The Functional Callan-Symanzik Equation for the Coulomb Gas

A non-perturbative scheme, based on the functional generalization of the Callan-Symanzik equation is developed to treat the Coulomb interaction in an electron gas. The one-particle irreducible vertex functions are shown to satisfy an evolution equation whose initial condition is given by means of the classical action and the final point corresponds to the physical system. This equation is truncated by expanding it in momenta and excitation energies, leaving the electric charge as an arbitrary, not necessarily small parameter. Exact coupled partial differential equations up to first order in the frequencies and excitation energies are derived. The numerical integration of these equations is left to a later stage. Nevertheless, in order to demonstrate the relation with the perturbation expansion the one-loop Lindhard function and screening are reproduced in the independent mode approximation of the evolution equation.     

Vinyl oxidative coupling as a synthetic route to catalytically active monovalent chromium

Reaction of the deprotonated form of cis-{(t-Bu)N(H)P[μ-N(t-Bu)](2)PN(H)(t-Bu)} with CrCl(3)(THF)(3) afforded the trivalent cis-{(t-Bu)NP[μ-N(t-Bu)](2)PN(t-Bu)}[Li (THF)])CrCl(2) (1). Subsequent reaction with 2 equiv of vinyl Grignard (CH(2)=CH)Mg Cl gave the butadiene derivative (cis-{(t-Bu)NP[μ-N(t-Bu)](2)PN(t-Bu)}[Li(THF)])Cr(cis-η(4)-butadiene) (3) formally containing the metal in its monovalent state. The presence of the monovalent state was thereafter confirmed by DFT calculations. The coordination of the butadiene unit appears to be rather robust since reaction with Me(3)P afforded cleavage of the dimeric ligand core but not its displacement. The reaction formed the new butadiene complex [(t-Bu)N-P-N(t-Bu)]Cr(cis-η(4)-butadiene)PMe(3) (4) containing a regular NPN monoanion. In agreement with the presence of monovalent chromium, complexes 3 and 4 act as single-component self-activating catalysts for selective ethylene trimerization and dimerization, respectively.     

Contribution of proton NMR relaxation to the investigation of molecular dynamics in columnar mesophases of discotic and polycatenar molecules

We present in this work a review concerning wide frequency rangeT ₁ proton NMR relaxation studies performed in compounds exhibiting columnar mesophases, namely the Col_ho mesophase in the case of a liquid crystal of discotic molecules and the ø_h mesophase in the case of a liquid crystal of biforked molecules. These NMR relaxation studies were performed combining conventional and fast field cycling NMR techniques in a frequency range between 100 Hz and 300 MHz. The possibility of probing such a large frequency range has provided a way to effectively distinguish the influence, on theT ₁ relaxation profiles, of the different molecular movements observed in this type of mesophases. In addition, we present a comparison between the molecular dynamics in columnar (ø_h) and lamellar (SmC) mesophases exhibited by the same biforked compound.     

Artificial neural network applied to solid state thermal decomposition

A multi-layer neural network is constructed to describe the thermal decomposition of rhodium acetate. Critical analysis of the residual, trained, interpolated and extrapolated errors, with the number of neurons, indicates the efficiency of the present approach. It was possible, within this framework, to improve the A _n model, with a better correlation between the results. A new value of the activation energy, E _a, and frequency factor, Z, are calculated for the decomposition process. Since the neural network is more precise than a particular model, the calculated values for these quantities are believed to be more precise. The computed values are E _a=194.0 kJ mol^-1 and Z=5.23·1016 s^-1. The neural network eliminates the step to decide, among the available models, the one that best fit the data. An agreement up to four significant figures can be achieved even for data not used in the training process, both in the interpolated and extrapolated regions. This method suggests, therefore, an important alternative tool for the experimentalists. The present approach can also be adapted to other systems and to data in two dimensions. This revised version was published online in July 2006 with corrections to the Cover Date.