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Showing papers on "Gaussian published in 1971"


Journal ArticleDOI
TL;DR: In this article, an extended basis set of atomic functions expressed as fixed linear combinations of Gaussian functions is presented for hydrogen and the first row atoms carbon to fluorine, where each inner shell is represented by a single basis function taken as a sum of four Gaussians and each valence orbital is split into inner and outer parts described by three and one Gaussian function, respectively.
Abstract: An extended basis set of atomic functions expressed as fixed linear combinations of Gaussian functions is presented for hydrogen and the first‐row atoms carbon to fluorine. In this set, described as 4–31 G, each inner shell is represented by a single basis function taken as a sum of four Gaussians and each valence orbital is split into inner and outer parts described by three and one Gaussian function, respectively. The expansion coefficients and Gaussian exponents are determined by minimizing the total calculated energy of the atomic ground state. This basis set is then used in single‐determinant molecular‐orbital studies of a group of small polyatomic molecules. Optimization of valence‐shell scaling factors shows that considerable rescaling of atomic functions occurs in molecules, the largest effects being observed for hydrogen and carbon. However, the range of optimum scale factors for each atom is small enough to allow the selection of a standard molecular set. The use of this standard basis gives theoretical equilibrium geometries in reasonable agreement with experiment.

8,551 citations


Book
01 Jan 1971
TL;DR: In underwater sonar systems, external acoustic noise is generated by waves and wind on the water surface, by biological agents (fish, prawns, etc.), and by man-made sources such as engine noise.
Abstract: Probability. Random Processes. Narrowband Signals. Gaussian Derived Processes. Hypothesis Testing. Detection of Known Signals. Detection of Signals with Random Parameters. Multiple Pulse Detection of Signals. Detection of Signals in Colored Gaussian Noise. Estimation of Signal Parameters. Extensions. References. Bibliography. Index.

1,421 citations


Journal ArticleDOI
TL;DR: A density approximation involving convex combinations of gaussian density functions is introduced and proposed as a meaningful way of circumventing the difficulties encountered in evaluating these relations and in using the resulting densities to determine specific estimation policies.

965 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the field of a Gaussian beam can be represented by a function G(P) =eikr/r, where r is the distance from the observation point P to a fixed point having a complex location.
Abstract: It is observed that the function G(P)=eikr/r, where r is the distance from the observation point P to a fixed point having a complex location, represents the field of a Gaussian beam. This can be used to justify, without further computation, the application of formulas of ordinary optics to the transformation of beams through optical systems. It can also be used to solve very simply some problems of diffraction and scattering of Gaussian beams.

557 citations


Journal ArticleDOI
TL;DR: In this paper, estimators for the scale parameter and characteristic exponent of symmetric stable distributions are proposed and Monte Carlo studies of these estimators are reported. And the powers of various goodness-of-fit tests of a Gaussian null hypothesis against non-Gaussian stable alternatives are also investigated.
Abstract: Building on results of an earlier article [6], estimators are suggested for the scale parameter and characteristic exponent of symmetric stable distributions, and Monte Carlo studies of these estimators are reported. The powers of various goodness-of-fit tests of a Gaussian null hypothesis against non-Gaussian stable alternatives are also investigated. Finally, a test of the stability property of symmetric stable variables is suggested and demonstrated.

529 citations


Journal ArticleDOI
TL;DR: Nonparametric normal range estimates are, for practical purposes, as accurate as estimates that assume the distribution of data to be gaussian or loggaussian when the distribution assumed is true.
Abstract: The choice of statistical method can greatly influence the calculated normal range apart from any biological or chemical considerations. Nonparametric normal range estimates are, for practical purposes, as accurate as estimates that assume the distribution of data to be gaussian or loggaussian when the distribution assumed is true. When data are not distributed as assumed, nonparametric estimates are more accurate.

506 citations


Journal ArticleDOI
01 Dec 1971
TL;DR: Results, both old and new, which will aid the reader in applying Volterra-series-type analyses to systems driven by sine waves or Gaussian noise are presented.
Abstract: Troublesome distortions often occur in communication systems. For a wide class of systems such distortions can be computed with the help of Volterra series. Results, both old and new, which will aid the reader in applying Volterra-series-type analyses to systems driven by sine waves or Gaussian noise are presented. The n-fold Fourier transform G n of the nth Volterra kernel plays an important role in the analysis. Methods of computing G n from the system equations are described and several special systems are considered. When the G n are known, items of interest regarding the output can be obtained by substituting the G n in general formulas derived from the Volterra series representation. These items include expressions for the output harmonics, when the input is the sum of two or three sine waves, and the power spectrum and various moments, when the input is Gaussian. Special attention is paid to the case in which the Volterra series consists of only the linear and quadratic terms.

479 citations


Journal ArticleDOI
TL;DR: In this article, the Hartree-Fock representation of the first row atoms and hydrogen was investigated by means of Hartree Fock calculations on the water and nitrogen molecules, and it was found that a two-term Gaussian expansion is required to adequately represent the 3D polarization functions of hydrogen and water, whereas a single Gaussian function is sufficient.
Abstract: The representation of polarization functions for the first row atoms and hydrogen is investigated by means of Hartree—Fock calculations on the water and nitrogen molecules. It is found that a two‐term Gaussian expansion is required to adequately represent the 3d polarization functions of the first row atoms, whereas for the 2p polarization functions of hydrogen, a single Gaussian function is sufficient. The interaction of oxygen and hydrogen polarization functions in the water molecule is briefly discussed.

330 citations


Journal ArticleDOI
TL;DR: The definition of fast fractional Gaussian noises, as sums of Markov-Gauss and other simple processes, fits the intuitive ideas that climate can be visualized as either unpredictably inhomogeneous or ruled by a hierarchy of variable regimes.
Abstract: By design fast fractional Gaussian noises (ffGn) have the following characteristics: The number of operations needed to generate them is relatively small. The long run statistical dependence that they exhibit is strong and has the form required for self similar hydrology. Their short run properties, as expressed by the correlations between successive or nearly successive yearly averages, are adjustable (within bounds) and can be fitted to the corresponding short run properties of records. Extension to the multidimensional (multisite) case should raise no essential difficulty. Finally, their definition, as sums of Markov-Gauss and other simple processes, fits the intuitive ideas that climate can be visualized as either unpredictably inhomogeneous or ruled by a hierarchy of variable regimes.

314 citations


Journal ArticleDOI
M. Gans1
TL;DR: In a fading channel, maximal ratio diversity combilling improves the average signal-to-noise ratio over that of a single branch in proportion to the number of diversity branches combined, however, its main advantage is the reduction of the probability of deep fades.
Abstract: In a fading channel, maximal ratio diversity combilling improves the average signal-to-noise ratio over thatof a single branch in proportion to the number of diversity branches combined. However, its main advantage is the reduction of the probability of deep fades. The effect of Gaussian errors in the combiner weighting factors on the probability distribution of the output signal-to-noise ratio is computed. The limits on allowable error for a specified probability of fades below any given level are indicated. The results are applied to a mobile radio example in which the weighting factor is determined from a pilot transmitted along with the signal. To keep the pilot from overlapping the signal, they are separated either in frequency or in time. In this case the Gaussian error is due to decorrelation of the pilot from the signal either because their frequency separation or their time separation is too large.

252 citations


Journal ArticleDOI
TL;DR: A Gaussian basis set consisting of 12s-type, 6p-type and 4d-type functions has been optimized for the third row atoms, together with a 9s, 5p, 3d set for the corresponding dipositive ion as mentioned in this paper.
Abstract: A Gaussian basis set consisting of 12s-type, 6p-type and 4d-type functions has been optimized for the third row atoms, together with a 9s, 5p, 3d set for the corresponding dipositive ion. The applicability of these atomic sets for molecular calculation is discussed.

Journal ArticleDOI
TL;DR: In this article, the use of rays to construct fields is illustrated by finding the field in the region z>0 when the field is given on the plane z = 0, where the calculation involves only complex rays.
Abstract: The use of rays to construct fields is illustrated by finding the field in the region z>0 when the field is given on the plane z = 0. This construction is valid for complex rays as well as real ones. The method is applied to a gaussian field in the plane z = 0, in which case a gaussian beam results. The calculation involves only complex rays. Exactly the same results are also obtained by applying the method of stationary phase to an integral representation of the field. However, the ray method is simpler than the stationary-phase method, and it is also applicable to problems for which the stationary-phase method cannot be used because no integral representation of the field is known.

Journal ArticleDOI
TL;DR: In this paper, an algebraic algorithm for the Gaussian quadrature of the one-dimensional product integral is described, where the weight function function is known through modified moments and the coefficients are polynomials of degreel satisfying 3-term recurrence relations with known coefficients.
Abstract: An algebraic algorithm, the long quotient- modified difference (LQMD) algorithm, is described for the Gaussian quadrature of the one-dimensional product integral ?f(x)w(x)dx when the weight function ?(x) is known through modified momentsv l =; theP l (x) are any polynomials of degreel satisfying 3-term recurrence relations with known coefficients. The algorithm serves to establish the co-diagonal matrix, the eigenvalues of which are the Gaussian abscissas. Applied to ordinary moments it requires far fewer divisions than the quotient-difference algorithm; if theP l (x) are themselves orthogonal with a kernelw 0 03F0;, there is no instability due to rounding errors. For smooth kernels ?(x) it is safe to use secondorder interpolation in determining the eigenvalues by Givens' method. The Christoffel weights can be expressed as ratios of two terms which are most easily calculated in a Sturm sequence beginning with the highest value ofl. A formula for the Christoffel weights applicable for rational versions of theQR algorithm is also derived. Convergence and the propagation of rounding errors are illustrated by several examples, and anAlgol procedure is given.

Book ChapterDOI
TL;DR: The following sections are included:INTRODUCTIONALTERNATIVE ESTIMATORSSAMPLING RESULTS Conclusion as mentioned in this paper, and the following sections include the following abstracts:
Abstract: The following sections are included:INTRODUCTIONALTERNATIVE ESTIMATORSSAMPLING RESULTSCONCLUSIONSREFERENCES (This abstract was borrowed from another version of this item.)

Journal ArticleDOI
TL;DR: Comparisons and applications are made of the various procedures which have been devised recently for obtaining bounds to dispersion force coefficients, and it is shown that these procedures are similar to the Gaussian quadrature and Pade methods.
Abstract: Comparisons and applications are made of the various procedures which have been devised recently for obtaining bounds to dispersion force coefficients. Particular attention is given to the methods based on approximate Gaussian quadratures and on Pade approximants, both of which require oscillator‐strength sum rule values for their implementation. The equivalence of the bounds obtained from the two procedures, when identical input information is employed, is demonstrated to all orders, and the improvements that can result from the introduction of additional information, in the form of experimental transition frequencies and other sum rules, is investigated in the course of numerical applications. Comparisons with alternative bounding methods, such as those based on variational principles and related inner projection techniques, show that these procedures, which can also employ sum rule values for their implementation, are similar to the Gaussian quadrature and Pade methods and provide identical results if ...

Journal ArticleDOI
TL;DR: In this paper, a generalization of the Navier-Stokes equation, valid for wavelengths and times of a molecular order of magnitude, is discussed on the basis of viscoelastic behavior of simple classical liquids.
Abstract: A generalization of the Navier-Stokes equation, valid for wavelengths and times of a molecular order of magnitude, is discussed on the basis of viscoelastic behavior of simple classical liquids. In this theory, transport coefficients are replaced by appropriate viscoelastic memory functions. The theory is verified by analyzing the data on current-correlation functions obtained from computer experiments. Three different models for the time dependence of the viscoelastic memory are investigated, namely, a single-exponential decay, a modified-exponential decay, and a Gaussian decay. It is obsrved that the memory functions are approximately Gaussian, at least for times of the order of one or two relaxation times. This is in agreement with a conjecture of Forster, Martin, and Yip. The wave-number dependence of the half-width of the Gaussian decay, and of the longitudinal- and shear-viscosity coefficients, are found from computer experiments. The extrapolated values of these transport coefficients, in the limit $k\ensuremath{\rightarrow}0$, are in good agreement with experiments on liquid argon.

Journal ArticleDOI
TL;DR: Dyson's method is adopted here for the so called Gaussian ensembles and confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensemble, the circular and the Gaussian ones.
Abstract: Dyson's method is adopted here for the so called Gaussian ensembles. Incidently this confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensembles, the circular and the Gaussian ones.

Journal ArticleDOI
TL;DR: In this paper, a series of SCF calculations for the ammonia molecule in various gemoetries have been performed, using large basis sets of STO's, and it is found that the height of the inversion barrier can be accurately predicted in the Hartree-Fock approximation, but that the basis set must be close to complete.
Abstract: A series of SCF calculations for the ammonia molecule in various gemoetries have been performed, using large basis sets of STO's. It is found that the height of the inversion barrier can be accurately predicted in the Hartree–Fock approximation, but that the basis set must be close to complete. Our largest calculations are in excellent agreement with previously reported calculations using Gaussian basis orbitals. A potential curve has been computed for the inversion motion using a near‐Hartree–Fock basis set. The computed spectrum is in qualitative agreement with experiment; quantitative accuracy is not obtained, probably because the computed NH bond distance is too short.

Journal ArticleDOI
TL;DR: In this paper, the entropy of a quantum-mechanical system whose statistical state is characterized by a Wigner distribution function is determined, and it is found to depend only on the variance of the distribution function.
Abstract: The entropy is determined of a quantum-mechanical system whose statistical state is characterized by a Wigner distribution function that is Gaussian in form. It is found to depend only on the variance of the distribution function. This enables us to study quantum mechanically the approach to equilibrium of coupled harmonic-oscillator chains, a system for which a classical treatment has been given recently by Robertson and Huerta.


Journal ArticleDOI
TL;DR: The state-vector a posteriori probabilities for prediction and smoothing are obtained via the 'partition theorem' and optimal linear smoothing algorithms are obtained in a new form for the special class of non-linear estimation problems with linear models excited by white gaussian noise.
Abstract: For the non-linear estimation problem with non-linear plant and observation models, white gaussian excitations and continuous data, the state-vector a posteriori probabilities for prediction and smoothing are obtained via the 'partition theorem'. Moreover, for the special class of non-linear estimation problems with linear models excited by white gaussian noise, and with non-gaussian initial state, explicit results are obtained for the a posteriori probabilities, the optimal estimates and the corresponding error-covariance matrices for filtering, prediction and smoothing. In addition, for the latter problem, approximate but simpler expressions are obtained by using a gaussian sum approximation of the initial state-vector probability density. As a special case of the above results, optimal linear smoothing algorithms are obtained in a new form.

Proceedings ArticleDOI
01 Dec 1971
TL;DR: For the nonlinear estimation problem with nonlinear plant and observation models, white gaussian excitations and continuous data, the state-vector a-posteriori probabilities for prediction, and smoothing are obtained via the "partition theorem", and optimal linear smoothing algorithms are obtained in a new form.
Abstract: For the nonlinear estimation problem with nonlinear plant and observation models, white gaussian excitations and continuous data, the state-vector a-posteriori probabilities for prediction, and smoothing are obtained via the "partition theorem". Moreover, for the special class of nonlinear estimation problems with linear models excited by white gaussian noise, and with nongaussian initial state, explicit results are obtained for the a-posteriori probabilities, the optimal estimates, and the corresponding error-covariance matrices for filtering, prediction, and smoothing. In addition, for the latter problem, approximate but simpler expressions are obtained by using a gaussian sum approximation of the initial state-vector probability density. As a special case of the above results, optimal linear smoothing algorithms are obtained in a new form.

Journal ArticleDOI
TL;DR: In this article, a study of gas phase flow in a three-dimensional cubic network of interconnected tubes is described and the anisotropy and connectivity of the model can be modified by elimination of tubes from each unit cubic cell.
Abstract: A study of gas phase flow, with and without an additional surface flux, in a three-dimensional cubic network of interconnected tubes is described. The anisotropy and connectivity of the model can be modified by elimination of tubes from each unit cubic cell. For a network in which all tubes have the same radius, the anisotropy can be related to gas phase structure factors. Computer calculations enable the properties of networks with Gaussian radius distributions to be investigated and it is established that estimates of structure factors for infinitely large networks can be made using networks of limited size. The effect of distribution breadth and of the amount of interconnection at a junction on the structure factors is discussed. It is found that radius distribution is less important in network systems than in the serial model described in an earlier paper. This difference is attributed to the dominance of constrictions in the former model in contrast to the multiple paths available at each junction in the latter.

Journal ArticleDOI
TL;DR: In this article, the concept of spline functions is introduced as a new tool for resolution error correction in small-angle scattering, and a general procedure using cubic splines is developed for the solution of the integral equations associated with the three distinct types of resolution errors and arbitrary weighting functions.
Abstract: The concept of spline functions is introduced as a new tool for resolution error correction in small-angle scattering. After discussing the approximating properties with respect to the estimation of experimental distributions and their derivatives from statistical data natural cubic splines are applied to the closed solution of the slit-height effect with Gaussian weighting functions including infinite slit height and to the correction of the slit-width effect based on Sauder's unfolding series. On the basis of an extended variational problem with one constraint taking adequately into account the statistical errors of the data a general procedure using cubic splines is developed for the solution of the integral equations associated with the three distinct types of resolution errors and arbitrary weighting functions. Computer simulations of small-angle scattering experiments have been performed to demonstrate the efficiency of the spline function algorithms in the different applications using scattering functions and slit-height and slit-width distributions of various kinds.

Journal ArticleDOI
TL;DR: In this article, the finite Gaussian expansion method for molecular integrals has been extended to the integrals of molecular properties, and integral formulas of so-called moment, field and field gradient integrals have been derived.
Abstract: The finite Gaussian Expansion method for molecular integrals proposed by Taketa, O-ohata and Huzinaga has been extended to the integrals of molecular properties. The integral formulas of so-called moment, field and field gradient integrals have been derived. It has been numerically shown that in order to evaluate the field and the field gradient integrals based on Slater type orbitals, eight- or ten-term Gaussian expansions are sufficient but this method fails to attain sufficient effective numbers for the moment integrals.

Journal ArticleDOI
TL;DR: In this paper, the authors analyzed the Breit-Wigner convolution of a particle group from a state in the continuum of an excited nucleus with a Gaussian detector.

Journal ArticleDOI
TL;DR: An optimized basis set of contracted gaussian type functions (CGTF's) for ethylene is obtained by scaling "best-atom" carbon and hydrogen CGTF exponents to minimize the ethylene molecular SCF energy as discussed by the authors.

Journal ArticleDOI
TL;DR: Approximations, accurate to ± 1 × 10−8, have been developed for the functions Fm(z) which arise in molecular calculations using Gaussian basis functions and are found to reduce integral computation time by about a factor of 5 over other methods.
Abstract: Approximations, accurate to ± 1 × 10−8, have been developed for the functions Fm(z) which arise in molecular calculations using Gaussian basis functions. These are found to reduce integral computation time by about a factor of 5 over other methods. Approximations are listed for 0 ≤ m ≤ 16.

Journal ArticleDOI
TL;DR: In this paper, a T-matrix method for determining the effective potential appropriate to the correlated Gaussian functions employed in lattice dynamics is developed, similar to those discussed by the authors in this paper.
Abstract: A T-matrix method for determining the effective potential appropriate to the correlated Gaussian functions employed in lattice dynamics is developed. The method is similar to those discussed by Guy...

Journal ArticleDOI
TL;DR: In this paper, the ground state of the permanganate ion was studied by an ab initio SCF MO calculation using a minimal basis set of contracted gaussian functions of near Hartree-Fock accuracy.