Showing papers on "Basis function published in 1979"
TL;DR: In this article, a mixed-basis method is developed for the calculation of the electronic structure of solids, which is shown to be capable of treating crystals with large complex unit cells, thus leading to a very efficient representation of systems which contain both highly localized (atomiclike) and delocalized (plane-wave-like) electrons.
Abstract: A mixed-basis method is developed for the calculation of the electronic structure of solids. The method is shown to be capable of treating crystals with large complex unit cells. A combined set of plane waves and Bloch sums of localized functions is employed as basis functions, thus leading to a very efficient representation of systems which contain both highly localized (atomiclike) and delocalized (plane-wave-like) electrons. The crystalline potential is determined in a fully self-consistent manner with no approximations made to its shape. The present method has the flexibility of being easily applicable to the study of many different systems (e.g., surface calculations with supercells). Specific application is made to bulk Nb and Pd to demonstrate the efficiency and accuracy of the method. Very good agreement with experimental results and with band structures calculated using other methods is obtained. It is found that, with a mixed basis, only a relatively small set of functions is needed to obtain convergent wave functions for the electrons.
273 citations
TL;DR: In this article, the application of a dissipative Galerkin scheme to the numerical solution of the Korteweg de Vries (KdV) and Regularised Long Wave (RLW) equations is investigated.
116 citations
TL;DR: In this paper, a technique for determining the modal propagation properties of a homogeneous cylindrical dielectric waveguide of arbitrary cross sectional shape and index n 1 embedded in a medium of index n 2 is presented.
Abstract: A technique is presented for determining the modal propagation properties of a homogeneous cylindrical dielectric waveguide of arbitrary cross sectional shape and index n1 embedded in a medium of index n2. Both the weakly guiding case in which n1 ≈ n2 and the general case of arbitrary index difference are discussed theoretically. In both cases the approach is to derive integral representations for appropriate components of E and B. These satisfy the appropriate Helmholtz equations inside and outside the guide and also guarantee that the boundary conditions are satisfied. On expansion of the components in certain sets of basis functions, the representations become a set of linear equations. The vanishing of the determinant of this set yields the propagation constants of the various modes. Numerical results are given for weakly guiding fibers of various shapes. Among these are rectangles and ellipses, which make comparisons with previous work possible.
97 citations
TL;DR: In this paper, a weighted least-squares approximation of a discrete function on the sphere in terms of surface spherical harmonics has been obtained, which has the property that waves are resolved uniformly on a sphere that is, if the discrete function is replaced by a tabulation of its spectral representation.
Abstract: Several topics are discussed which concern the representation of a discrete function on the sphere in terms of surface spherical harmonics Several methods are reviewed, with particular attention given to the recent work of Machenhauer and DaleyThe aliasing of the spherical harmonics is discussed and for a given grid a finite number of spherical harmonics are chosen as the discrete basis A harmonic is included in the basis if and only if it does not alias on the grid The number of basis functions is about half the number of grid points, with the result that the approximation may not fit all the function values Nevertheless, it is shown that a weighted least-squares approximation is obtained This approximation has the property that waves are resolved uniformly on the sphere That is, if the discrete function is replaced by a tabulation of its spectral representation, then the high frequencies which are artificially induced by the closeness of the grid points near the pole are removedThe approximation
85 citations
TL;DR: A class of feature-detection operations using orthonormal basis functions is introduced in this paper, where the basis functions are derived from the Karhunen-Loeve expansion of the local image data.
73 citations
TL;DR: In this paper, the linear hydrodynamic equations which describe the motion of the sea under the influence of an externally applied surface stress are solved using a finite difference grid in the horizontal and the Galerkin method through the vertical.
73 citations
TL;DR: In this article, two discrete-basis function approaches to the solution of the T-matrix equations for the scattering of electrons by atoms or molecules were discussed, one of which is based on the Schwinger variational principle.
Abstract: The authors discuss two discrete-basis-function approaches to the solution of the T-matrix equations for the scattering of electrons by atoms or molecules. Both methods, one of which is based on the Schwinger variational principle, have major advantages over the previously proposed T-matrix methods and do not require large-basis-set expansions. Results are reported for s- and p-wave scattering for helium.
62 citations
TL;DR: In this article, a correlated basis function is introduced for accurate evaluation of the combination of overlap and Hamiltonian matrix elements required in perturbation treatment of the uniform extended Fermi medium by the method of correlated basis functions.
58 citations
TL;DR: In this paper, the static polarizabilities and hyperpolarizabilities for the ground states of first row atoms, helium through neon, have been calculated solving the SCF equations of an atom in an electric field by numerical integration of coupled one dimensional differential equations using the Numerov method in matrix form.
Abstract: The static polarizabilities and hyperpolarizabilities for the ground states of first row atoms, helium through neon, have been calculated solving the SCF equations of an atom in an electric field by numerical integration of coupled one dimensional differential equations using the Numerov method in matrix form. The calculated polarizabilities agree within 2% with the values obtained by basis function methods. Heretofore only the hyperpolarizabilities of helium, lithium, beryllium, and neon have been calculated. For helium the hyperpolarizability is in good agreement with previous calculations, whereas for beryllium and neon the values obtained by basis function methods are scattered and significantly lower than the values we obtain. Comparison with experimental results for neon indicates that the Numerical Hartree Fock (NHF) method leads to a better representation of polarized orbitals than the use of basis function methods, which require tedious nonlinear optimization of large basis sets.
54 citations
TL;DR: In this article, a new method for achieving or speeding up SCF convergence in molecular ab initio calculations is presented, where gross populations are calculated in each cycle and extrapolated in order to derive an optimal damping factor for the density matrix in the basis of atomic orbitals.
45 citations
01 Jan 1979
TL;DR: Spectral methods are the extension of the standard technique of separation of variables to the solution of arbitrarily complicated problems as mentioned in this paper, and are particularly attractive for problems in several space dimensions in which high accuracy is required.
Abstract: Publisher Summary
This chapter presents some techniques that permit the efficient application of spectral methods to solve problems in nearly arbitrary geometries. The resulting methods are a viable alternative to finite difference and finite element methods for these problems. Spectral methods are particularly attractive for problems in several space dimensions in which high accuracy is required. These methods are based on representing the solution to a problem as a truncated series of smooth functions of the independent variables. Whereas finite element methods are based on expansions in local basis functions, spectral methods are based on expansions in global functions. Spectral methods are the extension of the standard technique of separation of variables to the solution of arbitrarily complicated problems. The chapter illustrates spectral methods for the simple one-dimensional heat equation. The chapter also discusses the difficulty caused by nontrivial boundary conditions and the difficulty of treating nonlinear and nonconstant coefficient terms. It then summarizes the properties of spectral methods for problems in simple geometries and explains how spectral methods can be extended to problems in complicated geometries.
TL;DR: In this article, a perturbation theory is described which provides a means of systematic solution of the matrix eigenvalue problem in a non-orthogonal correlated basis and general cluster-expansion algorithms are offered for evaluation of the required matrix elements in the correlated representation.
TL;DR: In this article, a modified Hermitian basis function is developed as a means of enhancing numerical solutions to the convection-dominated transport equation using orthogonal collocation, and an asymmetric approximation to the first-order spatial derivative is provided so that oscillations in the numerical solution are eliminated while introducing a minimum amount of numerical dispersion.
Abstract: A modified Hermitian basis function is developed as a means of enhancing numerical solutions to the convection-dominated transport equation using orthogonal collocation. An asymmetric approximation to the first-order spatial derivative is provided so that oscillations in the numerical solution are eliminated while introducing a minimum amount of numerical dispersion. The degree of assymetry in the basis function required to achieve a smooth solution is shown to be dependent upon the magnitude of the convective and dispersive terms.
TL;DR: In this paper, the authors demonstrate the inadequacy of the first order of the Hyperspherical Harmonic Expansion Method, the Lm approximation, for the calculation of the binding energies, charge form factors and charge densities of doubly magic nuclei like 16O and 40Ca.
TL;DR: By using random phase masks in the input and the filter planes of the correlator, this work has been able to pack many coefficients close together in the output plane, and thus take better advantage of the space-bandwidth product of the optical system.
Abstract: A coherent optical method capable of performing arbitrary two-dimensional linear transformations has recently been studied, in which transform coefficients are given by two-dimensional inner products of the input image and a set of basis functions. Since the inner product of two functions is equal to the value of their correlation when there is zero shift between the functions, it is possible to use an optical correlator to solve for the coefficients of the transform. By using random phase masks in the input and the filter planes of the correlator, we have been able to pack many coefficients close together in the output plane, and thus take better advantage of the space-bandwidth product of the optical system. Both the input random phase mask and the spatial filter are computer-generated holographic elements, created by a computer-controlled laser beam scanner. The system can be "pro-grammed" to perform arbitrary two-dimensional linear transformations. For demonstration, the set of two-dimensional Walsh functions was chosen as a transform basis. When the resolution of the Walsh functions was limited to 128 x 128, up to 256 transform coefficients were obtained in parallel. The signal-to-noise and accuracy of the transform coefficients were compared to the theory.
TL;DR: The Brueckner-Hartree-Fock approximation is the first order term of two different hole-line expansions, derived respectively from Green function theory and from Goldstone's linked cluster series as mentioned in this paper.
TL;DR: In this article, correlated basis functions are adapted to the nuclear-matter problem with two-nucleon potentials containing tensor as well as central components, and procedures for evaluating through three-body cluster order the energy expectation value with respect to a constrained trial ground-state wave function incorporating tensor and central correlations, and for calculating in twobody cluster approximation the second-order perturbation correction in a basis of likewise-correlated functions.
TL;DR: In this paper, the higher polarizabilities of H/sub 2/ using different-sized basis sets composed of James-Coolidge basis functions were calculated for different wave functions.
Abstract: The higher polarizabilities ..gamma../sub z/zzz, B/sub z/z:zz, and C/sub z/z:zz have been calculated for H/sub 2/ using different-sized basis sets composed of James-Coolidge basis functions. The most-accurate wave functions led to the values 674, -89.8, and 5.97 a.u., respectively. It is emphasized that these properties are critically sensitive to wave-function quality.
TL;DR: In this article, the Galerkin method with Hermite cubic polynomials as basis functions is viewed as a method of lines with mesh h. A Fourier analysis of the solution has two waves, one with speed 1 and another with speed -7.
Abstract: The Galerkin method for u/sub t/ + u/sub x/ = 0 with Hermite cubic polynomials as basis functions may be viewed as a method of lines with mesh h. A Fourier analysis of this method of lines shows that the solution has two waves, one with speed 1 and another with speed -7. If the initial data are smooth, then the amplitude of the backward wave is O(h/sup 4/). By filtering out the backward wave, an accuracy O(h/sup 6/) can be obtained.
TL;DR: In this article, the diatomics-in-molecules method is applied to calculate the ground state 1A′ potential energy surface for LiOH, and three different sets of polyatomic basis functions are investigated.
Abstract: The diatomics-in-molecules method is applied to calculate the ground state 1A′ potential energy surface for LiOH. Three different sets of polyatomic basis functions are investigated. An adequate description of the surface is obtained only with the set including both neutral and ionic atomic states.
TL;DR: In this paper, the results obtained by Kurten, Ristig and Clark in the framework of the correlated basis functions approach to nuclear matter are compared with those derived from two versions of the Brueckner-Hartree-Fock and of the renormalized BRHFock approximations.
TL;DR: In this paper, it was shown that the charge transfer problem can be most conveniently treated by using atomic-centered basis functions and expressing the Hamiltonian simultaneously with respect to two separate atomic origins.
Abstract: We briefly review the difficulties associated with the traditional adiabatic quantum treatment of low energy charge transfer collisions between atoms and suggest that the charge transfer problem can be most conveniently treated by using atomic‐centered basis functions and expressing the Hamiltonian simultaneously with respect to two separate atomic origins. As a result, we must solve the quantum nuclear equations of motion in a nonorthogonal basis. We demonstrate that this may be efficiently done by building the R‐matrix propagator in this basis. Flux is automatically conserved even though the R matrix is built up from a series of nonunitary transformations. A simple model problem designed to represent the radial coupling in the reaction Li++Na→Li+Na+ is solved to illustrate the technique.
TL;DR: In this paper, a scheme for calculating electronic energy states of infinite solid surface systems by a cluster approach under the framework of the method of linear combinations of atomic orbitals is presented.
TL;DR: The relationship between the Green's function and the Brueckner expansions for the binding energy of nuclear matter is discussed in this article, where the origin of spurious-looking graphs which appear in the green's function expansion is elucidated.
Abstract: We discuss the relationship between the Green's function and the Brueckner expansions for the binding energy of nuclear matter. The origin of spurious-looking graphs which appear in the Green's function expansion is elucidated. Numerical values are given in the case of the Hamman-Ho-Kim nucleon-nucleon interaction. We point out the existence of a striking analogy between the Green's function and the correlated basis functions approaches.
TL;DR: In this paper, spline moment functions are used to solve the singular integral equations that describe three-body scattering. But the spline moments approach is not suitable for the case where the basis functions are chosen to be a spline function.
12 Mar 1979
TL;DR: In this article, a finite element solution space consisting of piecewise biquadratics defined on the finite element discretization of the geometry of the duct is defined and compared to a Galerkin approach.
Abstract: : In this report, solutions of the equations which describe sound propagation in nonuniform ducts containing flow are computed with a finite element approach. A least squares approach is considered and compared to a Galerkin approach. The least squares problem is solved using an iterative method and compared with results obtained using direct Gaussian elimination. The accuracy of linear basis functions on triangles, bilinear basis functions on rectangles, and biquadratic basis functions on rectangles are compared. For the nonuniform ducts, the use of quadrilaterals as elements and an isoparametric map are considered. The biquadratics permit good approximation of curved boundaries and better convergence than the bilinear basis functions. Consequently, the finite element solution space consists of piecewise biquadratics defined on the finite element discretization of the geometry of the duct. Acoustic fields within uniform ducts both with and without flow have been computed and compare well with modal solutions and finite difference solutions. For nonuniform ducts without flow, the computed acoustic fields also compare well with exact or other computed solutions. For the case of two-dimensional flow within nonuniform ducts, sample calculations favorably compare with other solutions or limiting cases. (AUTHOR)
TL;DR: In this article, a general scheme is given to determine the matrix representation of the differential operator of the kinetic theory of gas mixtures, i.e., the streaming operator in the one-particle phase space.
Abstract: A general scheme is given to determine the matrix representation of the differential operator of the kinetic theory of gas mixtures, i.e., the streaming operator in the one‐particle phase space. The matrix elements are evaluated explicitly for the Burnett basis functions and a Lorentz type external field. The density, temperature, and mean velocity are treated as field quantities in order to assure the applicability of the results to inhomogeneous problems.
TL;DR: The set of harmonically-related nonorthogonal triangle waves is shown to form a basis spanning the same function space representable by Fourier (trigonometric) series, which is most attractive for digital signal representation.
Abstract: The set of harmonically-related nonorthogonal triangle waves is shown to form a basis spanning the same function space representable by Fourier (trigonometric) series. The triangle function set is, further, equivalent to the trigonometric series in important convergence-completeness properties. The weights of this series, and the weights of the finite series having minimum mean-square error, are calculated directly without resort to optimisation or other iterative techniques. This basis function set is most attractive for digital signal representation because these functions can be conveniently generated in a digital context. Unused `time slots? of time-shared digital filter sections are also easily diverted to real-time signal representation. Thus, depending on the application, triangle waves can provide ease of implementation while maintaining the convergence properties of trigonometric series. For coding applications, continuous-time and discrete-time triangular transforms for aperiodic and sampled signals can be enunciated. Several laboratory and computer-generated examples are given.
TL;DR: In this paper, the size and shape parameters for the core, bonding, and lone electron pairs of the ten-electron hydrides (CH4, NH3, H2O, HF) were determined from ab initio MO wave functions using various Gaussian basis sets.
Abstract: Size and shape parameters for the core, bonding, and lone electron pairs of the ten-electron hydrides (CH4, NH3, H2O, HF) were determined from ab initioMO wave functions using various Gaussian basis sets. The fundamental features of approximate electron pair loge representation are somewhat more sensitive to the quality of the basis functions than the molecular total energy. The total size of the molecular electron distribution is less affected by basis set variations than its components: the core, bonding, and lone pair sizes. There is an apparent tendency to “preserve” the total size of molecular distribution.