Showing papers on "Basis function published in 1977"
TL;DR: In this article, 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.
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.
1,820 citations
TL;DR: In this article, a coarse-mesh method for the solution of multidimensional neutron kinetics problems is presented that is based on the approximation of the desired solution by basis functions with local nonoverlapping supports corresponding to the volume elements of the spatial mesh.
Abstract: A coarse-mesh method for the solution of multidimensional neutron kinetics problems is presented that is based on the approximation of the desired solution by basis functions with local nonoverlapping supports corresponding to the volume elements of the spatial mesh. Integration of the approximating functions over their supports, and exploitation of continuity conditions for neutron flux and current, yields local seven-point difference operators with solution-dependent coupling coefficients. Due to the finite-difference (FD) structure of the resulting matrix equation, any technique developed for FD methods can be used for its solution. However, a novel (''almost implicit'') alternating direction explicit-implicit technique has been developed that is especially suited for coarse-mesh applications. Numerical examples that demonstrate the high efficiency of the method are presented. By using a spatial grid corresponding to the fuel element structure, it is possible to compute power distribution and its time history very accurately (at most, with a several percent error) at an economically tolerable expense.
137 citations
TL;DR: In this paper, the Hartree-Fock energy for the 1s orbital of helium is derived, and the accuracy depends on the number of points N+1 and the polynomial order 2s+1.
Abstract: Piecewise polynomials are examined as basis functions for electronic wavefunctions. The spline function method is a special case, which is shown to be less accurate, for a fixed set of mesh points, than a method based directly on Hermite’s interpolation formula. The determination of a suitable mesh is discussed both inductively and deductively, and a logarithmic formula for the 1s orbital of helium is ’’derived.’’ The accuracy is shown to depend on the number of points N+1 and on the polynomial order 2s+1, approximately according to the formula, δE∼N−4s−2, for appropriate meshes. A striking result is the possibility for systematically increasing the accuracy of the energy by systematically increasing the number of points, without encountering linear dependence problems, is demonstrated by calculations on the helium atom. With a 16‐point theoretically derived mesh, and with seventh order polynomials, we obtain a Hartree–Fock energy for helium of −2.8616799956122 a.u.
66 citations
TL;DR: A theoretical analysis and computational technique for constrained least squares image restoration using spline basis functions for space-variant or space-invariant point-spread function degradations has been studied.
Abstract: This paper presents a theoretical analysis and computational technique for constrained least squares image restoration using spline basis functions. A realistic continuous–discrete physical imaging model has been adopted throughout the formulation. The optical system is assumed to be incoherent, and the general problem of image restoration with space-variant or space-invariant point-spread function degradations has been studied.
46 citations
TL;DR: In this paper, a finite-element formulation for the vertical structure of primitive equation models has been developed, which is a variant of the Galerkin procedure in which the dependent variables are expanded in a finite, set of basis functions and then the truncation error is orthogonalized to each of the basis functions.
Abstract: A finite-element formulation for the vertical structure of primitive equation models has been developed. The finite-element method is a variant of the Galerkin procedure in which the dependent variables are expanded in a finite, set of basis functions and then the truncation error is orthogonalized to each of the basis functions. In the present case, the basis functions are Châpeau functions in sigma, the vertical coordinate. The procedure has been designed for use with a semi-implicit time discretization algorithm. Although this vertical representation has been developed for ultimate implementation in a three-dimensional finite-element model, it has been first tested in a spherical harmonic, baroclinic, primitive equations model. Short-range forecasts made with this model are very encouraging.
45 citations
TL;DR: In this paper, the Adams-Gilbert (AG) equation for nonorthogonal localized orbitals of a single-determinant wave function has been modified so as to enable one to compute wavefunctions of large polyatomic systems by the expansion method.
Abstract: Adams–Gilbert (AG) equation for nonorthogonal localized orbitals of a single‐determinant wavefunction has been modified so as to enable one to compute wavefunctions of large polyatomic systems by the expansion method. This equation is named as modified Adams–Gilbert (MAG) equation. One solves the AG or the MAG equation by each subsystem and, collecting all the orbitals obtained, one constructs wavefunction of the system. It is shown that when one employs the expansion method, one must actually use basis functions common to all the subsystems (common basis set) to solve the AG equation, while one can employ, by each subsystem, different basis functions appropriate to the subsystem (fluctuating basis set) to solve the MAG equation. An expansion method suitable for solving the AG and the MAG equations has been presented. Application of the method to HF, H2O, and CH4 has revealed that (1) the method proposed is workable, (2) actually so many basis functions are not needed for describing some subsystems, especially for core electrons, and (3) it is necessary to orthogonalize approximately, not necessarily rigorously, the orbitals in the system.
34 citations
TL;DR: In this paper, the second-order correlation energy calculations according to the Sinanogˇlu method are optimized with reference to rather crude SCF functions, which are then used in a one-step calculation of the correlation energy with respect to the near-Hartree-Fock reference State.
32 citations
IBM1
TL;DR: Algorithms which permit the construction of fully symmetry adapted Fock matrices are presented for cases where the basis functions and integrals are obtained using only the symmetry of a subgroup of the full point group of the system.
18 citations
TL;DR: In this article, a contracted Gaussian basis function is used to represent the lone-pair orbitals in rotation-barrier calculations on NH3, N2H2, H2O, and N2O2.
Abstract: Single Gaussian basis functions have been used to represent lone‐pair orbitals in contracted‐Gaussian basis calculations on NH3, N2H2, H2O, and H2O2. Both Pople’s STO‐4G and Dunning’s [4s3p/2s]← (9s5p/4s) contracted basis sets were used. The lone‐pair functions are located near the N and O nuclei, and their optimum positions and exponents are shown to be relatively insensitive to molecular environment. This indicates that lone‐pair functions may be explicitly included as part of a contracted basis set to represent polarization effects. This augmented, contracted‐Gaussian basis set is shown to yield significant improvements in computed values of rotational barriers and molecular geometries at relatively small cost in additional computation time.
17 citations
TL;DR: In this article, a quasi-static solution for the circular microstrip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived by using the Green's function approach.
Abstract: The quasi-static solution for the circular microstrip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived by using the Green's function approach. The basis functions are modified Legendre polynomials combined with a reciprocal square root to provide the correct singularity in charge density at the edge of the disk. The integrals involving the singular part of the Green's function are evaluated exactly, the remainder by using Gaussian quadrature. The method is compared in computational efficiency with recent methods based either on a Galerkin approach in the spectral domain, or the use of dual integral equations. Numerical results are given for charge distribution and capacitance; they are compared to exact results and those obtained by others, and the limitations of those methods are discussed. Closed form expressions are given for the capacitance of a disk based on two simple charge distributions.
17 citations
TL;DR: A new and effective technique for basis set augmentation within correlated wavefunction computations of intermolecular forces is illustrated and an illustrative application to He2 is described.
Abstract: A new and effective technique for basis set augmentation within correlated wavefunction computations of intermolecular forces is illustrated. The technique involves reducing or buffering the ’’ghost orbital’’ basis set borrowing by one species of the functions of a neighboring species when the basis set used to describe each species is not complete. This is achieved by employing a small set of Gaussians (s and higher angular momentum types) moved off the nuclei into the internuclear region. Provided these small sets are carried with each species as the interspecies distance is varied, reliable interspecies potential curves may be obtained. An illustrative application to He2 is described.
TL;DR: In this article, a collocation method is given for streamwise integration of stiff, nonlinear reactor models, which uses a small set of mesh points, and selects them to minimize global truncation errors.
01 May 1977
TL;DR: In this article, the Bubnov-Galerkin projective method with Lagrangian interpolation polynomials as basis functions was used to solve the Pocklington's integral equations.
Abstract: A theoretical-numerical technique is developed to solve for the current distributions and, hence, for the field patterns of perfectly conducting antenna structures composed of a rectangular plate and straight wires arranged in any fashion. Scattering by thin plates can also be solved. Electrical field integral equations of the Pocklington's type are formulated. These equations are solved by the Bubnov-Galerkin projective method with Lagrangian interpolation polynomials as basis functions. Numerical results are shown to give good agreement with those obtained by other methods.
TL;DR: In this article, a new orbital theory is proposed, in which general spin orbitals (GSO) are introduced in the spinoptimized (SO) SCF scheme, and the effective Hamiltonian for each orbital takes the form of a 2×2 matrix composed of the eigenfunctions for two-component spinors.
Abstract: A new orbital theory is proposed, in which general spin orbitals (GSO) are introduced in the spin‐optimized (SO) SCF scheme. In this SO–SCF–GSO theory, the effective Hamiltonian for each orbital takes the form of a 2×2 matrix composed of the eigenfunctions for two‐component spinors. It is found that the GSO’s thus defined should still satisfy a general form of Koopmans’ theorem. The SO–SCF GSO’s are to be obtained by solving two sets of coupled SCF equations for the spin coupling coefficients and the linear combination coefficients for basis functions. Using an STO‐6G basis set of the double ζ quality, sample calculations have been carried out for the doublet state of the linear H3 system for which the bond lengths are fixed at 1.470 and 2.984 bohr. The total energy obtained is ∼3 kcal/mole lower than the values which have resulted from the SO–SCF–DODS and the spin‐extended Hartree–Fock (SEHF) GSO calculations with the same basis set. The resulting orbitals are found to be more delocalized over the entire system than those obtained by the SO–SCF–DODS theory.
TL;DR: In this article, the minimization of the variance integral is used for the determination of scattering wave functions which uses discrete basis functions exclusively. But the choice of Gaussian basis functions for the expansion of the scattering wave function should make the method particularly applicable to electron-molecule scattering.
Abstract: We show that the minimization of the variance integral provides a method for the determination of scattering wave functions which uses discrete basis functions exclusively. By using a separable representation of the scattering potential only one new class of matrix elements appears in the evaluation of the variance integral which is not already required in the diagonalization of the Hamiltonian. The choice of Gaussian basis functions for the expansion of the scattering wave function should make the method particularly applicable to electron-molecule scattering. Some advantages and limitations of the method are discussed.
TL;DR: In this article, the finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry.
Abstract: The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries. Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies.
TL;DR: In this paper, the Hartree-Fock energy of two-electron atomic systems in their ground state was found for n = 3 (He −B3+) and n = 4 (H−).
Abstract: The two-parameter function, φ = (C1 + C2rn−1) exp (−ζr), (n = 2–5), has been used as a basis function to determine the independent particle model energy of two-electron atomic systems in their ground state. The best energy is found for n = 3 (He—B3+) and for n = 4 (H−). Our energy values are significantly close to Hartree-Fock results.
20 Jun 1977
TL;DR: In this article, the authors generalize an earlier result of the author's on Gauss interpolation formulas for the one-dimensional heat equation, which is characterized by the requirement that they be exact for as many basis functions as possible.
Abstract: This paper simplifies and generalizes an earlier result of the author's on Gauss interpolation formulas for the one-dimensional heat equation. Such formulas approximate a function at a point (x*, t*) in terms of a linear combination of its values on an initial-boundary curve in the (x, t) plane. The formulas are characterized by the requirement that they be exact for as many basis functions as possible. The basis functions are generated from a Tchebycheff system on the line t = 0 by an integral kernel K(x, y, t), in analogy with the way heat polynomials are generated from the monomials xi by the fundamental solution to the heat equation. The total positivity properties of K(x, y, t) together with the theory of topological degree are used to establish the existence of the formulas.
TL;DR: In this article, the Brillouin-Lowdin perturbation expansion was applied to the rigid rotator in an electric field, which has already been studied by Choi and Smith using the bracketing function of an intermediate Hamiltonian.
Abstract: One of the variants of Lowdin's partitioning technique that makes use of a Brillouin-type perturbation expansion for the study of lower bounds to the eigenvalues of a Hamiltonian has not been applied to any practical quantum mechanical problem so far. To illustrate how powerful this method can be, an application is made to the rigid rotator in an electric field, which has already been studied by Choi and Smith using the bracketing function of an intermediate Hamiltonian. It turns out that for a given order of the basis set of the Bazley space the Brillouin--Lowdin perturbation expansion gives closer bounds than the method of Choi and Smith, except for the case l = m, where the procedures can be shown to be mathematically equivalent. Especially for high l--m the number of basis functions needed to attain the same accuracy is by far larger for the method of Choi and Smith than for the Brillouin--Lowdin method.
01 Jan 1977
TL;DR: Weighted residuals as mentioned in this paper are general techniques for developing approximate solutions of operator equations, where the unknown solution is approximated by a set of local basis functions containing adjustable constants or functions.
Abstract: The methods of weighted residuals are general techniques for developing approximate solutions of operator equations. In these, the unknown solution is approximated by a set of local basis functions containing adjustable constants or functions. These constants or functions are chosen by various criteria to give the best approximation for the selected family. There are several variations of any of the assorted weighted residual methods—the interior, boundary, and mixed procedures. If trial solutions that satisfy the differential equation but not the boundary conditions are selected, the variant is called the boundary method. An intermediate situation exists in the mixed methods, where the trial solution does not satisfy either the equations or boundary conditions. The application of differential and integral inequalities has much in common with maximum and minimum principles, although it is of a greater generality.
TL;DR: In this article, the problem of minimizing the functional E(~) with respect to the expansion coefficients of ~ in N-electron basis functions with the additional condition that the calculated value of some physical quantity coincide with its exact value was solved.
Abstract: In calculating approximate N-electron functions P (xl, ..., x n) from the minimum principle of the total energy E(P) significant errors often occur in the expectation values of physical quantities whose operators do not commute with the system Hamiltonian [I]. At the same time, the values measured in accurate experiments or predicted by rigorous theoretical relations can be used to refine the (optimized) function P [i, 2]. This problem was solved in [i] by minimizing the functional E(~) with respect to the expansion coefficients of ~ in N-electron basis functions with the additional condition that the calculated value of some physical quantity coincide with its exact value. The method of [i] was further developed on the basis of perturbation theory [2].
TL;DR: In this paper, products of Mathieu, parabolic-cylinder and Lame functions appear as basis functions for the irreducible representations of the groups M(2), O(2,1) and SO(3).
TL;DR: In this paper, the necessary and sufficient conditions for isometry of the mesh function interpolation operators are proved in terms of Fourier transforms of the basis functions, which enables projection-difference operators with previously assigned properties to be obtained.
Abstract: Necessary and sufficient conditions for isometry of the mesh function interpolation operators are proved in terms of Fourier transforms of the basis functions. It is shown that the isometry property enables projection-difference operators with previously assigned properties to be obtained. More detailed consideration is given to the two-dimensional case, for which coordinate functions are constructed such that the projection-difference analogues of certain differential operators of 2nd and 4th orders are identical with their elementary difference analogues on minimal patterns.
TL;DR: In this paper, the authors evaluate the effect of higher anisotropy terms on domain nucleation, magnetoelastic interactions, and certain nonlinearities in micromagnetic variational problems.
TL;DR: In this paper, the Part I parameter optimization method of Part I is applied to the exponents of real STOS of first row atoms, and some independently optimized excited states are discussed in the case of Be.
Abstract: The parameter optimization method of Part I is applied to the exponents of real STOS of first row atoms. In addition to minimum basis ground states, some independently optimized excited states are discussed in the case of Be. Local minima on the energy versus parameter surface are found in 4-configuration functions for the ground state of N. They are not present in either the simpler minimum basis function or in a more complete 8-configuration function.
01 Jan 1977
TL;DR: In this article, a quasi-static solution for the (circular microstrip disk) was studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived using the Green's function approaclh.
Abstract: The quasi-static solution for the (circular micro:strip disk is studied using a GaIerkin solution to the Fredholm integral equation of the first kind derived hy using the Green's function approaclh. The basis functions are modified Legerrdre polynomials combined with a reciprocal square root to provide the correct singularity in charge density at the edge of the disk. The integrals involving the singular part of the Green's function are evaluated exactly, the remainder by using Gaussian quadra- ture. The method is compared in computational efficiency with recent methods based either on a Galerkin approach in the spectral domain, or the use of dual integral equations. Numerical rmdts are given for charge distribution and capacitance; they are compru'ed to exact results and those obtained by others, and the limitations of those methods are dis- cussed. Closed form expressions are given for the capacitance of a disk based on two simple charge distributions.