A Class of Orthogonal Polynomials Suggested by a Trigonometric Hamiltonian: Symmetric States

A new subclass of the Jacobi polynomials arising in the exact analytical solution of the one-dimensional Schrödinger equation with a trigonometric potential has been introduced. The polynomials which consist of a free parameter are not ultraspherical polynomials and have been simply named the -polynomials since they are generated by a trigonometric Hamiltonian. In certain sense, it is shown that the -polynomials can be regarded as a generalisation of the airfoil polynomials or the Chebyshev polynomials of the third kind. This paper is intended to discuss the basic properties of the polynomials so defined.     

On the special values of monic polynomials of hypergeometric type

Special values of monic polynomials y _n (s), with leading coefficients of unity, satisfying the equation of hypergeometric type

On the Inversion of Quantum Mechanical Systems: Determining the Amount and Type of Data for a Unique Solution

The inverse problem of extracting a quantum mechanical potential from laboratory data is studied from the perspective of determining the amount and type of data capable of giving a unique answer. Bound state spectral information and expectation values of time-independent operators are used as data. The Schrödinger equation is treated as finite dimensional and for these types of data there are algebraic equations relating the unknowns in the system to the experimental data (e.g., the spectrum of a matrix is related algebraically to the elements of the matrix). As these equations are polynomials in the unknown parameters of the system, it is possible to determine the multiplicity of the solution set. With a fixed number of unknowns the effect of increasing the number of equations on the multiplicity of solutions is assessed. In general, if the number of the equations matches the number of the unknowns, the solution set is denumerable. A result on the solvability of polynomial equations is extended to the case where the number of equations exceeds the number of unknowns. We show that if one has more equations than the number of unknowns, generically a unique solution exists. Several examples illustrating these results are provided.     

A numerical basis for the accurate representation of the continuum spectrum of atomic Hamiltonians

We present a method for the accurate calculation of the complete spectrum of the Schrödinger equation in terms of B-splines polynomial basis. The method is capable to represent numerically the bound and continuum spectrum of complex atomic systems. The theoretical method is discussed, and an application to hydrogenic Hamiltonian is given.AMS subject classification: 65705, 34L40     

On the representation of atoms and molecules as self-interacting field with internal structure

The nonlinear Schrödinger equation with Gaussian convolution kernel K₂ induces the group SU₃ with reference to the classification of the multiplet structure of the eigenstates. Such a field can be used to describe some atoms (where the outermost electrons are related tos-orbitals) as a self-interacting, extended particle with an internal structure. In the case of those atoms, where the valence electrons are described byp-orbitals, and almost all molecules the Gaussian kernel K₂ has to be generalized by Hermite polynomials. By that, we can formulate a nonlinear field theory, establishing the spatial symmetry of a system via basis structure functions. Thus the symmetry represents the most essential starting-point for treating molecules as quasi-particles with an internal structure. It will be shown that there is some connection with the concept of chirality functions and the Ginzburg — Landau theory of super-conductivity. The latter theory indicates that we can consider the nonlinear Schrödinger equation and its generalizations as a classical field theory being associated with phase transitions.     

A new method for approximate solution of one-dimensional Schrödinger equations

Summary A general method for approximate solution of one-dimensional Schrödinger equations with a wide range of square-integrable potentials is described. The potential is expanded in terms of either Jacobi or Bessel functions of argument exp(-r). This allows the Schrödinger equation to be solved by the Frobenius method. In the absence of super-computing power the input requirement of a large number of significant figures was handled by an algebraic computing package, for illustrative purposes. A sum of Gaussian wells and a Morse potential are treated as examples.     

A Family of P-stable Eighth Algebraic Order Methods with Exponential Fitting Facilities

A P-stable exponentially-fitted method of algebraic order eight for the approximate numerical integration of the Schrödinger equation is developed in this paper. Since the method is P-stable (i.e., its interval of periodicity is equal to (0,), large stepsizes for the numerical integration can be used. Based on this new method and on a sixth algebraic order exponentially-fitted P-stable method developed by Simos and Williams [1], a new variable step method is obtained. Numerical results presented for the coupled differential equations arising from the Schrödinger equation show the efficiency of the developed method.     

Bound and scattering states for a hyperbolic‐type potential in view of a new developed approximation

A new developed approximation is used to obtain the arbitrary l‐wave bound and scattering state solutions of Schrödinger equation for a particle in a hyperbolic‐type potential. For bound state, the energy eigenvalue equation and unnormalized wave functions in terms of Jacobi polynomials are achieved using the Nikiforov–Uvarov (NU) method. Besides, energy eigenvalues are calculated numerically for some states and compared with those given in the literature to check accuracy of our results. For scattering state, the wave function is found in terms of hypergeometric functions. Furthermore, scattering amplitude and phase shifts are achieved using scattering solutions. Also it is shown that the energy eigenvalue equation obtained from analytic property of scattering amplitude is same with one obtained using NU method. © 2015 Wiley Periodicals, Inc.     

Post-adiabatic approach to atomic and molecular processes: The van der Waals interactions of some open shell systems

Summary Various properties of post-adiabatic representations of multichannel Schrödinger equations are described in the general context of adiabatic and classical path approximations as used in atomic and molecular physics. The van der Waals interactions of fluorine, chlorine, and oxygen atoms with rare gases, hydrogen, methane, and hydrogen halides are considered: it is found that in some of these systems, the first-order post-adiabatic scheme exhibits a smaller coupling than the adiabatic representation, thus providing an appropriate choice of the basis functions for a decoupling approximation.     

A Generator and an Optimized Generator of High-Order Hybrid Explicit Methods for the Numerical Solution of the Schrödinger Equation. Part 2. Development of the Generator, Optimization of the Generator and Numerical Results

The generator of tenth-order hybrid explicit methods, the basic method of which has been developed in part 1, is constructed and also optimized, by maximization of the intervals of periodicity. The efficiency of the new methods is shown by their application to the coupled differential equations of the Schrödinger type.     

Electrochemical noise of a lithium electrode in organic electrolytes: A study by a correlation function method

Fluctuations of the potential of a lithium electrode in conditions of galvanostatic polarization in aprotic organic electrolytes are studied by a method of correlation functions. Computer-aided removal of heavy interference in the form of slow variation of the electrode potential proved to be possible to perform with use made of fifth-power polynomials. The time coordinate of the first zero in the correlation function weakly depends on the electrolyte type and lies within the limits 1.5–3 s. At the same time, the electrolyte type affects the dispersion of the electrode potential fluctuations in a substantial manner. In so doing, lithium systems that feature a high cycling efficiency possess a lower level of noise.     

Response dynamics of 2-D quantum dots in the presence of time-varying fields: Anharmonicity and pulse shape effects

We explore the pattern of time evolution of different observables in a harmonically confined single carrier 2-D quantum dot when an external time-varying electric field is switched on. A static transverse magnetic field is also present. For given strengths of the confining field, cyclotron frequency, intensity and oscillation frequency of the external field, and pulse shape parameters, the system reveals a long time dynamics that leads to a kind of localization in the unperturbed state space. The presence of cubic anharmonicity in the confining field brings in new features in the dynamics. Frequency dependent linear and non-linear response properties of the dot are analyzed.     

Construction of Trigonometrically and Exponentially Fitted Runge–Kutta–Nyström Methods for the Numerical Solution of the Schrödinger Equation and Related Problems – a Method of 8th Algebraic Order

General conditions are presented for an m-stage Runge–Kutta–Nyström fitting to exponential and trigonometric functions. As an example an 8th order Runge–Kutta–Nyström method is constructed. Numerical results on the numerical solution of the Schrödinger equation and related problems indicate that the new method is more accurate than the classical one (we call classical Runge–Kutta–Nyström method the corresponding method with constant coefficients).     

A numerical solution of the pair equation of a model two‐electron diatomic system

It has been well‐documented that about 90% of the total correlation energy of atomic systems can be obtained by solving so‐called pair equations. For atoms, this approach requires solving partial differential equations (PDE) in two variables. In case of a diatomic molecule, we face devising a method for treating PDEs in five variables. This article shows how a well‐established finite difference method used to solve Hartree–Fock equations for diatomic molecules can be extended to solve numerically a model two‐electron Schrödinger equation for such systems. We show that using less than 100 grid points in each variable, it is possible to obtain the total energy of the helium atom and hydrogen molecule with a chemical accuracy and the S energy of the helium atom and hydride ion as accurately as the best results available. © 2015 Wiley Periodicals, Inc.     

Towards a complete identification of orthogonal variation in multiple regression from a PLS1 modeling point of view: including OPLS by a change of orthogonal basis

It is well known that the predictions of the single response orthogonal projections to latent structures (OPLS) and the single response partial least squares regression (PLS1) regression are identical in the single‐response case. The present paper presents an approach to identification of the complete y ‐orthogonal structure by starting from the viewpoint of standard PLS1 regression. Three alternative non‐deflating OPLS algorithms and a modified principal component analysis (PCA)‐driven method (including MATLAB code) is presented. The first algorithm implements a postprocessing routine of the standard PLS1 solution where QR factorization applied to a shifted version of the non‐orthogonal scores is the key to express the OPLS solution. The second algorithm finds the OPLS model directly by an iterative procedure. By a rigorous mathematical argument, we explain that orthogonal filtering is a ‘built‐in’ property of the traditional PLS1 regression coefficients. Consequently, the capabilities of OPLS with respect to improving the predictions (also for new samples) compared with PLS1 are non‐existing. The PCA‐driven method is based on the fact that truncating off one dimension from the row subspace of X results in a matrix X _orth with y ‐orthogonal columns and a rank of one less than the rank of X . The desired truncation corresponds exactly to the first X deflation step of Martens non‐orthogonal PLS algorithm. The significant y ‐orthogonal structure of X found by PCA of X _orth is split into two fundamental parts: one part that is significantly contributing to correct the first PLS score toward y and one part that is not. The third and final OPLS algorithm presented is a modification of Martens non‐orthogonal algorithm into an efficient dual PLS1–OPLS algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

Inward Matrix Product Algebra and Calculus as Tools to Construct Space–Time Frames of Arbitrary Dimensions

In this study, inward matrix products are used to construct a theoretical framework where new space-time structures of arbitrary dimensions can be built up. The mathematical theory, based on inward matrix algebra, allows the derivation and integration of vectors and matrices composed by well-behaved functional elements. Every function element is associated at least to a linearly independent variable connected to such an element. As examples are discussed first the construction of general density functions, followed by the reformulation of the time-dependent Schrödinger equation. A general N-dimensional classical universe is presented, where not only space but also time, mass, energy and other related physical properties acquire an arbitrary hypermatrix structure. In this hypothetical framework scalar values related to physical quantities can be alternatively associated to cosine-like measures in the chosen spaces. Finally, simple problems on special relativity are briefly discussed from this point of view.     

Diagonalization of a real-symmetric Hamiltonian by genetic algorithm: A recipe based on minimization of Rayleigh quotient

A genetic algorithm-based recipe involving minimization of the Rayleigh quotient is proposed for the sequential extraction of eigenvalues and eigenvectors of a real symmetric matrix with and without basis optimization. Important features of the method are analysed, and possible directions of development suggested     

On the usefulness of generalised quantum chemical valence parameters in monitoring the course of a chemical reaction: A case study of the photochemical decomposition of HNO in excited states

A variant of the orthogonal gradient method of orbital optimization in the INDO-MCSCF framework has been used to study the photochemical decomposition of the HNO molecule into H + NO in the lowest^1.3A″ states. A complete geometry optimization has been carried out at all points of the reaction path which appears to be almost barrierless. The one-electron density matrix extracted from the optimized wavefunction at each point has been used to generate the relevant sets of quantum chemical valence parameters. A sharp transition is noted in the N-H bond order and hydrogen free valence index when plotted as functions of r_NH. This enables us to locate the transition region easily.