One - determinantal pure state versus ensemble Kohn-Sham solutions in the case of strong electron correlation: CH2 and C2

The possibility that the Kohn-Sham (KS) solution for a noninteracting auxiliary electron system is not the conventional one-determinantal pure state but a few-determinantal ensemble has been investigated. The KS solutions (the exchange-correlation potential v _xc and the orbitals) have not been approximated by local-density or density-gradient approximations but have been constructed from an accurate ab initio electron density. The lowest singlet states of the CH₂ and C₂ molecules have been selected for this investigation since for these cases the ground-state wave function Ψ is nondegenerate but has an essentially multideterminantal character (electron correlation is strong). For C₂ the dependence of the type of KS solution on the bond distance R(C–C) has been studied at the QZ level. For the shortest distance considered, R(C–C)=1.8 a.u., a pure-state KS solution has been obtained. For the equilibrium distance R _e(C–C)=2.348 a.u. and at larger distances ensemble solutions have been obtained with widely varying weights of the individual determinants, depending on the bond distance. For CH₂ the dependence of the type of KS solution on the basis has been studied: calculation in the triple zeta (TZ) basis for the KS orbitals yields an ensemble solution, while the pure-state KS solution has been obtained in the quadruple zeta (QZ) basis. The form of the KS orbitals has been compared with that of the natural orbitals (NOs). It has been shown for the model example of the stretched H₂ molecule as well as for CH₂ and C₂, that the KS orbitals of the pure state may be rather different from the corresponding NOs, while the occupied KS orbitals of the ensemble solution can be considered as plausible approximations to the corresponding NOs. Received: 10 February 1998 / Accepted: 10 June 1998 / Published online: 3 September 1998     

Effect of magnetic field on electroosmotic flow of viscoelastic fluids in a microchannel

Electroosmotic flow is an efficient transportation technology driven by applying an external electric field across the microchannel, which has a great potential for future application. This work is presented to study the unsteady electroosmotic flow of viscoelastic fluids combined with a constant pressure gradient and a vertical magnetic field through a parallel plate microchannel. For the reason that the upper and bottom walls of the parallel plate microchannel in microfluidic devices can be made of different materials, this leads to different hydrophobic properties, asymmetric zeta wall potentials, and different slip boundary conditions. The Navier slip model with different slip coefficients at walls is considered. The generalized Maxwell fluid with fractional derivative is adopted for the constitutive equation of the fluid. The analytical and numerical solutions of velocity are derived by employing the integral transform method and finite difference method, respectively. Excellent agreement is found between the numerical solutions and analytical solutions. Finally, the effects of fractional parameter , relaxation time , slip coefficients and , the ratio of wall zeta potentials , Hartmann number , and electrical field strength parameter on velocity profiles are interpreted graphically in detail.     

Abtrennung von Ceriterden aus Gemischen mit Yttererden durch fraktionierte Kristallisation der Magnesiumdoppelnitrate in großem Maßstab

Summary The fractional crystallisation of magnesium double nitrates was used to separate cerium earths from 112 kg yttrium earth mixtures; they were recovered in three different fractions. By using excess magnesium nitrate the crystallisation could be extended at least to gadolinium without using HNO₃ as solvent. In the resulting solutions (69 kg oxide) there remained yttrium and nearly all erbium and heavy yttrium earths.

     

Impact of thermal ageing on sorption and diffusion properties of PTMSP

Gas and vapour permeability in both freshly cast and aged poly(1-trimethylsilyl-1-propyne) (PTMSP) membranes were investigated in terms of solubility and diffusion coefficients for two probe molecules, a permanent gas (nitrogen) and an organic vapour (dichloromethane). To get reliable data for this study, we set up a fast and reproducible ageing procedure consisting of thermal treatment of the polymer films (100 °C during 24 h under vacuum). As expected, measurements recorded from time-lag experiments and isothermal sorption showed strong variations of the PTMSP transport properties before and after the thermal ageing procedure. Freshly cast membranes exhibited high permeability, whereas after ageing a 40–45% decrease of the permeability was recorded for both probes. The results demonstrated that only the glassy physical microstructure of PTMSP was affected by the ageing procedure, while the chemical structure was unchanged. Based on a dual-mode model for sorption and a Long's model for diffusion, the analysis of the data showed that the solubility and diffusion coefficients of the gas and the vapour were not affected in the same way. For nitrogen, only the diffusion coefficient decreased, whereas for dichloromethane, the thermal treatment mainly influenced the sorption coefficient. The lower permeability due to the combination of sorption and diffusion parameters could be attributed to a change of the PTMSP hole geometry or the hole connections.     

An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk

If (X _n ) _n ⁼¹ is a sequence of i.i.d. random variables in the Euclidean plane such that we compute the mean of the perimeter of theconvex hull ofX ₁++X _k; 0kn}.     

Convergence of a Dinkelbach-type algorithm in generalized fractional programming

The convergence of a Dinkelbach-type algorithm in generalized fractional programming is obtained by considering the sensitivity of a parametrized problem. We show that the rate of convergence is at least equal to (1+5)/2 when regularity conditions hold in a neighbourhood of the optimal solution. We give also a necessary and sufficient condition for the convergence to be quadratic (which will be verified in particular in the linear case) and an idea of its implementation in the convex case.

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Zusammenfassung Die Konvergenz eines Verfahrens i. S. von Dinkelbach zur Lösung verallgemeinerter Quotientenprogramme wird durch Untersuchung der Sensitivität eines parametrisierten Problems abgeleitet. Es wird gezeigt, daß die Konvergenzrate durch (1+5)/2 nach unten beschränkt ist, falls gewisse Regularitätsbedingungen in einer Umgebung der Optimallösung erfüllt sind. Ferner wird eine notwendige und hinreichende Bedingung zur quadratischen Konvergenz hergeleitet. Es wird gezeigt, wie diese im Falle konvexer Probleme implementiert werden kann.

     

Solution of the Fokker-Planck equation for the shock wave problem

An eigenexpansion solution of the time-independent Brownian motion Fokker-Planck equation is given for a situation in which the external acceleration is a step function. The solution describes the heavy-species velocity distribution function in a binary mixture undergoing a shock wave, in the limit of high dilution of the heavy species and negligible width of the light-gas internal shock. The diffusion solution is part of the eigenexpansion. The coefficients of the series of eigenfunctions are obtained analytically with transcendentally small errors of order exp(–1/M), whereM 1 is the mass ratio. Comparison is made with results from a hypersonic approximation.     

Low-temperature and long-time anomalies of a damped quantum particle

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI() at low frequencies, where > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on . For > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for < 1, an initially localized state remains localized for all times. For 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for 2 and at zero temperature for 1 2.     

Dual Fluctuation–Dissipation Relationships between Tri- and Bispectra of the Brownian Motion

A fluctuation–dissipation analysis of nonlinear noise accompanying Brownian motion (BM) in electric systems is carried out. It is shown that, for symmetrical BM, there exists a linear set of dual fluctuation–dissipation relationships linking an equilibrium trispectrum to a derivative of a bispectrum with respect to a direct electric current (or voltage), taken at equilibrium. Analysis of an electrochemical model of BM in the form of a symmetrical slow discharge yields a proportionality coefficient.     

Fractional Birkhoffian method for equilibrium stability of dynamical systems

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering.