New two stages high order symmetric six-step method with vanished phase–lag and its first,second and third derivatives for the numerical solution of the Schrödinger equation

In the present paper, we obtain and analyze, for the first time in the literature, a new two-stages high order symmetric six-step method. The specific characteristics of the new proposed method are the highest possible algebraic order, the elimination of the phase–lag and its first, second and third derivatives. Additionally, for the new method we give the analysis of the method (both error and stability and interval of periodicity analysis) and the comparison of the effectiveness of the new developed method with the effectiveness of well known methods and very recently produced methods in the literature. The comparison is based on the numerical solution of the Schrödinger equation. The theoretical achievements and the numerical results show the effectiveness of the new developed method in comparison with other well known or recently developed numerical methods.     

An efficient and computational effective method for second order problems

An efficient and computational effective algorithm is introduced, for the first time in the literature, in the present paper. The main properties of the scheme are: (1) the algorithm is a two-step scheme, (2) the algorithm is symmetric one, (3) it is a hight algebraic order scheme (i.e of eight algebraic order), (4) it is a three-stages algorithm, (5) the first layer of the new method is based on an approximation to the point \(x_{n-1}\), (6) the scheme has vanished phase-lag and its first, second and third derivatives, (7) the new proposed algorithm has an interval of periodicity equal to \(\left( 0, 9.8 \right) \). For the present new scheme we study: (1) its construction, (2) its error analysis (3) its stability analysis. Finally, the investigation of the effectiveness of the new algorithm leads to its application to systems of differential equations arising from the Schrödinger equation.     

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.     

A Study of the Influence of the Surrounding Gas on the Plasma Jet and Coating Quality During Plasma Spraying

Coating quality is affected by arc and plume instabilities during plasma spraying. In closed chamber plasma spraying, gradual drift is one of the intermediate instabilities, which is mainly due to the electrode erosion. This work focuses on the source of the gradual drift of the plasma jet and the influence on coating quality. The ambient state inside the chamber was controlled by a ventilation system and a vacuum system. The variation in the plasma jet was observed by a particle flux image device based on a CCD camera. The optical spectrum of the plasma plume was measured and analyzed through an optical spectrometer. The results indicated that the addition of hydrogen to plasma gas induced the change in the plasma jet length and width with changing rates depending on the chamber state and the ventilation power. With poor ventilation, the intensity of H_α emission was found to become gradually stronger while H_β and H_γ were found to become weaker. On closing the chamber and retaining enough ventilation power, it was observed that the ambient gas slowly turned red. Simultaneously, the coating weight and thickness were slightly decreased meanwhile the porosity ratio was obviously increased. The red ambient gas has been proved to be able to acidify the city water with pH value decreased from 7 to 1–3. Without hydrogen, the plasma jet was found to be stable without reddening and variation, but the plasma enthalpy was unfortunately low.     

Uncatalyzed synthesis of new substituted dihydro-2H-dipyrimido[1,2-a,4,5-d]pyrimidine-2,4(3H)-dione

Research on Chemical Intermediates - A convenient and new method has been developed for the synthesis of highly substituted dihydro-2H-dipyrimido[1,2-a,4,5-d]pyrimidine-2,4(3H)-dione derivatives by...     

On parallel Branch and Bound frameworks for Global Optimization

Branch and Bound (B&B) algorithms are known to exhibit an irregularity of the search tree. Therefore, developing a parallel approach for this kind of algorithms is a challenge. The efficiency of a B&B algorithm depends on the chosen Branching, Bounding, Selection, Rejection, and Termination rules. The question we investigate is how the chosen platform consisting of programming language, used libraries, or skeletons influences programming effort and algorithm performance. Selection rule and data management structures are usually hidden to programmers for frameworks with a high level of abstraction, as well as the load balancing strategy, when the algorithm is run in parallel. We investigate the question by implementing a multidimensional Global Optimization B&B algorithm with the help of three frameworks with a different level of abstraction (from more to less): Bobpp, Threading Building Blocks (TBB), and a customized Pthread implementation. The following has been found. The Bobpp implementation is easy to code, but exhibits the poorest scalability. On the contrast, the TBB and Pthread implementations scale almost linearly on the used platform. The TBB approach shows a slightly better productivity.     

Coprecipitation of calcium hydroxyapatite,graphene oxide,and chitosan from aqueous solutions

We determined the character of interactions between calcium hydroxyapatite Са₁₀(РO₄)₆(ОН)₂ (HA), graphene oxide (GO), and chitosan (С₆Н₁₁NO₄) _n (CHT) to yield HA/CHT/GO nanocomposites (NCs) in the СаС1₂–(NH₄)₂НРО₄–NH₃–Н₂О–(С₆Н₁₁NO₄) _n –GO system (25°С). A set of physicochemical methods helped us to elucidate composition–synthesis parameters–structure–particle size–properties correlations for the prepared NCs and to prove the feasibility to manufacture NCs with tailored HA, CHT, and GO contents, described by the bulk formula Са₁₀(РО₄)₆(ОН)₂ · х(С₆Н₁₁NO₄) _n · yGO · zН₂О, where х = 0.1, 0.2, 0.3; y = 0.6, 1.2, 2.4; and z = 6.0–7.4.     

Phase equilibria in the Cu–Cu<Subscript>2</Subscript>Se–As system

Phase equilibria in the Cu–Cu₂Se–As were investigated by differential thermal analysis and X-ray powder diffraction analysis. Informative plots describing this system were constructed, viz., the polythermal sections Cu_0.667Se_0.333–As, Cu_0.667Se_0.333–Cu_0.735As_0.265, and Cu_0.8Se_0.2–As, the isothermal section of the phase diagram at 300 K, and the projection of the liquidus surface. The obtained results differ from the published data in length of fields of primary crystallization of phases and in coordinates of a number of invariant equilibrium points.     

Kinetics and mechanism of the chain reaction between <Emphasis Type="Italic">N</Emphasis>-phenyl-1,4-benzoquinone monoimine and thiophenol

The kinetics of the reaction between N-phenyl-1,4-benzoquinone monoimine (quinone monoimine) and thiophenol is studied in chlorobenzene at 343 K. The reaction has the same mechanism proposed earlier for a similar reaction involving N,N'-diphenyl-1,4-benzoquinone diimine (quinone diimine). This mechanism has two paths: chain and nonchain. An important difference between the kinetics of the two reactions is the apparent reversible nature of the chain reaction in the quinone monoimine + thiophenol system. This nature reveals itself when the concentrations of thiophenol are comparable to or slightly higher than the concentrations of quinone imine. In light of this, kinetic research is conducted under conditions where the concentrations of thiophenol are significantly higher than those of quinone monoimine, allowing us to simplify the kinetic features and obtain interpretable data. The rate constants of the reaction’s elementary steps are estimated and found to be three to five times lower for the reaction involving quinone monoamine than for the one involving quinone diimine. Both reactions have relatively short chains whose lengths do not exceed several tens of units.     

Thermochemistry of С<Subscript>60</Subscript> fullerene solutions in benzene,toluene, <Emphasis Type="Italic">о</Emphasis>-xylene,and <Emphasis Type="Italic">о</Emphasis>-dichlorobenzene at 298.15 K

The enthalpies of dissolution of С₆₀ in benzene, toluene, о-xylene and о-dichlorobenzene are measured in a sealed high-sensitivity calorimeter at 298.15 K and at different concentrations of the solute. The standard enthalpies of dissolution of С₆₀ in these solvents are determined.