Showing papers on "Basis function published in 1970"
TL;DR: In this paper, the effects of contraction on the energies and one-electron properties of the water and nitrogen molecules were investigated, and the authors obtained principles which can be used to predict optimal contraction schemes for other systems without the necessity of such exhaustive calculations.
Abstract: The contraction of Gaussian basis functions for use in molecular calculations is investigated by considering the effects of contraction on the energies and one‐electron properties of the water and nitrogen molecules. The emphasis is on obtaining principles which can be used to predict optimal contraction schemes for other systems without the necessity of such exhaustive calculations. Using these principles, contractions are predicted for the first‐row atoms.
4,595 citations
TL;DR: In this article, the use of a linear combination of Gaussian type orbitals (CGTO) instead of an individual Gaussian-type orbital (GTO) as a unit of basis functions for large-scale molecular calculations is discussed.
Abstract: The use of a linear combination of Gaussian‐type orbitals (CGTO), instead of an individual Gaussian‐type orbital (GTO), as a unit of basis functions for large‐scale molecular calculations, is discussed. A systematic construction of the CGTO basis functions is attempted and the results for the atoms from Li through Ar are reported.
3,257 citations
TL;DR: In this paper, a rapidly converging difference method, based on harmonic analysis, is described, which can be applied to periodic or nonperiodic bound-state problems of the general Sturm-Liouville type.
Abstract: A rapidly converging difference method, based on harmonic analysis, is described. It can be applied to periodic or nonperiodic bound‐state problems of the general Sturm–Liouville type. Numerical examples for the Mathieu problem and for the harmonic oscillator show considerable accuracy. Advantages and disadvantages of the method are discussed in a comparison with Harris's matrix transformation technique and with direct integration methods. The set of difference equations representing a quantum‐mechanical problem constitutes a symmetric matrix eigenvalue problem which is approximately equivalent to the algebraic problem obtained by using a finite trigonometric basis. Basis functions associated with the difference method are related to the Dirichlet kernel. In an approximation which corresponds to the difference method, these basis functions can be treated in a similar way as Dirac's δ function.
192 citations
TL;DR: In this paper, the contraction of the (12s9p) primitive basis sets of Veillard for the second row atoms was investigated and the [7s, [6s], [5p], and [4p] basis sets were proposed.
114 citations
TL;DR: In this paper, a non-collinear magnetic structure for α-Mn is proposed, and the magnetic point group is isomorphic with either D 2 d of C 3 v depending on whether the principal axis is along either a or a, respectively.
Abstract: A systematic method of magnetic structure analysis is developed and applied to the case of α-Mn. Scalar, vector, and tensor quantities in a phenomenological thermodynamical potential are expanded in complete sets of basis functions which form irreducible representations of the space group, and on this basis, a possible magnetic structure for α-Mn is discussed. A non-collinear magnetic structure containing thirteen parameters is proposed. The magnetic point group is isomorphic with either D 2 d of C 3 v depending on whether the principal axis is along either a or a , respectively. Possible modes of magnetostrictive atom-displacement are also discussed.
32 citations
TL;DR: In this article, a tutorial review of the microscopic theory with emphasis on the method of correlated basis functions is presented, including potentials, distribution functions, consequences of the correspondence principle, working approximations for the three-particle distribution function, useful matrix elements, sum rules, the paired-phonon function space, trial forms for the ground state of the boson system, distribution of particles in momentum space, elementary excitations, basis functions, and matrix elements for liquid 3He and for the dilute solution of 3He in liquid 4He.
Abstract: A highly developed microscopic theory of the helium liquids exists based on the Schrodinger equation involving realistic forms of the interaction potential between pairs of neutral atoms. This report is a tutorial review of the microscopic theory with emphasis on the method of correlated basis functions. Among the topics considered are potentials, distribution functions, consequences of the correspondence principle, working approximations for the three-particle distribution function, useful matrix elements, sum rules, the paired-phonon function space, trial forms for the ground state of the boson system, distribution of particles in momentum space, elementary excitations, basis functions, and matrix elements for liquid 3He and for the dilute solution of 3He in liquid 4He.
17 citations
TL;DR: The one-dimensional Laplace transform of exp (− rx ): can be used to generate functions which could be useful as basis sets for atomic and molecular calculations as discussed by the authors, and these functions were used as basis functions for the helium isoelectronic series and accounted for 99.95% (H − ), 99.98% (He), 99.99% (Li + ) of Hartree-Fock energy.
13 citations
TL;DR: In this paper, the Hartree-Fock problem in a finite basis set is solved, which permits each orbital to be expanded in a different basis, and a calculation on the ground state of beryllium is performed using the nested procedure.
Abstract: A method for solving the Hartree–Fock problem in a finite basis set is derived, which permits each orbital to be expanded in a different basis. If the basis set for each orbital ϕi contains the basis functions for the preceding orbitals, ϕi−1, ϕi−2,… ϕ1, then the ϕi form an orthonormal set. One advantage over the standard Hartree–Fock method is that a different long range behavior for each orbital, as for example is required in the Hartree–Fock-Slater method, can be forced. A calculation on the ground state of beryllium is performed using the nested procedure. Very little energy is lost because of nesting, and the node in the 1s orbital disappears.
12 citations
TL;DR: In this paper, it is suggested that given two molecules potentially capable of being involved in the same reaction, one as a product and the other as a reactant, it seems reasonable that the calculated difference in en...
Abstract: It is suggested that given two molecules potentially capable of being involved in the same reaction, one as a product and one as a reactant, it seems reasonable that the calculated difference in en...
12 citations
TL;DR: In this paper, a general formulation of the problem is given, which takes into account both the static periodic potential of a model substrate and He-He interactions, leading to expressions of surface phonon states obtained recently by Jackson.
Abstract: We are concerned here with the physical adsorption of helium atoms on a crystal surface. A general formulation of the problem is first given, which takes into account both the static periodic potential of a model substrate and He-He interactions. This formulation leads in a natural way to expressions of surface phonon states obtained recently by Jackson. As the first step of carrying out our formalism, we extend an earlier single-particle calculation by Lennard-Jones and Devonshire. A set of basis functions consisting of products of Mathieu functions and eigenfunctions of a Morse Hamiltonian are defined, and a low-order perturbative calculation is performed. While the procedure provides a qualitative description of the energy bands, it appears to be quantitatively inadequate. To facilitate a basis for future work, which will include the effects of correlations between the helium atoms, we carry out variational calculations in this zero-coverage limit, using various combinations of Gaussian and Morse functions. A reasonably single combination results in a ground-state energy which is just 1% higher than that of Ricca, Pisani, and Garrone, who performed elaborate computations to solve a secular equation obtained with harmonic oscillator basis functions.
10 citations
TL;DR: In this paper, LiBeLi was determined to be a stable species, linear in its ground state, and the equilibrium bond distance was determined, and force constants and normal frequencies were obtained.
Abstract: A LCAO–MO–SCF investigation of LiBeLi utilizing Gaussian 985p basis functions established LiBeLi to be a stable species, linear in its ground state. The equilibrium bond distance was determined, and force constants and normal frequencies were obtained. Using a procedure for estimating the correlation energy similar to the one we employed successfully for BeH2, we determined the dissociation energy of LiBeLi. Our results imply that it may be possible to form an LiBeLi alloy which would be denser than either Li metal or Be metal themselves.
DOI•
01 Jan 1970
TL;DR: In this article, the development of triangular fractals that geometrically sum into an area whose boundary is a function, of a specific type, is used to expand the Complex Variable Boundary Element Method (or CVBEM) into a series.
Abstract: In this paper, the development of triangular fractals that geometrically sum into an area whose boundary is a function, of a specific type, is used to expand the Complex Variable Boundary Element Method (or CVBEM) into a series. The entire approximation effort can be written as a sum of Cauchy integrals of incremental changes in basis functions.
10 Dec 1970
TL;DR: An entirely new class of algorithms is obtained by translating the pattern recognition problem into the problem of minimizing a function of several variables and selecting suitable functions, which includes most known algorithms as special cases.
Abstract: : The M-class pattern recognition problem is to construct a set of discriminant functions hwhich partition a feature space into M regions, one region per pattern class. Each point in the feature space is a potential pattern and each pattern represents an object. Almost nothing is assumed about the origins of the patterns. Distributions are not associated with the pattern classes. A set of training patterns is to be generalized into a set of discriminant functions which classify the potential patterns. The fundamental algorithms developed here concern the situation where the origin of each training pattern is known. An extension to the unsupervised case is also given. Several new multi-class decision-making algorithms are proposed. An entirely new class of algorithms is obtained by translating the pattern recognition problem into the problem of minimizing a function of several variables and selecting suitable functions. This general formulation includes most known algorithms as special cases. The class of algorithms includes all procedures which approximate discriminant functions by linear combinations of basis functions. Several sucessful two-class algorithms are extended to the M-class problem. (Author)
DOI•
01 Jan 1970
TL;DR: In this article, a new formulation for transient radiative transport which promotes the use of orthogonal collocation is proposed, where Chebyshev polynomials of the first kind are used as the basis functions for the spatial variable while the temporal variable is resolved by an initial value method.
Abstract: A new formulation is offered for transient radiative transport which promotes the use of orthogonal collocation. An intermediate variable is introduced which permits the efficient and rapid development of accurate numerical results. Chebyshev polynomials of the first kind are used as the basis functions for the spatial variable while the temporal variable is resolved by an initial value method. Some a posteriori error estimates are presented illustrating the effectiveness of the approach. This new formulation has potential impact to the boundary element community with regard to nonlinear problems.
TL;DR: In this paper, a method for synthesizing a sub-optimal feedback controller which is sensitive to variations in plant parameters is presented, where the structure of the suboptimal control is a linear combination of suitably chosen basis functions multiplied by coefficients which are functions of varying plant parameters.
Abstract: In this paper a method for synthesizing a sub-optimal feedback controller which is sensitive to variations in plant parameters is presented. The structure of the sub-optimal control is a linear combination of suitably chosen basis functions multiplied by coefficients which are functions of the varying plant parameters. The coefficient multipliers are determined by the minimization of a mean-square error using data obtained from numerically computed optimal trajectories. The controller may be called adaptive since it uses identification of the varying plant parameters to modify the coefficient multipliers. Examples are included which show that with relative sensitivity as the criterion the method is superior to other methods of suboptimal control design.
TL;DR: A new method is presented, based on graph theory, to detect automatically the minimum number of modes guaranteeing a fixed spatial resolution for whatever considered structure, and results prove that a substantial speed-up is obtained with the use of the optimal modal set.
Abstract: The time and accuracy performances of a 3D rectangular waveguide ModeMatching (MM) simulator are strictly dependent on the appropriate selection of the basis functions used for the modal expansion of the fields. The number of selected modes determines the size of a linear system of equations whose solution is the computational core of the MM analysis. Moreover, the use of modes not suitable to the studied structures increases the risk of numerical instability. We present a new method, based on graph theory, to detect automatically the minimum number of modes guaranteeing a fixed spatial resolution for whatever considered structure. The method is general, and does not require any considerations on the geometric properties of the structure. Results prove that a substantial speed-up is obtained with the use of the optimal modal set.