Showing papers on "Basis (linear algebra) published in 1981"

Exact finite-dimensional filters for certain diffusions with nonlinear drift

Abstract: Let and be independent Wiener processes, and consider the task of estimating a diffusion solving the stochastic DE dx t =f(x t )dt+dw t on the basis of noisy observations defined bydy t =x t dt+db t . This problem is governed by the filtering equation for the unnormalized conditional density with A * the forwarded operator Theorem: if then the fundamental solution of the filtering equation can be written explicity in terms of a small number of statistics satisfying a matrixvector equation. The Lie algebraic interpretation of this result is studied and described. Extensions to many dimensions and applications to optimal stochastic control readily follow.

Theory of electronic transitions in slow atomic collisions

Abstract: This review deals with quantitative descriptions of electronic transitions in atom-atom and ion-atom collisions. In one type of description, the nuclear motion is treated classically or semiclassically, and a wave function for the electrons satisfies a time-dependent Schr\"odinger equation. Expansion of this wave function in a suitable basis leads to time-dependent coupled equations. The role played by electron-translation factors in this expansion is noted, and their effects upon transition amplitudes are discussed. In a fully quantum-mechanical framework there is a wave function describing the motion of electrons and nuclei. Expansion of this wave function in a basis which spans the space of electron variables leads to quantum-mechanical close-coupled equations. In the conventional formulation, known as perturbed-stationary-states theory, certain difficulties arise because scattering boundary conditions cannot be exactly satisfied within a finite basis. These difficulties are examined, and a theory is developed which surmounts them. This theory is based upon an intersecting-curved-wave picture. The use of rotating or space-fixed electronic basis sets is discussed. Various bases are classified by Hund's cases (a)-(e). For rotating basis sets, the angular motion of the nuclei is best described using symmetric-top eigenfunctions, and an example of partial-wave analysis in such functions is developed. Definitions of adiabatic and diabatic representations are given, and rules for choosing a good representation are presented. Finally, representations and excitation mechanisms for specific systems are reviewed. Processes discussed include spin-flip transitions, rotational coupling transitions, inner-shell excitations, covalent-ionic transitions, resonant and near-resonant charge exchange, fine-structure transitions, and collisional autoionization and electron detachment.

The choice of Gaussian basis sets for molecular electronic structure calculations

Abstract: The choice of Gaussian type basis sets for electronic structure calculations of molecules is discussed in detail for treatments on the SCF and Cl level. This article is organized in the following sections : I. Introduction, II. Mathematical foundation of the LCAO-MO method, III. Basis sets of first and second row atoms in SCF calculations, IV. Transition metals, V. Beyond-Hartree-Fock calculations, VI. Summary.Detailed proposals are made for the choice of basis sets at various levels of computational expense.

Application of discrete-basis-set methods to the Dirac equation

Abstract: Variational solutions to the Dirac equation in a discrete ${L}^{2}$ basis set are investigated. Numerical calculations indicate that for a Coulomb potential, the basis set can be chosen in such a way that the variational eigenvalues satisfy a generalized Hylleraas-Undheim theorem. A number of relativistic sum rules are calculated to demonstrate that the variational solutions form a discrete representation of the complete Dirac spectrum including both positive-and negative-energy states. The results suggest that widely used methods for constructing ${L}^{2}$ representations of the nonrelativistic electron Green's function can be extended to the Dirac equation. As an example, the relativistic basis sets are used to calculate electric dipole oscillator strength sums from the ground state, and dipole polarizabilities.

A systematic preparation of new contracted Gaussian‐type orbital sets. V. From Na through Ca

Abstract: Minimal compact contracted Gaussian basis sets are constructed for the atoms from Na to Ca. They give satisfactory valence shell orbital energies, although they are minimal-type basis sets. Split-type basis sets are also derived from the minimal Gaussian basis sets in order to enhance the flexibility of the basis sets for molecular calculations.

Algorithms and Data Structures for Sparse Symmetric Gaussian Elimination

Abstract: In this paper we present algorithms and data structures that may be used in the efficient implementation of symmetric Gaussian elimination for sparse systems of linear equations with positive definite coefficient matrices. The techniques described here serve as the basis for the symmetric codes in the Yale Sparse Matrix Package.

Gaussian basis sets for the transition metals of the first and second series

Abstract: Gaussian basis sets consisting of, respectively, (13s, 7p, 5d) and (14s, 8p, 7d) Gaussian functions have been optimized for the transition metal atoms of the first and second series. The optimization criteria and the applicability of these atomic sets for molecular calculations are discussed.

On the matrix elements of the U(n) generators

Abstract: A straightforward derivation of the matrix elements of the U(n) generators is presented using algebraic infinitesimal techniques. An expression for the general fundamental Wigner coefficients of the group is obtained as a polynomial in the group generators. This enables generalized matrix elements to be defined without explicit reference to basis states. Such considerations are important for treating groups such as Sp(2n) whose basis states are not known.

Basis set error in quadrupole moment calculations

Abstract: For a series of first row diatomic molecules, quadrupole moments obtained from restricted Hartree-Fock level calculations with various basis sets are compared with exact values obtained from numerical solution of the Hartree-Fock equations. Significant basis set errors are frequently observed, even with large and/or exponent optimized basis sets. Estimates of the correlation errors in the quadrupole moments lead to the conclusion that basis set error is probably just as serious as correlation error for most molecular quadrupole moment calculations.

Reduced basis technique for collapse analysis of shells

Abstract: A hybrid finite-element Rayleigh-Ritz technique is used to predict the collapse behavior of shells. In this hybrid technique, the modeling versatility of the finite-element method is preserved, and a significant reduction in the number of degrees of freedom is achieved by expressing the nodal displacement vector as a linear combination of a small number of basis vectors. A Rayleigh-Ritz technique is used to approximate the finite-element equations of the discretized shell by a reduced system of nonlinear algebraic equations. A scalar function is introduced to measure the degree of nonlinearity of the structure for the case of loading applied by means of axial end shortening. Also, a quantitative measure for the error of the reduced system of equations is proposed. Some insight is given as to why and when the reduced basis technique works, and the effectiveness of the technique for predicting the collapse behavior of shells is demonstrated by means of a numerical example of elastic collapse of an axially compressed pear-shaped cylinder.

Source power estimation method associated with high resolution bearing estimator

Abstract: In the eigenvalue - eigenvector decomposition of the spectral density matrix of the signals received on a passive array, two sets of eigenvectors are found. The first set contains eigenvectors which are asymptotcally orthogonal to the sources direction vectors : from them a high resolution bearing estimator has been deduced. The other set contains eigenvectors which are asymptotically a basis for the sources direction vectors space. It is shown in this paper that from this set and the corresponding eigenvalues, an estimator for the sources spectral densities can be derived. Simulation results are given.

Molecular wave functions in momentum space

Abstract: Momentum-space calculations exhibit two kinds of advantages over position space: First, the numerical solution of Hartree-Fock equation is feasible without expansion of the wave functions in a particular basis. Equations only exhibit one avoidable singularity even for the multicenter case. Several mathematical techniques are presented, including standard fast Fourier-transform (FFT) techniques and numerical calculation of the involved convolutions. Second, momentum representation contributes in an original way to a better understanding of several physical problems arising in quantum chemistry. The two-body density matrix involving the electronic correlation are examined in both position and momentum space. If an expansion in Gaussian functions is used, momentum space renders feasible the obtainment of a multidimensional fully correlated wave function, starting from the Hartree-Fock solution.

A Nested Partitioning Procedure for Numerical Multiple Integration

Abstract: : An algorithm is presented for adaptively partitioning a multidimensional coordinate space based on optimization of a scalar function of the coordinates. The goal is to construct a set of hyperrectangular regions, such that the variation of function values within each region is small. These regions are then used as the basis for a stratified sampling estimate of the definite integral of the function. (Author)

Computer for calculating the similarity between patterns

Abstract: A computer for calculating the similarity between first and second patterns, each represented by a sequence of feature vectors, comprises a calculating circuit for calculating a weighting factor from feature vectors in the first pattern, a weighting circuit for applying the weighting factor to the feature vectors of the second pattern to calculate an estimated vector for the second pattern, and a similarity calculating circuit for calculating and determining the similarity between the first and the second patterns on the basis of the feature vector of the first pattern and of the estimated vector of the second pattern.

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 approximate solution concerning strong evaporation into a half space

Abstract: Evaporation of a liquid into a vacuum occupying a half space is investigated on the basis of the one-dimensional BGK model linearized about a drifting Maxwellian distribution. TheF N method is used to deduce accurate numerical results for the perturbations in the density and temperature.

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.

Irreducible Components of the Density Matrix

Abstract: As discussed in Chapters 1 and 2 it is often useful to expand ρ in terms of a conveniently chosen operator set Q i . This method has two main advantages. First of all, it gives a more satisfactory definition of ρ (see, for example, Section 1.1.7), and secondly by using explicitly the algebraic properties of the basis operators the calculations are often greatly simplified (see Section 2.5). The usefulness of this method depends on the choice of the basis operator set. When the angular symmetries of the ensemble of interest are important it is convenient to expand ρ in terms of irreducible tensor operators. This method provides a well-developed and efficient way of using the inherent symmetry of the system. It also enables the consequences of angular momentum conservation to be simply allowed for and enables dynamical and geometrical factors in the equation of interest to be separated from each other.

On the Fundamental Solution of Delay-Differential Equations in

Abstract: Eq. (1.1) is reduced to The study of (1.2) is rather classic and a great number of research papers and monographs exist (see [5, 9, 10, 15, 161 and their references). In these works, various types of existence, uniqueness, differentiability and continuous dependence theorems are established on the basis of the construction and the representation of the semi-group T(t) relating to (1.2). Our purpose here is to give two representations of the fundamental solution of (1.1) in terms of Z(t) and