Hydroxide-promoted core conversions of molybdenum-iron-sulfur edge-bridged double cubanes: oxygen-ligated topological PN clusters

The occurrence of a heteroatom X (C, N, or O) in the MoFe7S9X core of the iron-molybdenum cofactor of nitrogenase has encouraged synthetic attempts to prepare high-nuclearity M-Fe-S-X clusters containing such atoms. We have previously shown that reaction of the edge-bridged double cubane [(Tp)2Mo2Fe6S8(PEt3)4] (1) with nucleophiles HQ- affords the clusters [(Tp)2Mo2Fe6S8Q(QH)2](3-) (Q = S, Se) in which HQ- is a terminal ligand and Q(2-) is a mu2-bridging atom in the core. Reactions with OH- used as such or oxygen nucleophiles generated in acetonitrile from (Bu3Sn)2O or Me3SnOH and fluoride were examined. Reaction of 1 with Et4NOH in acetonitrile/water generates [(Tp)2Mo2Fe6S9(OH)2]3- (3), isolated as [(Tp)2Mo2Fe6S9(OH)(OC(=NH)Me)(H2O)](3-) and shown to have the [Mo2Fe6(mu2-S)2(mu3-S)6(mu6-S)] core topology very similar to the P(N) cluster of nitrogenase. The reaction system 1/Et4NOH in acetonitrile/methanol yields the P(N)-type cluster [(Tp)2Mo2Fe6S9(OMe)2(H2O)](3-) (5). The system 1/Me3SnOH/F- affords the oxo-bridged double P(N)-type cluster {[(Tp)2Mo2Fe6S9(mu2-O)]2}5- (7), convertible to the oxidized cluster {[(Tp)2Mo2Fe6S9(mu2-O)]2}4- (6), which is prepared independently from [(Tp)2Mo2Fe6S9F2(H2O)](3-)/(Bu3Sn)2O. In the preparations of 3-5 and 7, hydroxide liberates sulfide from 1 leading to the formation of P(N)-type clusters. Unlike reactions with HQ-, no oxygen atoms are integrated into the core structures of the products. However, the half-dimer composition [Mo2Fe6S9O] relates to the MoFe7S9 constitution of the putative native cluster with X = O. (Tp = hydrotris(pyrazolyl) borate(1-)).     

Spinorial Formulations of Graph Problems

Beginning with an arbitrary finite graph, various spinor spaces are constructed within Clifford algebras of appropriate dimension. Properties of spinors within these spaces then reveal information about the structure of the graph. Spinor polynomials are introduced and the notions of degrees of polynomials and Fock subspace dimensions are tied together with matchings, cliques, independent sets, and cycle covers of arbitrary finite graphs. In particular, matchings, independent sets, cliques, cycle covers, and cycles of arbitrary length are all enumerated by dimensions of spinor subspaces, while sizes of maximal cliques and independent sets are revealed by degrees of spinor polynomials. The spinor adjacency operator is introduced and used to enumerate cycles of arbitrary length and to compute graph circumference and girth.     

Synthesis and hydroamination catalysis with 3-aryl substituted pyrrolyl and dipyrrolylmethane titanium(IV) complexes

Using an iridium-catalyzed borylation/Suzuki-Miyaura coupling sequence several 3-aryl-pyrroles were accessed; in addition, dipyrrolylmethanes incorporating these 3-arylpyrroles were synthesized. Using the monodentate pyrrolyls, Ti(NMe(2))(2)(HNMe(2))(pyr(3,5-CF3))(2) (1) and a 2,4-diarylpyrrolyl complex Ti(NMe(2))(3)(pyr(Ar/Ar')) (2) were prepared and structurally characterized. Titanium species bearing the new dipyrrolylmethane ligands Ti(NMe(2))(2)(NHMe(2))(dpm(3,5-CF3)) (3) and Ti(NMe(2))(2)(NHMe(2))(dpm(F3)) (4) were also generated. Kinetics under pseudo-first order conditions with 3 and 4 showed them to be measurably more active than the parent derivative without the electron-withdrawing aryl groups.     

Development of a formal catalytic asymmetric [4+2] addition of ethyl-2,3-butadienoate with acyclic enones

Allene esters are unique not only as excellent electrophiles but also because of their ability for subsequent reactivity after the initial nucleophilic attack. A mechanistically inspired cyclization using ethyl-2,3-butadienoate and acyclic enones to provide dihydropyrans in excellent yields and enantioselectivity under solvent-free conditions at room temperature is reported.     

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L²(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L²(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L²(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.     

Connected components and evolution of random graphs: an algebraic approach

Questions about a graph’s connected components are answered by studying appropriate powers of a special “adjacency matrix” constructed with entries in a commutative algebra whose generators are idempotent. The approach is then applied to the Erd?s–Rényi model of sequences of random graphs. Developed herein is a method of encoding the relevant information from graph processes into a “second quantization” operator and using tools of quantum probability and infinite-dimensional analysis to derive formulas that reveal the exact values of quantities that otherwise can only be approximated. In particular, the expected size of a maximal connected component, the probability of existence of a component of particular size, and the expected number of spanning trees in a random graph are obtained.     

Crystal structures of copper(II) complexes of 2-formylpyridine substituted thiosemicarbazones; the first example of a coordinated thiosemicarbazone with a thiol function

The crystal structures of two 5-coordinate copper(II) complexes containing neutral, tridentate 2-formylpyridine N(4)-substituted thiosemicarbazones have been determined. 2-Formylpyridine N(4)-cyclohexylthiosemicarbazone, HFo4CHex coordinates via the pyridine nitrogen, imine nitrogen and thione sulfur with two chloro ligands to produce [Cu(HFo4CHex)Cl₂]. Similarly, 2-formylpyridine 3-(4-methylpiperazine)thiosemicarbazone, HFoppz4M, produces [Cu(HFoppz4M)Cl₂] with one major difference; rather than the expected thione form of the thiosemicarbazone moiety, it coordinates as the thiol isomer. Both complexes are close to a square pyramid structure with axial and equatorial chloro ligands and the thiosemicarbazone moieties of both ligands nearly planar. Also included is the crystal structure of N-cyclohexylthiosemicarbazide, CHextsc.