Showing papers on "Basis function published in 1982"

Atoms in molecules

Abstract: A set of spherical density basis functions for atoms sodium through argon have been obtained by least squares fitting of the corresponding electron density function for the ground state. These density functions, conveniently scaled, can be used in electron population analysis of molecules. To make this population analysis more flexible the density basis functions are split into two inner (K and L shell) parts and one outer (M shell) part. This technique gives populations which are relatively independent of the orbital bases used in spanning the molecular wavefunction. These density functions can be easily converted to scattering factors to be used in similar population analysis of accurate X-ray diffraction data.

An efficient algorithm for calculating ab initio energy gradients using s, p Cartesian Gaussians

Abstract: A method is presented for the calculation of analytical first derivatives of the two electron integrals over s‐ and p‐type Cartesian Gaussian basis functions. Formulas are developed for derivatives with respect to the positions of the nuclei and basis functions (for use in geometry optimization) and with respect to the exponents of the primitive Gaussians (for use in basis set optimization). Full use is made of the s = p constraint on the Gaussian exponents. Contributions from an entire shell block are computed together and added to the total energy derivative directly, avoiding the computation and storage of the individual integral derivatives. This algorithm is currently being used in the ab initio molecular orbital program gaussian 80.

A general Chebyshev complex function approximation procedure and an application to beamforming

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. 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.

An Accurate, Unified Solution to Various Fin-Line Structures, of Phase Constant, Characteristic Impedance, and Attenuation

Abstract: The analysis of several fin-line configurations (unilateral fin-line, bilateral fin-line, antipodal fin-line, and coupled fin-lines) has been completed accurately. In this unified method, propagation constant is achieved via the generalized spectral domain technique where the basis functions for the bounded and unbounded fields are chosen to be trigonometric functions and Legendre polynomials, respectively. The conduction loss and dielectric loss solution for the first time are found through a perturbation method. The conductor loss so derived is believed to be sufficiently accurate for practical purposes. Characteristic impedances of these transmission lines using tentative definitions have been presented. The CPU time on an IBM 360/65 for calculation of the mentioned parameters does not exceed five seconds if the fourth-order solution in the spectral analysis gives the required accuracy. The programs are also capable of detection of higher order modes.

Resonance states by the generalized complex variational method

Abstract: The resonance states are presented as the complex stationary solutions of generalized secular equations. The study of the analytical behaviour of the complex stationary solutions of these equations as a function of a coupling parameter λ in the hamiltonian yields the following. 1. (1)Criteria to distinguish between the complex stationary solutions that describe the resonances and the complex solutions that may be obtained as a result of the restrictions on the basis set. 2. (2)Criteria and a computational procedure for judging the stability of results obtained within the framework of the complex coordinate method. On this basis it is pointed out that the enhanced stability of the resonant eigenvalue when a complex basis function is added to the real basis set is due to the fact that the expectation value of the second derivative of the hamiltonian with respect to the scaling parameter can be negative while for a real basis set it is equal to the kinetic energy and therefore gives positive values only.

Image classification by an optical implementation of the fukunaga-koontz transform.

Abstract: The Fukunaga–Koontz (F–K) transform is a linear transformation that performs image-feature extraction for a two-class image classification problem. It has the property that the most important basis functions for representing one class of image data (in a least-squares sense) are also the least important for representing a second image class. We present a new method of calculating the F–K basis functions for large dimensional imagery by using a small digital computer, when the intraclass variation can be approximated by correlation matrices of low rank. Having calculated the F–K basis functions, we use a coherent optical processor to obtain the coefficients of the F–K transform in parallel. Finally, these coefficients are detected electronically, and a classification is performed by the small digital computer.

A criterion for the applicability of the method of diatomics‐in‐molecules to potential surface calculations. I. Selection of the DIM basis

Abstract: A self‐consistency criterion is applied to the method of diatomics‐in‐molecules in order to determine, in a resasonable manner, the minimum DIM polyatomic basis set needed to describe a given state of a polyatomic system. The method requires only the calculation of overlap integrals of the DIM basis functions with ab initio wave functions for the diatomic fragment states. The procedure is applied to the reaction Be+HF→BeF+H (1A′ ground state), for which extensive DIM model calculations are available.

Eulerian-Lagrangian Methods for Advection-Dispersion

Abstract: Three different Eulerian-Lagrangian schemes are used to solve the one-dimensional advection-dispersion equation. Advection is formally decoupled from dispersion in a manner analogous to Neuman (1981) which does not leave room for ambiguity. The resulting advection problem is solved by a novel approach called the method of reverse streaklines, and by the more conventional method of continuous particle tracking. Dispersion is handled by implicit finite elements on a fixed grid, using linear and quadratic basis functions. The results are compared with a third Eulerian-Lagrangian method in which the concentration function remains undecomposed. Preliminary results suggest that the first two methods may, after further improvement, work well for a wide range of Peclet numbers. The third method suffers from numerical dispersion and overshoot when Peclet numbers are large.

Waveform Feature Extraction Based on Tauberian Approximation

Abstract: A technique is presented for feature extraction of a waveform y based on its Tauberian approximation, that is, on the approximation of y by a linear combination of appropriately delayed versions of a single basis function x, i.e., y(t) = ?M i = 1 aix(t - ?i), where the coefficients ai and the delays ?i are adjustable parameters. Considerations in the choice or design of the basis function x are given. The parameters ai and ?i, i=1, . . . , M, are retrieved by application of a suitably adapted version of Prony's method to the Fourier transform of the above approximation of y. A subset of the parameters ai and ?i, i = 1, . . . , M, is used to construct the feature vector, the value of which can be used in a classification algorithm. Application of this technique to the classification of wide bandwidth radar return signatures is presented. Computer simulations proved successful and are also discussed.

Shape Design, Representation, and Restoration with Splines

Abstract: The purpose of the present paper is to demonstrate several applications of interpolation and approximation methods such as splines to picture engineering In particular, three types of applications, i.e., shape design, shape representation, and shape restoration are discussed. Shape design is concerned with construction of visually beautiful curves and interactive modification of the curves. The method of B-splines gives a physically natural (minimum curvature] curve which is locally modifiable, since it is a linear combination of basis functions (B-splines) having local support. Bezier’s method generates a smooth curve (polynomial) by means of Bezier polygon and is appropriate for interactive use, although a local modification may propagate throughut the whole interval. Shape representation approximates a given irregular shape, where a method suitable for use depends on a particular way in which data are given. For surface approximation Coons methods are used as a standard tool, where an entire surface is subdivided into pieses whose boundary data are used to construct an approximating surface for the corresponding piece. When the boundary data are given at a selected set of points, tensor product of splines has been known to be usable. For surfaces which can be adequately described in terms of one-parameter family of curves, the technique known as lofting is applicable. If two families of parametric curves must be mixed to define a given surface, one may effectively use boolean sum approximation. Shape restoration is an approximation of an existing object, where the data are noisy or incomplete. In this case an approximation will depend on smoothing property and the nature of data given. Moreover, in some cases, irregularly distributed data points must be incorporated. In the image restoration the convolution property of B-splines is useful for the restoration of space- invariant degradations. As an additional example, we mention the pattern generation of air pollution, where spline under tension is prefered to the ordinary cubic spline in view of the accuracy of the approximation.

Rapid convergence of V–V energy transfer calculated using adiabatic basis functions. I. An accurate two‐state model for low‐energy resonant V–V energy transfer. II.

Abstract: We report a series of calculations designed to show the efficiency of adiabatic basis sets for V–V energy transfer in diatom–diatom collisions. We do find that these basis sets are very efficient for both symmetric and nonsymmetric cases. For resonant V–V energy transfer at low energies we obtain quantitative accuracy with a two‐state model. The only dynamical calculations required for this model are two phase shifts.

Time‐dependent scattering by a bounded obstacle in three dimensions

Abstract: In the prsent paper we introdue a new method of treating the scattering of transient fields by a bounded obstacle in three‐dimensional space. The method is a generalization to the time domain of the null field approach first given by Waterman. We define new sets of time‐dependent basis functions, and use these to expand the free space Green’s function and the incoming and scattered fields. The scattering problem is then reduced to the problem of solving a system of ordinary differential equations. One way of solving these equations is by means of Fourier transformation, and this leads to an efficient way of obtaining the natural frequencies of the obstacle. Finally, we have calculated the natural frequencies numerically for both a spheroid and a peanut‐shaped obstacle for various ratios of the axes.

Choice of Basis for Chebyshev Approximation

Abstract: In polynomial Chebyshev approximat ion on an interval by the Remez algorithm, one can use a power basis or a Chebyshev polynomial basis for polynomials. More generally, in rat ional Chebyshev approximat ion by the F r a s e r H a r t R e m e z algorithm, one can use a power basis or a Chebyshev polynomial basis for polynomials. We consider in this pape r the stabil i ty of the sys tem of leveling equat ions for the two choices of basis. I t should be noted t ha t Fraser and H a r t [6] and " C o m p u t e r Approximat ions" [8, pp. 55-56] advocate the Chebyshev polynomial basis, whereas the ALGOL procedure of Cody, Fraser, and H a r t [3] uses a power basis. T h e Chebyshev polynomials Tk are discussed in the texts of Cheney [1], Davis [4], and Rivlin [9], and by Clenshaw [2]. T h e y are used for approximat ion on [ 1 , 1]. For approximat ion on [0, 1], the shifted Chebyshev polynomials T~ , given by Clenshaw [2], are used instead. We will assume tha t approximat ion problems have been converted, by change of variable if necessary, to approximat ion on a s tandard interval. For a square matr ix A the condition number cond(A) is defined to be [[A[[ [[ A -1 [[, where [[ ][ is a matr ix norm. Some relat ions suggesting tha t the stabil i ty of solution of a l inear sys tem with matr ix A depends on cond(A) are given by Wilkinson [11, formulas (102) and {107)]. T h e L1 and L~ mat r ix norms are given by Wilkinson [11, formulas (12) and (13)].

The most efficacious one‐electron bases for determining and representing correlated molecular electronic wave functions. Unity in seemingly disparate electron correlation methods

Abstract: Coefficient matrices and associated operator matrices are being used increasingly in various large‐scale correlation methods. These matrices are used to find and represent the wave function directly in terms of one‐electron basis functions. They eliminate serious redundancies in computation and provide for the use of different sets of nonorthogonal external orbitals to improve convergence. These features are shown to be independent of the choice of a one‐electron basis, and illustrative calculations are presented for N2H2, HCN, and HNC.

Simple results for one-dimensional periodic potentials

Abstract: A simple solution is derived for an arbitrary one-dimensional potential in terms of basis functions in a unit cell and Chebyshev polynomials. The effect on the allowed energy bands of a commensurate disturbance is also discussed.

Calculation of S-matrix poles using the Schwinger variational principle

Abstract: The analytic properties of the Schwinger variational expression for the T matrix are discussed. It is shown that the Schwinger principle provides a variational approximation for the poles of the analytically continued T matrix (or S matrix). The method is applied to calculate bound states, virtual states and resonances of a simple model problem (s-wave scattering by an attractive square-well potential) where it yields accurate results with few basis functions. The Schwinger principle appears to be a useful alternative to other methods for the calculation of complex S-matrix poles such as the complex coordinate or complex basis function expansion methods.

Electron-ion collisions at medium energies. I. L 2 basis and resonance averaging

Abstract: Treatment of the continuum spectrum of a scattering system by a discrete set of pseudostates generated by the diagonalization of the Hamiltonian in square-integrable basis functions results in spurious resonances. An energy-averaging procedure is formulated using the Stieltjes mean values for the spectral density. The connection between this L/sup 2/ basis approach to scattering problems and the stabilization method is discussed, and an improved form of the latter is derived.

Geometry of collective motions

Abstract: A general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL+(3,R) and the quotient set GL+(3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. It is proposed that an appropriate basis function for the GL+(3,R) collective model is provided by an irreducible representation of the boson SU(6) group.

Some applications of geometry in numerical analysis

Abstract: Surface interpolation is used to produce basis functions for two-dimensional curved finite elements. The connection between rational surfaces and isoparametric methods is discussed with the occurrence of the Steiner surface being highlighted. A corollary to Max Noether's intersection theorem is used to produce high order stable bases. Parametric cubic curves are discussed from a geometrical viewpoint and this viewpoint is used to develop transfinite blending functions for a wide variety of shapes.

Generalization of the Yvon-Born-Green theory to polyatomic liquids

Abstract: The YBG-equation for the pair-correlation function is generalized to polyatomic fluids. The expansion of the interaction potential and of the correlation function with respect to angle-dependent basis functions as well as the connection of macroscopic quantities with the correlation function are treated in full generality for axisymmetric molecules. For the special case of Stockmayer potential, the generalized YBG-equation is solved numerically and the radial dependence of the various angular parts of the pair-correlation function is displayed graphically. Furthermore, the resulting equation of state and Kirkwood's g -factor are calculated.

A variational formulation of multiple scattering in muffin-tin alloys

Abstract: The authors develop a method for the band calculation of muffin-tin alloys which is as simple as the Korringa-Kohn-Rostoker (KKR) method for pure crystals. They begin by postulating Bloch-like basis functions, minimise the energy and arrive at a secular matrix with a formal resemblance to the secular matrix of the average t-matrix approximation (ATA) method. The density of energy states is derived from an argument based on the number of orthogonal basis functions. Using a similar argument, they derive an expression for the energy variance of the Bloch-like states. The advantage of the method is its extreme simplicity: an alloy calculation actually becomes as simple as a band structure calculation for a pure crystal. The method is also extended to situations in which it is feasible to increase the precision of the calculation by working with unit cells with more than one atomic site. Finally, the method is applied to a unidimensional alloy and compared favourably to a CPA and to a Monte Carlo calculation.

Method of correlated basis functions and FHNC theory

Abstract: A method for the accurate evaluation of the diagonal and off-diagonal matrix elements of the Hamiltonian operator within the correlated basis function scheme is presented. Considering a basis consisting of Slater determinants, correlated with a Jastrow correlation factor, the expectation value of the kinetic energy operator has been expressed in the Jackson-Feenberg form. A diagrammatic analysis is presented for the construction and evaluation of the cluster expansions of the required quantities incorporating the Fermi hypernetted chain theory.