Asymptotic behaviour of solutions for second order nonlinear autonomous differential equations

For a class of second order nonlinear autonomous differential equations asymptotic properties of the solutions as are studied. The investigation is done in the phase plane and the phase portrait of the equivalent system is obtained by using the zero-isocline function. Some suffcient conditions are presented which for the case of Liénard equation become also necessary. Received December 21, 1995     

The Lamb-dip spectrum of the J + 1 ← J (J = 0, 1, 3-8) transitions of HCN: The nuclear hyperfine structure due to H, C, and N

The pure rotational J + 1 ← J transitions, with J = 0, 1, 3-8, of H¹³CN have been observed in the millimeter- and submillimeter-wave region using the Lamb-dip technique to resolve the hyperfine structure due to H, ¹³C, and ¹⁴N. The present observations allow us to provide for the first time the spin-rotation constant of ¹³C and the spin-spin interaction constant S₁₂ (between H and ¹³C) as well as to remarkably improve the quadrupole coupling and spin-rotation constants of ¹⁴N. In addition, a good empirical estimation of C_I(H), based on ab initio calculations, has also been provided. Furthermore, our frequencies together with previous data permit to determine the most accurate ground state rotational parameters known up to now.     

Self‐Aggregated Dinuclear Lanthanide(III) Complexes as Potential Bimodal Probes for Magnetic Resonance and Optical Imaging

Homodinuclear lanthanide complexes (Ln=La, Eu, Gd, Tb, Yb and Lu) derived from a bis‐macrocyclic ligand featuring two 2,2′,2′′‐(1,4,7,10‐tetraazacyclododecane‐1,4,7‐triyl)triacetic acid chelating sites linked by a 2,6‐bis(pyrazol‐1‐yl)pyridine spacer (H₂L³) were prepared and characterized. Luminescence lifetime measurements recorded on solutions of the Eu^III and Tb^III complexes indicate the presence of one inner‐sphere water molecule coordinated to each metal ion in these complexes. The overall luminescence quantum yields were determined (?=0.01 for [Eu₂(L³)] and 0.50 for [Tb₂(L³)] in 0.01 M TRIS/HCl, pH 7.4; TRIS=tris(hydroxymethyl)aminomethane), pointing to an effective sensitization of the metal ion by the bispyrazolylpyridyl unit of the ligand, especially with Tb. The nuclear magnetic relaxation dispersion (NMRD) profiles recorded for [Gd₂(L³)] are characteristic of slowly tumbling systems, showing a low‐field plateau and a broad maximum around 30 MHz. This suggests the occurrence of aggregation of the complexes giving rise to slowly rotating species. A similar behavior is observed for the analogous Gd^III complex containing a 4,4′‐dimethyl‐2,2′‐bipyridyl spacer ([Gd₂(L¹)]). The relaxivity of [Gd₂(L³)] recorded at 0.5 T and 298 K (pH 6.9) amounts to 13.7 mM ^?1 s^?1. The formation of aggregates has been confirmed by dynamic light scattering (DLS) experiments, which provided mean particle sizes of 114 and 38 nm for [Gd₂(L¹)] and [Gd₂(L³)], respectively. TEM images of [Gd₂(L³)] indicate the formation of nearly spherical nanosized aggregates with a mean diameter of about 41 nm, together with some nonspherical particles with larger size.     

A Flexible Approach to Strained Sandwich Compounds: Chiral [1]Ferrocenophanes with Boron,Gallium, Silicon,and Tin in Bridging Positions

The enantiomerically pure dibromoferrocene 3 [(S_p,S_p)‐1,1′‐dibromo‐2,2′‐di(isopropyl)ferrocene], equipped with two iPr groups in α positions, was prepared using known “Ugi amine” chemistry. Species 3 was targeted in order to gain access to new [1]ferrocenophanes ([1]FCPs) to be used as monomers for ring‐opening polymerization. The iPr groups on the sandwich unit were introduced to stabilize bridging moieties, as well as to increase solubilities of targeted metallopolymers. The planar chiral dibromide 3 can quantitatively be lithiated at 0 °C [2 equiv nBuLi, hexanes/thf (9:1), 30 min]. Salt‐metathesis reactions with respective element dichloride species gave chiral [1]FCPs with a variety of bridging moieties [ER_x=Ga[2‐(Me₂NCH₂)C₆H₄] ( 4 a ), SiMe₂ ( 4 b ), SntBu₂ ( 4 c ), BNiPr₂ ( 4 d )]. The new [1]FCPs were fully characterized including single‐crystal X‐ray analysis. The stabilizing iPr groups on the Cp rings increase the thermal stabilities of 4 b – d compared to known [1]FCPs, equipped with the same bridging moieties. All three compounds 4 b – d are volatile and could be isolated by vacuum sublimation. Our new approach to [1]FCPs has the potential to overcome many of the existing difficulties in ferrocenophane chemistry, such as limited stability of starting monomers and low solubilities of resulting polyferrocenes.     

Tetraaryl ZnII Porphyrinates Substituted at β‐Pyrrolic Positions as Sensitizers in Dye‐Sensitized Solar Cells: A Comparison with meso‐Disubstituted Push–Pull ZnII Porphyrinates

A facile and fast approach, based on microwave‐enhanced Sonogashira coupling, has been employed to obtain in good yields both mono‐ and, for the first time, disubstituted push–pull Zn^II porphyrinates bearing a variety of ethynylphenyl moieties at the β‐pyrrolic position(s). Furthermore, a comparative experimental, electrochemical, and theoretical investigation has been carried out on these β‐mono‐ or disubstituted Zn^II porphyrinates and meso‐disubstituted push–pull Zn^II porphyrinates. We have obtained evidence that, although the HOMO–LUMO energy gap of the meso‐substituted push–pull dyes is lower, so that charge transfer along the push–pull system therein is easier, the β‐mono‐ or disubstituted push–pull porphyrinic dyes show comparable or better efficiencies when acting as sensitizers in DSSCs. This behavior is apparently not attributable to more intense B and Q bands, but rather to more facile charge injection. This is suggested by the DFT electron distribution in a model of a β‐monosubstituted porphyrinic dye interacting with a TiO₂ surface and by the positive effect of the β substitution on the incident photon‐to‐current conversion efficiency (IPCE) spectra, which show a significant intensity over a broad wavelength range (350–650 nm). In contrast, meso‐substitution produces IPCE spectra with two less intense and well‐separated peaks. The positive effect exerted by a cyanoacrylic acid group attached to the ethynylphenyl substituent has been analyzed by a photophysical and theoretical approach. This provided supporting evidence of a contribution from charge‐transfer transitions to both the B and Q bands, thus producing, through conjugation, excited electrons close to the carboxylic anchoring group. Finally, the straightforward and effective synthetic procedures developed, as well as the efficiencies observed by photoelectrochemical measurements, make the described β‐monosubstituted Zn^II porphyrinates extremely promising sensitizers for use in DSSCs.     

Towards a Comprehensive Hydride Donor Ability Scale

Rates of hydride transfer from several hydride donors to benzhydrylium ions have been measured at 20 °C and used for the determination of empirical nucleophilicity parameters N and s_N according to the linear free energy relationship log k_{20 °C}=s_N(N+E). Comparison of the rate constants of hydride abstraction by tritylium ions with those calculated from the reactivity parameters s_N, N, and E showed fair agreement. Therefore, it was possible to convert the large number of literature data on hydride abstraction by tritylium ions into N and s_N parameters for the corresponding hydride donors, and construct a reactivity scale for hydride donors covering more than 20 orders of magnitude.