Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig Relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.     

Perturbations produced by electrospray ionization mass spectrometry in the speciation of aluminium(III)/1,6‐dimethyl‐4‐hydroxy‐3‐pyridinecarboxylate aqueous solutions

Electrospray ionization mass spectrometry (ESI‐MS) is very often employed to study metal/ligand equilibria in aqueous solution. However, the ionization process can introduce perturbations which affect the speciation results in an unpredictable way. It is necessary to identify these perturbations in order to correctly interpret the ESI‐MS speciation results. Aluminium(III)/1,6‐dimethyl‐4‐hydroxy‐3‐pyridinecarboxylate (DQ716) aqueous solutions at various pH were analysed by ESI‐MS, and speciation results were compared with those obtained by equilibrium techniques. Differences observed were both qualitative and quantitative. The ESI‐MS spectral changes due to different settings of the following instrumental parameters were analyzed: the solution flow rate (F_S), the nebulizer gas flow rate (F_G), the potential applied at the entrance capillary (E_C), and the temperature of the drying gas (T_G). The effects produced by F_S and E_C on the spectra strongly suggest the key role of surface activity in determining the relative fraction of the ions reaching the detector. The experimental effects of F_S and T_G were interpreted considering the presence of at least two reactions in the gas phase and a dimerization occurring in the droplets. These perturbations cannot be generalized because they appear to be chemical system‐related and instrument‐dependent. Therefore, the identification of perturbations is a required task for any metal‐ligand equilibrium study performed by ESI‐MS. Our results indicate that perturbations can be identified by evaluating the effects produced in the spectra by a change of instrumental parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

Synthesis of E-aryl ethenesulfonamides: a simple one-pot, two-step procedure from 1-hydroxy-1-arylalkanes

The unusual reactivity of 1-phenyl-1-ethanesulfonic acid in thionyl chloride was investigated. Mechanistic considerations led us to set up a new and efficient synthesis of E-arylethenesulfonamides starting from 1-hydroxy-1-arylalkanes. The easy availability of the starting materials and the straightforward, one-pot procedure make this process an attractive method for the preparation of these compounds currently largely employed in chemical and pharmaceutical fields.     

Allylstannation: II. A total “cis-preference” in the addition of n-Bu2ClSnCH(CH3)CHCH2 to aldehydes

1-Buten-3-yldi-n-butylchlorotin, formed by redistribution of ($\text{E}\text{Z}$)-2-butenyltri-n-butyltin and Bu₂SnCl₂, reacts readily with neat RCHO (R  C₂H₅, C₂H₅(CH₃)CH, (CH₃)₂CH, (CH₃)₃C and C₆H₅) to give high yields (80–100%) of alcohols of the type RCH(OH)CH₂CHCHCH₃ only in the Z-configuration. This appears to be the first example of total “cis-preference” in the addition of Grignard-like reagents to carbonyl compounds. The results are discussed in terms of steric requirements around the tin centre which is probably five-coordinate in the transition state.     

Development and conformational analysis of a pseudoproline-containing turn mimic

The liquid-phase synthesis and the conformational analysis of a small library of fully protected tetramers containing L-pyroglutamic acid (L-pGlu), (4S,5R)-4-methyl-5-carboxybenzyloxazolidin-2-one (L-Oxd), or (4R,5S)-4-methyl-5-carboxybenzyloxazolidin-2-one (D-Oxd) as residue i + 1 are reported to test the tendency of these oligomers to assume a -hairpin conformation. The most promising molecule is Boc-L-Val-D-Oxd-Gly-L-Ala-OBn, which assumes a preferential -turn conformation in CDCl3, as shown by IR and 1H NMR analysis. These findings have been confirmed by DFT calculations, which provide an interpretation for the available experimental data and agree with the reported observations.     

An NMR and X-ray study of the structure of the azo coupling product of 4-dimethylaminopent-3-en-2-one and benzenediazonium-tetrafluoroborate

4-Dimethylaminopent-3-en-2-one reacts with two molecules of benzenediazonium-tetrafluoroborate to give compound 1. The structure of this compound was determined by means of X-ray analysis of its crystal and 1H, 13C and 15N NMR spectra of its solution in CDCl3. The molecule of this compound contains one azo group and one hydrazone group. The substance exists, both in crystal form and in solutions of concentrations above 0.1 mol l(-1), in the form of a dimer, in which the pair of molecules are bound by two hydrogen bonds N-H...N. On diluting the solution, the dimers decompose, the two forms being in an equilibrium that is rapid on the NMR time scale.     

Regioselective chlorocarbonylation of polybenzyl cores and functionalization using dendritic and organometallic nucleophiles

Regiospecific chlorocarbonylation of the polybenzyl cores PhCH2CH2Ph, C6(CH2CH2Ph)6, 7, and CH(CH2Ph)24-1,2,4,5-C6H2, 8, in the para position of the benzyl groups gives the chlorocarbonyl derivatives 2, 9, and 10, respectively, in good yields. The octachlorocarbonyl derivative 10 reacts with Newkome's aminotripod NH2C(CH2OCH2CH2CN)3 to give the 24-nitrile dendrimer 13 which is characterized by its molecular peak in the MALDI TOF mass spectrum and with (5-aminopentyl)-1-ferrocene to give the octaferrocene complex 14. Reactions of 2, 9, and 10 with sodium methanolate in methanol gives the methyl esters 3, 15, and 16 which are reduced by LiAlH4 to the primary alcohols 4, 17, and 18; reactions of these alcohols with NaI and BF3.Et2O yield the iodomethyl derivatives 5, 19, and 20. The organoiron nucleophile [FeIICp(eta 5-C6Me5CH2)], 1, reacts with 5, 19, and 20 leading to C-C bond formation and recovery of the aromatic structure of the ligand. This reaction with 5 yields a soluble complex, [FeIICp(eta 6-C6Me5CH2CH2C6H4CH2-)]2, 6, in which the two redox groups, separated by 14 carbon atoms, are independent, being reversibly reduced at approximately the same potential in an overall two-electron wave recorded by cyclic voltammetry. The analogous reaction with 19 and 20, however, gave almost insoluble hexa- and octa-iron complexes 21 and 22 with mediocre purities.     

Solution and solid state structure and tautomerism of azo coupled enaminone derivatives of benzoylacetone

The reaction of 4-substituted benzenediazonium tetrafluoroborates with 3-amino-1-phenylbut-2-en-1-one, 4-amino-4-phenylbut-3-en-2-one and their N-aryl derivatives 1a-1g has been used to prepare the respective azo coupling products i.e. compounds 2-5 from enaminone 1a, compounds 6-9 from enaminone 1c, compound 10 from enaminone 1d, compound 11 from enaminone 1e, compounds 12, 13 from enaminone 1f, compounds 14, 15 from enaminone 1b and compound 16 from enaminone 1g. Tautomerism of the azo coupling products prepared has been investigated in CDCl3 solutions by means of 1H, 13C and 15N NMR spectra. Crystal structures of selected products have also been investigated by means of X-ray diffraction.