Showing papers on "Basis function published in 1986"

Basis Set Selection for Molecular Calculations

Abstract: The quantum chemistry literature containa references to a plethora of basis seta, currently numbering almost 100. While professional quantum chemists might become familiar with several dozen of these in a lifetime of calculations, the OCeasionaJ user of ab initio programs probably wishes to ignore all but the two or three sets which, through habitual use, have become personal favorites. Unfortunately, this attitude has its drawbacks. Intelligent reading of the literature requires a t least a cursory knowledge of the limitations of other basis sets. Information conceming the likely aceuracy of a specific basis for a particular property is essential in order to judge the adequacy of the computational method and, hence, the soundness of the results. Occasionally, for reasons of economy or computational feasibility, a basii set is selected for which the computed results are nearly without significance. In light of the large number of publications reporting new basis sets or detailing the performance of existing sets the task of remaining informed has become very diffteult for experts and nonexperts alike. The existence of such a vast multitude of basis sets is attributable, a t least in part, to the difficulty of finding a single set of functions which is flexible enough to produce 'good" results over a wide range of molecular geometries and is still small enough to leave the problem computationally tractible and economically within reason. The driving force behind much of the research effort in small basis sets is the fact that the computer time required for some parts of an ab initio calculation is very strongly dependent on the number of basis functions. For example, the integral evaluation goes as the fourth power of the number of Gaussian primitives. Fortunately this is the only step which explicitly depends on the number of primitives. All subsequent steps depend on the number of contracted functions formed from the primitives. The concept of primitive and contracted functions will be discussed later. Consider a collection of K identical atoms, each with n doubly occupied orbitals and N unoccupied (or virtual) orbitals. The SCF step increases as (n + N'K', while the full transformation of the integrals over the original basis functions to integrals over molecular orbitals goes as (n + N5P. Methods to account for correlation effects vary greatly. Only a few of the popular ones will be considered here. Second order Moller-Plesset (MP2) perturbation theory goes as n2NX4 but still requires an nN4F integral transformation. MP3 goes as n2N4P, while a Hartree-Fock singles and doubles CI will have n2WK4 configurations, (n2NW')' hamiltonian matrix elements of which n2N4P will be nonzero. Pople and co-workers' have proposed

Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions

Abstract: In many applications one encounters the problem of approximating surfaces from data given on a set of scattered points in a two-dimensional domain. The global interpolation methods with Duchon's “thin plate splines” and Hardy's multiquadrics are considered to be of high quality; however, their application is limited, due to computational difficulties, to $ \sim 150$ data points. In this work we develop some efficient iterative schemes for computing global approximation surfaces interpolating a given smooth data. The suggested iterative procedures can, in principle, handle any number of data points, according to computer capacity. These procedures are extensions of a previous work by Dyn and Levin on iterative methods for computing thin-plate spline interpolants for data given on a square grid. Here the procedures are improved significantly and generalized to the case of data given in a general configuration.The major theme of this work is the development of an iterative scheme for the construction of a smooth surface, presented by global basis functions, which approximates only the smooth components of a set of scattered noisy data. The novelty in the suggested method is in the construction of an iterative procedure for low-pass filtering based on detailed spectral properties of a preconditioned matrix. The general concepts of this approach can also be used in designing iterative computation procedures for many other problems.The interpolation and smoothing procedures are tested, and the theoretical results are verified, by many numerical experiments.

Highly excited vibrational levels of "floppy" triatomic molecules: A discrete variable representation - Distributed Gaussian basis approach

Abstract: A novel, efficient, and accurate quantum method for the calculation of highly excited vibrational levels of triatomic molecules is presented. The method is particularly well suited for applications to ‘‘floppy’’ molecules, having large amplitude motion, on potential surfaces which may have more than one local minimum. The discrete variable representation (DVR) for the angular, bend coordinate is combined with the distributed (real) Gaussian basis (DGB) for the expansion of other, radial coordinates. The DGB is tailored to the potential, covering only those regions where V(r)

Comparison of the propagation of semiclassical frozen Gaussian wave functions with quantum propagation for a highly excited anharmonic oscillator

Abstract: The propagation of an initially highly excited localized wave packet in an anharmonic oscillator potential is studied within the frozen Gaussian approximation. Comparison is made to quantum mechanical basis set calculations. The frozen Gaussian approximation involves the expansion of the initial wave function in terms of an overcomplete Gaussian basis set. The wave function evolution is evaluated by allowing each Gaussian to travel along a classical trajectory with its shape held rigid. A Monte Carlo algorithm is employed in the selection of the initial Gaussian basis functions. The frozen Gaussian results are very good for times on the order of a few vibrational periods of the oscillator and remain qualitatively correct for the entire length of the calculations which is 12 vibrational periods. The dependence of the calculations on the width of the Gaussian basis functions is investigated and the effect of a simplifying approximation for the prefactor of the Gaussians is tested.

Modal estimation of a wave front from difference measurements using the discrete Fourier transform

Abstract: It is shown that a wave front, or in general any scalar two-dimensional function, can be reconstructed from its discrete differences by a simple multiplicative filtering operation in the spatial-frequency domain by using complex exponentials as basis functions in a modal expansion. Various difference-sampling geometries are analyzed. The difference data are assumed to be corrupted by random, additive noise of zero mean. The derived algorithms yield unbiased reconstructions for finite data arrays. The error propagation from the noise on the difference data to the reconstructed wave fronts is minimal in a least-squares sense. The spatial distribution of the reconstruction error over the array and the dependence of the mean reconstruction error on the array size are determined. The algorithms are computationally efficient, noniterative, and suitable for large arrays since the required number of mathematical operations for a reconstruction is approximately proportional to the number of data points if fast-Fourier-transform algorithms are employed.

Gaussian basis sets and the nuclear cusp problem

Abstract: Variational calculations of the H-atom are performed with different types of Gaussian basis sets, (a) Huzinaga basis sets of 1s-Gaussians, (b) ns basis sets with n = 1, 2, 3, 4 ⋯ and optimized common exponents for each set, (c) ns basis sets with n = 1, 3, 5 ⋯ (d) ns basis sets with n = 1, 2, 3, 5, 7 ⋯ (e) ns basis sets with n = 1, 2, 3, 4 ⋯ but with all orbital exponents optimized. The number of basis functions in a set is N . For the sets (a) the errors of the energy (upper or lower bound) and of the expectation values r k , k = −2, −1, ⋯ 5 appear to converge as ≈ exp (— b √ N ), for sets (b) and (e) as ≈exp (— bN ), for set (c) the error of the upper bound converges as ≈N −2 and for set (d) as ≈ N −4 . For medium accuracy the Huzinaga sets are surprisingly good. They are poorly suited only for representing the cusp at the nucleus and the variance of the energy. For high accuracy basis sets (b) and (e) are superior. They represent the cusp correctly. For them the energy variance converges rapidly to zero, and all other quantities converge finally much faster. Basis sets with only even powers of r (or with just 2 s included) and a common (optimized) orbital exponent are, although formally “complete in the energy space”, of no practical use. Basis sets (e) yield by far the fastest convergence. The results are analyzed and their consequences are discussed.

A common structure for recursive discrete transforms

Abstract: This paper presents a common framework for the recursive implementation of arbitrary discrete transformations. The transform coefficients to be applied are periodically time-varying and can be derived from the discrete basis functions of the transforms. The method is based on Hostetter's dead-beat observer approach to signal processing [1], [2], but instead of the ongoing calculation of the transform coefficients, explicit expressions are derived. The proposed structure can be efficiently used even for FIR and IIR filtering operations.

Energy levels of a neutral hydrogen-like system in a constant magnetic field of arbitrary strength

Abstract: The variational method is employed to calculate the energy levels of a hydrogenlike system in a constant magnetic field of arbitrary strength. The general approach is similar to that of Aldrich and Greene (1979), but uses a different set of basis functions, enabling smaller matrices to be used. Results are obtained for energy levels up to n=4, and are of high accuracy judged by comparison with other published data.

Study of a high-dimensional chaotic attractor

Abstract: The nature of a very high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high-dimensional attractor in the phase space. We introduce two complementary bases from which the attractor is observed, one the Lyapunov basis composed of the Lyapunov vectors and the another the Fourier basis composed of the Fourier modes. We introduce the “exterior” subspaces on the basis of the Lyapunov vectors and observe the chaotic motion projected onto these exteriors. It is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace “goes out” of the attractor. The origin of such a phenomenon is attributed to more fundamental features of our attractor, which become manifest when the attractor is observed from the Lyapunov basis. A counterpart of the phenomenon can be observed also on the Fourier basis because there is a statistical one-to-one correspondence between the Lyapunov vectors and the Fourier modes. In particular, a statistical property of the high-pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor.

Densities, operators, and basis sets.

Abstract: Relationships between density matrices and densities or between operators and local potentials are considered for model problems defined by the introduction of basis sets. Some properties depend only on the space spanned by the basis while others depend on a particular choice of basis functions. Linear-dependency conditions play a critical role. In a model problem defined by a basis with all products linearly independent, the effect of any operator can be reproduced by a local potential, but any complete basis must have linearly dependent products. A one-electron density matrix or single-determinant wave function can be determined from the density (or experimental measurements sensitive only to the density) in the model problem defined by a basis with linearly independent products, but not otherwise. A simple example illustrates some of the general results.

Relativistic perturbation theory: II. One-electron variational perturbation calculations

Abstract: For pt.I see ibid., vol.19, p.149 (1986). The effect of the approximate character of the non-relativistic wavefunction on the second-order relativistic energy correction for the ground state of hydrogen-like atoms is illustrated by means of prototype calculations using Gaussian-type basis functions. Generalisation of the concept of even-tempered basis sets for use in relativistic calculations is proposed. Applications using Gaussian-type and exponential-type functions are presented and the convergence of the second-order relativistic correction with increasing size of basis set is examined. Some attention is paid to the possibility of computation of the third-order energy correction using the first-order correction to the wavefunction.

Continuity of the null space basis and constrained optimization

Abstract: Many constrained optimization algorithms use a basis for the null space of the matrix of constraint gradients. Recently, methods have been proposed that enable this null space basis to vary continuously as a function of the iterates in a neighborhood of the solution. This paper reports results from topology showing that, in general, there is no continuous function that generates the null space basis of all full rank rectangular matrices of a fixed size. Thus constrained optimization algorithms cannot assume an everywhere continuous null space basis. We also give some indication of where these discontinuities must occur. We then propose an alternative implementation of a class of constrained optimization algorithms that uses approximations to the reduced Hessian of the Lagrangian but is independent of the choice of null space basis. This approach obviates the need for a continuously varying null space basis.

EEG spatial filter and method

Abstract: Method and apparatus for producing enhanced EEG or MEG information related to a selected brain activity in a subject. The apparatus includes a two-dimensional array of at least about 16 sensors for recording EEG or MEG traces from the subject. Control and test traces recorded before and during an interval in which the brain activity is occurring, respectively, are each decomposed into a series of basis functions which may be analytic components such as temporal frequency components, generated by spectral decomposition of an ensemble average of the recorded traces, or principal components determined by principal component analysis. The control and test traces are then represented as a sum of the products of the individual basis functions times a spatial domain matrix which represents the spatial pattern of amplitudes of that basis function, as measured by the individual sensors in the array. The signals can be extracted by filtering spatially and/or temporally to remove basis function components which are not related to the selected brain activity (clutter), and to remove spatial frequencies inherent in the spatial domain matrices to optimize the contrast between control and and corresponding test matrices, for each selected basis function.

Finite Element Algorithms for Simulating Three‐Dimensional Groundwater Flow and Solute Transport in Multilayer Systems

Abstract: A finite element formulation is developed for the simulation of groundwater flow and solute transport in multilayer systems of several aquifers and aquitards. This formulation is general, flexibile, and capable of taking full advantage of the nature of flow in such multilayer systems. A fully three-dimensional spatial representation can be performed for certain aquifers or for the entire flow system if needed. Those parts of the system that require three-dimensional spatial discretization are handled effectively by combining two-dimensional basis functions in the x-y space and one-dimensional basis functions in the z space. Furthermore, the formulation has a desirable option to perform one-dimensional representation of flow and transport in aquitards and areal representation of flow and transport in certain aquifers that do not require the fully three-dimensional discretization. When the one-dimensional representation of an aquitard is used, coupling of adjacent aquifer and aquitard layers is handled using a convolution integral approach. A general solution strategy is also developed to allow systematic time stepping and cost-effective matrix handling schemes. For those parts of the system that require three-dimensional discretization, an algorithm referred to as the alternate sublayer and line sweep procedure is presented for decomposing three-dimensional matrix equations. This algorithm can accomodate several thousand nodal unknowns without requiring excessive core storage and CPU time. Four examples of transient flow and transport problems are provided. These examples show verification of numerical results and demonstrate that the present finite element models are far more cost-effective than earlier three-dimensional models that are based on the conventional Galerkin approach and direct matrix solution algorithms.

The efficient evaluation of configuration interaction analytic energy second derivatives: Application to hydrogen thioperoxide, HSOH

Abstract: We present an efficient reformulation of the analytic configuration interaction (CI) energy second derivative. Specifically, the Z‐vector method of Handy and Schaefer is used to avoid solving the second order coupled perturbed Hartree–Fock (CPHF) equations. We have incorporated translational–rotational invariance into the new method. We present a more efficient method for the evaluation of the Y matrix contribution. The procedure which has been implemented can accommodate very large basis sets and CI expansions for any general restricted Hartree–Fock (RHF) reference wave function. As a test case, we apply the new procedure to the HSOH molecule using a double zeta plus polarization basis set. This leads to 50 contracted Gaussian basis functions and 116 403 configurations in the CI expansion. Harmonic vibrational frequencies and infrared intensities are predicted for HSOH and its deuterated isotopomers. The analytic method described herein requires only 56% of the central processor unit time used by a numerical method.

Stable highly excited vibrational eigenvalues without the variational principle

Abstract: The energy spectrum in the stochastic region for a model Hamiltonian of two strongly coupled modes is calculated by diagonalizing small matrices which do not provide the Hylleraas–Undheim–MacDonald variational energy upper bounds. In addition, a method for selecting the most important basis functions by artificial intelligence algorithms is utilized. The energy convergence is determined by the Hazi–Taylor stabilization method and by the nearest‐neighbor‐spacing distribution function which measures the local fluctuations in the spectrum.

Recursion, non-orthogonal basis vectors, and the computation of electronic properties

Abstract: The authors present a more general and systematic formulation of the recursion method, from which one can simply compute any matrix element of the Green function as a continued fraction. The generalisations are important if one uses non-orthogonal basis functions, such as overlapping atomic orbitals. They investigate the significance of these generalisations for computation of densities of states and of the electrical conductivity of disordered metals. The corrections are unimportant for liquid Mn, but they are very important for liquid Co.

Finite element method for charged-particle calculations

Abstract: A new method for solving the Boltzmann-Fokker-Planck equation is presented. Following the finite element technique, the solution is projected onto a space defined by linear discontinuous basis functions. Three approaches for the angular flux are derived and compared: the first two for a coupled energy-position discretization and the third one for the coupled energy-position-angle discretization. The last was specifically developed for highly anisotropic problems, such as ion beams impinging on an inertial confinement fusion target. Numerical results show clearly that the finite element approaches are higher order approximations. The convergence rate, stability, and performance compared with other methods are examined.

Practical method for highly accurate large-scale surface calculations.

Abstract: An accurate and efficient film linearized muffin-tin orbital (FLMTO) technique for surface electronic-structure calculations is presented which uses only 60-70 basis functions, as opposed to the 300 functions used in the linear augmented plane-wave method. Calculations for three different (3d and 4d) transition-metal films resulted in high quality results for five-layer slabs of Cu(001), Fe(001), and Ru(001), in addition to good results for the work functions and projected density of states. By retaining the LMTO small basis size, computer time and memory are reduced, making practical the study of systems with a larger number of atoms in the two-dimensional unit cell.

A new algebraic approach to the eigenvalue problems of linear differential operators without integrations

Abstract: In this work, a new approximation scheme based on the evaluation of the pointwise expectation of the Hamiltonian (H) via a conveniently chosen basis set is proposed. This scheme does not necessitate integration; however, physical and mathematical considerations in choosing the basis set are considerably important when very precise and rapidly convergent results are desired. In this method, the best linear combination of “well-selected” basis functions are sought in a way such that H ψ / ψ is flat in the neighborhood of a conveniently chosen point in the domain of H. This yields an algebraic eigenvalue problem. Some concrete applications that have already been realized confirm the efficiency of this approach.

Variation-difference schemes for problems in the mechanics of ideally elastoplastic media

Abstract: A general scheme is presented for extending variational problems to nonreflexive spaces. This is suitable both for problems in ideal plasticity as well as for variational problems associated with non-parametric surfaces. To find an effective approximation to discontinuous solutions on the basis of the extended formulation, a variation-difference method is constructed in which the proposed features are introduced directly into the basis functions. The convergence of the method is proved.

Geometry of density matrices. VI. Superoperators and unitary invariance

Abstract: We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one-particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group-theoretic sense, for the group of unitary transformations of the set of one-electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle-number-conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non-particle-number-conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.

Structure and properties of transition-metal compounds. A systematic study of basis set effects in ab initioSCF calculations

Abstract: The recently developed Gaussian basis functions [2] were used in calculations on the ground electronic states of molecules containing transition-metal atoms: ScF3, TiCl4, ZrCl4, Cr(CO)6, Ni(CO)4, CuF, CuCl, Zn(CH3)2, and Cd(CH3)2. The usefulness of minimal basis sets, the importance of splitting of the valence part of the minimal basis sets, the role of the triple splitting of the d-block functions, and the need for p-, d-, and f-type polarization functions were discussed in the context of the geometrical structure and the firstorder electronic properties of the transition-metal atom compounds.

Spin and charge densities from Hiller–Sucher–Feinberg identities. Test calculations for He, B, C, N, O, and F atoms and H2O, N2, and CH3 molecules

Abstract: Spin and charge densities for several atoms and molecules can be calculated by the use of the Hiller–Sucher–Feinberg identities in the Hartree–Fock (HF) or the unrestricted Hartree–Fock (UHF) level with Gaussian basis sets. Previously presented formulas for the necessary integrals are improved for rapid computations when the d-type Gaussian functions are added to the basis set. For charge density, a reliable description is obtained even for the HF or the UHF wave function, despite the use of the noncusp basis functions. The lack of the cusp in the Gaussian functions can be overcome also for spin density.

Universal basis sets of elliptical functions. Applications to simple diatomic molecules

Abstract: The use of universal basis sets of elliptical functions in molecular electronic structure calculations for diatomic systems is investigated. It is demonstrated that the same integrals over elliptical basis functions can be employed in calculations at a range of internuclear separations, for low-lying excited states and for a range of nuclear charges. Universal basis sets of elliptical functions are shown to lead to accurate energy gradients. Applications to one- and two-electron systems are reported together with a calculation on the helium dimer.

Relativistic Gaussian basis sets for the atoms hydrogen to neon

Abstract: Gaussian basis sets for use in relativistic molecular calculations are developed for atoms and ions with one to ten electrons. A relativistic radial wavefunction coupled to an angular function of l-symmetry is expanded into a linear combination of spherical Gaussians of the form r l exp (−αr 2). One set of basis functions is used for all large and small components of the same angular symmetry. The expansion coefficients and the orbital exponents have been determined by minimizing the integral over the weighted square of the deviation between the Dirac or Dirac-Fock radial wavefunctions and their analytical approximations. The basis sets calculated with a weighting function inversely proportional to the radial distance are found to have numerical constants very similar to those of their energy-optimized non-relativistic counterparts. Atomic sets are formed by combining l-subsets. The results of relativistic and non-relativistic calculations based on these sets are analyzed with respect to different criteria, e.g. their ability to reproduce the relativistic total energy contribution and the spin-orbit splitting. Contraction schemes are proposed.