Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: I. Time-independent case

A new method for the exact solution of the interaction of an isolated state with an infinite dimensional quantum system §_∞ ^b containing several one-parameter eigenvalue bands is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction between the state and the system . It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the formalism of the perturbation expansion method. In particular, due to the interaction with the system , eigenvalue E of the state shifts to a new position. In addition, if this eigenvalue is embedded inside the range of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much more complicated behavior     

Volumetric and Viscosimetric Measurements for Methanol + CH 3 –O–(CH 2 CH 2 O) n –CH 3 ( n = 2, 3, 4) Mixtures at (293.15–303.15) K and Atmospheric Pressure: Application of the ERAS Model

Densities, $$\rho$$, and kinematic viscosities, $$\nu$$, have been determined at atmospheric pressure and at 293.15–303.15 K for binary mixtures formed by methanol and one linear polyether of the type CH3–O–(CH2CH2O)n–CH3 (n = 2, 3, 4). Measurements on $$\rho$$ and $$\nu$$ were carried out, respectively, using an Anton Paar DMA 602 vibrating-tube densimeter and an Ubbelohde viscosimeter. The $$\rho$$ values were used to compute excess molar volumes, $$V_{{\text{m}}}^{{\text{E}}}$$, and, together with the $$\nu$$ results, dynamic viscosities ($$\eta$$). Deviations from linear dependence on mole fraction for viscosity, $$\Delta \eta$$, are also provided. Different semi-empirical equations have been employed to correlate viscosity data. Particularly, the equations used are the: Grunberg–Nissan, Hind, Frenkel, Katti–Chaudhri, McAllister and Heric. Calculations show that better results are obtained from the Hind equation. The $$V_{{\text{m}}}^{{\text{E}}}$$ values are large and negative and contrast with the positive excess molar enthalpies, $$H_{{\text{m}}}^{{\text{E}}}$$, available in the literature, for these systems. This indicates that structural effects are dominant. The $$\Delta \eta$$ results are positive and correlate well with the difference in volumes of the mixture compounds, confirming the importance of structural effects. The temperature dependences of $$\eta$$ and of the molar volume have been used to calculate enthalpies, entropies and Gibbs energies, $$\Delta G^{*}$$, of viscous flow. It is demonstrated that $$\Delta G^{*}$$ is essentially determined by enthalpic effects. Methanol + CH3–O–(CH2CH2O)n–CH3 mixtures have been treated in the framework of the ERAS model. Results for $$H_{{\text{m}}}^{{\text{E}}}$$ are acceptable, while the composition dependence of the $$V_{{\text{m}}}^{{\text{E}}}$$ curves is poorly represented. This has been ascribed to the existence of strong dipolar and structural effects in the present solutions.     

Cluster Oxalate Complexes [W3(μ3-Q)(μ2-Q)3 (C2O4)3(H2O)3]2? (Q=S, Se): Mechanochemical Synthesis and Crystal Structure

Mechanochemical reaction of cluster coordination polymers (Q=S, Se) with solid leads to the cluster core excision with the formation of anionic complexes . Extraction of the reaction mixture with water followed by crystallization gives crystalline (main product) and (1) (minor product). In the case of the Se cluster, the complex could not be isolated, and the treatment of the aqueous extract with PPh₃ gave (2) in a low yield. Alternatively, it was obtained from and in high yield. Both 1 and 2 were characterized by X-ray structure analysis. Dedicated to Academician I. I. Moiseev on the occasion of his 75th birthday and in recognition of his outstanding contribution to cluster chemistry.     

High-rate Deposition of Polycrystalline Diamond Film Using Time-series Exposure of Modulated/Non-modulated Induction Thermal Plasmas at Different Flow Rates of Carbon Source Gas

Plasma Chemistry and Plasma Processing - In this paper $$\hbox {CH}_4$$ / $$\hbox {H}_2$$ gas flow rate effect on polycrystalline diamond deposition was studied using a time-series exposure...     

The variance of a consensus

In proficiency tests the consensus of the participants' results is often used as the assigned value to calculate z-scores. Where the consensus is quantified as the robust mean of n results, the standard error of the assigned value is often taken to be , where is the robust standard deviation estimated from the same data ¹ . As some of the results are downweighted in robust estimation, is too large a denominator, so that tends to have a somewhat low bias. This bias is shown to be inconsequential for proficiency testing purposes. However, an unbiased estimate can be obtained by using the bootstrap.

---

     

Electrostatic-thermodynamic-kinetic (ELTHEKI) modeling of the coupled $$\text {Ni}$$ / $$\text {Ni}\text {O}$$ /water system,under physico-chemical conditions of pressurized water reactors

Journal of Solid State Electrochemistry - In the present paper, we develop an electrostatic-thermodynamic-kinetic (ELTHEKI) model for studying the behavior of the specific $$\text {Ni}$$ / $$\text...     

Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: II. time-dependent case

Time-dependent properties of a state that interacts with an infinite dimensional quantum system containing several one-parameter eigenvalue bands are considered. This is done by a new mathematical method that produces correct results, however strong the interaction between the state and the system . It is shown that in the case of the weak interaction one obtains standard results that are usually obtained within the formalism of the perturbation expansion method. In particular, if the eigenvalue E of the state is embedded inside the range of the unperturbed eigenvalues, time evolution of the state that is initially prepared in the state has typical exponential decay behavior. One also reproduces standard results concerning probabilities of the transition of the state at infinite time (t=∞) into various eigenvalue bands. However, if the interaction is strong, one finds much more complex and much more complicated behavior.     

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

Determination of ferrous and ferric iron from total iron content and thermogravimetric analysis

Journal of Thermal Analysis and Calorimetry - Ferrous ( $$\hbox {Fe}^{2+}$$ ) and ferric ( $$\hbox {Fe}^{3+}$$ ) iron content in mineral samples was determined from total iron (as obtained, for...     

Direct and exchange contributions to inner and outer radii in many-electron atoms

Within the Hartree-Fock framework, the spinless two-electron density function Γ (r ₁, r ₂) consists of direct Γ_di (r ₁, r ₂) and exchange Γ_ex (r ₁, r ₂) parts. Accordingly, the inner and outer radii in many-electron systems are rigorously separated into the direct and exchange contributions, i.e., and . It is generally shown that and , where is the usual average radius of an electron. Numerical examinations of the direct and exchange contributions for the 102 atoms from He to Lr in their ground states find that the electron exchange works to decrease and increase . However, the exchange parts are very small and the direct parts essentially govern the inner and outer radii.     

Solvent effects on the dielectric dispersion of poly(vinyl pyrrolidone)-poly(ethylene glycol) blends

The complex relative dielectric function , loss tangent , complex electric modulus and alternating current electrical conductivity dispersion behaviour of liquid poly(vinyl pyrrolidone)-poly(ethylene glycol) (PVP–PEG) blends in water, ethyl alcohol and 1,4–dioxane solvent over the entire volume mixture concentration range has been investigated in the frequency range of 20 Hz to 1 MHz at 25 °C. The PVP–PEG blends show the dielectric dispersion corresponding to the micro-Brownian motion of the PVP chain in the upper frequency region, whereas in the lower frequency region, dielectric dispersion is due to ionic conduction and electrode polarization phenomena. Results show that the conductivity values of these blends in water and dioxane can be monitored with the change in the solvent concentration, whereas it has a small variation with ethyl alcohol concentration. The comparative dielectric dispersion shape study confirms that the chain dynamics of polymers blend is influenced by heterogeneous interactions and solvent polar strength.     

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.     

On some multiplicative product of kth derivative of Dirac's delta in x0+|x| and x0−|x|

In this paper we give a sense to the products $${{\left| x \right|^{(n - 2)/2} }} \cdot \frac{{\delta ^{(k - 1)} (x_0 + \left| x \right|)}} {{\left| x \right|^{(n - 2)/2} }}$$ and $\delta ^{(k - 1)} (x_0 - \left| x \right|) \cdot \delta ^{(k - 1)} (x_0 + \left| x \right|)$ . The first of them is a generalization of the product $${{\left| x \right|^{(n - 2)/2} }} \cdot \frac{{\delta (x_0 + \left| x \right|)}} {{\left| x \right|^{(n - 2)/2} }}{\text{ }}$$ given in [1, p. 158].     

On the largest eigenvalues of trees with perfect matchings

Let λ₁ (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let be the set of trees with perfect matchings on 2m vertices, and . Among the trees in , we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when . Furthermore, it is proved that, for two trees T ₁ and T ₂ in (m≥ 4), if and Δ (T ₁) > Δ (T ₂), then λ₁ (T ₁) > λ₁ (T ₂).     

Two Group 8 Carbonyl Clusters Containing a Naked {\varvec{\mu}}_5-Sb Atom

The synthesis and structural characterisation of two novel clusters, 2, and 3, are presented. They are the first examples of osmium and ruthenium clusters containing a naked atom.     

Generating functions for oscillator matrix elements

Two general harmonic oscillator elements \documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle m \right|\hat x^s \hat p\left| n \right\rangle $$\end{document} and \documentclass{article}\pagestyle{empty}\begin{document}$$ \left\langle m \right|\exp \left[ { - \alpha \left( {\hbar /m*\omega } \right)\hat x^2 } \right]\exp \left( {\beta \hat x} \right)\left| n \right\rangle $$\end{document} are derived by a generating function method using operator techniques which contain practically all one- and two-centre integrals with equal frequencies of chemical physics.     

Komplexe zwischen Eisen(II)- und Tartrat-Ion

The formation of complexes between iron(II) and tartrate ion (L) has been studied at 25° C in 1m-NaClO₄, by using a glass electrode. The e.m.f. data are explained with the following equilibria: $$\begin{gathered} Fe^{2 + } + L \rightleftarrows FeL log \beta _1 = 1,43 \pm 0,05 \hfill \\ Fe^{2 + } + 2L \rightleftarrows FeL_2 log \beta _2 = 2,50 \pm 0,05 \hfill \\\end{gathered} $$ The protonation constants of the tartaric acid have been determinated: $$\begin{gathered} H^ + + L \rightleftarrows HL logk_1 = 3,84 \pm 0,03 \hfill \\ 2H^ + + L \rightleftarrows H_2 L logk_2 = 6,43 \pm 0,02 \hfill \\\end{gathered}$$ .     

The graphical representation of equilibrium for the isothermal and isobaric reactions of ideal gases

The equilibrium for the isothermal and isobaric reactions of ideal gases is investigated in virtue of the intuitionistic figure. The curve is similar to the curve of tangential function which has one inflection and two vertical asymptotes. The equation only has one root ξ _e and it is suitable to find ξ _e by dichotomy. For non-inert substance, when or x_i^0 $$" align="middle" border="0"> , to increase substance i will make an equilibrium shift in the direction to deplete substance i; when {\nu_i} \mathord{\left/ {\vphantom {{\nu_i} {\sum_i {\nu _i}}}} \right. \kern-\nulldelimiterspace} {\sum _i {\nu _i}}> 0$$" align="middle" border="0"> , to increase substance i will make an equilibrium shift in the direction to produce more substance i.     

Lattice parameters and thermal expansion of δ-VN1−x from 298–1000 K

The thermal expansion of VN_1?x was determined from measurements of the lattice parameters in the temperature range of 298–1000 K and in the composition range of VN_0.707–VN_0.996. Within the accuracy of the results the expansion of the lattice parameter with temperature is not dependent on the composition. The lattice parameter as a function of composition ([N]/[V]=0.707?0.996) and temperature (298–1000 K) is given by $$\begin{gathered} a([N]/[V],T) = 0.38872 + 0.02488([N]/[V]) - \hfill \\ - (1.083 \pm 0.021) \cdot 10^{ - 4} T^{1/2} + (6.2 \pm 0.1) \cdot 10^{ - 6} T. \hfill \\ \end{gathered} $$ . The coefficient of linear thermal expansion as a function of temperature (in the same range) is given by $$\alpha (T) = a([N]/[V],T)^{ - 1} [( - 5.04 \pm 0.01) \cdot 10^5 T^{ - 1/2} + (6.2 \pm 0.1) \cdot 10^{ - 6} ].$$ . The average linear thermal expansion coefficient is $$\alpha _{av} = 9.70 \pm 0.15 \cdot 10^{ - 6} K^{ - 1} (298 - 1 000K).$$ . The data are compared with those of several fcc transition metal nitrides collected and evaluated from the literature.     

Investigation on enthalpies of combustion and heat capacities for 2-aminomethylpyridine derivatives

Journal of Thermal Analysis and Calorimetry - The molar energies of combustion $$\left( {\Delta_{\text{c}} U_{\text{m}} } \right)$$ for 2-aminomethylpyridine (AMP), tert-butyl...