Showing papers on "Basis function published in 1981"

A CNDO/INDO molecular orbital formalism for the elements H to Br. theory

Abstract: A CNDO and INDO formalism is presented that can be used for any atom combination up to bromine under inclusion of the first transition metal series. The semiempirical parameters were chosen to reproduce results ofab initio calculations on metalorganic compounds. The calculational results are invariant to rotations of the coordinate system but not to a general transformation into other basis functions. The one-center Coulomb-expressions were selected in order to include intraatomic correlation contributions. Within the CNDO model this could be achieved by the scaled monopole termF 0, while in the INDO framework the one-center Coulomb integrals are given as a sum of the monopole-contributionF 0 and higher multipole contributions expressed as a linear combination of Slater-Condon parameters. The invariance problem in the case of local rotations within the INDO approximation was solved by considering the combination of one-center Coulomb and exchange integrals as a function ofl but independent ofm. The two-center electron-electron interaction terms were calculated via the Dewar-Sabelli, Ohno-Klopman relation. Penetration effects were treated according to Fischer and Kollmar. For the resonance integralH μv AB parameters are used which carry information related to the directed nature of the chemical bond by using optimized Klondyke functions. The core-core repulsion is constructed as a superposition of a soft potential function, describing polarization effects of the atomic cores, and a hard repulsion function, avoiding the collapse of the atomic cores with decreasing distance.

On a dimensional reduction method. I. The optimal selection of basis functions

Abstract: : This paper is the first in a series of three, which analyze an adaptive approximate approach for solving (n+1)- dimensional boundary value problems by replacing them with systems of equations in n-dimensional space. In this approach the unknown functions of (n+1)- variables are projected onto finite linear combinations of functions of just n-variables. This paper shows how the coefficients of these linear combinations can be chosen optimally. (Author)

Analysis of a circular microstrip disk antenna with a thick dielectric substrate

Abstract: The problem of a circular microstrip disk excited by a probe is solved using rigorous analysis. The disk is assumed to have zero thickness, and the current on the probe is taken to be uniform. Using vector Hankel transforms the problem is formulated in terms of vector dual-integral equations, from which the unknown current can be solved for. Due to the singular nature of the current distribution arising from probe excitation, the direct application of Galerkin's basis function expansion method gives a slowly convergent result. Therefore the singular part of the current is removed since the singularity is known a priori. The unknown current to be solved for is then regular and tenable to Galerkin's method of analysis. It is shown that this analysis agrees with the single-mode approximation when the dielectric substrate layer is thin, and that it deviates from the single-mode approximation when the substrate layer is thick. Excellent agreement of both the computed real and imaginary parts of the input impedance with experimental data is noted. The radiation patterns and the current distributions on the disk are also-presented.

Efficient moment method analysis of radiating slots in a thick-walled rectangular waveguide

Abstract: A narrow slot in the broad wall of a rectangular waveguide is analysed using a moment method which takes account of both the wave guide wall thickness, and the influence of side walls. In essence, the method of moments is used to solve a pair of coupled-integral equations, derived from the electromagnetic boundary conditions using Green's function techniques, to find the tangential electric fields on the upper and lower surfaces of the slot. These fields are represented by a sinusoidal Fourier set of basis functions, and it is shown that only a few terms are required to cause the series to converge. The necessity of inverting large, full matrices, which can be a disadvantage of this method, is thus avoided. Theoretical results are presented for the scattering parameters of slots and these are compared with experiment. A computed curve showing the functional form of the tangential electric field in the slot is also included and is compared with some earlier calculated and trial slot-field distribution.

Redundancy reduction for improved display and analysis of body surface potential maps. II. Temporal compression.

Abstract: This paper describes use of the Karhunen-Loeve expansion to identify and reduce temporal redundancy in electrocardiographic body surface potential maps (192 body surface leads recorded simultaneously at 1 kHz/channel for approximately 600 msec). Temporal data compression of about 20 to 1 was obtained with accurate representation of the original data. Use of separate sets of orthonormal basis functions for QRS and ST-T provided a more accurate representation than the basis derived from QRST. Combined with the spatial compression described in the preceding paper, overall map data compression of about 320 to 1 was obtained without significant loss of accuracy of representation or map appearance. With both spatial and temporal compression the 100,000 numbers which typically comprise a single cardiac complex were accurately represented by 216 coefficients. Using basis functions derived from a single cardiac complex were accurately represented by 216 coefficients. Using basis functions derived from a training s...

An approximately divergence‐free 9‐node velocity element (with variations) for incompressible flows

Abstract: Explicit basis functions are constructed for 9-node biquadratic velocity fields which guarantee that a weak form of the continuity equation is satisfied. The corresponding pressure approximations are either piecewise constant, piecewise linear or piecewise bilinear. These results are extended to give bases for bilinear velocity/piecewise constant pressure elements and also to some three-dimensional brick elements.

Basis functions for general Hsieh‐Clough‐Tocher triangles, complete or reduced

Abstract: In this note, we give the basis functions (shape functions) for Hsieh-Clough-Tocher triangles, complete or reduced. In order to take the best advantage of symmetry and for simplicity, we derive these functions for a general triangle by using simultaneously the ‘area co-ordinates’ and the so-called ‘eccentricity parameters’. We conclude with some remarks concerning the implementation.

The Polarizabilities of Electrically Small Apertures of Arbitrary Shape

Abstract: Numerical procedures, based on a method of moments approach, are given for computing the magnetic and electric polarizabilities of electrically small apertures of arbitrary shape. The magnetic polarizability density is determined through the use of pulse expansion functions defined over quadrilateral subdomains, while the electric polarizability density is obtained by using basis functions, each of which consists of a piecewise arrangement of simple linear functions defined over triangles and having an area coordinate representation. All the subdomains are generated automatically by applying either the Gordon-Hall or the Zienkiewicz-Phillips subdivision techniques. Computed results are obtained for several aperture shapes, including the circle and the ellipse. The calculations for the last two cases are in excellent agreement with the exact values.

Method and apparatus for transmitting data over limited bandwidth channels

Abstract: A method of developing waveforms having the energy therein concentrated within a limited bandwidth and modems for generating, transmitting, and receiving digital data utilizing such waveforms as carriers. A set of mutually orthogonal basis signals are selected to maximize their energies within the channel, each is binary coded and stored in the modem. A set of optimized coefficients is stored and utilized to weight each basis function in a coding arrangement to define bytes of digital data. The coded and weighted basis functions for each successive byte are summed to form a composite signal and transmitted over the channel as an analog waveform. Stored basis signals at the receiving end are correlated with the received waveform to thereby extract the coefficients which are decoded to reproduce the transmitted bytes. Both coding gain and reduction of intersymbol interference is achieved.

Scaling in second-order electron correlation calculations using systematic sequences of even-tempered basis sets

Abstract: Improved results can often be obtained from second-order Rayleigh-Schrodinger perturbation calculations of electron correlation energies using large basis sets by introducing a scaling factor in the zero-order Hamiltonian. The scaling parameter may be determined from full third-order calculations using a smaller basis set. This scaling procedure can be applied in a systematic fashion by employing a sequence of even-tempered basis sets. Calculations illustrating this approach for the beryllium atom and the neon atom are presented. The scaling procedure is also employed in conjunction with a universal systematic sequence of basis functions. Calculations illustrating this Correlation energy — Mang-body perturbation theory.

Gaussian Basis Sets

Abstract: All calculations performed at the school are of the LCAO-MO-SCF or LCAO-MO-SCF-CI type. The calculations were performed using a program (MONSTERGAUSS (1)) which uses as atomic orbital s (AO) or basis functions, the Gaussian function.

An examination of the use of vibrationally adiabatic basis functions for the calculation of molecular collision cross sections

Abstract: Comparison is made between the use of two different types of vibrational basis functions for the expansion of the total wave function in a vibrationally inelastic scattering problem. The calculations are performed within the framework of the sudden approximation for the rotational motion of the molecular fragments. The different basis functions that are compared are a vibrationally adiabatic set and the standardly used set of diabatic vibrational basis functions. The adiabatic vibrational basis functions are chosen so as to approximately diagonalize the matrix representation of the interaction potential at each value of the scattering coordinate. Nevertheless, they permit the formulation of analytic expressions for the nonadiabatic coupling terms of the kinetic energy operator that are present when an adiabatic basis is used. In order to provide a reference against which to judge the two different bases, the sets of coupled differential equations which arise in the rotational sudden approximation are solved for the He+H2 system, and fixed‐angle S matrices are calculated at several scattering energies and different values of the total angular momenta. It is shown that if the customary diabatic basis is used in conjunction with first order distorted wave perturbation theory to calculate the fixed‐angle S matrices, these do not agree well with the exactly computed S matrices to which they should correspond. In contrast, if an adiabatic vibrational basis is used, the distorted wave approximation yields fixed‐angle S matrices which are in good agreement (within 15% or better) with the fully converged exact calculations.

Complex-coordinate calculations with complex basis sets

Abstract: Two proposals have recently been made for improving the convergence of complex-coordinate calculations of electron-atom scattering resonances. The first is that complex (Siegert-typej functions be added to the usual basis of real functions, and the second is that bound-state basis functions not be rotated to the complex p1ane. In this paper we present calculations which test these suggestions for the lowest e-He+ resonance, comparing the results with previous studies. The stabilization length criterion is employed to assess the convergence and stability of the variational energies. We conclude that neither of these suggestions satisfactorily removes the difficulties associated with complex-coordinate calculations of resonance lifetimes.

Basis set generation for the SCF calculation

Abstract: A method for basis set generation for SCF calculations is proposed. Using SCF orbitals and orbital energies obtained in the extended basis set the Fock operator can be expressed as its spectral resolution. The sum of differences between occupied orbital energies and corresponding eigenvalues obtained by the diagonalization of this operator in the new smaller basis set is a criterion of the quality of this new set. The present method consists of the minimization of this sum by changing the parameters that determine the new basis functions. An example of the optimization of the different Gaussian basis sets for the LiH molecule is described.

A new method of constructing symmetry-adapted linear combinations for a finite group: II. Correspondence theorem and basis operators

Abstract: A correspondence theorem is introduced to describe the relation between an elementary basis function and a symmetry-adapted linear combination belonging to the same irreducible representation of a point group By means of basis operators, the general expressions are given for the symmetry-adapted linear combinations, the coupling coefficients, the lattice harmonics and the hybrid atomic orbitals Illustrative examples are discussed to demonstrate the effectiveness of the formalism

On the possibility of deriving equivalent localized basis sets from symmetry adapted ones

Abstract: It is shown that, given a symmetry adapted set of basis functions, for certain systems of high symmetry the generation of an equivalent localized set of basis functions is not always possible if the localized set is subject to the following requirements: 1) be linearly independent; 2) be partitionable into disjoint subsets each belonging to a localized site and transform according to the irreducible representations of the corresponding site symmetry; 3) symmetrically equivalent sites have equivalent subsets. The conclusion can be drawn that for highly symmetrical systems the description in terms of linearly independent localized single-particle functions may not be suitable.

A new iterative method for finding the normal modes of a laterally heterogeneous body

Abstract: We present a new iterative method for finding the normal modes of a laterally heterogeneous body. The number of operations used by our method is proportional to the square of the number of basis functions used, while the effort required by the variational method increases as the cube of the number of basis functions. A numerical example shows that, at least for lower order modes, our iterative method yields results which are as accurate as the variational method.

Linear Chebyshev Complex Function Approximation.

Abstract: : A new computational technique is described for the Chebyshev, or minimax, approximation of a given complex valued function by means of linear combinations of given complex valued basis functions The domain of definition of all functions can be any finite set whatever Neither the basis functions nor the function approximated need satisfy any special hypotheses beyond the requirement that they be defined on a common domain Theoretical upper and lower bounds on the accuracy of the computed Chebyshev error are derived These bounds permit both a priori and a posteriori error assessments Efforts to extend the method to functions whose domain of definition is a continuum are discussed Numerical examples and a FORTRAN program listing are included An application is presented involving re-shading a 50-element antenna array to minimize the effects of a 10% element failure rate, while maintaining full steering capability and mainlobe beamwidth (Author)

Comparison of the R-matrix and Hulthén-Kohn methods for a model multichannel scattering problem

Abstract: Exact asymptotic wave functions are included in the basis set for variational calculations of the K matrix in a multichannel model problem. Rapid convergence to exact results is demonstrated for both the variational R-matrix method and for the restricted-interpolated-anamoly-free (RIAF) variant of the Hulthen-Kohn method. If ''irregular'' asymptotic functions are omitted, convergence of the R-matrix calculations is much less satisfactory. By a particular choice of the form of discrete basis functions, vanishing at a fixed radius r/sub 1/ in the model problem, the main computational steps of both methods are made to be identical.

A new method of computing geopotential fields

Abstract: Summary A new method is proposed for the geopotential field computation and gravitational attraction modelling The usual method is to use a uniform density discrete numerical integration to represent either the gravitational potential or the gravitational attraction from a given density configuration In this paper, an interpolation scheme is explained, using a piecewise continuous basis function to represent the arbitrarily varying density configuration in one, two and three dimensions This new approach greatly simplifies the potential integrations and, in certain cases where symmetry exists, analytical evaluation of the integrals is also possible Numerical tests and examples are given for a hypothetical salt dome, a vertical dyke with varying density structure and the hydrostatic ellipticity of earth model 1066B The numerical error in this method is limited to the analytical approximation and interpolation errors in each case This new approach can also be used as efficiently for other potential field studies

A variational method for multichannel scattering

Abstract: A multichannel variational method is proposed that allows use of basis functions extending to infinity, not truncated at a finite boundary, although the method uses R-matrix formalism. Exact asymptotic wavefunctions are used as a continuum basis. Two-channel model calculations demonstrate the validity of the method.

The contracted density product (CDP) concept as a new way to simplify molecular orbital calculations

Abstract: It is shown that invariant MO–CI schemes, approximate in the sense that they neglect part of the electron–electron interaction, can be derived in a systematic way from rigid ab initio methods. Two electron integrals over basis functions can be divided into independent groups on account of the number of centers over which they and the individual electron densities extend. Entire such groups can be neglected, retained as they are, or retained in space averaged form by contracting their corresponding products of density matrices. For irreducible tensors as basis functions, full contraction distinguishes integrals which have a vanishing space average up to 3p orbitals from integrals with a nonvanishing space average. The latter can be replaced by a set of four parameters, three of which are linearly dependent wherever the Coulomb and exchange terms are interdependent in the unified atom limit.

Complete orthogonal functions for strip conductor in stratified dielectrics

Abstract: A new method for systematically constructing basis functions for two-dimensional field problems containing a strip conductor buried in stratified dielectrics is proposed. By applying a simplifying averaging process to the associated static problem, the structure of the specific kernel or Green's function for the problem can be determined approximately. An integral form of continuity equation is derived; general sets of quasi-eigenfunctions are established, which are orthogonal and complete in Hubert spaces L2(Ω, w). By substituting the specific information into the latter, the required quasi-eigenfunctions for the problem can be obtained. Hence, with both the edge condition and the continuity equation being satisfied simultaneously and with the inhomogeneity of dielectric constants being taken into account, these resultant functions form a particularly suitable expansion basis for the series solution to such physical quantities as scalar potential, induced surface charge, and surface current.

Development of improved finite elements formulation for shallow water equations.

Abstract: : The basic principles of the Galerkin finite element method are discussed and applied to two different formulations; one using different basis functions and the other using the vorticity-divergence form of the shallow water equations. Each formulation is compared to the primitive form of the equations developed by Kelley (1976). The testing involves a comparison of three finite element prediction models using variable size elements. Equilateral elements significantly improve the solution and are used in most of the comparisons. The formulation using different basis functions produces poorer results than the primitive formulation. The vorticity-divergence formulation produces superior results while executing faster than the primitive model. However, it does require more storage and the relaxation parameters are sensitive to the domain geometry. The computer implementation for the vorticity-divergence model is discussed and the source listing is included.

Optical recognition of statistical patterns

Abstract: Optical implementation of the Fukunaga-Koontz transform (FKT) and the Least-Squares Linear Mapping Technique (LSLMT) is described. The FKT is a linear transformation which performs image feature extraction for a two-class image classification problem. The LSLMT performs a transform from large dimensional feature space to small dimensional decision space for separating multiple image classes by maximizing the interclass differences while minimizing the intraclass variations. The FKT and the LSLMT were optically implemented by utilizing a coded phase optical processor. The transform was used for classifying birds and fish. After the F-K basis functions were calculated, those most useful for classification were incorporated into a computer generated hologram. The output of the optical processor, consisting of the squared magnitude of the F-K coefficients, was detected by a T.V. camera, digitized, and fed into a micro-computer for classification. A simple linear classifier based on only two F-K coefficients was able to separate the images into two classes, indicating that the F-K transform had chosen good features. Two advantages of optically implementing the FKT and LSLMT are parallel and real time processing.