Identification of source terms in wave equation with dynamic boundary conditions

This paper studies an inverse hyperbolic problem for the wave equation with dynamic boundary conditions. It consists of determining some forcing terms from the final overdetermination of the displacement. First, the Fréchet differentiability of the Tikhonov functional is studied, and a gradient formula is obtained via the solution of an associated adjoint problem. Then, the Lipschitz continuity of the gradient is proved. Furthermore, the existence and the uniqueness for the minimization problem are discussed. Finally, some numerical experiments for the reconstruction of an internal wave force are implemented via a conjugate gradient algorithm.     

Very long-distance magnetic coupling in a dicopper(II) metallacyclophane with extended π-conjugated diphenylethyne bridges

Self-assembly of the rigid rodlike ligand N,N'-4,4'-diphenylethynebis(oxamate) (dpeba) and Cu(2+) ions affords a novel dinuclear copper(II) metallacyclophane (nBu(4)N)(4)[Cu(2)(dpeba)(2)]·4MeOH·2Et(2)O (1) featuring a very long intermetallic distance (r = 15.0 ?). Magnetic susceptibility measurements for 1 reveal a moderately weak but nonnegligible intramolecular antiferromagnetic coupling between the two metal centers across the double para-substituted diphenylethynediamidate bridge (J = -3.9 cm(-1); H = -JS(1)S(2), where S(1) = S(2) = S(Cu) = (1)/(2)). Density functional electronic structure calculations on 1 support the occurrence of a spin polarization mechanism.     

Unprecedented stereoselective synthesis of catalytically active chiral Mo3CuS4 clusters

Cluster excision of polymeric {Mo3S7Cl4}n phases with chiral phosphane (+)-1,2-bis[(2R,5R)-2,5-(dimethylphospholan-1-yl)]ethane ((R,R)-Me-BPE) or with its enantiomer ((S,S)-Me-BPE) yields the stereoselective formation of the trinuclear cluster complexes [Mo3S4{(R,R)-Me-BPE}3Cl3]+ ([(P)-1]+) and [Mo3S4{(S,S)-Me-BPE}3Cl3]+ ([(M)-1]+), respectively. These complexes possess an incomplete cuboidal structure with the metal atoms defining an equilateral triangle and one capping and three bridging sulfur atoms. The P and M symbols refer to the rotation of the chlorine atoms around the C3 axis, with the capping sulphur atom pointing towards the viewer. Incorporation of copper into these trinuclear complexes affords heterodimetallic cubane-type compounds of formula [Mo3CuS4{(R,R)-Me-BPE}3Cl4]+ ([(P)-2]+) or [Mo3CuS4{(S,S)-Me-BPE}3Cl4]+ ([(M)-2]+), respectively, for which the chirality of the trinuclear precursor is preserved in the final product. Cationic complexes [(P)-1]+, [(M)-1]+, [(P)-2]+, and [(M)-2]+ combine the chirality of the metal cluster framework with that of the optically active diphosphane ligands. The known racemic [Mo3CuS4(dmpe)3Cl4]+ cluster (dmpe = 1,2-bis(dimethylphosphanyl)ethane) as well as the new enantiomerically pure Mo3CuS4 [(P)-2]+ and [(M)-2]+ complexes are efficient catalysts for the intramolecular cyclopropanation of 1-diazo-5-hexen-2-one (3) and for the intermolecular cyclopropanation of alkenes, such as styrene and 2-phenylpropene, with ethyl diazoacetate. In all cases, the cyclopropanation products were obtained in high yields. The diastereoselectivity in the intermolecular cyclopropanation of the alkenes and the enantioselectivity in the inter- or intramolecular processes are only moderate.     

Mechanism of triplet photosensitized Diels-Alder Reaction between indoles and cyclohexadienes: theoretical support for an adiabatic pathway

Diels-Alder reactions between indoles (InHs) and 1,3-cyclohexadienes (CHDs) were achieved by using aromatic ketones as photosensitizers. For instance, irradiation of deaerated dichloromethane solutions containing benzoylthiophene (BT, 1 mM), indole (20 mM), and phellandrene (40 mM), in the presence of an acylating agent, led to the N-acetylated Diels-Alder cycloadduct in 46% yield (endo:exo ratio of 1.8:1). Energy transfer from the BT triplet to the CHD gave rise to diene dimers as byproducts. Several combinations of CHDs, InHs, and aromatic ketones were tested; the Diels-Alder reaction was found to be a general process, except when the indole nucleus was substituted at position 2 or 7 and when aromatization of the CHD was favored. Theoretical calculations support a stepwise mechanism involving a triplet ternary complex TC(T1), arising from a nearly barrierless reaction between CHD and the 3(BT...HIn) exciplex. All subsequent steps proceed downhill in the triplet excited state, leading to a triplet cycloadduct-sensitizer CA-BT(T1) radical pair. Attempts to detect this species, which is basically an aminyl/BT ketyl radical pair, were performed by laser excitation of a solution containing BT, phellandrene, and indole. The observed transient absorption spectra could be compatible with the added spectra of the expected components of the radical pair.     

Synergistic assembly of hyperbranched polyethylenimine and fatty acids leading to unusual supramolecular nanocapsules

Self-assembly of hyperbranched polyethylenimine (PEI) and fatty acids leads to supramolecular inverted micellar structures that are able to irreversibly transfer water-soluble guest molecules into organic solvents.     

Photocatalysis within hyperbranched polyethers with a benzophenone core

Quenching and product studies have been performed to demonstrate the suitability of hyperbranched polyethers with a tetrafunctionalized benzophenone core as photocatalysts. The triplet photosensitized transformation of an unsaturated diazo compound has been used as the model reaction. The polymer with highest molecular weight led to a similar product distribution even after several catalytic cycles, which evidences its excellent photostability under prolonged irradiation time. We attribute this to the stabilizing effect of the hyperbranched polymer shell.     

Insight into the mechanism of diazocompounds transformation catalyzed by hetero cuboidal clusters [Mo3CuQ4(MeBPE)3X4]+, (Q = S,Se; X = Cl,Br): The catalytically active species

Two enantiomerically pure trinuclear compounds of formula (P)-[Mo₃S₄{(R,R)-Me–BPE}₃Br₃]Br and (P)-[Mo₃Se₄{(R,R)-Me–BPE}₃Cl₃]Cl, (P)-1b.Br and (P)-1c.Cl, respectively, have been synthesized in a good yield and a stereospecific manner by excision of polymeric [Mo₃Q₇X₄]_n (Q = S or Se, X = Cl or Br) phases with (R,R)-Me–BPE{1,2-bis[(2R,5R)-2,5-(dimethylphospholan-1-yl)]ethane}. They have been transformed into chiral hetereo cuboidal compounds [Mo₃S₄{(R,R)-Me–BPE}₃Br₃]PF₆, (P)-2b.PF₆, and [Mo₃Se₄{(R,R)-Me–BPE}₃Cl₃]PF₆, (P)-2c.PF₆, by reaction with copper salts. All these compounds have been characterized by ³¹P NMR, IR, UV–Vis, mass spectrometry, elemental analysis, and chiral dichroism. The catalytic potential of tetranuclear cuboidal compounds has been assessed in the paradigm intermolecular cyclopropanation reaction of styrene with ethyl diazoacetate. Results are compared with those obtained for the analogue [Mo₃S₄{(R,R)-Me–BPE}₃Cl₃]PF₆, (P)-2a.PF₆. The catalytic data demonstrate that the Se derivative (P)-2c.PF₆ is less reactive than the S analogues, but it leads to a similar product distribution as the sulfide analogue (P)-2a.PF₆. By contrast, exchange of chlorine by the bulky bromine gives rise to a catalyst which makes the carbene dimerization more competitive. These data agree with temporal breaking of one of the Cu–Q bonds to generate an active catalytic species.     

Ligand effects on the structures and magnetic properties of tricyanomethanide-containing copper(II) complexes

The preparation, crystal structure and magnetic properties of four heteroleptic copper(II) complexes with the tricyanomethanide (tcm(-)) and the heterocyclic nitrogen donors 3,6-bis(2-pyridyl)pyridazine (dppn), 2,5-bis(2-pyridyl)pyrazine (2,5-dpp), 2,3-bis(2-pyridyl)pyrazine (2,3-dpp) and 2,3-bis(2-pyridyl)quinoxaline (2,3-dpq) are reported, {[Cu(2)(dppn)(OH)(tcm)(2)] x tcm}(n) (1), {[Cu(2,5-dpp)(tcm)] x tcm}(n) (2), {[Cu(2)(2,3-dpp)(2)(tcm)(3)(H(2)O)(0.5)] x tcm x 0.5H(2)O}(n) (3) and [Cu(2,3-dpq)(tcm)(2)](n) (4). 1 has a ladder-like structure with single mu-1,5-tcm ligands forming the sides and a bis-bidentate dppn and a single mu-hydroxo providing the rung. Each copper atom in 1 exhibits a distorted square pyramidal CuN(4)O surrounding: the basal plane is built by the hydroxo-oxygen, a nitrile-nitrogen atom from a tcm group and one pyrazine and a pyridyl nitrogen atoms from the dppn whereas the apical position is filled by a nitrile-nitrogen atom from a symmetry-related tcm ligand. The structures of 2-4 consists of zig-zag (2 and 3)/linear (4) chains of copper(II) ions which are bridged by either bis-bidentate 2,5-dpp (2) and 2,3-dpp (3) molecules or single mu-1,5-tcm (4) groups. The copper atoms in 2 and 4 are five coordinated with distorted trigonal bipyramidal (2) and square pyramidal (4) CuN(5) surroundings. The axial positions in 2 are occupied by two pyridyl-nitrogen atoms from two 2,5-dpp ligands whereas the trigonal plane is built by a nitrile-nitrogen from a terminally bound tcm group and two pyrazine nitrogen atoms from two 2,5-dpp molecules. The basal plane in 4 is defined by a pyridyl and a pyrazine nitrogen atoms from the bidentate 2,3-dpq ligand and two nitrile nitrogen atoms from two tcm groups (one terminal and the other bridging) whereas the apical position is filled by a nitrile nitrogen from another tcm ligand. The crystallographically independent copper atoms in 3 [Cu(1) and Cu(2)] exhibit elongated octahedral geometries being defined by four nitrogen atoms from two 2,3-dpp groups [Cu(1) and Cu(2)] either two terminally bound tcm ligands [Cu(1)] or a water molecule and a monodentate tcm ligand [Cu(2)] in cis positions. Magnetic susceptibility measurements for 1-4 in the temperature range 1.9-295 K reveal the occurrence of strong [J ca.-1000 cm(-1) (1); H = -JS(A) x S(B)] and weak [J = -0.13 (2), -0.67 (3) and -0.18 cm(-1) (4); H = -J Sigma(I)S(i) x S(i+1)] antiferromagnetic interactions in agreement with the different nature of the exchange pathways involved, diazine and single mu-hydroxo (1) and the extended 2,5-dpp (2), 2,3-dpp (3) and single mu-1,5-tcm (4) bridges with copper-copper separations of 3.363(8) (1), 7.111(1) (2), 6.823(1) and 7.056(1) (3) and 7.446(1) A (4).