Finitely Generated Groups Are Universal

Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as universal as possible, given that they are finitely generated: for every finitely generated structure, there is a finitely generated group which has the same computability-theoretic properties. The same is not true for finitely generated fields. We apply the results of this investigation to quasi Scott sentences

1-MeOH.cif from Arene guest selectivity and pore flexibility in a metal-organic framework with semi-fluorinated channel walls

CIF for 1-MeO

Metal Hydrides Form Halogen Bonds: Measurement of Energetics of Binding

The formation of halogen bonds from iodopentafluorobenzene and 1-iodoperfluorohexane to a series of bis­(η⁵-cyclopentadienyl)­metal hydrides (Cp₂TaH₃, <b>1</b>; Cp₂MH₂, M = Mo, <b>2</b>, M = W, <b>3</b>; Cp₂ReH, <b>4</b>; Cp₂Ta­(H)­CO, <b>5</b>; Cp = η⁵-cyclopentadienyl) is demonstrated by ¹H NMR spectroscopy. Interaction enthalpies and entropies for complex <b>1</b> with C₆F₅I and C₆F₁₃I are reported (Δ<i>H</i>° = −10.9 ± 0.4 and −11.8 ± 0.3 kJ/mol; Δ<i>S</i>° = −38 ± 2 and −34 ± 2 J/(mol·K), respectively) and found to be stronger than those for <b>1</b> with the hydrogen-bond donor indole (Δ<i>H</i>° = −7.3 ± 0.1 kJ/mol, Δ<i>S</i>° = −24 ± 1 J/(mol·K)). For the more reactive complexes <b>2</b>–<b>5</b>, measurements are limited to determination of their low-temperature (212 K) association constants with C₆F₅I as 2.9 ± 0.2, 2.5 ± 0.1, <1.5, and 12.5 ± 0.3 M^–1, respectively

Fe(III) Protoporphyrin IX Encapsulated in a Zinc Metal–Organic Framework Shows Dramatically Enhanced Peroxidatic Activity

Two MOFs, [H₂N­(CH₃)₂]­[Zn₃(TATB)₂­(HCOO)]·HN­(CH₃)₂·DMF·6H₂O (<b>1</b>) and Zn-HKUST-1 (<b>2</b>), were investigated as potential hosts to encapsulate Fe­(III) heme (Fe­(III) protoporphyrin IX = Fe­(III)­PPIX). Methyl orange (MO) adsorption was used as an initial model for substrate uptake. MOF <b>1</b> showed good adsorption of MO (10.3 ± 0.8 mg g^–1) which could undergo <i>in situ</i> protonation upon exposure to aqueous HCl vapor. By contrast, MO uptake by <b>2</b> was much lower (2 ± 1 mg g^–1), and PXRD indicated that structural instability on exposure to water was the likely cause. Two methods for Fe­(III)­PPIX-<b>1</b> preparation were investigated: soaking and encapsulation. Encapsulation was verified by SEM-EDS and showed comparable concentrations of Fe­(III)­PPIX on exposed interior surfaces and on the original surface of fractured crystals. SEM-EDS results were consistent with ICP-OES data on bulk material (1.2 ± 0.1 mass % Fe). PXRD data showed that the framework in <b>1</b> was unchanged after encapsulation of Fe­(III)­PPIX. MO adsorption (5.8 ± 1.2 mg g^–1) by Fe­(III)­PPIX-<b>1</b> confirmed there is space for substrate diffusion into the framework, while the UV–vis spectrum of solubilized crystals confirmed that Fe­(III)­PPIX retained its integrity. A solid-state UV–vis spectrum of Fe­(III)­PPIX-<b>1</b> indicated that Fe­(III)­PPIX was not in a μ-oxo dimeric form. Although single-crystal XRD data did not allow for full refinement of the encapsulated Fe­(III)­PPIX molecule owing to disorder of the metalloporphyrin, the Fe atom and pyrrole N atoms were located, enabling rigid-body modeling of the porphine core. Reaction of 2,2′-azino-bis­(3-ethylbenzothiazoline)-6-sulfonic acid (ABTS) with H₂O₂, catalyzed by Fe­(III)­PPIX-<b>1</b> and -<b>2</b>, showed that Fe­(III)­PPIX-<b>1</b> is significantly more efficient than Fe­(III)­PPIX-<b>2</b> and is superior to solid Fe­(III)­PPIX-Cl. Fe­(III)­PPIX-<b>1</b> was used to catalyze the oxidation of hydroquinone, thymol, benzyl alcohol, and phenyl ethanol by <i>tert</i>-butyl-hydroperoxide with <i>t</i>_1/2 values that increase with increasing substrate molecular volume