Accuracy of transfer matrix approaches for solving the effective mass Schr\"{o}dinger equation

The accuracy of different transfer matrix approaches, widely used to solve the stationary effective mass Schr\"{o}dinger equation for arbitrary one-dimensional potentials, is investigated analytically and numerically. Both the case of a constant and a position dependent effective mass are considered. Comparisons with a finite difference method are also performed. Based on analytical model potentials as well as self-consistent Schr\"{o}dinger-Poisson simulations of a heterostructure device, it is shown that a symmetrized transfer matrix approach yields a similar accuracy as the Airy function method at a significantly reduced numerical cost, moreover avoiding the numerical problems associated with Airy functions

A comparative study of two molecular mechanics models based on harmonic potentials

We show that the two molecular mechanics models, the stick-spiral and the beam models, predict considerably different mechanical properties of materials based on energy equivalence. The difference between the two models is independent of the materials since all parameters of the beam model are obtained from the harmonic potentials. We demonstrate this difference for finite width graphene nanoribbons and a single polyethylene chain comparing results of the molecular dynamics (MD) simulations with harmonic potentials and the finite element method with the beam model. We also find that the difference strongly depends on the loading modes, chirality and width of the graphene nanoribbons, and it increases with decreasing width of the nanoribbons under pure bending condition. The maximum difference of the predicted mechanical properties using the two models can exceed 300% in different loading modes. Comparing the two models with the MD results of AIREBO potential, we find that the stick-spiral model overestimates and the beam model underestimates the mechanical properties in narrow armchair graphene nanoribbons under pure bending condition.Comment: 40 pages, 21 figure

High-order numerical methods for 2D parabolic problems in single and composite domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests-with respect to accuracy and convergence-for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin

High-temperature surface diffusion of copper on the (112) face of tungsten under condition of film-layer growth of adsorbed film

The phase state of the epitaxial Cu film on W(112) face has been investigated by the method of the contact difference of potentials under condition of film-layer growth. We have determined desorption heat, critical temperature and critical coverage experimentally. The phase diagram has been plotted as well as the temperature dependence of heat of a two-dimensional phase transition “liquid – gas” has been obtained. An exponent of order parameter has been found. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/2097

Molecular double core-hole electron spectroscopy for chemical analysis

We explore the potential of double core hole electron spectroscopy for chemical analysis in terms of x-ray two-photon photoelectron spectroscopy (XTPPS). The creation of deep single and double core vacancies induces significant reorganization of valence electrons. The corresponding relaxation energies and the interatomic relaxation energies are evaluated by CASSCF calculations. We propose a method how to experimentally extract these quantities by the measurement of single and double core-hole ionization potentials (IPs and DIPs). The influence of the chemical environment on these DIPs is also discussed for states with two holes at the same atomic site and states with two holes at two different atomic sites. Electron density difference between the ground and double core-hole states clearly shows the relaxations accompanying the double core-hole ionization. The effect is also compared with the sensitivity of single core hole ionization potentials (IPs) arising in single core hole electron spectroscopy. We have demonstrated the method for a representative set of small molecules LiF, BeO, BF, CO, N2, C2H2, C2H4, C2H6, CO2 and N2O. The scalar relativistic effect on IPs and on DIPs are briefly addressed.Comment: 35 pages, 6 figures. To appear in J. Chem. Phys

The Theory of Difference Potentials in the Three-Dimensional Case

The method of difference potentials can be used to solve discrete elliptic boundary value problems, where all derivatives are approximated by finite differences. Considering the classical potential theory, an integral equation on the boundary will be investigated, which is solved approximately by the help of a quadrature formula. The advantage of the discrete method consists in the establishment of a linear equation system on the boundary, which can be immediately solved on the computer. The described method of difference potentials is based on the discrete Laplace equation in the three-dimensional case. In the first step the integral representation of the discrete fundamental solution is presented and the convergence behaviour with respect to the continuous fundamental solution is discussed. Because the method can be used to solve boundary value problems in interior as well as in exterior domains, it is necessary to explain some geometrical aspects in relation with the discrete domain and the double-layer boundary. A discrete analogue of the integral representation for functions in will be presented. The main result consists in splitting the difference potential on the boundary into a discrete single- and double-layer potential, respectively. The discrete potentials are used to establish and solve a linear equation system on the boundary. The actual form of this equation systems and the conditions for solvability are presented for Dirichlet and Neumann problems in interior as well as in exterior domain

Local-basis Difference Potentials Method for elliptic PDEs in complex geometry

We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allow unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically