number can be seen.     

Iterative Arylation of Amino Acids and Aliphatic Amines via δ‐C(sp3)−H Activation: Experimental and Computational Exploration

Directed C?H functionalization has been realized as a complementary tool to the traditional approaches for a straightforward access of non‐proteinogenic amino acids; albeit such a process is restricted mostly up to the γ‐position. In the present work, we demonstrate the diverse (hetero)arylation of amino acids and analogous aliphatic amines selectively at the remote δ‐position by tuning the reactivity controlled by ligands. An organopalladium δ‐C(sp³)?H activated intermediate has been isolated and crystallographically characterized. Mechanistic investigations carried out experimentally in conjunction with computational studies shed light on the difference in the mechanistic picture depending on the substrate structure.     

On overpartition pairs into odd parts modulo powers of 2

In 2012, Lin (Electron. J. Combin. 19(2) (2012) #P17) investigated the 2 and 3-divisibility properties for ${\overline{pp}}_o\left(n\right)$, the number of overpartition pairs into odd parts. Using modular forms, he proved that for a fixed positive integer $k$, ${\overline{pp}}_o\left(n\right)$ is almost always divisible by $2^k$. In this paper, we prove several congruences for ${\overline{pp}}_o\left(n\right)$ modulo higher powers of 2 in an elementary way.     

Oxasmaragdyrin-ferrocene and oxacorrole-ferrocene conjugates: synthesis, structure, and nonlinear optical properties

Ferrocenyl macrocyclic conjugates involving 22 pi oxasmaragdyrins and 18 pi oxacorroles have been synthesized and characterized. The direct covalent linkage of the ferrocenyl moiety to the meso position of the macrocycle is achieved by simple oxidative coupling of appropriate precursors with trifluoroacetic acid as catalyst. The electronic coupling between the ferrocenyl moiety and the macrocyclic pi system is apparent from: a) the red shifts (293-718 cm(-1)) of the Soret and Q-bands in the electronic absorption spectra of ferrocenyl conjugates; b) the shift of oxidation potentials (50-130 mV) of both the ferrocene and the corrole rings to the positive potentials; and c) considerable shortening of the C-C bond which connects the ferrocene and the meso-carbon atom of the macrocycle. The single-crystal X-ray structure of oxasmaragdyrin-ferrocene conjugate 9 reveals the planarity of the 22 pi skeleton with very small deviations of the meso-carbon atoms. The meso-ferrocenyl substituent has a small dihedral angle of 38 degrees, making way for mixing of the molecular orbitals of the ferrocene and the macrocycle. However, the other two meso substituents are almost perpendicular to the mean plane, defined by the three meso carbon atoms. Classical C-H...O and nonclassical C-H...pi interactions lead to a two-dimensional supramolecular network. Ferrocene-smaragdyrin conjugate 9 bonds to a chloride ion in the protonated form and a rhodium(i) ion in the free base form. Nonlinear optical measurements reveal a larger nonlinear refractive index (-5.83 x 10(-8)cm(2)W(-1)) and figure of merit (2.28 x 10(-8)cm(3)W(-1)) for the rhodium smaragdyrin-ferrocene conjugate 19 than for the others, suggesting its possible application in optical devices.