The convergence properties of direct energy minimisation with respect to linear coefficients in the MC-LCAO-MO-SCF approach

It is shown that in the LCAO-MO-MC-SCF problem, if the molecular orbital orthonormality constraints are introduced in the manner suggested by Kari and Sutcliffe or indeed by any similar method then the Hessian of the problem with respect to the linear coefficients is singular. The nature of this singularity is analysed and it is shown that in general it is possible to remove it in a level-shifting-like scheme, but that only in certain special cases is this procedure likely to be quickly convergent.     

How accurate is your two-dimensional numerical simulation? Part 1. An introduction

Numerical simulations of the diffusion processes at electrode surfaces are subject to three sources of error: those arising in the calculation of the concentration values, those arising in the numerical approximation of the flux at the electrode surface, and those arising from the integration of the flux over the electrode surface. In this paper we investigate the effects of each type of error on the accuracy of numerical simulation at the microdisc electrode by solving the steady state problem (for which the analytical solution is known). We are able to show that the major source of error is due to the boundary singularity at the electrode edge. By introducing a simple model problem, we demonstrate that the theoretical rates of convergence of the standard finite difference schemes can be attained in the absence of a boundary singularity, but these rates are destroyed by the presence of the singularity when solving for the electrode problem. Finally, we show that it is not possible to recover accuracy using n-point flux calculations or spline functions at the electrode edge.     

Exact and approximate Hartree–Fock calculations for extended metallic systems

Starting out with the electron gas, we make a survey of the reasons for the singularity in the derivative of the orbital energy with respect to the wave number at the Fermi level for a realistic extended metallic system. Some properties of the occupation function are reviewed and it is pointed out that the direct reason for the singularity resides in a divergent lattice sum originating in the exchange part of the orbital energy. Numerical aspects are discussed, in particular with reference to the difficulty in detecting this singularity in actual computations.     

Electric fields in an electrolyte solution near a strip of fixed potential

Electrostatic fields produced by flat electrodes are often used to manipulate particles in solution. To study the field produced by such an electrode, we consider the problem of an infinite strip of width 2a with imposed constant potential immersed in an electrolyte solution. Sufficiently close to the edge of the strip, the solution is determined by classical electrostatics and results in a field singularity. We examine two limiting cases, (a) when strip width a<1k, the Debye screening length, and (b) when strip width is much greater than the Debye screening length, a>1k. We present exact results for the two cases in the limit of small potentials where the Poisson-Boltzmann equation can be linearized. By drawing on an analogy with antiplane shear deformations of solids, and by employing the path-independent J integral of solid mechanics, we present a new method for determining the strength of the edge singularity. The strength of the singularity defines an exact near-field solution. In the far field the solution goes to that of a line of charge. The accuracy of the solution is demonstrated by comparison with the numerical solutions of the Poisson-Boltzmann equation using the finite element method.     

Avoiding singularity problems associated with meta-GGA (generalized gradient approximation) exchange and correlation functionals containing the kinetic energy density

Convergence problems of meta-GGA (generalized gradient approximation) XC (exchange and correlation) functionals containing a self-interaction correction term are traced back to a singularity of the latter that occurs at critical points of the electron density. This is demonstrated for the bond critical point of equilibrium and stretched H2. A simple remedy is suggested that cures meta-XC functionals such as VSXC, TPSS, M05, M06, and their derivatives without extra cost.     

MOOC资源的有效利用 ——基于SPOC的无机及分析化学教学改革

In view of the increasing quantity of MOOC curriculum construction and the utilization of MOOC resources, this paper puts forward how to efficiently make use of MOOC resources through deepening curriculum teaching reform. In this paper, the problems in MOOC teaching are summarized, and then the SPOC course is introduced. Along with introducing SPOC course, taking the course of inorganic and analytical chemistry in Huazhong Agricultural University as an example, four main functions of SPOC are expounded emphatically which include solving the problem of cultivating students' autonomous learning ability, solving the problem of obtaining knowledge from MOOC in a piecemeal way, solving the concern about the effect of online learning and solving the low participation of MOOC. It is concluded that the course of SPOC is an important way to make good use of MOOC. It is suggested that the co-construction and sharing of MOOC should be strengthened to enhance the efficiency of MOOC.     

The theory of local degeneracy

The concept of a locally degenerate Hamiltonian is defined. It is shown that such a Hamiltonian is, with respect to a particular quantum number, degenerate in a single level only. Some examples of single-particle systems suggested by a study of the stereographic projection are examined, a comparison being made with the Kepler problem. It is found, in these examples, that the degenerate level lies at the limiting value of the potential energy at large distances and the condition for the existence and the properties of these states are deduced. It is also shown that for such states certain operators may have eigenfunctions in common with the Hamiltonian and for this reason they are called local constants of the motion. Like ordinary constants of the motion, they help to simplify the eigenvalue problem and, in particular, the singular inverse square law potential is studied by this approach. Finally, the application of the results to resonance scattering (by bound states near the continuum) and to the calculation of the number of bound states is indicated.     

A new computational method for estimating X-ray spectral distributions

A new computational method is described for estimating the exposure-rate spectral distributions of X-rays from attenuation data measured with various filtrations. The estimation problem of X-ray spectra is formulated as the numerical computation of solving a set of linear equation with an ill-conditional nature. In this paper, the singular-value decomposition technique, which differs from the iterative method, is applied to this singular numerical computation problem. The principle of the analysis method is based on that the response matrix of filtrations can be decomposed into some inherent component matrices. X-ray spectral distributions are then represented in a simple combination of some component curves, so that the estimation process can be systematically constructed. The singularity in its computation is removed by selecting the components of the combination, and a performance index is also presented for the optimal selection. The feasibility of the proposed method is studied in detail in a computer simulation using a hypothetical X-ray spectrum produced by assuming experimental conditions. The application results are also shown about the spectral distribution from a 140 kV constant voltage X-ray source.     

Derivation of the electronic nonadiabatic coupling field in molecular systems: an algebraic-vectorial approach

In this Communication it is suggested that various elements of the nonadiabatic coupling matrix, tau(jk)(s) are created by the singular nonadiabatic coupling terms of the system. Moreover, given the spatial distribution of these coupling terms in the close vicinity of their singularity points yields, according to this approach, the integrated intensity of the field at every point in the region of interest. To support these statements we consider the conical intersections of the three lower states of the H+H(2) system: From an ab initio treatment we obtain the nonadiabatic coupling terms around each conical intersection separately (at its close vicinity) and having those, create the field at every desired point employing vector-algebra. This approach is also used to calculate the intensity of the Curl of those matrix elements that lack their own sources [tau(13)(s) in the present case]. The final results are compared with relevant ab initio calculations.     

Multifractal analysis of dynamic potential surface of ion-conducting materials

A multifractal analysis using singularity spectra [T.C. Halsey et al., Phys. Rev. A 33, 1141 (1986)] provides a general tool to study the temporal-spatial properties of particles in complex disordered materials such as ions in ionically conducting glasses and melts. Obtained by molecular-dynamics simulations, the accumulated positions of the particles dynamically form a structural pattern called the dynamical potential surface. In this work, the complex dynamical potential surfaces of Li ions in the lithium silicates were visualized and characterized by the multifractal analysis. The fractal dimensions and strength of the singularity related to the spatial intermittency of the dynamics are examined, and the relationship between dynamics and the singularity spectra is discussed.     

Phenomenological description of transport properties of three-phase composites

The possibility is discussed of estimating the values of conductivity and dielectric permeability of three-phase composites using the modified mixing equation similar to the equation suggested earlier for two-phase systems. It is shown that the choice of theoretical mixing equation parameters for three-phase composites of the conductor–electrolyte–insulator type can be controversial; variants of solving this problem are suggested. According to the equation, three percolation thresholds can exist in this three-phase system; their position can be estimated on the basis of the maximum of dielectric permeability. The value of the maximum is determined by the ratio of conductivities of the conducting and dielectric phases. The equation can be extended to the case when interaction exists between the individual phases of the composite and a highly conducting layer is formed at the interface between these phases. The calculation of three-phase composites of the conductor–composite solid electrolyte–insulator type is calculated for the first time. It is shown that the mixing equation also allows calculating electric properties of composites in this case.     

Branch point singularities in the energy of the delta-function model of one-electron diatoms

Two different perturbation series (the polarization expansion and a united-atom expansion) of the ground state energy of the delta-function model for one-electron diatoms are studied and the radii of convergence are determined. For both expansions the singularity in the energy which limits the radius of convergence is a branch point with exponent one-half. The physical significance of the branch point is that for particular values of the perturbation parameter, two different energy eigenvalues coalesce. The positions of the branch points are computed as a function of the internuclear separation R. For all values of R, both series converge for all physical values of the perturbation parameters. A lower bound to the radius of convergence of the polarization expansion has been computed previously by Claverie. It is proved in the present paper that the lower bound calculation is in fact an exact determination of the radius of convergence. The results of the model study are applied to real one-electron diatoms to suggest the possible location of a branch point singularity in the energy of the ground state.     

Influence of thermodynamic conditions of polymerization on supramolecular and molecular structure of the polymers

The problems of the formation of the supramolecular and molecular structure of polymers are considered. Based on kinetic investigations of trioxane polymerization, the thermodynamic approach to the problem of regulating the supramolecular and molecular structure of polymers during synthesis is formulated. A method for producing polymers with the given supramolecular structure is suggested. It is noted that the application of this method allows one to regulate the structure during the synthesis of three-dimensional cross-linked polymer systems as single-component, filled, reinforced, and other composite materials.     

Critical parameters from power series expansions

A simple and quite general method is developed for calculating critical para meters from power series expansions. The series coefficients of the problem function are introduced into a properly parametrized recurrence relationship. As a result, sequences are obtained that converge towards the critical parameters characterizing the closest singularity to the origin. The low-temperature series for the thermodynamic functions of spin-1/2 Ising models are discussed. Results are shown for the interfering, nonphysical singularities of the face-centred cubic and body-centred cubic lattices. The determination of these singularities would allow their factorization previous to the study of the physically important ones.     

Adaptive finite element methods in electrochemistry

In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the edge effect. Our approach to overcoming this problem has involved the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm, allowing calculation of the quantity of interest (the current) to within a prescribed tolerance. We illustrate the generic applicability of the approach by considering a broad range of steady-state applications of the technique.     

Adaptative finite element simulation of currents at microelectrodes to a guaranteed accuracy. Application to a simple model problem

In this series of papers we consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the ‘edge-effect’. Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current that can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. We begin by demonstrating the power of the method for a simple model problem — an E reaction mechanism at a microdisc electrode — for which the analytical solution is known. In this paper we give the background to the problem, and show how an a posteriori error bound can be used to drive an adaptive mesh-generation algorithm. We then use the algorithm to solve our model problem and obtain very accurate results on comparatively coarse meshes in minimal computing time. We give the technical details of the background theory and the derivation of the error bound in the accompanying paper.     

A new correlation effect in the Helmholtz and surface potentials of the electrical double layer

The restricted primitive model of an electrical double layer around a spherical macroparticle is studied by using integral equation theories and Monte Carlo simulations. The resulting theoretical curves for the Helmholtz and surface potentials versus the macroparticle charge show an unexpected positive curvature when the ionic size of uni- and divalent electrolyte species is increased. This is a novel effect that is confirmed here by computer experiments. An explanation of this phenomenon is advanced in terms of the adsorption and layering of the electrolytic species and of the compactness of the diffuse double layer. It is claimed that the interplay between electrostatic and ionic size correlation effects, absent in the classical Poisson-Boltzmann view, is responsible for this singularity.     

Interface crack tip singularity with contact and friction

The interface crack tip singularity is numerically investigated within the generalised plane strain framework, with anisotropic substrates. Under the assumption that there is a contact zone along the crack front, the frictionless contact case and Coulomb's friction law are examined. The slip direction being prescribed, it is then shown that friction acts like a crack tip blunting, and the singularity is weaker than in the frictionless contact or crack opening cases.