The multiplicative product (\delta (x_0 - \left| x \right|)/\left| x \right|^{(n - 2)/2} )(\delta (x_0 + \left| x \right|)/\left| x \right|^{(n - 2)/2} )

In this paper we give a sense to the distributional multiplicative product $(\delta (x_o - \left| x \right|)/\left| x \right|^{(n - 2)/2} )(\delta (x_o + \left| x \right|)/\left| x \right|^{(n - 2)/2} )$ . For the case n=4 we obtain a formula which is used for a perturbative calculation of Green function in quantum field theories. As an application of our product, we show the calculation of the self-energy of the theory of $\lambda \emptyset ^4 $ .     

Electrogenerated chemiluminescence sensor for glutathione with Ru\left( {bpy} \right)_3^{2+ } -doped silica nanoparticles

An electrogenerated chemiluminescence (ECL) sensor for reduced glutathione was developed based on $ \mathrm{Ru}\left( {\mathrm{bpy}} \right)_3^{2+ } $ -doped silica nanoparticles-modified gold electrode (Ru-DSNPs/Au). These uniform Ru-DSNPs (about 58?+?4 nm) were prepared by a water-in-oil microemulsion method and characterized by transmission electron microscope and scanning electron microscope. With such a unique immobilization method, a considerable $ \mathrm{Ru}\left( {\mathrm{bpy}} \right)_3^{2+ } $ was immobilized three dimensionally on the electrode, which could greatly enhance the ECL response and thus result in an increased sensitivity. The ECL analytical performances of this sensor for reduced glutathione based on the quenched ECL emission of $ \mathrm{Ru}\left( {\mathrm{bpy}} \right)_3^{2+ } $ have been investigated in detail. Under the optimum condition, the ECL intensity was linear with the reduced glutathione concentration in the range of 1.0?×?10^?9 to 1.0?×?10^?4?mol?L^?1 (R?=?0.9971). This method has been successfully applied for the determination of reduced glutathione in serum samples with satisfactory results. The as-prepared ECL sensor for the determination of reduced glutathione displayed good sensitivity and stability.     

Free ions Pr IV (4\text{ f }^{2}) and Tm IV (4\text{ f }^{12}): intermediate versus LS coupling scheme

The intermediate and LS-coupling schemes for the free lanthanide ions $\text{ Pr }^{3+}$ Pr 3 + and $\text{ Tm }^{3+}$ Tm 3 + have been compared by the matrix elements of the tensor operator ${{\varvec{U}}}^{({\varvec{k}})}, \text{ k } = 2, 4, 6$ U ( k ) , k = 2 , 4 , 6 . The necessary eigenvectors and eigenvalues have been computed with the aid of four parameters, $\text{ F }_{2}, \text{ F }_{4}, \text{ F }_{6}$ F 2 , F 4 , F 6 , and $\zeta _{4\mathrm{f}}$ ζ 4 f , known from free-ion spectra of the same ions. It has been found that both coupling types for each ion lead to close values of ${\vert }{{\varvec{U}}}^{({\varvec{k}})}{\vert }^{2}$ | U ( k ) | 2 only for transitions from the ground level to certain lower-lying energy levels within the $4\text{ f }^\mathrm{N}$ 4 f N configuration.     

A CASPT2//CASSCF study of the quartet excited state \tilde{a}^{4}A^{\prime\prime} of the HCNN radical

Complete active space self-consistent field and second-order multiconfigurational perturbation theory methods have been performed to investigate the quartet excited state ${\tilde{a}}^{4}{A^{\prime\prime}}$ potential energy surface of HCNN radical. Two located minima with respective cis and trans structures could easily dissociate to CH $({\tilde{a}}^{4}\Sigma^{-})$ and $N_{2} ({\tilde{X}}^{1}\Sigma_{\rm g}^{+})$ products with similar barrier of about 16.0 kcal/mol. In addition, four minimum energy crossing points on a surface of intersection between ${\tilde{a}}^{4}A^{\prime\prime}$ and X ( $X={\tilde{X}}^{2}A^{\prime\prime}$ and ${\tilde{A}}^{2}A^{\prime}$ ) states are located near to the minima. However, the intersystem crossing ${\tilde{a}}^{4}A^{\prime\prime} \rightarrow X$ is weak due to the vanishingly small spin–orbit interactions. It further indicates that the direct dissociation on the ${\tilde{a}}^{4}{A^{\prime\prime}}$ state is more favored. This information combined with the comparison with isoelectronic HCCO provides an indirect support to the recent experimental proposal of photodissociation mechanism of HCNN.     

Analytical and numerical treatment of auxiliary function $${B_{m^\prime m}^j \left(\beta\right)}$$ for finite rotation matrix elements

In this study, a new method evaluate the auxiliary function \({B_{m^\prime m}^j \left( { \beta }\right)}\) which the function appears in the matrix elements \({d_{m^\prime ,m}^j \left( \beta \right)}\) are formulated. Also, the generating functions, Rodrigues’ formula, and orthogonality relationships for the \({B_{m^\prime m}^j \left( { \beta }\right)}\) function are presented. To analyze their formal mathematical structure, \({B_{m^\prime m}^j \left( { \beta } \right)}\) functions are expressed in terms of the Jacobi, Gegenbauer, Legendre, and Chebyshev polynomials. \({B_{m^\prime m}^j \left( { \beta }\right)}\) functions and their linear combinations are calculated numerically for large values of the indices j, m′, m quantum numbers and β angles by using generating function. Finally, evaluating numerical values for them are checked with obtained control expressions and results of Öztekin and Özcan (J Math Chem 44:28, 2008).     

Evaluation of intermolecular interaction energy using one-range addition theorems for {\psi ^\alpha} – ETO and Coulomb-Yukawa like correlated interaction potentials with integer and noninteger indices

Using one-range addition theorems for complete orthonormal sets of ${\psi^\alpha }$ – exponential type orbitals (α = 1, 0, ?1, ?2, . . .) intruduced by the author, the series expansion relations are established for the energy of interaction between molecules which have any number of closed and open shells. The final results are expressed through the linear combination coefficients of molecular orbitals, the parameters of Coulomb-Yukawa like correlated interaction potentials with integer and noninteger indices and the multicenter overlap integrals. The formulas obtained are useful for the study of interaction between atomic-molecular systems when the ${\psi^\alpha}$ – exponential type orbital basis functions in Hartree-Fock-Roothaan and explicitly correlated methods are employed.     

Asymptotics (p\rightarrow \infty ) of L_p-norms of hypergeometric orthogonal polynomials

The determination of the weighted $L_p$ norms of the real orthogonal polynomials of hypergeometric type $\left\{ y_n(x)\right\} $ is not only a very important problem per se in the theory of special functions, but also because of their recent entropic characterization and applications in quantum chemistry, quantum physics and information theory. Indeed, they essentially describe the $p$ th-order Rényi and Tsallis entropies of the numerous quantum systems whose wavefunctions are controlled by these polynomials. Moreover, for different values of $p$ , up to a constant factor, these norms characterize various fundamental and experimentally accessible quantities of many-electron systems. As well, the $L_p$ norms have been used to develop and interpret all energy components in the density-functional theory of the ground-state of atoms and molecules. The asymptotics of these quantities when $n \rightarrow \infty $ and $p>0$ have been recently calculated for Hermite polynomials, although not yet for Laguerre and Jacobi polynomials. Here, we determine the asymptotics ( $p\rightarrow \infty $ , $n$ fixed) of the weighted $L_p$ norms for general orthogonal polynomials in terms of the weight function and the coefficients of the second-order hypergeometric differential equation that they satisfy, and we apply it to the three classical families of real orthogonal polynomials. Moreover we analyse and discuss the monotonicity of this asymptotics, and we carry out a detailed numerical study of it.     

Transport, thermomechanical, and electrode properties of perovskite-type {\left( {{\hbox{L}}{{\hbox{a}}_{0.{75} - x}}{\hbox{S}}{{\hbox{r}}_{0.{25} + x}}} \right)_{0.{95}}}{\hbox{M}}{{\hbox{n}}_{0.{5}}}{\hbox{C}}{{\hbox{r}}_{0.{5} - x}}{\hbox{T}}{{\hbox{i}}_x}{{\hbox{O}}_{{3} - }}_\delta \left( {x = 0 - 0.{5}} \right)

Increasing Sr²⁺ and Ti⁴⁺ concentrations in perovskite-type $ {\left( {{\hbox{L}}{{\hbox{a}}_{0.{75} - x}}{\hbox{S}}{{\hbox{r}}_{0.{25} + x}}} \right)_{0.{95}}}{\hbox{M}}{{\hbox{n}}_{0.{5}}}{\hbox{C}}{{\hbox{r}}_{0.{5} - x}}{\hbox{T}}{{\hbox{i}}_x}{{\hbox{O}}_{{3} - }}_\delta \left( {x = 0 - 0.{5}} \right) $ results in slightly higher thermal and chemical expansion, whereas the total conductivity activation energy tends to decrease. The average thermal expansion coefficients determined by controlled-atmosphere dilatometry vary in the range (10.8?C14.5)?×?10^?6?K^?1 at 373?C1,373?K, being almost independent of the oxygen partial pressure. Variations of the conductivity and Seebeck coefficient, studied in the oxygen pressure range 10^?18?C0.5?atm, suggest that the electronic transport under oxidizing and moderately reducing conditions is dominated by p-type charge carriers and occurs via a small-polaron mechanism. Contrary to the hole concentration changes, the hole mobility decreases with increasing x. The oxygen permeation fluxes through dense ceramic membranes are quite similar for all compositions due to very low level of oxygen nonstoichiometry and are strongly affected by the grain-boundary diffusion and surface exchange kinetics. The porous electrodes applied onto lanthanum gallate-based solid electrolyte exhibit a considerably better electrochemical performance compared to the apatite-type La₁₀Si₅AlO_26.5 electrolyte at atmospheric oxygen pressure, while Sr²⁺ and Ti⁴⁺ additions have no essential influence on the polarization resistance. In H₂-containing gases where the electronic transport in $ {\left( {{\hbox{L}}{{\hbox{a}}_{0.{75} - x}}{\hbox{S}}{{\hbox{r}}_{0.{25} + x}}} \right)_{0.{95}}}{\hbox{M}}{{\hbox{n}}_{0.{5}}}{\hbox{C}}{{\hbox{r}}_{0.{5} - x}}{\hbox{T}}{{\hbox{i}}_x}{{\hbox{O}}_{{3} - }}_\delta $ perovskites becomes low, co-doping deteriorates the anode performance, which can be however improved by infiltrating Ni and $ {\hbox{Ce}}{{\hbox{O}}_{{\rm{2}} - }}_\delta $ v into the porous oxide electrode matrix.     

Preparation and electrochemical properties of {\hbox{N}}{{\hbox{d}}_{{{2} - x}}}{\hbox{S}}{{\hbox{r}}_x}{\hbox{Fe}}{{\hbox{O}}_{{{4} + \delta }}} cathode materials for intermediate-temperature solid oxide fuel cells

Cathodic materials $ {\hbox{N}}{{\hbox{d}}_{{{2} - x}}}{\hbox{S}}{{\hbox{r}}_x}{\hbox{Fe}}{{\hbox{O}}_{{{4} + \delta }}} $ (x?=?0.5, 0.6, 0.8, 1.0) with K₂NiF₄-type structure, for use in intermediate-temperature solid oxide fuel cells (IT-SOFCs), have been prepared by the glycine?Cnitrate process and characterized by XRD, SEM, AC impedance spectroscopy, and DC polarization measurements. The results have shown that no reaction occurs between an $ {\hbox{N}}{{\hbox{d}}_{{{2} - x}}}{\hbox{S}}{{\hbox{r}}_x}{\hbox{Fe}}{{\hbox{O}}_{{{4} + \delta }}} $ electrode and an Sm_0.2Gd_0.8O_1.9 electrolyte at 1,200?°C, and that the electrode forms a good contact with the electrolyte after sintering at 1,000?°C for 2?h. In the series $ {\hbox{N}}{{\hbox{d}}_{{{2} - x}}}{\hbox{S}}{{\hbox{r}}_x}{\hbox{Fe}}{{\hbox{O}}_{{{4} + \delta }}} $ (x?=?0.5, 0.6, 0.8, 1.0), the composition $ {\hbox{N}}{{\hbox{d}}_{{{1}.0}}}{\hbox{S}}{{\hbox{r}}_{{{1}.0}}}{\hbox{Fe}}{{\hbox{O}}_{{{4} + \delta }}} $ shows the lowest polarization resistance and cathodic overpotential, 2.75????cm² at 700?°C and 68?mV at a current density of 24.3?mA?cm^?2 at 700?°C, respectively. It has also been found that the electrochemical properties are remarkably improved the increasing Sr content in the experimental range.     

On the anomaly of the quasiclassical product distributions of the \hbox{OH} +\hbox{CO} \rightarrow\hbox{H} +\hbox{CO}_2 reaction

A grid empowered molecular simulator (GEMS) embodying in a single workflow the ab initio treatment of elementary chemical processes has been extended to four atom reactions. GEMS has been used to carry out a massive quasiclassical investigation for the $\hbox{OH} +\hbox{CO} \rightarrow\hbox{H} +\hbox{CO}_2$ reaction on the most recently proposed potential energy surface. The type of potential energy surface used and the possibility of running the simulations on the grid have allowed us to keep the error of the order of a few percent at all values of the collision energy and to estimate accurately the dependence of the reaction cross section on the collision energy. The accuracy of the calculations has allowed to unequivocally single out the fact that the calculated center-of-mass angular distribution is clearly isotropic and radically differs from the asymmetric forward?Cbackward structure obtained from the experiment. However, when the laboratory frame analogues are compared, the difference almost vanishes.     

Reactions of/n\pi ^{\rlap{--} x} / and/\pi \pi ^{\rlap{--} x} / states of triplet acetophenone with hydroperoxide moleculesstates of triplet acetophenone with hydroperoxide molecules

The photochemical interaction of triplet acetophenone /^TRO/ with α-phenyl ethyl hydroperoxide /HROOH/ has been studied in CCl₄ and CH₃CN. In CCl₄ only the \(/n\pi /^{\rlap{--} x}\) state of^TRO is active abstracting hydrogen from HROOH molecules. In CH₃CN both hydrogen abstraction and homolysis of HROOH take place, that is both \(/n\pi /^{\rlap{--} x}\) and \(/\pi \pi /^{\rlap{--} x}\) states of triplet RO play active role.     

Volumetric and Viscometric Studies in Binary Liquid Mixtures of 2-Methyl-2-propanol with Some Ketones at Different Temperatures

Densities, ??, and viscosities, ??, of binary mixtures of 2-methyl-2-propanol with acetone (AC), ethyl methyl ketone (EMK) and acetophenone (AP), including those of the pure liquids, were measured over the entire composition range at 298.15, 303.15 and 308.15?K. From these experimental data, the excess molar volume $V_{\mathrm{m}}^{\mathrm{E}}$ , deviation in viscosity ????, partial and apparent molar volumes ( $\overline{V}_{\mathrm{m},1}^{\,\circ }$ , $\overline{V}_{\mathrm{m},2}^{\,\circ }$ , $\overline{V}_{\phi ,1}^{\,\circ}$ and $\overline{V}_{\phi,2}^{\,\circ} $ ), and their excess values ( $\overline{V}_{\mathrm{m},1}^{\,\circ \mathrm{E}}$ , $\overline{V}_{\mathrm{m,2}}^{\,\circ \mathrm{ E}}$ , $\overline {V}_{\phi \mathrm{,1}}^{\,\circ \mathrm{ E}}$ and $\overline{V}_{\phi \mathrm{,2}}^{\,\circ \mathrm{ E}}$ ) of the components at infinite dilution were calculated. The interaction between the component molecules follows the order of AP > AC > EMK.     

Infrared Spectroscopy of \hbox {CH}_4/\hbox {N}_{2} and \hbox {C}_{2}\hbox {H}_{m}/\hbox {N}_{2} ({m = 2, 4, 6}) Gas Mixtures in a Dielectric Barrier Discharge

Fourier transform infrared spectroscopy of \(\hbox {CH}_{4}/\hbox {N}_{2}\) and \(\hbox {C}_{2}\hbox {H}_{m}/\hbox {N}_2\) ( \(m = 2, 4, 6\) ) gas mixtures in a medium pressure (300 mbar) dielectric barrier discharge was performed. Consumption of the initial gas and formation of other hydrocarbon and of nitrogen-containing HCN and \(\hbox {NH}_{3}\) molecules was observed. \(\hbox {NH}_{3}\) formation was further confirmed by laser absorption measurements. The experimental result for \(\hbox {NH}_{3}\) is at variance with simulation results.     

Positive ions of SiH4: Jahn-Teller-like distortions for {{\rm SiH}_{4}^{+}} and eventually molecular collapse on further ionisation.

We have recently studied by quantum-chemical calculations some fifteen tetrahedral and octahedral molecules. These seemingly disparate numerical tabulations were, it was demonstrated, pulled together by comparison with model semiclassical scaling laws for (i) nuclear–nuclear repulsion energy at the equilibrium geometry and (ii) total energies likewise. Here, we again appeal to such model scaling predictions, but now for positive ions of SiH₄. We then report Hartree-Fock equilibrium geometries and MP2 corrections. For ${{\rm SiH}_{4}^{+}}$ , we assumed the symmetry to be C_2v , by analogy with ${{\rm CH}_{4}^{+}}$ for which experimental confirmation of this symmetry is available. Larger distortions, but still for C_2v symmetry, are found from our quantum-chemical studies in the case of ${{\rm SiH}_{4}^{2+}}$ . But for ${{\rm SiH}_{4}^{3+}}$ there is a marked tendency to return to a configuration quite close to tetrahedral symmetry. But non-convergence is found for ${{\rm SiH}_{4}^{4+}}$ . Finally relations to the admittedly simplistic semiclassical geometry scaling predictions of Lawes and March are conjectured.     

The effective resonance energy\overline {E_r } as a parameter for the correction of resonance integrals in 1/E1+α epithermal neutron spectra; tabulation of\overline {E_r } -values for 96 isotopesfor 96 isotopes

The present paper shows that the effective resonance energy \(\overline {E_r } \) introduced by RYVES, is a useful parameter for the correction of resonance integrals in non-ideal epithermal neutron spectra, which can be approximated by a 1/E^1+α-flux distribution. The definition, the characteristics and the calculation of \(\overline {E_r } \) are discussed thoroughly. Tabulations are included, giving \(\overline {E_r } \) for 96 isotopes.     

Short- and long-range behavior of the inner and outer densities

We report the short- and long-range behavior of the inner $ \rho_<(r) $ and outer $ \rho_>(r) $ densities, which result from a rigorous partitioning of the spherically averaged one-electron density $ \rho(r) $ in many-electron atoms. It is found that for a small r, $ \rho_<(r) $ has one-electron nature but $ \rho_>(r) $ has two-electron nature. For a large r, however, the opposite is true.     

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers. II. The partial-cycle-index method of the USCI approach combined with the stereoisogram approach

The symmetry-itemized enumeration of quadruplets of stereoisograms is discussed by starting from a tetrahedral skeleton, where the partial-cycle-index (PCI) method of the unit-subduced-cycle-index approach (Fujita in Symmetry and combinatorial enumeration of chemistry. Springer, Berlin, 1991) is combined with the stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004, Tetrahedron 60:11629–11638, 2004). Such a tetrahedral skeleton as contained in the quadruplet of a stereoisogram belongs to an RS-stereoisomeric group denoted by $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , where the four positions of the tetrahedral skeleton accommodate achiral and chiral proligands to give quadruplets belonging to subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ according to the stereoisogram approach. The numbers of quadruplets are calculated as generating functions by applying the PCI method. They are itemized in terms of subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , which are further categorized into five types. Quadruples for stereoisograms of types I–V are ascribed to subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , where their features are discussed in comparison between RS-stereoisomeric groups and point groups.     

Volumetric properties of a stoichiometric mixture of K+ and Br− ions in H/D isotope-substituted aqueous methanol at 278.15–318.15 K: I. The H2O-CH3OH-KBr system

The densities of potassium bromide solutions in aqueous methanol mixtures have been measured with an error of at most ±(1 × 10^?5) g/cm³ for methanol mole fractions x ₂ of 0.06, 0.1, 0.3, or 0.6 and for the potassium bromide mole fractions up to about 2.65 × 10^?2 at 278.15, 288.15, 298.15, 308.15, and 318.15 K. Limiting partial molar volumes $\overline V _3^\infty $ , excess molar volumes $\overline V _3^{E, \infty } $ , and expansibilities $\overline E _{p, 3}^\infty $ have been calculated for a stoichiometric mixture of solvated K⁺ and Br^? ions in the mixed solvents. In the region of x ₂ ≈ 0.25, $\overline E _{p, 3}^\infty $ changes its sign from positive to negative. The $\overline V _3^{E, \infty } $ (x ₂) trend, on the whole, reflects the topologic features of the molecular structure of aqueous methanol associated through H-bonding.     

Synthesis and reactivity of(\mu - H)(\mu _3 - \eta ^2 - H_2 )_4 )Os_3 (CO)_9 : An unusually reactive triosmium cluster

The synthesis of $(\mu - H)(\mu - \eta ^2 - H_2 )_4 )Os_3 (CO)_{10} $ (4) from piperidine and Os₃(CO)₁₀(CH₃CN)₂ and its solid state structure are reported. The room temperature reactions of the decarbonylation product of4, $(\mu - H)(\mu _3 - \eta ^2 - H_2 )_4 )Os_3 (CO)_9 $ (3), with P(C₆H₅)₃, CNCH₃, HCl and H₂ are reported. Overall, the products obtained closely resemble those obtained from the analogous, $(\mu - H)(\mu _3 - \eta ^2 - H_2 )_3 )Os_3 (CO)_9 $ (1). The isomerizations of the phosphine addition products $(\mu - H)(\mu - \eta ^2 - H_2 )_n )Os_3 (CO)_9 P(C_6 H_5 )_3 $ (n = 3,6a;n = 4,5a) have been studied by¹H-NMR techniques and the initial rearrangement was shown to be an intramolecular process. Slower conversion to the complex $(\mu - H)(\mu _3 - \eta ^2 - H_2 )_4 )Os_3 (CO)_8 P(C_6 H_5 )_3 $ (8) was observed and the solid state structure of this product is reported and compared with a related compound containing an ethyl,n-propylμ ₃-imidoyl ligand. Compound4 crystallizes in the triclinic space group Pl (#2) withZ= 2, and unit cell parametersa = 9.294(3) Å,b = 15.758(5) Å,c = 7.406(2) Å,a = 81.10(2)°,β=76.47(2)°,y =74.88(2)°, andV =1013(l) ų. Least-squares refinement of 2677 reflections gave a final discrepancy factor ofR = 0.054 (R _w = 0.066). Compound8 crystallizes in the space group C2/c with unit cell parametersa = 24.818(3) Å,b = 16.389(3) Å,c = 18.111(3) Å,β= 120.94(2)°,V = 6318(4) ų, andZ = 8. Least squares refinement of 3439 reflections gave a final discrepancy factor ofR = 0.039 (R _w =0.047).     

Thermodynamic study of metal corrosion and inhibitor adsorption processes in copper/N-1-naphthylethylenediamine dihydrochloride monomethanolate/nitric acid system: part 2

N-1-Naphthylethylenediamine dihydrochloride monomethanolate (N-NEDHME) was tested as a corrosion inhibitor for copper in 2 M HNO₃ solution using the standard gravimetric technique at 303–343 K. N-NEDHME acts as an inhibitor for copper in an acidic medium. Inhibition efficiency increases with increase in concentration of N-NEDHME but decreases with a rise in temperature. Thermodynamic parameters such as adsorption heat ( $ \Updelta H_{\text{ads}}^\circ $ ), adsorption entropy ( $ \Updelta S_{\text{ads}}^\circ $ ) and adsorption free energy ( $ \Updelta G_{\text{ads}}^\circ $ ) were obtained from experimental data of the temperature studies of the inhibition process at five temperatures ranging from 303 to 343 K. Kinetic parameters activation such as $ E_{a} $ , $ \Updelta H_{\text{a}}^\circ $ , $ \Updelta S_{\text{a}}^\circ $ and pre-exponential factors have been calculated and are discussed. Adsorption of N-NEDHME on the copper surface in 2 M HNO₃ follows the Langmuir isotherm model.