Linear Variational Method Research Articles

A two-dimensional harmonic oscillator confined in a circle in the presence of a constant electric field: an informational approach

In this work, we study an electron subjected to a harmonic oscillator potential confined in a circle of radius r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document} and in the presence of a constant electric field. We obtain energies and eigenfunctions for three different confinement radii as a function of the electric field strength. We have used the linear variational method by constructing the trial function as a linear combination of two-dimensional confined harmonic oscillator wave functions. We calculate the radial standard deviation as a measure of the dispersion of the probability density. We also computed the Shannon entropy and Fisher information, in configuration and momentum spaces, as localization-delocalization measures for three different confinement radii and as a function of the electric field strength. We find that Shannon entropy and Fisher information are more reliable than variance in determining electron location. The behaviour of Shannon entropy and Fisher information curves is shown to depend on each specific state under study.Graphical abstract

Cosmic string influence on a 2D hydrogen atom and its relationship with the Rytova–Keldysh logarithmic approximation in semiconductors

A two-dimensional hydrogen atom offers a promising alternative for describing the quantum interaction between an electron and a proton in the presence of a cosmic string. Reducing the hydrogen atom to two dimensions enhances its suited to capture the cylindrical/conical symmetry associated with the cosmic string, providing a more appropriate description of the physical system. After solving Schrd̈inger’s equation, we calculate the eigenenergies, probability distribution function, and expected values for the hydrogen atom with logarithmic potential under the influence of the topological defect. The calculations for the 2D hydrogen atom are performed for the first time using the Finite Difference Method. The results are presented through graphics, tables, and diagrams to elucidate the system’s physical properties. We have verified that our calculations agree with a linear variational method result. Our model leads to an interesting analogy with excitons in a two-dimensional monolayer semiconductor located within a specific semiconductor region. To elucidate this analogy, we present and discuss some interaction potentials and their exciton eigenstates by comparing them with the results from the literature.

The one-dimensional hydrogenic impurity states confined at one end of the InAs quantum well

ABSTRACT The one-dimensional (1-D) confined hydrogenic impurity states confined in the InAs quantum-well has been studied due to its wide application in manufacturing high-electron-mobility transistors in semiconductor physics. We calculate the energy levels and wave functions of this system using the linear variational method. Choosing the wave function of a particle confined in a 1-D infinitely deep quantum well as a basis set, we obtain the numerical solutions of the energy levels and wave functions of the hydrogenic impurity states confined at one end of the InAs quantum-well by solving the Hamiltonian matrix equation. It is found when the quantum well is very narrow, the eigen-energy approaches to the exact solution with a small number of basis set functions; whereas for a relatively wide quantum well, it is necessary to use a large number of basis set functions. The confinement effect of the quantum well on the energy level and binding energy of the hydrogenic impurity state has been studied in great detail. Although the linear variational method used in this work has only been applied to the hydrogenic impurity state confined in the quantum well, for the non-hydrogenic confined systems, this method is still applicable. Therefore, our work provides a new method to study the energy level structure of the impurity state confined in the external environment and has some practical applications in the fields of semiconductor physics, material science, etc.

Entropy in Li- and Mn-Rich Layered Oxides Measured Via Linear Temperature Variation

For Li-ion battery development, Li- and Mn-rich layered oxide cathodes (LMRNCM, Li1+x[NiCoMn]1-xO2 with typically 0.1 < x < 0.2), are currently under development. Their exceptionally high gravimetric capacity is, however, accompanied by a significant voltage hysteresis, which complicates the SOC management and reduces the round-trip energy efficiency of the cell.1 In particular, a significant part of the voltage hysteresis is still present during open circuit conditions (OCV) being independent of the applied current and thus a material-specific property (see Figure 1a). The hysteresis is not only expressed in the voltage profile but also in other parameters, such as a path dependent cathode resistance2 and a hysteresis of the LMRNCM lattice parameters3.To investigate the OCV hysteresis, potentiometric entropy measurements were conducted to calculate the changes in the partial molar free entropy of the Li (de-)intercalation during charge and discharge, and from this, the reversible heat Qrev. The entropy is accessible via the temperature dependence of the OCV. To minimize errors from temperature history4 and relaxation5, a linear temperature variation method was established based on the work by Liebmann et al. 6 Using this method, entropy changes within the LMRNCM were measured as a function of state of charge (SOC) to reveal a path dependence between charge and discharge (see Figure 1b). This path dependence vanishes when the entropy is correlated to the OCV of the respective SOC indicating that the structural changes within the LMRNCM are rather a function of OCV than SOC, as can be seen in Figure 1c. This is in agreement with a previous study by Strehle et al.3 where a similar behavior was found for the lattice parameters of LMRNCM by diffraction methods. However, the herein conducted measurements did not reveal the entropy as a cause of the path-dependence (or hysteresis) but rather demonstrated it to be another indication of this phenomenon. Ultimately, the reversible heat Qrev will be compared to the energy loss correlating to the OCV hysteresis indicating that the latter heat term is significantly larger and independent from the entropy changes within the LMRNCM. Figure 1: (a) Voltage hysteresis and (b,c) changes in the partial molar free entropy for LMRNCM. A LMRNCM/Li Swagelok T-cell (equipped with a Li reference electrode) was cycled at C/10 at 25°C between 2.0-4.7 V. For the OCV curve, intermediate OCV phases of 1h were applied every ≈10% SOC (a). The entropy was measured in various T-cells via linear temperature variation after relaxation of the OCV at different SOCs during charge and discharge and is shown as a function of SOC (b) and OCV (c) with trendlines as guide for the eye.Acknowledgements:We want to acknowledge BASF SE for the support within the frame of its scientific network on electrochemistry.

Entropy Measurements of Cells with Li- and Mn-Rich Layered Oxides Measured Via Linear Temperature Variation

For Li-ion battery development, Li- and Mn-rich layered oxide cathodes (HE-NCM, Li1+x[M]1-xO2 with M = Ni, Mn, Co and typically 0.1 < x < 0.2) are currently under development. Their exceptionally high gravimetric capacity is, however, accompanied by a significant voltage hysteresis which reduces the round-trip energy efficiency of the cell.1 In particular, a significant part of the voltage hysteresis is still present during measurements at open circuit voltage (OCV hysteresis, see Figure 1a) and hence leads to an energy loss, which is independent of the applied current.To investigate this OCV hysteresis, potentiometric entropy measurements were conducted in half-cells to calculate the overall entropy changes ΔS of Li (de-)intercalation during charge and discharge, and from this, the reversible heat Qrev. ΔS is accessible via the temperature dependence of the OCV. To minimize errors from temperature history2 and relaxation3, a linear temperature variation method was established based on the work by Liebmann et al. 4 Using this method, entropy changes of Li/Mn-rich NCM half-cells were measured as a function of state of charge (SOC), revealing a path dependency of ΔS between charge and discharge, which can only be ascribed to entropy changes of the cathode active material (see Figure 1b). This path dependency vanishes when the entropy is correlated to the OCV of the respective Mn-rich NCM SOC, indicating that the structural changes within the cathode are rather a function of OCV than SOC, as can be seen in Figure 1c. This is in agreement with a previous study by Strehle et al.,5 where a similar behavior was found for the hysteresis in the evolution of the Mn-rich NCM lattice parameters between the charge and the discharge direction. However, the measurements conducted in the present study did not reveal the entropy as a cause of the path-dependency (or hysteresis), but rather demonstrated it to be another indication of this phenomenon. Ultimately, the reversible heat Qrev will be compared to the energy loss correlating to the OCV hysteresis, QOCV, indicating that the latter heat term is significantly larger and independent of the entropy changes within the Mn-rich NCM.Acknowledgements:We want to acknowledge BASF SE for the support within the frame of its scientific network on electrochemistry.

Dilemma of the “Best Wavefunction”: Comparing Results of the STO-NG Procedure versus the Linear Variational Method

The aim of this article is to present a set of simple exercises that help the students of a second course in quantum chemistry to understand that some molecular properties do not improve even when the molecular energy improves with an increasing number of basis functions in the expansion of the wavefunction. The essential idea is to create a dilemma for the students by showing that a wavefunction that gives a better energy (i.e., lower, according to the variational principle) may not yield a better value for another molecular property. The first exercise is to calculate the ground-state energy of the hydrogen atom for the contracted 1s STO-2G basis function; the second is to compute the ground-state energy for the STO-2G basis function reformulated as a linear variational function; the third is to calculate the expectation value of the electron–nuclear distance for the two approximate wavefunctions. Comparison of the approximate results with the exact one reveals starkly the dilemma of the “best wavefunction”. The set of exercises presented here are interesting to any quantum mechanics course.

Second-Order Chaos Indicators MEGNO2 and OMEGNO2: Theory

Modifications of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) linear variational method called MEGNO2 and OMEGNO2 indicators are introduced. The modifications are based on taking into account not only the linear, but also the nonlinear part of the increment of the phase flow in the divergence among nearby trajectories according to the second-order formulas. The new indicators allow one to determine more quickly the nature of the orbits under study in dynamical systems with zero or small Lyapunov exponents in comparison with the first-order variational indicators. They improve the analysis of regular regions and, in particular, periodic orbits as well as prevent the appearance of spurious structures in the resulting mappings.

The confined hydrogen atom: a linear variational approach

We study the size effect on the confinement of a hydrogen atom in a spherical box of impenetrable walls. We compute the energy of the ground and a few excited states as a function of the box radius Rc. To obtain the energy eigenvalues and eigenfunctions we utilize the linear variational method via a basis set of free-particle-in-a-box wave functions. For small values of Rc convergence is attained with a small number of basis set functions, whereas for Rc ≥ 5.0 au, it is necessary to use over 100 terms in the expansion.

Some problems in applications of the linear variational method

The linear variational method is a standard computational method in quantum mechanics and quantum chemistry. As taught in most classes, the general guidance is to include as many basis functions as practical in the variational wave function. However, if it is desired to study the patterns of energy change accompanying the change of system parameters such as the shape and strength of the potential energy, the problem becomes more complicated. We use one-dimensional systems with a particle in a rectangular or in a harmonic potential confined in an infinite rectangular box to illustrate situations where a variational calculation can give incorrect results. These situations result when the energy of the lowest eigenvalue is strongly dependent on the parameters that describe the shape and strength of the potential. The numerical examples described in this work are provided as cautionary notes for practitioners of numerical variational calculations.

Robust stability analysis for discrete and distributed time-delays Markovian jumping reaction-diffusion integro-differential equations with uncertain parameters

In this paper, the authors employ Lyapunov stability theory, and the M-matrix, H-matrix, and linear matrix inequality (LMI) techniques and variational methods to obtain the LMI-based stochastically exponential robust stability criterion for discrete and distributed time-delays Markovian jumping reaction-diffusion integro-differential equations with uncertain parameters, whose background of physics and engineering is bidirecional associative memory (BAM) neural networks. It is worth mentioning that an LMI-based stability criterion can easily be computed by the Matlab toolbox which has high efficiency and other advantages in large-scale engineering calculations. Since using the M-matrix and H-matrix methods is not easy in obtaining the LMI criterion conditions, the methods employed in this paper improve those of previous related literature to some extent. Moreover, a numerical example is presented to illustrate the effectiveness of the proposed methods.

Polarizabilities and Hyperpolarizabilities for Lithium Atoms in Debye Plasmas

The effects of plasma environments on energies, oscillator strengths, polarizabilities and hyperpolarizabilities for lithium atom have been calculated by combining the 1-dependent model potential of free lithium atom and linear variation method based on B-spline basis functions. The influence of plasma on lithium atom is represented by the Debye screened potential, which describes effectively the averaged effect of the plasma environment on atomic spectra. The results are in agreement with other reported ones.

Variational Approach to Molecular Kinetics.

The eigenvalues and eigenvectors of the molecular dynamics propagator (or transfer operator) contain the essential information about the molecular thermodynamics and kinetics. This includes the stationary distribution, the metastable states, and state-to-state transition rates. Here, we present a variational approach for computing these dominant eigenvalues and eigenvectors. This approach is analogous to the variational approach used for computing stationary states in quantum mechanics. A corresponding method of linear variation is formulated. It is shown that the matrices needed for the linear variation method are correlation matrices that can be estimated from simple MD simulations for a given basis set. The method proposed here is thus to first define a basis set able to capture the relevant conformational transitions, then compute the respective correlation matrices, and then to compute their dominant eigenvalues and eigenvectors, thus obtaining the key ingredients of the slow kinetics.

High-Order Dispersion Coefficients for Alkali-metal Atoms

High-order dispersion coefficients C9, C11, C12, and C13 for the ground-state alkali-metals were calculated by combining the l-dependent model potential of alkali-metal atoms and linear variation method based on B-spline basis functions. The results were compared.

Stochastic stability criteria with LMI conditions for Markovian jumping impulsive BAM neural networks with mode-dependent time-varying delays and nonlinear reaction–diffusion

In this paper, the problem of stochastic stability for Markovian jumping impulsive BAM neural networks with mode-dependent time-varying delays and nonlinear diffusion has been investigated. The non-linear p-Laplace diffusion gives a great difficulty in playing its role about judging the stability of stochastic system. By employing the Lyapunov technique, the linear matrix inequality (LMI) approach and variational methods in Sobolev space W1,p(Ω), a series of new criteria are obtained. The new criteria have advantages over some previous ones due to less conservatism and higher computational efficiency. Besides, since linear Laplace reaction–diffusion is only the special example of p-Laplacian (in the case of p=2), the new criteria and the numerical example’s methods are more general than some existing results. A numerical example is given to illustrate the effectiveness of the proposed methods.

The hydrogen atom confined in both Debye screening potential and impenetrable spherical box

Abstract The ground state energy, some low-lying excited state energies and oscillator strengths for a hydrogen atom confined in both a Debye screening potential and finite impenetrable spherical box have been calculated. These have been calculated using a linear variational method based on B-spline basis functions. The results have been compared with those of other authors. The evaluated energies and oscillator strengths with respect to different plasma screening parameters with a certain confinement radii are discussed.

Using the Screened Coulomb Potential To Illustrate the Variational Method

The screened Coulomb potential, or Yukawa potential, is used to illustrate the application of the single and linear variational methods. The trial variational functions are expressed in terms of Slater-type functions, for which the integrals needed to carry out the variational calculations are easily evaluated in closed form. The variational energies of the first two states of the screened Coulomb potential are calculated by minimizing them with respect to the exponential parameters included in the Slater functions using elementary minimization algorithms. The project allows students to concentrate on the fundamentals and the performance of the variational method at its increasingly accurate levels of application to realistic systems.

Statistics and Correlation Properties of Diamagnetic High Rydberg Hydrogen Atom

The spectra of Rydherg hydrogen atom in magnetic fields have been calculated using linear variational method with B-splines basis functions [Acta Phys. Sin. 55 (2006) 3380]. Based on these calculations we have done some statistics analysis about the high Rydberg energy levels. The nearest-neighbor energy spacing distribution and the 3-statistics have been shown about diamagnetic Rydberg hydrogen atom with the magnetic field being 0.6 T and 6 T. The phenomena of multiply crossing, multiply anti-crossing, and the mixed of crossing and anti-crossing of energy levels have appeared in this paper. For both cases, in range of lower energy, the energy level statistics properties close to Poisson distribution. With the increasing of the energy, the energy level statistics properties are away to Poisson distribution and tend to Wigner distribution step by step.

A linear variational exercise with a simple non-orthogonal basis for the particle-in-the-box problem

The particle-in-the-box, with or without an additional potential, is proposed as an excellent laboratory to teach and explore the details of the linear variational method using a non-orthogonal basis. The xn(a − x)n and xn(a/2 − x)(a − x)n polynomials are shown to form a complete basis for the even and odd states, respectively, of the particle confined to the x ∊ [0, a] interval. A short and simple Octave code is presented as the natural extension to the hand calculations when the basis set grows in size.

Recent developments for treating non-muffin-tin potentials and charge densities in band structure and cluster calculations

An increasing interest in both more adequate one-electron model Hamiltonians and more precise calculations of eigenvalues and total energies for clusters and solids has led to renewed efforts to devise efficient computational schemes capable of treating general potentials with high accuracy. In this communication, a linear variational method for generalizing the Korringa-Kohn-Rostoker (KKR) method to treat general one-electron potentials and charge densities is briefly reviewed and compared with several other recent approaches to this problem.

Estimate on the Dimension of Global Attractor for Nonlinear Dissipative Kirchhoff Equation

In this paper,we estimate the dimension of the global attractor for nonlinear dissipative Kirchhoff equation in Hilbert spaces H01×L2(Ω) and D(A)×H01(Ω). Using rescaling technology and linear variation method, we obtain the upper bound for its Hausdorff and fractal dimensions.