Pseudospectral methods of solution of the Schrödinger equation

Several different pseudospectral methods of solution of the Schrödinger equation are applied to the calculation of the eigenvalues of the Morse potential for I₂ and the Cahill–Parsegian potential for Ar₂ [Cahill, Parsegian, J. Chem. Phys. 121, 10839 (2004)]. The calculation of the eigenvalues for the Woods–Saxon potential are also considered. The convergence of the eigenvalues with a quadrature discretization method is found to be very fast owing to the judicious choice for the weight function, basis set and quadrature points. The weight function used is either related to the exact ground state wavefunction, if known, or an approximation to it from some reference potential. We compare several different pseudospectral methods.     

Extending the kinetic solution of the classic Michaelis–Menten model of enzyme action

The principal aim of studies of enzyme-mediated reactions has been to provide comparative and quantitative information on enzyme-catalyzed reactions under distinct conditions. The classic Michaelis–Menten model (Biochem Zeit 49:333, 1913) for enzyme kinetic has been widely used to determine important parameters involved in enzyme catalysis, particularly the Michaelis–Menten constant (K _M ) and the maximum velocity of reaction (V _max ). Subsequently, a detailed treatment of the mechanisms of enzyme catalysis was undertaken by Briggs–Haldane (Biochem J 19:338, 1925). These authors proposed the steady-state treatment, since its applicability was constrained to this condition. The present work describes an extending solution of the Michaelis–Menten model without the need for such a steady-state restriction. We provide the first analysis of all of the individual reaction constants calculated analytically. Using this approach, it is possible to accurately predict the results under new experimental conditions and to characterize and optimize industrial processes in the fields of chemical and food engineering, pharmaceuticals and biotechnology.     

Electrodeposition mechanism of Ni-Mo-P alloy in the solution of ammoniac citrate

The induced codeposition mechanism of Mo, P and Ni from the solution of ammoniac citrate was studied by means of steady-state polarization, AC impedance and X-ray Photoelectron Spectroscopy (XPS). The result of electrochemical measurements proved that [NiCit(NH3)2]- is the electro-active species of nickel, though nickel ions exist mainly as [NiCit(NH3)3]- in ammoniac citrate. XPS experiments proved the existence of tetravalent molybdenum corresponding to MoO2 on the surface of some deposits. The intermediate product, MoO2, was probably reduced to Mo in the alloy deposit by atomic hydrogen adsorbed on the induced metal nickel. The reduction of H2PO2- occurs through two distinctive steps with PH3 as an intermediate, which subsequently reacts with atomic hydrogen to form P in the alloy deposit. The electrodeposition mechanism was proposed in this paper.     

An economical eighth-order method for the approximation of the solution of the Schrödinger equation

In this paper we introduce, for the first time in the literature, a three-stages two-step method. The new algorithm has the following characteristics: (1) it is a two-step algorithm, (2) it is a symmetric method, (3) it is an eight-algebraic order method (i.e of high algebraic order), (4) it is a three-stages method, (5) the approximation of its first layer is done on the point \(x_{n-1}\) and not on the usual point \(x_{n}\), (6) it has eliminated the phase–lag and its derivatives up to order two, (7) it has good stability properties (i.e. interval of periodicity equal to \(\left( 0, 22 \right) \). For this method we present a detailed analysis : development, errorand stability analysis. The new proposed algorithm is applied to systems of differential equations of the Schrödinger type in order to examine its efficiency.     

What is the acidity of a plutonium solution?

The determination of the pH of a plutonium solution has traditionally depended on an electrode or a titration in the presence of a complexing agent. A new approach uses the equilibrium distribution of the Pu oxidation states to estimate the hydrogen ion concentration. The method is used to estimate the equilibrium constant of the first hydrolysis reaction of tetravalent plutonium.     

Deconstructing the excited-state dynamics of β-carotene in solution

The femtosecond to nanosecond dynamics of the all-trans β-carotene carotenoid dissolved in 3-methylpentane is characterized and dissected with excitation-wavelength and temperature-dependent ultrafast dispersed transient absorption signals. The kinetics measured after red-edge (490 nm) and blue-edge (400 nm) excitation were contrasted under fluid solvent (298 K) and rigid glass (77 K) conditions. In all four measured data sets, the S* population kinetics was resolved prompting the development of a modified multicompartment model. The temperature-dependent and excitation wavelength-dependent S* quantum yield is ascribed to a competition of population surmounting a weak (55 cm(-1)) energy barrier on the S(2) state to favor S(1) generation and rapid internal conversion that favors S* generation. When cooled from room temperature to 77 K, the S* decay time scale shifted significantly from 30 to 400 ps, which is ascribed to small-scale structural relaxation with a 115 cm(-1) energy barrier. For the first time under low-energy excitation conditions, the triplet state is observed and confirmed to not originate from S* or S(1), but from S(2). The interconnectivity of the S* and S(1) populations is discussed, and no observed population flow is resolved between S* and S(1). Comparison of samples obtained from different laboratories with different purity levels demonstrates that sample contamination is not the primary origin of the S* state.     

Calculation of the molecular exchanging energy of binary surfactants system on the surface monolayer of aqueous solution

By using the binary anionic/cationic surfactants system CH3(CH2)nOSO_3/CH3(CH2)nN (CH3)3 as an ex-ample, the molecular exchanging energy (ε) of adsorption on the surface monolayer of aqueous solu-tion has been studied. ε can be obtained with two methods. One is from the relationship between ε and the molecule interaction parameter (β). This relationship is founded by considering that the adsorption of mixed surfactants on the surface monolayer of solution satisfies the dimensional crystal model condition under which β can be obtained by testing the surface tension of solution. The other is directly from the molecular structure of surfactants with the Lennard-Jones formula. The results for the studied system show that these two methods coincide well.     

A family of multiderivative methods for the numerical solution of the Schrödinger equation

A family of multiderivative methods with minimal phase-lag are introduced in this paper, for the numerical solution of the Schrödinger equation. The methods are called multiderivative since uses derivatives of order two, four or six. Numerical application of the new obtained methods to the Schrödinger equation shows their efficiency compared with other similar well known methods of the literature.Active Member of the European Academy of Sciences and Arts     

A study of the thermal analysis of the solid solution (AgxK1−xNO3

The effect of partial replacement of K⁺ by Ag⁺ in the mixed system of KNO₃ and AgNO₃ during a set of DSC heating cycles was studied by means of a modern computerized DSC system. Thermal analysis was performed in the vicinity of the phase transition II I for pure KNO₃. The results revealed a large change in enthalpy and the phase transition temperature close to the morphotropic boundary x=0.5–0.6. At this boundary, the transformation enthalpy reached the maximum value of 101 J g^–1. On the basis of the data obtained from this accurate thermal analysis work, a model is suggested for the energy barrier of rotation of the nitrate ion in the mixed nitrate system.     

Behaviour of β2-microglobulin in the electrical double layer of the mercury/solution interface

β₂-Microglobulin (β₂-m) is a small globular protein (12000 Da) which exists as a monomer in numerous body fluids. Data published on the electrochemical behaviour of β₂-m are scarce. In this paper an attempt was made to ge more information on the structure of the double layer at mercury/solution interface in the presence of β₂-m. Using phase selective ac polarography the capacitive current-potential (C-E) curve was analysed as the function of various β₂-m concentration, pH of medium, temperature, denaturation agent concentration etc. The data obtained are discussed particularly from the aspect of folding/unfolding state of the protein studied.     

An efficient six-step method for the solution of the Schrödinger equation

In this paper we develop an efficient six-step method for the solution of the Schrödinger equation and related problems. The characteristics of the new obtained scheme are:

• It is of twelfth algebraic order.
• It has three stages.
• It has vanished phase-lag.
• It has vanished its derivatives up to order two.
• All the stages of the scheme are approximations on the point \(x_{n+3}\).

This method is developed for the first time in the literature. A detailed theoretical analysis of the method is also presented. In the theoretical analysis, a comparison with the the classical scheme of the family (i.e. scheme with constant coefficients) and with recently developed algorithm of the family with eliminated phase-lag and its first derivative is also given. Finally, we study the accuracy and computational effectiveness of the new developed algorithm for the on the approximation of the solution of the Schrödinger equation. The above analysis which is described in this paper, leads to the conclusion that the new algorithm is more efficient than other known or recently obtained schemes of the literature.

     

The mechanism and pathway of the ozonation of 4-chlorophenol in aqueous solution

Chlorophenols (CPs) have been widely used in dif- ferent formulations as preservatives, herbicides, insec- ticides, bactericides and solvents. Parts of chlorophe- nols were released to the natural environment during the usage. As a result, many water sources were con- taminated with CPs[1,2]. Furthermore, they also can be formed during the disinfection of phenol containing water by chlorination. Several CPs are recognized as the priority pollutants by the United States EPA (En- vironmenta…     

A new methodology for the development of numerical methods for the numerical solution of the Schr?dinger equation

In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr?dinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications.     

Parsing of the free energy of aromatic–aromatic stacking interactions in solution

We report an analysis of the energetics of aromatic–aromatic stacking interactions for 39 non-covalent reactions of self- and hetero-association of 12 aromatic molecules with different structures and charge states. A protocol for computation of the contributions to the total energy from various energetic terms has been developed and the results are consistent with experiment in 92% of all the systems studied. It is found that the contributions from hydrogen bonds and entropic factors are always unfavorable, whereas contributions from van-der-Waals, electrostatic and/or hydrophobic effects may lead to stabilizing or destabilizing factors depending on the system studied. The analysis carried out in this work provides an answer to the questions “What forces stabilize/destabilize the stacking of aromatic molecules in aqueous-salt solution and what are their relative importance?”     

Do orientation effects contribute to the molecular weight dependence of the free solution mobility of DNA?

The free solution mobility of DNA increases with increasing molecular weight and then levels off and becomes constant at molecular weights above approximately 400 bp (Stellwagen, N. C., Gelfi, C., Righetti, P. G., Biopolymers 1997,42, 687-703). To investigate whether the increase in mobility could be attributed to an increased orientation of the larger DNA molecules in the electric field, the free solution mobility of DNA was measured by capillary electrophoresis as a function of electric field strength. Mixtures containing 20-, 118- and 422-bp DNA molecules, and 20-, 422- and 2116-bp DNAs, were studied. If the larger DNA molecules in each mixture were oriented by the electric field, their mobilities should increase with electric field strength faster than the mobility of the 20-bp oligomer, which is too small to be oriented by the electric fields used in this study. Instead, the ratios of the mobilities of the 118-, 422- and 2116-bp fragments to the mobility of the 20-bp oligomer were independent of electric field strength. Hence, orientation effects are not important for DNA molecules up to 2 kbp in size, in electric fields up to 500 V/cm in amplitude. An explanation is suggested.     

The stable planar conformation of the γ-butyrolactone ring in solution

<正>The conformations ofγ-butyrolactone ring in solution were deduced on the basis of ~1H NMR spectra of geminal protons of the butyrolactone ring.A series of optically pure(Z)-(-)-4-(1'-alkoxyl-1'-carbalkoxy-methylene)-5(R)[(1R)-menthyloxy]-γ-butyr-olactones with a stable planar conformation of γ-butyrolactone ring were found.     

Embedded eighth order methods for the numerical solution of the Schrödinger equation

A new method for the approximate numerical integration of the radial Schrödinger equation is developed in this paper. Phase-lag and stability analysis of the new method is included. The new method is called the embedded method because of a simple natural error control mechanism. Numerical results obtained for the phase-shift problem of the radial Schrödinger equation show the validity of the developed theory.     

A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation

A dissipative exponentially fitted method is constructed in this paper for the numerical integration of the Schr?dinger equation. We note that the present method is a nonsymmetric multistep method (dissipative method) An application to the bound-states problem and the resonance problem of the radial Schr?dinger equation indicates that the new method is more efficient (i.e. more accurate and more rapid) than the classical dissipative method and other well-known methods. Based on the new method and the method of Raptis and Allison(19) a new variable-step method is obtained. The application of the new variable-step method to the coupled differential equations arising from the Schr?dinger equation indicates the efficiency of the new approach.     

A variable-step Numerov method for the numerical solution of the Schrödinger equation

Numerovs method is one of the most widely used algorithms for solving second-order ordinary differential equations of the form y = f(x,y). The one-dimensional time-independent Schrödinger equation is a particular example of this type of equation. In this article we present a variable-step Numerov method for the numerical solution of the Schrödinger equation.