Lorenz integrable system moves \`a la Poinsot

A transformation is derived which takes Lorenz integrable system into the well-known Euler equations of a free-torque rigid body with a fixed point, i.e. the famous motion \`a la Poinsot. The proof is based on Lie group analysis applied to two third order ordinary differential equations admitting the same two-dimensional Lie symmetry algebra. Lie's classification of two-dimensional symmetry algebra in the plane is used. If the same transformation is applied to Lorenz system with any value of parameters, then one obtains Euler equations of a rigid body with a fixed point subjected to a torsion depending on time and angular velocity. The numerical solution of this system yields a three-dimensional picture which looks like a "tornado" whose cross-section has a butterfly-shape. Thus, Lorenz's {\em butterfly} has been transformed into a {\em tornado}.Comment: 14 pages, 6 figure

Pump and Probe of Ultrafast Charge Reorganization in Small Peptides: A Computational Study through Sudden Ionizations

The ultrafast migratory dynamics of the nonstationary hole resulting from a sudden ionization of the small tetrapeptides, Trp-(Leu)₃ and Tyr-(Ala)₃, is studied using as input a high level quantum chemistry description of the electronic structure for extended conformers computed for frozen nuclei. The sudden ionization process prepares a localized electronic wavepacket that is a superposition of a few stationary states of the cation that are energetically allowed. The superposition evolves field-free until a second ionization to the dication. The wavelength and polarization of the first ultrashort VUV ionizing pulse can be used to tailor the amplitudes on the states of the cation and the initial localization of the hole. For these molecular chains that extend over 15 Å, the most efficient mechanism for charge migration is sequential, involving coherent transitions between neighbor and next neighbor amino-acid subunits. The migration of the hole is probed by a second sudden ionization leading to a dication peptide. Its time scale is in the range of a few to a dozen of femtoseconds depending on the initial state of the cation built by the ionization process. The computed angular distributions provide a clear signature of the field-free dynamics between the two sudden ionization processes. Our results are consistent with the experimental observation that the charge transfer is activated, meaning that an excess energy above the ionization potential of the cation is required for facile migration of charge

Influence of the Preparation Procedure of Vanadium-Containing SiBEA Zeolites on Their Catalytic Activity in Propene Epoxidation

Two series of V-containing BEA zeolite catalysts, V<i>_x</i>SiBEA­(I) and V<i>_x</i>SiBEA­(II), were prepared by a two-step postsynthesis preparation procedure which consists, in the first step, in the dealumination of TEABEA zeolites by a treatment with nitric acid solution to obtain SiBEA zeolites with a Si/Al atomic ratio of 1000 and then, in the second step, in bringing SiBEA into contact with an aqueous NH₄VO₃ solution with different concentrations at a pH of 2.7. After 3 days, the solids were recuperated from the reaction mixtures: (1) on a sinter funnel and washed several times with distilled water to obtain the V<i>_x</i>SiBEA­(I) series and (2) in a rotating evaporator under vacuum via a membrane pump to obtain the V<i>_x</i>SiBEA­(II) series. The combined use of H₂-temperature-programmed reduction, diffuse reflectance (DR) UV–vis, Fourier transform infrared (FTIR), ⁵¹V magic-angle spinning (MAS) NMR, electron paramagnetic resonance (EPR), and X-ray photoelectron spectroscopy allows determining the nature and environment of vanadium in both series of V-containing SiBEA zeolites after different treatments. The characterization of V<i>_x</i>SiBEA­(I) and V<i>_x</i>SiBEA­(II) series reveals the formation of various forms of vanadium species depending on the V content and conditions applied upon preparation of each series of V-containing SiBEA zeolites. As evidenced by DR UV–vis and ⁵¹V MAS NMR in the V<i>_x</i>SiBEA­(I) series, vanadium was present mainly as mononuclear framework pseudo-tetrahedral V­(V) species. In contrast, in the V<i>_x</i>SiBEA­(II) series, vanadium was present as mononuclear framework pseudo-tetrahedral and polynuclear extra-framework pseudo-octahedral V­(V) species. As shown by EPR, the oxidation state of V species easily changes upon calcinations in oxygen, outgassing under vacuum at 773 K and treatment with hydrogen at a high temperature (873 K). The presence of Brønsted and Lewis acidic centers was evidenced in both V<i>_x</i>SiBEA­(I) and V<i>_x</i>SiBEA­(II) series by FTIR spectroscopy with pyridine used as a probe molecule. The catalytic activity tests in propene epoxidation revealed that the highly dispersed mononuclear framework pseudo-tetrahedral V­(V) species are responsible for high selectivity to propene oxide, whereas polynuclear extra-framework pseudo-octahedral V­(V) species catalyzed mainly total oxidation. The V<i>_x</i>SiBEA­(I) series of catalysts with vanadium, present mainly as mononuclear framework pseudo-tetrahedral V­(V) species, show lower turn over frequency values than the V<i>_x</i>SiBEA­(II) series in which vanadium is present as both mononuclear framework pseudotetrahedral V­(V) species and polynuclear extra-framework pseudo-octahedral V­(V). It thus demonstrates that apart from highly dispersed isolated vanadium species, the availability of vanadium species to reagents also plays an important role in the gas-phase propene epoxidation