Coulombic and ring-shaped potentials treated in a unified way via a nonbijective canonical transformation

This paper is concerned with the three-dimensional potentialV _q =² (2a ₀/r–q a ₀ ²/r ² sin² ) ₀ which comprises as particular cases the ring-shaped potential (q = 1) and the Coulomb potential (q = 0). The Schrödinger equation for the potentialV _q is transformed via a nonbijective canonical transformation, viz., the Kustaanheimo-Stiefel transformation, into a coupled pair of Schrödinger equations for two-dimensional harmonic oscillators with inverse-square potentials. As a consequence, the discrete spectrum for the potentialV _q is obtained in a straightforward way. A special attention is paid to the caseq = 0. In particular, the coupled pair of Schrödinger equations for two-dimensional harmonic oscillators is tackled in the situations where the spectrum for the potentialV ₀ is discrete, continuous, or reduced to the zero point. Finally, some group-theoretical questions about the potentialV _q are mentioned as well as a connection, via the Kustaanheimo-Stiefel and the Levi-Civita transformations, between the quantum-mechanical problems for the potentialV _q and the Sommerfeld and Kratzer potentials.     

Supersymmetric treatment of a particle subjected to a ring-shaped potential

The ring-shaped Hartmann potential was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, fundamental concepts of supersymmetric quantum mechanics (SUSYQM) are discussed. The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM. © 1996 John Wiley & Sons, Inc.     

Exact solutions of a new Coulomb ring-shaped potential

A new Coulomb ring-shaped potential is proposed, which results from adding a potential proportional to(cos θ/r ² sin² θ)to a Coulomb potential. The Schrödinger equation with this new model potential is separated into angular and radial components. The exactly wavefunctions and the spectrum equation for bound state are presented by the standard approach.     

The surface potential of a spherical colloid particle: functional theoretical approach

By using the iterative method in functional analysis, the potential of the electrical double layer of a spherical colloid particle, which is represented by the so-called Poisson-Boltzmann (PB) equation, has been solved analytically under general potential conditions. With the help of the diagram method in mathematics, the surface potential of the particle has been defined from the second iterative solution. The influence of the parameters included in the solutions on the surface potential has been studied. The results show that the surface potential of the particle increases as the temperature of the system, the aggregation number, and the concentration of ions increase, but decreases with an increase in the dielectric constant and the valence of the ions. The corresponding space charge density also has been illustrated in this work.     

Wigner function approach to the quantum Brownian motion of a particle in a potential

Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.     

Nucleation theory in Langevin's approach and lifetime of a Brownian particle in potential wells

The multivariable theory of nucleation suggested by Alekseechkin [J. Chem. Phys. 124, 124512 (2006)] is further developed in the context of Langevin's approach. The use of this approach essentially enhances the capability of the nucleation theory, because it makes possible to consider the cases of small friction which are not taken into account by the classical Zel'dovich-Frenkel theory and its multivariable extensions. The procedure for the phenomenological determination of the nucleation parameters is described. Using the similarity of the Kramers model with that of nucleation, the lifetime of a Brownian particle in potential wells in various dimensionalities is calculated with the help of the expression for the steady state nucleation rate.     

Extended implementation of canonical transformation theory: parallelization and a new level-shifted condition

The canonical transformation (CT) theory has been developed as a multireference electronic structure method to compute high-level dynamic correlation on top of a large active space reference treated with the ab initio density matrix renormalization group method. This article describes a parallelized algorithm and implementation of the CT theory to handle large computational demands of the CT calculation, which has the same scaling as the coupled cluster singles and doubles theory. To stabilize the iterative solution of the CT method, a modification to the CT amplitude equation is introduced with the inclusion of a level shift parameter. The level-shifted condition has been found to effectively remove a type of intruder state that arises in the linear equations of CT and to address the discontinuity problems in the potential energy curves observed in the previous CT studies.     

Comment on "The surface potential of a spherical colloid particle: functional theoretical approach"

In a work published in this journal by Z.W. Wang, G.Z. Li, D.R. Guan, X.Z. Yi, and A.J. Lou [J. Colloid Interface Sci. 246 (2002) 302], an iterative method for the determination of the potential around a colloidal particle is presented. It is claimed that successive terms of the iteration series converge to the exact solution of the Poisson-Boltzmann equation. This claim seems to be unfounded when the analytical expressions of the iteration terms are compared with well established numerical data.     

Zeta potential of particle bilayers on mica: a streaming potential study

The streaming potential of mica covered by bilayers of latex particles was measured using the parallel-plate channel cell. The size of the first latex (A500) bearing amidine charged groups was 503 nm and the second latex (L800) bearing sulfonate groups was 810 nm (at pH 5.5 and an ionic strength of 10(-2)M). The A500 latex exhibited an isoelectric point at pH 10.5, whereas the L800 latex was strongly negative at all pH. Mica sheets were precovered first by the A500 latex particles under diffusion transport conditions. The coverage of this supporting layer was regulated between 0.02 and 0.5 by changing the bulk concentration of latex and the deposition time. Then, the second layer of the L800 latex of regulated coverage up to 0.55 was deposited under the diffusion transport. The coverage of particles and their distributions in both layers were determined by a direct enumeration of particles by optical microscopy under wet conditions and by AFM. It was shown that the structure of the L800 particle layers and the maximum coverage were in accordance with theoretical simulations performed according to the random sequential adsorption (RSA) model. After forming bilayers of desired composition and structure, streaming potential measurements were carried out. The influence of the mica substrate, the supporting layer coverage, and its zeta potential on the apparent zeta potential of bilayers was systematically studied. It was established that for a bilayer coverage exceeding 0.20, the net zeta potential became independent of the substrate and the supporting layer zeta potentials. Then, the asymptotic values of the zeta potential of the bilayer approach 1/√2=0.71 of the bulk zeta potential of the particles forming the external (second) layer. This behavior was interpreted theoretically in terms of the electrokinetic model derived previously for monolayers. It was also concluded that results obtained in this work can be exploited for interpretation of polyelectrolyte film formation in the layer by layer (LbL) processes and protein adsorption pertinent to the antigen/antibody interactions.     

A treatment of the torsional wave equation with an internal rotation constant depending on a torsional angle via point canonical transformation

A numerical method using a canonical point transformation has been developed for determining the energy levels of the internal rotation of asymmetric rotors. An internal rotation constant is assumed to be a function of a torsional angle.     

Trapping of a particle in a short-range harmonic potential well

Eigenstates of a particle in a localized and unconfined harmonic potential well are investigated. Effects due to the variation of the potential parameters as well as certain results from asymptotic expansions are discussed.     

Motion of a spherical particle in a cylindrical channel using arbitrary Lagrangian-Eulerian method

A finite element particle transport model, consisting of Navier-Stokes and continuity equations defined in arbitrary Lagrangian-Eulerian (ALE) kinematics, is employed to describe the motion of a rigid uncharged spherical particle in a cylindrical channel of uniform cross-section. The wall correction factors for the spherical particle moving with a fluid confined in an infinitely long cylindrical channel, as well as in finite length channels are presented. Two finite channel effects are considered, namely, motion of the particle at the entrance and exit of an open channel, and the motion of a particle toward the capped end of the channel. The numerical model demonstrates good agreement with many existing analytical results for infinite channels in the Stokes flow regime. Simple correlations for the hindrance factors are presented.     

Magneto-optical properties of a spinless particle moving in a harmonic potential

Exact solutions to the quantum mechanical problem of an anisotropic oscillator in a one-dimensional magnetic field are obtained. These solutions (eigenenergies and wave functions) are then applied to the problem of calculating the magneto-optical properties of a charged spinless particle constrained to move in a harmonic potential field. General expressions for the dipole strengths and rotational strengths associated with radiation induced transitions between the eigenstates of this model system are developed, and these quantities are further related to observables of magneto-optical absorption spectroscopy and Faraday effect studies.     

A quantum particle in a moving finite harmonic potential

In this paper a quantum particle trapped in a finite harmonic oscillator confined in the moving interval is investigated. With a help of a suitable unitary transform, the wave functions and time-dependent energy levels are obtained.     

Brownian motion of an asymmetrical particle in a potential field

It is well known that a free ellipsoidal Brownian particle exhibits anisotropic diffusion for short times which changes to isotropic at long times, and, that the long-time diffusion coefficient is an average of the translational diffusion coefficients along the different semiaxes of the particle. We show analytically that in the presence of external forces, the long-time diffusion coefficient is different from that of a free particle. The magnitude of the difference in the two diffusion coefficients is found to increase proportionately with the particle's asymmetry, being zero only for a perfectly spherical Brownian particle. It is also found that, for asymmetrical particles, the application of external forces can amplify the non-Gaussian character of the spatial probability distributions which consequently delays the transition to the classical behavior. We illustrate these phenomena by considering the quasi-two-dimensional Brownian motion of an ellipsoidal rigid particle in linear and harmonic potential fields. These two examples provide insight into the role played by particle asymmetry in electrophoresis and microconfinement due to a laser trap or due to intracellular macromolecular crowding.     

Dissipative particle dynamics simulations in the grand canonical ensemble: applications to polymer brushes.

We have used the dissipative particle dynamics (DPD) method in the grand canonical ensemble to study the compression of grafted polymer brushes in good solvent conditions. The force-distance profiles calculated from DPD simulations in the grand canonical ensemble are in very good agreement with the self-consistent field (SCF) theoretical models and with experimental results for two polystyrene brush layers grafted onto mica surfaces in toluene.     

Donnan potential and surface potential of a spherical soft particle in an electrolyte solution

A simple numerical method, which does not involve numerical integration of the Poisson-Boltzmann equations, is presented for obtaining the relationship between the Donnan potential and surface potential of a spherical soft particle (i.e., a polyelectrolyte-coated particle) in a symmetrical electrolyte solution. We assume that a soft particle consists of the particle core of radius a covered with an ion-penetrable surface layer of polyelectrolytes of thickness d and that ionized groups of valence Z are distributed at a uniform density of N in the polyelectrolyte layer and the relative permittivity takes the same value in the regions outside and inside the polyelectrolyte layer. The Donnan potential and surface potential are determined by the values of a, d, Z, N, and the Debye-Hückel parameter kappa of the electrolyte solution. Numerical results obtained by the present method are in excellent agreement with exact results obtained by solving the nonlinear spherical Poisson-Boltzmann equations for the both regions inside and outside the polyelectrolyte layer.     

An approach to isoindole skeleton via ortho palladation

An acid-induced cyclisation of the products of vinylation of ortho palladated NN-dimethylbenzylamines by 3-buten-2-one (II) leads to isoindolinium derivatives (IV) as confirmed by an X-ray structural study.     

Electrical potential in a cylindrical double layer: a functional theory approach

Employing an iterative method in functional theory, the electrical potential distribution for the case of a cylindrical surface is solved. Although the analytical result derived is of an iterative nature, the second-order solution is found to be sufficiently accurate under conditions of practical significance. For the case of constant surface potential, the radius and the surface potential of a cylindrical surface can be estimated based on the extreme of the electrical potential distribution. The effects of the key parameters, including the number and the valence of the ions on a surface, the length of a particle, the relative permittivity of the liquid phase, the temperature, and the concentration of electrolyte on the surface potential, are examined. The general behavior of these effects is similar to that for a spherical surface, except that the surface potential of a cylindrical surface is independent of the electrolyte concentration. The present approach is also applicable to the case where a cylindrical surface remains at a constant charge density.