Electrochemical noise spectroscopy: Method of secondary Chebyshev spectrum

A new method of electrochemical noise diagnostics is presented: the method of the secondary Chebyshev spectrum based on the splitting of an individual spectral line in the primary Chebyshev spectrum with formation of a system of spectral lines of the secondary Chebyshev spectrum. Algorithm for calculation of the secondary Chebyshev spectrum is developed. The suggested method based on analysis of noises measured in a specific electrochemical system is tested. It is shown that the new method allows determining the differences in the state of the electrochemical system more reliably, than the method of primary Chebyshev noise spectra.     

Analyzing electrochemical noise with Chebyshev’s discrete polynomials

The principles underlying a novel method intended for analyzing experimental data obtained when studying fluctuation processes are considered. The method in question is Chebyshev’s spectroscopy. The application of this method allows one to determine statistic characteristics of steady-state electrochemical noise against the background of severe deterministic interference without invoking the procedure of the fitting of the initial data. The potentialities of this novel method, which is intended for treating noise experiment, are demonstrated by examining model examples and analyzing the electrochemical noise generated by a lithium electrode placed in an aprotic organic electrolyte.     

Discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise

A discrete version of Wiener-Khinchin theorem for Chebyshev’s spectrum of electrochemical noise is developed. Based on the discrete version of Wiener-Khinchin theorem, the theoretical discrete Chebyshev spectrum for the Markov random process is calculated. It is characterized by two parameters: the dispersion and the relaxation frequency (or relaxation time). The noise of corrosion process and the noise of recording equipment are measured. Using the theoretical Chebyshev spectrum, the Markov parameters were found both for the noise of the corrosion process and for the noise of the measuring equipment.     

Analysis of the topological dependency of the characteristic polynomial in its chebyshev expansion

The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z _G ) was discussed. Operated for the U.S. Department of Energy by ISU under contract no. W-ENG-7405-82. Supported in part by the Office of Director     

Electrophoresis simulations using Chebyshev pseudo-spectral method on a moving mesh

We present the implementation and demonstration of the Chebyshev pseudo-spectral method coupled with an adaptive mesh method for performing fast and highly accurate electrophoresis simulations. The Chebyshev pseudo-spectral method offers higher numerical accuracy than all other finite difference methods and is applicable for simulating all electrophoresis techniques in channels with open or closed boundaries. To improve the computational efficiency, we use a novel moving mesh scheme that clusters the grid points in the regions with poor numerical resolution. We demonstrate the application of the Chebyshev pseudo-spectral method on a moving mesh for simulating nonlinear electrophoretic processes through examples of isotachophoresis (ITP), isoelectric focusing (IEF), and electromigration-dispersion in capillary zone electrophoresis (CZE) at current densities as high as 1000 A/m. We also show the efficacy of our moving mesh method over existing methods that cluster the grid points in the regions with large concentration gradients. We have integrated the adaptive Chebyshev pseudo-spectral method in the open-source SPYCE simulator and verified its implementation with other electrophoresis simulators.     

Inelastic scattering with Chebyshev polynomials and preconditioned conjugate gradient minimization

We describe and test an implementation, using a basis set of Chebyshev polynomials, of a variational method for solving scattering problems in quantum mechanics. This minimum error method (MEM) determines the wave function Psi by minimizing the least-squares error in the function (H Psi - E Psi), where E is the desired scattering energy. We compare the MEM to an alternative, the Kohn variational principle (KVP), by solving the Secrest-Johnson model of two-dimensional inelastic scattering, which has been studied previously using the KVP and for which other numerical solutions are available. We use a conjugate gradient (CG) method to minimize the error, and by preconditioning the CG search, we are able to greatly reduce the number of iterations necessary; the method is thus faster and more stable than a matrix inversion, as is required in the KVP. Also, we avoid errors due to scattering off of the boundaries, which presents substantial problems for other methods, by matching the wave function in the interaction region to the correct asymptotic states at the specified energy; the use of Chebyshev polynomials allows this boundary condition to be implemented accurately. The use of Chebyshev polynomials allows for a rapid and accurate evaluation of the kinetic energy. This basis set is as efficient as plane waves but does not impose an artificial periodicity on the system. There are problems in surface science and molecular electronics which cannot be solved if periodicity is imposed, and the Chebyshev basis set is a good alternative in such situations.     

An efficient wavelet based spectral method to singular boundary value problems

In this paper, we have established an efficient wavelet based approximation method to nonlinear singular boundary value problems. To the best of our knowledge, until now there is no rigorous shifted second kind Chebyshev wavelet (S2KCWM) solution has been addressed for the nonlinear differential equations in population biology. With the help of shifted second kind Chebyshev wavelets operational matrices, the nonlinear differential equations are converted into a system of algebraic equations. The convergence of the proposed method is established. The power of the manageable method is confirmed. Finally, we have given some numerical examples to demonstrate the validity and applicability of the proposed wavelet method.     

The discrete Chebyshev algorithm for nonparametric estimation of autocorrelation function of electrochemical random time series

Journal of Solid State Electrochemistry - The discrete Chebyshev algorithm for nonparametric estimation of autocorrelation function of electrochemical random time series is presented. The algorithm...     

Density matrix based time-dependent density functional theory and the solution of its linear response in real time domain

A density matrix based time-dependent density functional theory is extended in the present work. Chebyshev expansion is introduced to propagate the linear response of the reduced single-electron density matrix upon the application of a time-domain delta-type external potential. The Chebyshev expansion method is more efficient and accurate than the previous fourth-order Runge-Kutta method and removes a numerical divergence problem. The discrete Fourier transformation and filter diagonalization of the first-order dipole moment are implemented to determine the excited state energies. It is found that the filter diagonalization leads to highly accurate values for the excited state energies. Finally, the density matrix based time-dependent density functional is generalized to calculate the energies of singlet-triplet excitations.     

Ergodic Analysis of Three-Dimensional Chebyshev Spectrum of Electrochemical Noise

Russian Journal of Electrochemistry - The ergodicity of electrochemical noise of corrosion process with respect to the three-dimensional Chebyshev spectrum is studied. The electrochemical noise of...     

Analysis of discrete spectra of electrochemical noise of lithium power sources

Journal of Solid State Electrochemistry - The Fourier, Daubechies, and Chebyshev transforms are used to analyze discrete spectra of electrochemical noise of lithium power sources under the...     

Real-time linear response for time-dependent density-functional theory

We present a linear-response approach for time-dependent density-functional theories using time-adiabatic functionals. The resulting theory can be performed both in the time and in the frequency domain. The derivation considers an impulsive perturbation after which the Kohn-Sham orbitals develop in time autonomously. The equation describing the evolution is not strictly linear in the wave function representation. Only after going into a symplectic real-spinor representation does the linearity make itself explicit. For performing the numerical integration of the resulting equations, yielding the linear response in time, we develop a modified Chebyshev expansion approach. The frequency domain is easily accessible as well by changing the coefficients of the Chebyshev polynomial, yielding the expansion of a formal symplectic Green's operator.     

Minimum-error method for scattering problems in quantum mechanics: Two stable and efficient implementations

We examine here, by using a simple example, two implementations of the minimum error method (MEM), a least-squares minimization for scattering problems in quantum mechanics, and show that they provide an efficient, numerically stable alternative to Kohn variational principle. MEM defines an error-functional consisting of the sum of the values of (HPsi - EPsi)2 at a set of grid points. The wave function Psi, is forced to satisfy the scattering boundary conditions and is determined by minimizing the least-squares error. We study two implementations of this idea. In one, we represent the wave function as a linear combination of Chebyshev polynomials and minimize the error by varying the coefficients of the expansion and the R-matrix (present in the asymptotic form of Psi). This leads to a linear equation for the coefficients and the R-matrix, which we solve by matrix inversion. In the other implementation, we use a conjugate-gradient procedure to minimize the error with respect to the values of Psi at the grid points and the R-matrix. The use of the Chebyshev polynomials allows an efficient and accurate calculation of the derivative of the wave function, by using Fast Chebyshev Transforms. We find that, unlike KVP, MEM is numerically stable when we use the R-matrix asymptotic condition and gives accurate wave functions in the interaction region.     

A vector processing algorithm of auxiliary integral evaluation for two-electron Gaussian integrals

A six-term auxiliary integral expression for the two-electron Gaussian integral is derived on the basis of the Chebyshev polynomial approximation instead of the seven-term Taylor expansion. This expression and the related recurrence formula enable us to perform a high-speed calculation on a vector processing computer.     

On the number of spanning trees in alternating polycyclic chains

In this paper we generalize and unify results of several recent papers by presenting explicit formulas for the number of spanning trees in a class of unbranched polycyclic polymers. From these formulas we immediately deduce the asymptotic behavior of the number of spanning trees, and, as a consequence, we obtain combinatorial proofs of some identities for Chebyshev polynomials of the second kind.     

Planar polycyclic graphs and their Tutte polynomials

We consider several classes of planar polycyclic graphs and derive recurrences satisfied by their Tutte polynomials. The recurrences are then solved by computing the corresponding generating functions. As a consequence, we obtain values of several chemically and combinatorially interesting enumerative invariants of considered graphs. Some of them can be expressed in terms of values of Chebyshev polynomials of the second kind.     

“Neglect of Diatomic Differential Overlap” in nonempirical quantum chemical orbital theories. IV. An examination of the justification of the neglect of diatomic differential overlap (NDDO) approximation

The Neglect of Diatomic Differential Overlap approximation is examined in terms of a polynomial expansion in Γ. The expansion is based upon the Legendre or Chebyshev approximation as developed in Part II. Analogous to the theorems of Chandler and Grader, NDDO cannot be justified for one-electron integrals and only partially for the two-electron repulsion integrals. © 1995 John Wiley & Sons, Inc.     

Generation of the eigenvectors of the topological matrix from graph theory

The technique of describing the characteristic polynomial of a graph is here extended to construction of the eigenvectors. Recurrence relations and path tracing are combined to generate eigenvector coefficients as polynomial functions of the eigenvalues. The polynomials are expressed as linear functions of Chebyshev polynomials in order to simplify the computational effort. Particular applications to the Hückel MO theory, including heteroatom effects, are shown.     

Minimax approximation for the decomposition of energy denominators in Laplace-transformed Møller-Plesset perturbation theories

We implement the minimax approximation for the decomposition of energy denominators in Laplace-transformed Moller-Plesset perturbation theories. The best approximation is defined by minimizing the Chebyshev norm of the quadrature error. The application to the Laplace-transformed second order perturbation theory clearly shows that the present method is much more accurate than other numerical quadratures. It is also shown that the error in the energy decays almost exponentially with respect to the number of quadrature points.     

Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studies methods based on Krylov subspaces for computing Brownian noise vectors. These methods are related to Chebyshev polynomial approximations, but do not require eigenvalue estimates. We show that only low accuracy is required in the Brownian noise vectors to accurately compute values of dynamic and static properties of polymer and monodisperse suspension models. With this level of accuracy, the computational time of Krylov subspace methods scales very nearly as O(N(2)) for the number of particles N up to 10 000, which was the limit tested. The performance of the Krylov subspace methods, especially the "block" version, is slightly better than that of the Chebyshev method, even without taking into account the additional cost of eigenvalue estimates required by the latter. Furthermore, at N = 10 000, the Krylov subspace method is 13 times faster than the exact Cholesky method. Thus, Krylov subspace methods are recommended for performing large-scale Brownian dynamics simulations with hydrodynamic interactions.