Topic

# Gaussian function

About: Gaussian function is a(n) research topic. Over the lifetime, 7073 publication(s) have been published within this topic receiving 171197 citation(s). The topic is also known as: Gaussian.

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Abstract: Two extended basis sets (termed 5–31G and 6–31G) consisting of atomic orbitals expressed as fixed linear combinations of Gaussian functions are presented for the first row atoms carbon to fluorine. These basis functions are similar to the 4–31G set [J. Chem. Phys. 54, 724 (1971)] in that each valence shell is split into inner and outer parts described by three and one Gaussian function, respectively. Inner shells are represented by a single basis function taken as a sum of five (5–31G) or six (6–31G) Gaussians. Studies with a number of polyatomic molecules indicate a substantial lowering of calculated total energies over the 4–31G set. Calculated relative energies and equilibrium geometries do not appear to be altered significantly.

11,910 citations

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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.

7,677 citations

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Abstract: Contracted Gaussian basis sets for molecular calculations are derived from uncontracted (12,8) and (12,9) sets for the neutral second row atoms, Z=11–18, and for the negative ions P−, S−, and Cl−. Calculations on Na...2p63p, 2P and Mg...2p63s3p, 3P are used to derive contracted Gaussian functions to describe the 3p orbital in these atoms, necessary in molecular applications. The derived basis sets range from minimal, through double‐zeta, to the largest set which has a triple‐zeta basis for the 3p orbital, double‐zeta for the remaining. Where necessary to avoid unacceptable energy losses in atomic wave functions expanded in the contracted Gaussians, a given uncontracted Gaussian function is used in two contracted functions. These tabulations provide a hierarchy of basis sets to be used in designing a convergent sequence of molecular computations, and to establish the reliability of the molecular properties under study.

7,306 citations

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Abstract: t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical operations (all logarithms in this paper are for base 2, thus tog 7 ~ 2.8; the usual method requires approximately 2n 3 arithmetical operations). The algorithm induces algorithms for invert ing a matr ix of order n, solving a system of n linear equations in n unknowns, comput ing a determinant of order n etc. all requiring less than const n l°g 7 arithmetical operations. This fact should be compared with the result of KLYUYEV and KOKOVKINSHCHERBAK [1 ] tha t Gaussian elimination for solving a system of l inearequations is optimal if one restricts oneself to operations upon rows and columns as a whole. We also note tha t WlNOGRAD [21 modifies the usual algorithms for matr ix multiplication and inversion and for solving systems of linear equations, trading roughly half of the multiplications for additions and subtractions. I t is a pleasure to thank D. BRILLINGER for inspiring discussions about the present subject and ST. COOK and B. PARLETT for encouraging me to write this paper. 2. We define algorithms e~, ~ which mult iply matrices of order m2 ~, by induction on k: ~ , 0 is the usual algorithm, for matr ix multiplication (requiring m a multiplications and m 2 ( m t) additions), e~,k already being known, define ~ , ~ +t as follows: If A, B are matrices of order m 2 k ~ to be multiplied, write

2,366 citations

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Max Planck Society

^{1}, National University of Singapore^{2}, Alcatel-Lucent^{3}, Massachusetts Institute of Technology^{4}, AT&T^{5}TL;DR: The results show that on the United States postal service database of handwritten digits, the SV machine achieves the highest recognition accuracy, followed by the hybrid system, and the SV approach is thus not only theoretically well-founded but also superior in a practical application.

Abstract: The support vector (SV) machine is a novel type of learning machine, based on statistical learning theory, which contains polynomial classifiers, neural networks, and radial basis function (RBF) networks as special cases. In the RBF case, the SV algorithm automatically determines centers, weights, and threshold that minimize an upper bound on the expected test error. The present study is devoted to an experimental comparison of these machines with a classical approach, where the centers are determined by X-means clustering, and the weights are computed using error backpropagation. We consider three machines, namely, a classical RBF machine, an SV machine with Gaussian kernel, and a hybrid system with the centers determined by the SV method and the weights trained by error backpropagation. Our results show that on the United States postal service database of handwritten digits, the SV machine achieves the highest recognition accuracy, followed by the hybrid system. The SV approach is thus not only theoretically well-founded but also superior in a practical application.

1,278 citations