Showing papers on "Basis function published in 1972"
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TL;DR: In this article, 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.
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.
13,036 citations
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TL;DR: A heuristically optimal saturating nonlinearity is introduced and analyzed and two methods of compensating an arbitrary saturating device to obtain this optimal characteristic are presented.
Abstract: There are many instances in communication systems where bandpass signals are passed through nonlinear devices, such as traveling wave tubes, which exhibit both amplitude and phase nonlinearities. When the input signal is narrow band, the device may be characterized by measurements of its single-carrier amplitude and phase transfer functions. A sufficient model for such a device is a quadrature structure that includes two nonlinearities each of which, acting on its own, would exhibit only amplitude distortion. The outputs of the two halves of this model are linearly independent for arbitrary narrow-band input signals so that their power spectra add. Consequently, almost all previously published results for amplitude nonlinearities can be readily applied to the analysis of the general device. Emphasis is laid on practical procedures for analysis based directly on measured device characteristics rather than analytic approximations and accuracy is checked by comparison of certain intermodulation results with previous results and with measurements. A new result is the performance of an Intelsat IV tube for a large number of independent equal-power-density signals. A heuristically optimal saturating nonlinearity is introduced and analyzed and two methods of compensating an arbitrary saturating device to obtain this optimal characteristic are presented. Two methods of inverting the Chebyshev transform are used in this paper and the choice of basis functions for obtaining series representations of the measured device characteristics is discussed.
158 citations
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TL;DR: In this paper, the authors used Rayleigh-Schroedinger perturbation theory with the symmetric sum of Hartree-Fock operators for H 0.3428/a.
Abstract: The calculation of electron correlation energy in closed‐shell atoms and molecules is approached using Rayleigh‐Schroedinger perturbation theory with the symmetric sum of Hartree‐Fock operators for H0. The alleged advantages of using a VN−1 potential are questioned. Variational equations for first‐order pair correlation functions are computed for He, Be, B+ and Ne by expansion in linear combinations of correlated Gaussian‐type geminal basis functions containing r122 in the exponent. Such functions form mathematically complete sets, have convenient symmetry properties, and are integrable in closed form. An extensive search for optimum exponential parameters yielded trial functions for each of the ss orbital pairs giving better than 99% of the limiting second‐order pair energy using only five basis functions per pair. Similar but less thorough studies of sp and pp pairs in neon are also reported. Careful attention is paid to computational accuracy. An upper bound of −0.3428 a.u. is established on the second...
68 citations
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TL;DR: In this paper, a new model potential for one-valence-electron atoms is introduced, which consists of a core Coulomb potential modified by the addition of a Gaussian screened Coulomb Potential.
Abstract: A new model potential for one‐valence‐electron atoms is introduced. It consists [Eq. (6)] of a core Coulomb potential modified by the addition of a Gaussian screened Coulomb potential. This potential has the desired features outlined for a model potential, and it has particularly convenient and simple mathematical properties when used with Gaussian basis functions. Since the smooth valence orbitals are sought, Gaussian functions are a good basis set because their deficiencies in nuclear regions do not enter the problem. The potential is calibrated to experimental energies for the 2s and 2p states of Li and 3s and 3p states of Na, using extended basis sets. The model Hamiltonians so defined are used with a variety of more modest basis sets to determine pseudowavefunctions. Based on comparisons with ab initio orbitals, energy analyses, radial density calculations, and overlap and expectation value calculations, the conclusion is that good valence pseudowavefunctions can be obtained by this approach with relatively small basis sets. The approach looks promising for valence‐only molecular studies.
38 citations
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TL;DR: In this article, the first principle Korringa-Kohn-Rostoker functional and basis functions are transformed directly into the hybrid nearly-free-electron tight binding (H-NFE-TB) form for the case of the transition metals in which the valence electrons behave like virtual bound states or resonances.
Abstract: The first principle Korringa-Kohn-Rostoker functional and basis functions are transformed directly into the hybrid nearly-free-electron tight binding (H-NFE-TB) form for the case of the transition metals in which the valence electrons behave like virtual bound states or resonances. It is shown that the resulting basis functions are indeed consistent with the NFE-TB picture, because the TB Bloch sums are rapidly convergent over the first few nearest neighbours since the oscillating tails of the resonant wavefunctions have been completely taken care of by the conduction basis set. Further, the H-NFE-TB matrix elements are accurate to second order in the width of the d band and, therefore, contain terms that are directly equivalent to the orthogonality and three-centre integrals of the conventional TB scheme. Finally, the basis functions are orthonormalized and the corresponding energy independent model Hamiltonian presented. The accuracy of this H-NFE-TB representation is illustrated by comparison with the first principle wavefunctions and band structure of iron.
34 citations
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TL;DR: In this article, it was shown that the Fredholm determinant for the Lippmann-Schwinger equation can be computed for complex energies using only ${L}^{2}$ basis functions, showing that it is not necessary to explicitly enforce asymptotic boundary conditions for numerical scattering computations.
Abstract: It is shown that the Fredholm determinant for the Lippmann-Schwinger equation may be computed for complex energies using only ${L}^{2}$ basis functions. Analytic continuation to the real axis in the $E+i\ensuremath{\epsilon}$ limit gives elastic scattering phase shifts over a continuous range of energies, showing that it is not necessary to explicitly enforce asymptotic boundary conditions for numerical scattering computations, and suggesting that elastic electron-atom or -ion scattering information may be obtained using standard bound-state configuration-interaction methods.
31 citations
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TL;DR: In this paper, the Hartree-Fock functions of the separated molecules are used as one particle basis function, the initial set of states being truncated and non-orthogonal.
Abstract: In order to treat the interaction energy of two molecules a standard Rayleigh-Schrodinger perturbation theory is developed. The Hartree–Fock functions of the separated molecules are used as one particle basis functions, the initial set of states being truncated and non-orthogonal. The non-orthogonality is included into the Hamiltonian by orthogonalization of the basis set. The unperturbed Hamiltonian is chosen so that it possesses the correct symmetry properties with respect to the electron permutations between different molecules. The procedure of this kind automatically results in the appearance of charge transfer states.
A graphical technique is elaborated which is a modified version of the Feynman–Goldstone technique and provides a convenient representation of the interaction energy contributions of any order. As an example the first- and the second-order diagrams are considered.
A correct expression for the dispersion energy is obtained which differs by a factor from that of the theory using a nonsymmetrical zero approximation.
30 citations
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TL;DR: In this paper, a one-valence-electron theory was developed, which assumes nuclear nuclear repulsions to be between net core atomic charges, and that the potential seen by the valence electron is a superposition of atomic model potentials.
Abstract: A one‐electron theory for one‐valence‐electron molecules is developed. It assumes nuclear‐nuclear repulsions to be between net core atomic charges, and that the potential seen by the valence electron is a superposition of atomic model potentials. The atomic model potentials are the core Coulomb potentials modified by Gaussian screened Coulomb potentials, as found in earlier atomic studies. Such model potentials are quite convenient for the use of Gaussian basis functions, which are also taken from the atomic studies, and for which potential energy matrix elements are simply calculated from physically based rules. The theory is applied to the ground states of Li2+, Na2+, and LiH+, using a variety of basis sets. Increasing the basis set flexibility lowers the energy, as in ordinary all‐electron variational calculations. Binding energies and internuclear distances from the most extended basis calculations are: Li2+: De=1.23 eV, Re = 5.8 a.u.; Na2+: De=0.97 eV, Re = 6.7 a.u.; LiH+: De = 0.090 eV, Re = 4.5 a.u...
28 citations
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TL;DR: The first part of an investigation into the possibility of finding a set of simple functions for the construction of molecular wave functions which satisfies two essential requirements: that the number of basis functions required to produce a given accuracy be comparable to the number required to give the same accuracy, and that the molecular multicentre integrals should be considerably easier to evaluate than those over Slater functions as mentioned in this paper.
Abstract: The present paper describes the first part of an investigation into the possibility of finding a set of simple functions for the construction of molecular wave functions which satisfies two essential requirements: that the number of basis functions required to produce a given accuracy be comparable to the number of Slater functions required to give the same accuracy, and that the molecular multicentre integrals should be considerably easier to evaluate than those over Slater functions. It is shown that a basis made up of 1s gaussians supplemented by a single ‘cusp function’ of the form may be expected to satisfy the requirements for the s-orbitals in atoms other than hydrogen. Calculations of the ground-state orbital approximations of the helium and beryllium atoms show that an inner-shell 1s orbital can be satisfactorily described by the cusp function and two gaussians, whilst subsequent s-orbitals should require not more than two or three gaussians per orbital to produce better than ‘double-zeta’ accuracy.
24 citations
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TL;DR: In this article, mixed sets of functions consisting of 1s Slater orbitals and floating spherical Gaussians are proposed as expansion functions for ab initio variational type calculations for small molecules.
Abstract: Mixed sets of functions consisting of 1s Slater orbitals and floating spherical Gaussians are proposed as expansion functions for ab initio variational type calculations for small molecules. After the discussion of the basic computational methods the results of some trial calculations for the ground state of the helium atom are presented.ZusammenfassungFür ab initio Rechnungen vom Variationstyp werden im Falle kleiner Moleküle gemischte Funktionssätze, die aus 1s Slater-Orbitalen bestehen, vorgeschlagen. Nach einer Diskussion der grundlegenden Rechenmethoden werden die Resultate von einigen Proberechnungen für den Grundzustand des Heliumatoms geschildert.
19 citations
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TL;DR: In this article, a series of calculations for the ground state of the H2 molecule were reported, using full configuration interaction method and a mixed orbital basis of 1s Slater and floating spherical Gaussian functions.
Abstract: A series of calculations for the ground state of the H2 molecule are reported, using full configuration interaction method and a mixed orbital basis of 1s Slater and floating spherical Gaussian functions. The results obtained compare favourably with the results of previous H2 calculations using pure Slater type orbital bases.
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TL;DR: In this article, a linear combination of basis functions which are not mutually orthogonal is used to express the electronic density of states and electrical conductivity of a system in this representation and the differences in these properties between perfect and imperfect crystals, and between different types of crystal, are examined.
Abstract: Formulae are derived for the electronic Green function of a system, and in particular of a crystal, when this is expressed as a linear combination of basis functions which are not mutually orthogonal. Expressions are derived for the electronic density of states and the electrical conductivity of a system in this representation, and the differences in these properties between perfect and imperfect crystals, and between different types of crystal, are examined. Application of the method to a simple cubic crystal with the Mathieu potential shows that, as for a one dimensional system, the use of optimized in place of fixed atomic orbitals as basis functions leads to a great increase in accuracy.
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TL;DR: In this article, the ground state of an electron gas is studied by means of the method of correlated basis functions developed by Wu and Feenberg, and the optimum form of the Bijl-Dingle-Jastrow-type wave function of a charged-boson gas obtained with the application of the paired-phonon analysis is used as the correlation factor in a trial wave function describing the electron gas.
Abstract: The ground state of an electron gas is studied by means of the method of correlated basis functions developed by Wu and Feenberg. The optimum form of the Bijl-Dingle-Jastrow-type wave function of a charged-boson gas obtained with the application of the paired-phonon analysis is used as the correlation factor in a trial wave function describing the electron gas. Numerical results are given for the correlation energy per particle and the radial distribution function at metallic densities. The high-density limit of the system is briefly discussed.
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01 Jan 1972
TL;DR: In this article, the concept of conjugate approximation functions is generalized to mixed formulations of linear boundary value problems and Biorthogonal basis functions are generated using two distinct finite element models of the domain of a given function.
Abstract: The concept of conjugate approximation functions is generalized to mixed formulations of linear boundary value problems. Biorthogonal basis functions are generated using two distinct finite-element models of the domain of a given function. Various special cases are considered. It is shown that four distinct models can be developed corresponding to a single linear operator. A comparison of the concepts of the hypercircle and conjugate approximation is given.
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01 Jan 1972TL;DR: In this article, a finite element procedure using basis functions consisting of piecewise bicubic Hermite polynomials defined on a mesh which is refined in a well-defined manner in a neighborhood of each corner is discussed.
Abstract: Publisher Summary This chapter focuses on the computational aspects of the finite element method. The finite element procedure discussed in the chapter uses basis functions consisting of piecewise bicubic Hermite polynomials defined on a mesh which is refined in a well-defined manner in a neighborhood of each corner. The coefficients and right-hand side of the resulting linear algebraic system of equations involve integrals over two-dimensional rectangular elements that are approximated by the local nine-point product Gaussian quadrature scheme —the tensor product of the one-dimensional three-point Gaussian quadrature schemes. Finally, the approximate linear algebraic system of equations is symmetric and positive definite and is solved by either the band Cholesky or profile Cholesky decomposition procedure. The chapter presents theoretical justifications for the procedure used in the chapter. It is shown that asymptotically the procedure used in the chapter is far more efficient than the combination of the five-point central difference approximation and successive overrelaxation (SOR).
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TL;DR: In this paper, a general expression that relates the matrix elements of position and momentum allowing for the possibility of a mixed basis is derived, where the Born-Oppenheimer wavefunctions for two different electronic states are used as basis functions.
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TL;DR: In this article, the ground state HH+ and HH interactions were used as model interactions for investigating the feasibility of using Gaussian basis sets for representing charge overlap effects in intermolecular forces.
Abstract: The ground state HH+ and HH interactions are used as model interactions for investigating the feasibility of using Gaussian basis sets for representing charge overlap effects in intermolecular forces. The non-expanded charge-induced dipole energy and the non-expanded dipole-dipole dispersion energy, respectively, for these interactions are calculated using two types of Gaussian basis functions to represent the first order wave function, Ψ(1). Very good results for these interaction energies, which include charge overlap effects, are obtained for all interatomic separations by using small Gaussian basis sets to represent the interaction, that is Ψ(1), and/or the isolated atoms (the zeroth order wave function).
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TL;DR: In this paper, a consistent procedure for introducing an external single particle potential into the cluster expansion of the energy expectation value with respect to a correlated wave function has been developed, and the usual Brueckner reaction-matrix approximation may be extracted from the correlated-basis-functions theory by a special treatment of the external potential and an associated special choice of two-body correlation factor.
Abstract: We develop a consistent procedure for introducing an external single-particle potential into the cluster expansion of the energy expectation value with respect to a correlated wave function. It is shown that the usual Brueckner reaction-matrix approximation may be extracted from the correlated-basis-functions theory by a special treatment of the external potential and an associated special choice of two-body correlation factor.
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TL;DR: In this paper, a method to calculate relativistic energy bands and wave functions is presented, in which rapidly converging basis sets are created from linearly combined RAPW functions, by constructing other basis sets, convergence of the band structure of complex materials (e.g. compounds) can be studied with a matrix of moderate size.
Abstract: A method to calculate relativistic energy bands and wave functions is presented, in which rapidly converging basis sets are created from linearly combined RAPW functions. By constructing other basis sets, convergence of the band structure of complex materials (e.g. compounds) can be studied with a matrix of moderate size. Results of an application to fcc praesodymium show the usefulness of the method and indicate a considerable potential for future developments.
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TL;DR: In this article, a generalization of the Integral Transform method for particle transport in homogeneous material systems having plane and spherical symmetry has been proposed, with the explicit inclusion of anisotropic scattering (higher Legendre components of the scattering kernel).
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TL;DR: A series of calculations have been carried out for the linear symmetric H3 molecule using configuration interaction wavefunctions constructed from a mixed set of 1s Slater and floating spherical Gaussian functions as mentioned in this paper.
Abstract: A series of calculations have been carried out for the linear symmetric H3 molecule using configuration interaction wavefunctions constructed from a mixed set of 1s Slater and floating spherical Gaussian functions. The results, although they do not surpass the best CI results reported in the literature, are sufficiently good to encourage further work along these lines.ZusammenfassungEs wurde eine Reihe von Rechnungen für das lineare, symmetrische H3-Molekül durchgeführt, bei denen Wellenfunktionen mit Konfigurations-Wechselwirkung verwendet werden, die aus einem gemischten Satz von 1s Slater- und “floating” sphärischen Gauß-Funktionen konstruiert wurden. Obwohl die Resultate die besten CI Ergebnisse, über die in der Literatur berichtet wird, nicht übertreffen, sind sie so gut, daß sie zu weiterer Arbeit nach dieser Methode ermutigen.
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TL;DR: The basis functions are displayed in closed form for piecewise polynomial approximation of degreen over a triangulation of the plane in terms of the pyramid functions for linear approximation.
Abstract: In many applications of the finite element method, the explicit form of the basis functions is not known. A well-known exception is that of piecewise linear approximation over a triangulation of the plane, where the basis functions are pyramid functions. In the present paper, the basis functions are displayed in closed form for piecewise polynomial approximation of degreen over a triangulation of the plane. These basis functions are expressed simply in terms of the pyramid functions for linear approximation.
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TL;DR: In this paper, a polynomial Schauder basis for the space of continuous functions is constructed, similar to the Schauder base but with better differentiability properties, and the bases are applied to the problem of the order of growth of the degrees of the degree.
Abstract: In this paper, new bases for the space of continuous functions are constructed, similar to the Schauder basis but having better differentiability properties. The bases constructed are applied to the problem of the order of growth of the degrees of a polynomial basis of the space . It is proved that for any nondecreasing sequence of natural numbers satisfying the condition it is possible to construct a polynomial basis with order of growth , .Bibliography: 16 items.
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TL;DR: In this article, a series of calculations have been carried out for the linear system of four equidistant hydrogen atoms for an internuclear separation of 1.7 a.u. The configuration interaction technique was used, the orbital basis consisting of a mixed set of 1s Slater and floating spherical Gaussian functions.
Abstract: A series of calculations have been carried out for the linear system of four equidistant hydrogen atoms for an internuclear separation of 1.7 a.u. The configuration interaction technique was used, the orbital basis consisting of a mixed set of 1s Slater and floating spherical Gaussian functions. The results obtained are encouraging although the effects of unoptimized non-linear parameters are noticeable.ZusammenfassungEs wurde eine Reihe von Rechnungen für ein lineares System von vier äquidistanten Wasserstoffatomen (Kernabstand jeweils 1.7 a.u.) durchgeführt. Dabei benutzt man die Konfigurationswechselwirkungs-Technik, die Orbitalbasis bestand aus einem gemischten Satz von 1s Slater- und “floating” sphärischen Gauß-Orbitalen. Die Resultate sind ermutigend, obwohl die Effekte der nicht optimierten nichtlinearen Parameter merklich sind.
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TL;DR: The quality of an atomic basis set in molecular calculations can be characterized by the deviation of the valence-shell orbital energies from their values in a Hartree-Fock limit calculation.
Abstract: The quality of an atomic basis set in molecular calculations can be characterized by the deviation of the valence-shell orbital energies from their values in a Hartree-Fock limit calculation. For atoms of a certain row of the periodic system, this quality appears to depend not only on the number of basis functions used in the calculations, as is usually accepted, but also on the number of valence shell electrons of the various atoms.
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TL;DR: In this paper, a method of generating a set of basis functions which satisfy the condition of strong orthogonality was developed, which can be used to generate a strong basis function.
Abstract: A method of generating a set of basis functions, which satisfy the condition of “strong orthogonality”, is developed.
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01 Jan 1972
TL;DR: A review of double expansions of relativistic amplitudes can be found in this article, where the general theory of the expansion of functions stipulated in both the time-like and space-like domains is presented.
Abstract: The present paper is a review of research on double expansions of relativistic amplitudes. In the beginning of the review the general theory of the expansion of functions stipulated in both the time-like and space-like domains is presented; then the general theory of the expansion of a scalar function and a function with spin is presented. After that various coordinate systems and matrices for the transition from certain basis functions to other ones are indicated. The review likewise considers certain corollaries deriving from the existence of relativistic expansions.
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TL;DR: In this paper, a trigonometric expression for the eigenfunctions and eigenvalues of 2 x 2 secular problems including overlap is presented, which is a convenient expression for 2 × 2 problems.
Abstract: A convenient trigonometric expression for the eigenfunctions and eigenvalues of 2 x 2 secular problems including overlap is presented.
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TL;DR: It is shown that for a large class of functions exact computation may be carried out by an algorithm whose computation time is independent of the length of the sequences.
Abstract: A special case of the problem discussed in this paper occurs in connection with non-parametric classification and is introduced from this point of view. The special case concerns the computation of expectations of statistical functions of the “distance” between pairs of fixed-length sequences over a binary alphabet with given a priori state transition probabilities. The general problem involves an extension to alphabets of arbitrary order and the comparison of an arbitrary number of fixed-length sequences. Given a set of sequences, it is shown that for a large class of functions exact computation may be carried out by an algorithm whose computation time is independent of the length of the sequences. It is further shown that results for all functions of this class may be derived from a small number of basis functions. Two methods for computing basis functions are given. Basis functions for the commonly encountered special case involving pairs of binary sequences are given explicitly.