Showing papers on "Basis function published in 1971"
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
TL;DR: In this article, a general computational technique for deriving fine histogram representations of singular k-dependent integrals for any crystalline solid is presented, and detailed consideration is given to the errors involved in such integrals, and in their representations.
88 citations
71 citations
70 citations
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.
30 citations
TL;DR: In this article, it was argued that the translation factor in the basis functions of the perturbed stationary state approximation must, in the separated atoms limit, involve v.r rather than Rr where v is the relative velocity of the nuclei, r is the position vector of the active electron, and R is the distance between the nodes.
Abstract: It is argued that the translation factor in the basis functions of the perturbed stationary state approximation must, in the separated atoms limit, involve v.r rather than Rr where v is the relative velocity of the nuclei, r is the position vector of the active electron, and R is the distance between the nuclei. The adoption of the incorrect translation factor leads to the appearance of spurious terms, increasing indefinitely with R, in the transition matrix elements. This has an insignificant effect on capture in slow H+-H(1s) collisions.
22 citations
TL;DR: In this article, the authors used Gaussian basis functions to calculate long-range intermolecular forces for simple H−H+ and H-H interactions, and showed that they can be obtained by using small Gaussian base sets to represent the interaction, that is Ψ(1), and/or the zeroth order wavefunction.
Abstract: The charge‐induced dipole induction energy and the dipole–dipole dispersion energy for the ground state H–H+ and H–H interactions are used as model energies for investigating the feasibility of using Gaussian basis functions to calculate long‐range intermolecular forces. The use of various types of Gaussian basis sets to represent the first‐order wavefunction, Ψ(1), in the calculation of the interaction energies and the errors introduced into the calculations by using Gaussian representations of the exact zeroth‐order wavefunctions for the isolated atoms are considered. Satisfactory long‐range forces, at least for the simple interactions considered here, can be obtained by using small Gaussian basis sets to represent the interaction, that is Ψ(1), and/or the zeroth‐order wavefunction.
14 citations
TL;DR: In this paper, a theory of the damping and frequency shift of phonons and of the ground-state energy corrections due to interactions between phonons in quantum crystals with singular forces is presented.
Abstract: A theory is presented of the damping and frequency shift of phonons and of the ground-state energy corrections due to interactions between phonons in quantum crystals with singular forces. The technique begins with the adoption of a trial ground-state wave function of the Jastrow form, together with trial excited-state wave functions constructed to represent one-, two-, and three-phonon excitations. The Hamiltonian matrix in this restricted basis is diagonalized, and the basis is optimized by minimizing the lowest eigenvalue with respect to variational phonon parameters. Using a lowest-order cluster expansion, the unambiguous prescription is obtained that a specific effective potential, softened by the Jastrow correlation function, replaces everywhere the true potential in the existing self-consistent theory of phonon damping applicable to nonsingular forces. Close analogies are drawn with the correlated basis function treatment, of superfluid liquid helium.
12 citations
TL;DR: Group theory is used to analyze arbitrary network symmetries and is incorporated into a time-domain analysis scheme that excites solutions displaying a specific symmetry characteristic.
Abstract: Group theory is used to analyze arbitrary network symmetries and is incorporated into a time-domain analysis scheme. Basis-function solutions are cataloged according to symmetry type, and inputs are constructed which excite only those solutions displaying a specific symmetry characteristic.
11 citations
TL;DR: In this article, accurate analytical selfconsistent field (SCF) wavefunctions were calculated for the dysprosium laseractive ion, namely for Dy2+, (Z = 66), 4f10, 5I.
Abstract: Accurate analytical self‐consistent‐field (SCF) wavefunctions were calculated for the dysprosium laseractive ion, namely for Dy2+, (Z = 66), 4f10, 5I. These ab initio calculations were done by the analytical SCF expansion method, with full exchange effects for all the 64 electrons included. All of the basis function exponents in the analytical expansions were exhaustively optimized. These results represent the only analytical wavefunctions for Dy2+ available at this time. Our usual accuracy criteria were satisfied in these calculations.
5 citations
TL;DR: In this article, the authors construct explicit basis functions of the representation spaces of the SL(2, C) group, in which the subgroup diagonalized is that one which in the four-dimensional representation leaves invariant the 4-vector pμ = (½(1 + v), 0, 0, ½(1 − v)) for an arbitrary real value of Pμ2 = v. The formalism is of importance in physical theories which postulate analyticity requirements and Lorentz invariance simultaneously.
Abstract: Unitary and nonunitary representations of the SL(2, C) group are investigated in such a basis, in which the subgroup diagonalized is that one which in the four‐dimensional representation leaves invariant the 4‐vector pμ = (½(1 + v), 0, 0, ½(1 − v)) for an arbitrary real value of Pμ2=v. The split of the representation space into irreducible subspaces changes smoothly when varying the value of v. The formalism is of importance in physical theories which postulate analyticity requirements and Lorentz invariance simultaneously (e.g., Regge and Lorentz pole theory). In this paper we construct explicit basis functions of the representation spaces.
TL;DR: In this article, Hartree-Fock wave functions for the He and Be isoelectronic sequences of ions are calculated using orbitals which are linear combinations of simple exponential functions.
Abstract: Hartree-Fock wave functions for the He and Be isoelectronic sequences of ions are calculated using orbitals which are linear combinations of simple exponential functions. By a full optimization of the exponents and coefficients close approximations to the HartreeFock energies were obtained. To the same order of accuracy the resulting Hartree–Fock orbitals require fewer basis functions than used previously. A number of difficulties which arise in the numerical procedures as the size of the basis set is increased are analysed in detail. Similar results are obtained for the Li sequence using the Unrestricted HartreeFock method with and without projection.
TL;DR: In this article, the authors considered the optimization of time-varying multivariable control systems with quadratic cost function which may have time-changing weighting matrices.
Abstract: This paper considers the optimization of time-varying multivariable control systems with quadratic cost function which may have time-varying weighting matrices. The problem is first formulated in the context of functional analysis which leads to the optimal control being denned by a functional equation. An approximate solution of this equation is then obtained by replacing the kernel of the equation by a double Fourier series expansion. It is shown that the resulting sub-optimal control may be made to approach the absolute optimum as closely as desired by extending the double Fourier series, and bounds on the norm of the error in the control and the cost function are derived for any finite expansion. Application of the method to a particular system shows that Chebyshev basis functions lead to a better approximation than the classic sinusoidal functions. Finally, the use of the method as an adjunct to iterative optimization based on the contraction mapping algorithm is discussed briefly.
01 Jan 1971
TL;DR: This chapter focuses on SQUARS, which is an algorithm for least-squares approximation, and provides an overview of the mathematical procedures used in the algorithm.
Abstract: Publisher Summary This chapter focuses on SQUARS, which is an algorithm for least-squares approximation. It provides an overview of the mathematical procedures used in the algorithm. The main features of the algorithm are as follows: (1) exact integration is replaced by approximate integration using the trapezoidal rule or the rectangular rule; (2) the polynomials can be replaced by an arbitrary set of basis functions supplied by the user; (3) a weight function can be specified in the integral, and, in particular, a weight function providing relative error approximation is supplied upon request; (4) a trend function can be inserted to multiply the approximating function; and (5) a desired error of approximation can be specified and the degree is increased until that error is achieved. There are six points in the algorithms where the numerical procedures are used: (1) numerical integration, (2) sum of squares, (3) linear dependence test for basis functions, (4) attainment of specified accuracy, (5) iterative refinement of the solution, and (6) reorthogonalization for the Gram–Schmidt process.
TL;DR: In this article, it was shown that one can set up a composite wave method with energy independent matrix elements by separating out the augmenting components for the lower l-values and including these as additional basis functions, but the resulting convenience of diagonalizing only once per k-point is obtained at the price of increased matrix size.
01 Jan 1971
TL;DR: In this paper, the Slater-Koster tight-binding interpolation method was extended to the linear combination of orthogonalized atomic orbitals (LCOAO) method, where the independent tight binding energy integrals that occur in the Hamiltonian matrix are treated as disposable parameters which are chosen to fit the results of accurate band calculations.
Abstract: Most applications of the Slater-Koster tight-binding interpolation method1 utilize basis functions formed from orthogonalized atomic orbitals. We shall refer to this scheme as the LCOAO method or the linear-combination-of-orthogonalized-atomic-orbitals method. In such applications, the independent tight-binding energy integrals that occur in the Hamiltonian matrix are treated as disposable parameters which are chosen to fit the results of accurate band calculations at symmetry points in the Brillouin zone.
TL;DR: In this article, the authors considered the possibility of constructing strictly localizable fields using generalized S-type spaces as spaces of basis functions, and the restrictions imposed on the asymptotic behavior of the amplitudes coincide with the well-known restrictions found by Jaffe.
Abstract: The possibility of constructing strictly localizable fields is considered, using generalized S-type spaces as spaces of basis functions. The restrictions imposed on the asymptotic behavior of the amplitudes coincide with the well-known restrictions found by Jaffe. Contrary to previous results, the spatial amplitudes and momenta are regular, and not singular functions, in the case considered. The possibility of formulating spectral conditions is investigated.
01 Jan 1971
TL;DR: In this article, a variant of the APW basis function set is described, which is based on the variational formulation for the solution of the periodic potential problem, which has been known for some time.
Abstract: The variational formulation for the solution of the periodic potential problem has been known for some time1. Thus, the development of a new band structure method usually consists of creating a basis function set which has some convenient features. What is to be described here is a variant of the APW basis function set.