Proof of the Fukui conjecture via resolution of singularities and related methods. I

The present article is the preliminary part of a series devoted to extending the foundation of the Asymptotic Linearity Theorems (ALTs), which prove the Fukui conjecture concerning the additivity problem of the zero-point vibrational energies of hydrocarbons. In this article, we establish a theorem, referred to as the Boundedness Theorem, through which one can easily form a chain of logical implications that reduces a proof of the Fukui conjecture to that of the Piecewise Monotone Lemma (PML). This chain of logical implications serves as a basis throughout this series of articles. The PML, which has been indispensable for demonstrating any version of the ALTs and has required for its proof a mathematical language not generally known to chemists, is directly related to the theory of algebraic curves. Proofs of the original and enhanced versions of the PML are obtainable via resolution of singularities and related methods.Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Repeat Space Theory Applied to Carbon Nanotubes and Related Molecular Networks. I

The present article is the first part of a series devoted to extending the Repeat Space Theory (RST) to apply to carbon nanotubes and related molecular networks. Four key problems are formulated whose affirmative solutions imply the formation of the initial investigative bridge between the research field of nanotubes and that of the additivity and other network problems studied and solved by using the RST. All of these four problems are solved affirmatively by using tools from the RST. The Piecewise Monotone Lemmas (PMLs) are cornerstones of the proof of the Fukui conjecture concerning the additivity problems of hydrocarbons. The solution of the fourth problem gives a generalized analytical formula of the pi-electron energy band curves of nanotube (a, b), with two new complex parameters c and d. These two parameters bring forth a broad class of analytic curves to which the PMLs and associated theoretical devices apply. Based on the above affirmative solutions of the problems, a central theorem in the RST, called the asymptotic linearity theorem (ALT) has been applied to nanotubes and monocyclic polyenes. Analytical formulae derived in this application of the ALT illuminate in a new global context (i) the conductivity of nanotubes and (ii) the aromaticity of monocyclic polyenes; moreover an analytical formula obtained by using the ALT provides a fresh insight into Hückel’s (4n+2) rule. The present article forms a foundation of the forthcoming articles in this series. The present series of articles is closely associated with the series of articles entitled ‘Proof of the Fukui conjecture via resolution of singularities and related methods’ published in the JOMC.     

Proof of the Fukui conjecture via resolution of singularities and related methods. II $$^{\star}$$

The present article is a direct continuation of the first part of this series. We reduce a proof of the Fukui conjecture (concerning the additivity problem of the zero-point vibrational energies of hydrocarbons) to that of a proposition related to the theory of algebraic curves, so that we can focus on the key mechanism of the additivity phenomena. Namely, by establishing what is called the Basic Piecewise Monotone Theorem (BPMT), we reduce a proof of the Fukui conjecture to that of a proposition, called the Local Analyticity Proposition, Version 1 (LAP1), which admits a proof via resolution of singularities. By LAP1, the essential part of the mechanism of the asymptotic linearity phenomena is extracted and is elucidated by using tools from the mathematical theory of algebraic curves, whose language is of vital importance in analyzing the crux of the additivity mechanism. Dedicated to the memory of Prof. Kenichi Fukui (1918–1998).     

Emulsion polymerization of styrene by horseradish peroxidase-mediated initiation

A new horseradish peroxidase (HRP)-catalyzed redox initiating system was successfully applied in the emulsion polymerization of styrene, producing stable polymer colloids and nanospheres. The particle size was about 30–50 nm with a diameter distribution (CV) 14–20% obtained by SEM. The molecular weight, was around 10⁵, and was around 10⁶. All the results were well consistent with the control, potassium persulfate (KPS)-initiating polymerization. The obtained experimental results supported a micelle mechanism similar to that of conventional emulsion polymerization of styrene.     

Proof of the Fukui conjecture via resolution of singularities and related methods. IV

The present article is a direct continuation of the previous part III of this series of articles, which have been devoted to cultivating a new interdisciplinary region between chemistry and mathematics. In the present part IV, we develop two sets of fundamental theoretical tools, using methods from the field of resolution of singularities and analytic curves. These two sets of tools are essential in structurally elucidating the assertion of the Fukui conjecture (concerning the additivity problems) and the crux of the functional asymptotic linearity theorem (functional ALT) that proves the conjecture in a broad context. This conjecture is a vital guideline for a future development of the repeat theory (RST)—the central unifying theory in the First and the Second Generation Fukui Project.     

Proof of the Fukui conjecture via resolution of singularities and related methods. V

The present article is a direct continuation of part IV of this series. The Local Analyticity Proposition (LAP1), which admits a proof via resolution of singularities is a major key to proving the Fukui conjecture via resolution of singularities and related methods. By LAP1, the essential part of the mechanism of the “asymptotic linearity phenomena” is extracted and is elucidated by using tools from the theory of algebraic and analytic curves. Here in the present article, we complete the proof of the LAP1 by using fundamental tools developed in parts III and IV of this series, thus completing the proof of the Fukui conjecture via resolution of singularities and related methods. This series of articles I-V establishes, for the first time, a new linkage between (i) the mathematical field of resolution of singularities and (ii) the chemical field of additivity problems tackled and solved in a unifying manner via the repeat space theory (RST), which is the central theory in the First and Second Generation Fukui Project. A new development called the Matrix Art Program in the Second Generation Fukui Project has also been expounded with a graphical representation of energy band curves of a carbon nanotube.     

Understanding the chemical reactivity of phenylhalocarbene systems: an analysis based on the spin-polarized density functional theory

Global and local indices based on the spin-polarized density functional theory (SP-DFT) have been used to rationalize the philicity power and spin polarization pattern of a family of singlet substituted phenylhalocarbenes, (pYPhXC, Y = –NO₂ , –CN, –CHO, –F, –H, –CH₃ , –OH, –OCH₃ , –NH₂ ; X = –F, –Cl, –Br). The local reactivity may be traced out by the simple condensed-to-atoms model for the SP-DFT Fukui functions, namely and . For the addition of some singlet phenylhalocarbenes on tetramethylethylene a linear correlation among the global and local electrophilicity index , and the observed rate constants were found. This result supports a mechanistic model where the carbene adds to the olefin in a single step that is controlled by the carbene electrophilicity. These results emphasize the usefulness of general SP-DFT philicities in the rationalization of chemical reactivity at initial stages of reactions that could involve both charge transfer and spin polarization processes.     

Binuclear cobalt(II/II) and cobalt(II/III) chelates with O<Subscript>2</Subscript>N<Subscript>2</Subscript> ligands: synthesis,structure and magnetic studies

Abstract  The title complexes and have been synthesized in excellent yields by reacting Co(OAc)₂·4H₂O with H₂L¹ and H₂L², respectively, in acetonitrile solution. Here, [L¹]²⁻ and [L²]²⁻ are the deprotonated forms of N,N-bis(2-hydroxybenzyl)-N′,N′-dimethylethylenediamine and N,N-bis(2-hydroxybenzyl)-2-picolylamine, respectively. The crystal structures of and were determined by x-ray crystallography. In , each cobalt atom has distorted trigonal bipyramid geometry, while in , each cobalt atom has distorted octahedral geometry. Variable temperature magnetic moment measurements show weak antiferromagnetic interaction in . The magnetic characterization for is in agreement with the presence of Co(II) and Co(III) centers. Graphical Abstract  The title complexes and have been synthesized in excellent yields by reacting Co(OAc)₂·4H₂O with dianionic N₂O₂ coordinating ligands. In complex 1, each cobalt atom has distorted trigonal bipyramid geometry, while in complex 2, each cobalt atom has distorted octahedral geometry. Variable temperature magnetic moment measurements show weak antiferromagnetic interaction in complex 1. The magnetic characterization for complex 2 is in agreement with the presence of Co(II) and Co(III) centers. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Mechanism of the catalyzed by cysteine electroreduction of the insoluble in water cobalt(II) salt as a component of carbon paste electrode

The mechanism of the Co(II) catalytic electroreduction of water insoluble CoR₂ salt in the presence of cysteine was developed. CoR₂ = cobalt(II) cyclohexylbutyrate is the component of a carbon paste electrode. Electrode surface consecutive reactions are: (a) fast (equilibrium) reaction of the complex formation, (b) rate-determining reversible reaction of the promoting process of CoR(Ac⁺) complex formation, (c) rate-determining irreversible reaction of the electroactive complex formation with ligand-induced adsorption, and (d) fast irreversible reaction of the electroreduction. Reactions (a,b) connected with CoR₂ dissolution and reactions (c,d) connected with CoR₂ electroreduction are catalyzed by . Regeneration of (reactions “b,d”) and accumulation of atomic Co(0) (reaction “d”) take place. Experimental data [Sugawara et al., Bioelectrochem Bioenergetics 26:469, 1991]: i _a vs E (i _a is anodic peak, E is cathodic accumulation potential), i _a vs , and i _a vs pH have been quantitatively explained.     

Synthesis and magneto-structural correlation of a new maleato bridged copper(II) coordination polymer

A one-dimensional (1-D) copper(II) coordination polymer [Cu(maleate)(2,2′-bipyridyl)] _n ·2H₂O has been synthesised. Single crystal X-ray diffraction study reveals that maleate ion bridges two adjacent copper(II) centres along the chain in a syn–anti fashion. A complete cryomagnetic investigation of the title complex correlates well with the distorted square pyramidal geometry of the central copper(II) ion and bridging nature of the maleate. A τ value of 0.26 indicates the distortion towards tbp coordination allowing the magnetic orbital to acquire some character leading to a weak antiferromagnetic interaction having J = −0.26 cm⁻¹. The complex has also been firmly established from several other instrumental techniques like Fourier transform infrared (FT-IR) and EPR spectroscopies.     

Synthesis and characterization of poly(lactic acid) and glucosamine-formaldehyde/dodecylbenzenesulfonate composite films

The composite films of poly(lactic acid) (PLA) doped with glucosamine(Gluc)-formaldehyde(FA) polymer/sodium dodecylbenzenesulfonate (SDBS) complexes at 1–5 wt% were synthesized to demonstrate striking improvement of their structural and mechanical properties. The polymer complexes were obtained by the hydrothermal polymerization of Gluc and FA at a molar ratio of 1:2 in the presence of SDBS. The atomic ratios of S in to N in (=S/N) in the polymer complexes limitedly range from 0.52 to 0.69, indicating that the complexation develops through the nonstoichiometric reaction between groups of (Gluc-FA) polymer and ones of SDBS and 31–48% of the groups remain unbound. The PLA composite film doped with 1 wt% (Gluc-FA)/SDBS showed the elongation-at-break of as large as 194% compared with 37% for PLA film, together with an appreciable increase of the crystallites size (D ₂₀₀) of PLA from 21.8 to 33.3 nm.     

Photoelectrode characteristics of an organic bilayer in water phase containing a redox molecule

We have recently reported that the organic bilayer of 3,4,9,10-perylenetetracarboxyl-bisbenzimidazole (PTCBI, n-type semiconductor) and 29H,31H-phthalocyanine (H₂Pc, p-type semiconductor), which is a part of a photovoltaic cell, acts as a photoanode in the water phase (Abe et al., ChemPhysChem 5:716, [2004]); in that case, the generation of the photocurrent involving an irreversible thiol oxidation at the H₂Pc/water interface took place to be coupled with hole conduction through the H₂Pc layer, based on the photophysical character of the bilayer. In the present work, the photoelectrode characteristics of the bilayer were investigated in the water phase containing a redox molecule , where the photo-induced oxidation and reduction for the couple were found to take place at the bilayer. The photoanodic current involving the oxidation efficiently occurred at the interface of H₂Pc/water, similar to the previous example. In the view of the voltammograms obtained, it was noted that there are pin-holes in the H₂Pc layer of the bilayer, leading to a cathodic reaction with at the PTCBI surface especially in the dark; that is, the band bending at the PTCBI/water interface can essentially be reduced by applying a negative potential [e.g., < ∼ 0 V (vs Ag/AgCl)] to the PTCBI, when the cathodic reaction may take place through the conduction band of the PTCBI. Moreover, under that applied potential condition of irradiation, the photogenerated electron carrier part can move to the PTCBI surface, thus enhancing the reduction of .     

Preparation and conductivity of tungstovanadogermanic heteropoly acid polyethylene glycol (PEG) hybrid material

In this work, the polyethylene glycol (PEG) hybrid material composited with tungstovanadogermanic heteropoly acid was prepared. Infrared (IR) spectrum revealed that the Keggin structure characteristic of the anion was present in PEG hybrid material. At room temperature (20°C), the conductivity of the sample is 4.07×10⁻³ S cm⁻¹. The results indicated that it is a new kind of excellent high-proton conductor. According to the experimental results, we proposed a possible mechanism of the proton conduction of the hybrid materials.     

Kinetics and Equilibria for Complex Formation between Palladium(II) and Chloroacetate

Kinetics and equilibria for the formation of a 1:1 complex between palladium(II) and chloroacetate were studied by spectrophotometric measurements in 1.00 mol HClO₄ at 298.2 K. The equilibrium constant, K, of the reaction

Recent developments in Ruddlesden–Popper nickelate systems for solid oxide fuel cell cathodes

Motivated by recent work on the Ruddlesden–Popper material, which was shown to be a superior oxide-ion conductor than conventional solid-oxide fuel cell cathode perovskite materials, we undertook A- and B-site doping studies of the Ruddlesden–Popper nickelate series in an attempt to identify other candidates for cathode application. In this paper, we summarize our most significant results for the and systems and more recently, the higher-order Ruddlesden–Popper phases La _n+1Ni _n O_3n+1 (n=2 and 3), which show greater promise as cathode materials than the n=1 compositions.     

Interaction of a finite quantum system with an infinite quantum system that contains a single one-parameter eigenvalue band

Interaction of a finite quantum system that contains ρ eigenvalues and eigenstates with an infinite quantum system that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system is developed. This system contains embedded eigenstates with continuous eigenvalues , and, in addition, it may contain isolated eigenstates with discrete eigenvalues . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system that interacts with the system can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however strong the interaction between quantum systems and . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of a two-dimensional system that interacts with the infinite system that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied, however strong the interaction between those two systems. This provides a strong verification of the suggested method.     

Molecular iodine selective membrane for iodate determination in salt samples: chemical amplification and preconcentration

A molecular iodine selective membrane has been used for preconcentration of I₂ generated in situ by iodometric reaction of with excess I⁻ in acidic medium (pH 1–2). This iodometric reaction amplifies the iodine content six times resulting in enhancement of analytical response ranging from three times for molecular methods to six times for elemental methods. The chemical conditions of this iodometric reaction were optimized for quantitative generation and subsequent sorption of I₂ in the membrane samples (96 ± 3%). The homogeneous transparent membrane was prepared by immobilizing I₂-complexing polyvinylpyrrolidone (PVP) in the plasticized cellulose triacetate matrix. Four different analytical methods were examined for quantitative determination of in iodized salt samples by preconcentrating it as I₂ in the membrane matrix. These methods were: (1) spectrophotometry of the PVP-I₂ complex formed in the membrane matrix, (2) a radiotracer method using I⁻ tagged with ¹³¹I radiotracer, (3) instrumental neutron activation analysis (INAA), and (4) energy-dispersive X-ray fluorescence (EDXRF) analysis. The contents thus determined in the iodized salt samples by the membrane-based radiotracer method were compared with the total iodine determined in salt samples by epithermal instrumental neutron activation analysis (EINAA). The membrane-based method for iodate determination in salt samples has advantages over conventional analytical methods, for example preconcentration and chemical amplification, and is free from interference from anions. Figure A molecular iodine selective membrane was used for the quantitative preconcentration of I₂ generated in situ by iodometric reaction of with excess I^aˆ’ in acidic medium, which amplifies iodine content six times     

Luminescence Behavior of Polynuclear Alkynylcopper(I) Phosphines

A series of soluble trinuclear and tetranuclear copper(I) complexes containing ₃-^l acetylides , and have been synthesized and shown to exhibit rich photoluminescent behavior at room temperature. The electrochemistry of the trinuclear Cu(I) acetylide complexes and the excited-state redox properties of have been investigated. The X-ray crystal structures of and have been determined.     

The First Structurally Characterized Mn(III) Substituted Sandwich-type Polyoxotungstates

The new polyoxotungstates H₂O (1), · 28H₂O (2) and H₂O (3) were synthesized in aqueous solution and characterized by IR and Raman spectroscopy, energy dispersive X-ray fluorescence and single-crystal X-ray analysis. The anions in 1 and 2 are the first structurally characterized sandwich-type polyoxoanions which contain trivalent manganese atoms. The manganese atoms are coordinated by four oxygen atoms of two Keggin fragments and one water molecule, forming a square pyramid. The manganese(II) containing anions in 3 are linked via Mn–O–W-bonds, forming a two-dimensional network.Dedicated to Prof. M.T. Pope on the occasion of his retirement.