Showing papers on "Gaussian published in 1977"

Gaussian basis sets for molecular calculations. The representation of 3d orbitals in transition‐metal atoms

Abstract: Augmented (4d) and (5d) Gaussian basis sets are presented which provide a balanced description of the 4s23dn−2, 4s3dn−1, and 3dn configurations of transition−metal atoms. When compared to accurate Hartree–Fock calculations, the (4d) and (5d) expansions in the literature, which have been optimized for the 4s23dn−2 configuration, can lead to errors of several eV for states of the latter configurations where the 3d orbitals become more diffuse. The augmented sets reported here consist of a single diffuse basis function added to the original sets, where the orbital exponent of the added function was optimized for the 3dn configuration. Basis sets and contraction schemes are presented for the atoms Sc through Cu.

Methods of Electronic Structure Theory

Abstract: 1. Gaussian Basis Sets for Molecular Calculations.- 2. The Floating Spherical Gaussian Orbital Method.- 3. The Multiconfiguration Self-Consistent Field Method.- 4. The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree-Fock Wave Functions.- 5. Pair Correlation Theories.- 6. The Method of Configuration Interaction.- 7. The Direct Configuration Interaction Method from Molecular Integrals.- 8. A New Method for Determining Configuration Interaction Wave Functions for the Electronic States of Atoms and Molecules: The Vector Method.- 9. The Equations of Motion Method: An Approach to the Dynamical Properties of Atoms and Molecules.- 10. POLYATOM: A General Computer Program for Ab Initio Calculations.- 11. Configuration Expansion by Means of Pseudonatural Orbitals.- Author Index.

Gaussian Basis Sets for Molecular Calculations

Abstract: In the following chapters the electronic structure of molecules will be discussed and the techniques of electronic structure calculations presented. Without exception the molecular electronic wave functions will be expanded in some convenient, but physically motivated, set of one-electron functions. Since the computational effort strongly depends on the number of expansion functions (see, e.g., the following chapters), the set of functions must be limited as far as possible without adversely affecting the accuracy of the wave functions. This chapter will discuss the choice of such functions for molecular calculations.

Path integrals for waves in random media

Abstract: The problem of wave propagation in a random medium is formulated in terms of Feynman’s path integral. It turns out to be a powerful calculational tool. The emphasis is on propagation conditions where the rms (multiple) scattering angle is small but the log‐intensity fluctuations are of order unity—the so‐called saturated regime. It is shown that the intensity distribution is then approximately Rayleigh with calculable corrections. In an isotropic medium, the local or Markov approximation which is commonly used to compute first and second (at arbitrary space–time separation) moments of the wave field is explicitly shown to be valid whenever the rms multiple scattering angle is small. It is then shown that in the saturated regime the third and higher moments can be obtained from the first two by the rules of Gaussian statistics. There are small calculable corrections to the Gaussian law leading to ’’coherence tails.’’ Correlations between waves of different frequencies and the physics of pulse propagation are studied in detail. Finally it is shown that the phenomenon of saturation is physically due to the appearance of many Fermat paths satisfying a perturbed ray equation. For clarity of presentation much of the paper deals with an idealized medium which is statistically homogeneous and isotropic and is characterized by fluctuations of a single typical scale size. However, the extension to inhomogneous, anisotropic, and multiple scale media is given. The main results are summarized at the beginning of the paper.

Bayesian Methods in Nonlinear Digital Image Restoration

Abstract: Prior techniques in digital image restoration have assumed linear relations between the original blurred image intensity, the silver density recorded on film, and the film-grain noise. In this paper a model is used which explicitly includes nonlinear relations between intensity and film density, by use of the D-log E curve. Using Gaussian models for the image and noise statistics, a maximum a posteriori (Bayes) estimate of the restored image is derived. The MAP estimate is nonlinear, and computer implementation of the estimator equations is achieved by a fast algorithm based on direct maximization of the posterior density function. An example of the restoration method implemented on a digital image is shown.

Spectral Moment Estimates from Correlated Pulse Pairs

Abstract: Estimates statistics of the first two power spectrum moments from the pulse pair covariance are analyzed. The input signal is assumed to be colored Gaussian and the noise, white Gaussian. Perturbation formulas for the standard deviation of both mean frequency and spectrum width are applied to a Gaussian shaped power spectrum, and so is a perturbation formula for the bias in the width estimate. Mean frequency estimation from interlaced pulse pairs is presented. Throughout this study, estimators from independent, spaced, and contiguous pulse pairs are compared to provide a continuum of statistics from equispaced tightly correlated to statistically independent pulse pairs.

Gaussian basis sets for the atoms gallium through krypton

Abstract: Hartree‐Fock calculations are reported on the atoms gallium through krypton using (13s9p5d) and (14s11p5d) Gaussian basis sets. (AIP)

Eulerian and Lagrangian renormalization in turbulence theory

Abstract: Systematic renormalized perturbation expansions for turbulence and turbulent convection are constructed which are invariant at each order under random Galilean transformations. Two types of expansion are developed whose lowest truncations give, respectively, the Lagrangian-history direct-interaction approximation and the abridged Lagrangian-history direct-interaction approximation. These approximations previously were derived as heuristic modifications of the Eulerian direct-interaction approximation (Kraichnan 1965). The techniques used involve reversion of primitive perturbation expansions for the generalized velocity field u(x, t/s), defined as the velocity measured at time s in the fluid element which passes through x at time t. The new expansions are illustrated by application to a random linear oscillator, to passive-scalar convection by a random velocity and to the Lagrangian velocity covariance. The lowest term of the expansion for the passive scalar gives Taylor's (1921) exact result for dispersion of fluid elements, and higher terms describe the deviations of the particle-displacement distribution from Gaussian form. In all the applications the assumed underlying statistics are more general than Gaussian statistics, which appear as a special case.

A stochastic variational method for few-body systems

Abstract: A detailed description of the stochastic variational method (SVM) suggested recently for solving the few-body problem in quantum mechanics is given. Results obtained with the SVM in a randomized gaussian basis are compared with the analogous results of other authors. It is shown that the matrix elements of the Hamiltonian for all conventional forms of the N-N potentials (and for the Coulomb potential too) in the randomized gaussian basis are evaluated fully analytically. The asymptotic behaviour of the variational functions obtained is studied.

The generalized Langevin equation with Gaussian fluctuations

Abstract: It is shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two‐point correlation function. The resulting description corresponds with a stationary, Gaussian, non‐Markovian process. Fokker–Planck‐like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is nonstationary, Gaussian, and Markovian.

Statistical properties of laser speckle produced in the diffraction field

Abstract: The statistical properties (i.e., the probability density function and the average contrast) of laser speckle produced by a weak diffuse object in the diffraction field have been theoretically and experimentally studied with the assumption of the Gaussian statistics for the formation of speckles. The general formulas for the probability density function and the average contrast, which are valid for an entire range of object surface and for the whole diffraction field, are introduced, and their special cases, which have been studied in the past, are derived and discussed. These formulas for the probability density function and the averagae contrast are actually evaluated in the diffraction field of a weak diffuse object illuminated by the Gaussian laser beam. The experimental study has been conducted to verify the above theoretical study, and its results are in good agreement with the theoretical ones. The circularity and noncircularity of the speckle statistics in the diffraction field are discussed on the basis of the theoretical study.

Theory of far-infrared generation by optical mixing

Abstract: The theory of far-infrared generation by optical mixing of focused Gaussian beams is developed, taking into account the effects of diffraction, absorption, double refraction, and multiple and total reflections at the boundary surfaces. Results of numerical calculations are presented. It is shown that focusing of the pump beams appreciably enhances the far-infrared output despite the strong far-infrared diffraction. In a 1-cm-long crystal, the optimum focal-spot size is approximately equal to or smaller than the far-infrared wavelength for an output at frequency less than 100 cm/sup -1/. Double refraction of the pump beams is relatively unimportant. Both far-infrared absorption and boundary reflections have major effects on the far-infrared output and its angular distribution. The former is often the factor which limits the output power. We show that a simple model treating the nonlinear polarization as a constant-1/e-radius Gaussian distribution of radiating dipoles is a good approximation to the problem. We also compare the results of our calculations with those for second-harmonic generation.

A direct computation of gravity and magnetic anomalies caused by 2- and 3-dimensional bodies of arbitrary shape and arbitrary magnetic polarization by equivalent-point method and a simplified cubic spline

Abstract: A computational method, which combines the Gaussian quadrature formula for numerical integration and a cubic spline for interpolation in evaluating the limits of integration, is employed to compute directly the gravity and magnetic anomalies caused by 2-dimensional and 3-dimensional bodies of arbitrary shape and arbitrary magnetic polarization. The mathematics involved in this method is indeed old and well known. Furthermore, the physical concept of the Gaussian quadrature integration leads us back to the old concept of equivalent point masses or equivalent magnetic point dipoles: namely, the gravity or magnetic anomaly due to a body can be evaluated simply by a number of equivalent points which are distributed in the “Gaussian way” within the body. As an illustration, explicit formulas are given for dikes and prisms using 2 × 2 and 2 × 2 × 2 point Gaussian quadrature formulas. The basic limitation in the equivalent‐point method is that the distance between the point of observation and the equivalent poin...

Generalized Gaussian beams in free space

Abstract: A new solution of the paraxially approximated wave equation in free space is presented. It has the form of a Hermite-Gaussian function, where the argument of the Hermite polynomial is generally complex. The usual Gaussian beams where such argument is real and those recently described by Siegman where it is complex but coinciding with that of the Gaussian function are particular cases of the general solution presented here. The application of these generalized Gaussian beams to the analysis of the beam emerging from a complex graded-index medium is discussed in detail.

Analysis of Nonstationary, Gaussian and Non-Gaussian, Generalized Langevin Equations Using Methods of Multiplicative Stochastic Processes

Abstract: Using the methods of multiplicative stochastic processes, a thorough analysis of “non-Markovian,” generalized Langevin equations is presented. For the Gaussian case, these methods are used to show that the nonstationary Fokker-Planck equation already found by Adelman and others is also obtainable from van Kampen's lemma for stochastic probability flows. Here, results applicable to an arbitraryn-component process are obtained and the specific two-component case of the Brownian harmonic oscillator is presented in detail in order to explicitly exhibit the matrix algebraic methods. The non-Gaussian case is presented at the end of the paper and shows that the methods already used in the Gaussian case lead directly to results for the non-Gaussian case. In order to use the methods of multiplicative stochastic processes analysis, it is necessary to transform the “non-Markovian,” generalized Langevin equation using a stochastic extension of a transformation discussed by Adelman. This transformation removes the “memory kernel” term in the usual generalized Langevin equation and in the Gaussian case leads to the result that the original process was in fact not “non-Markovian” but actually nonstationary,Markovian.

The Generation of Correlated Weibull Clutter for Signal Detection Problems

Abstract: A method is discussed for generating Weibull vectors with a desired correlation matrix and specified parameters. Such vectors may represent samples of a correlated clutter signal. The presented method makes use of a suitable nonlinear transformation of random Gaussian vectors with correlated components. Computational aspects of the method are also discussed.

Multiple Gaussian Targets: The Track-on-Jam Problem

Abstract: When a radar with amplitude comparison monopulse arithmetic encounters signals from multiple Gaussian sources it will "point" to the centroid of the incident radiation. The probability density function (pdf) of the monopulse ratio when N independent samples of difference and sum signals are processed in a maximum likelihood receiver is derived. For finite jam-to-noise ratio the estimate has a bias which is independent of N. The variance in the estimate does however depend upon N. Central moments of order less than or equal 2N - 2 exist and are given by a simple formula. Plots of the pdf and its bias and variance for various jam-to-noise ratios, locations of the centroid with respect to the boresight direction, and number of samples processed are presented in the accompanying figures.

Uniform quality Gaussian basis sets

Abstract: A direct optimization technique was applied to determine uniformly balanced, optimum (4s2p), (6s3p), (8s4p), and (10s5p) Gaussian basis sets for the first row atoms. These bases formally correspond to 2G, 3G, 4G, and 5G function representations per symmetry type per shell. The basis sets are throroughly balanced and all satisfy a rather rigorous quality criterion in terms of the local properties of the energy hypersurface over the space of the orbital exponents.

Uniform quality constrained gaussian basis sets

Abstract: Uniformly balanced (6S3P), (7S3P), and (8S4P) gaussian basis sets with identical exponent sets for functions describing the 2s and 2p subshells have been obtained for the first row atoms. The basis sets have been determined using a direct optimization technique; they are thoroughly balanced and satisfy a rigorous quality criterion. These uniform quality constrained basis sets were designed for applications in ab initio programs of the type of the GAUSSIAN 70 program system, that may utilize the integration-time saving constraint α2s = α2p.