Showing papers on "Basis (linear algebra) published in 1977"

Gaussian Basis Sets for Molecular Calculations

Abstract: In the following chapters the electronic structure of molecules will be discussed and the techniques of electronic structure calculations presented. Without exception the molecular electronic wave functions will be expanded in some convenient, but physically motivated, set of one-electron functions. Since the computational effort strongly depends on the number of expansion functions (see, e.g., the following chapters), the set of functions must be limited as far as possible without adversely affecting the accuracy of the wave functions. This chapter will discuss the choice of such functions for molecular calculations.

Axiomatic basis for spaces with noninteger dimension

Abstract: Five structural axioms are proposed which generate a space SD with ’’dimension’’ D that is not restricted to the positive integers. Four of the axioms are topological; the fifth specifies an integration measure. When D is a positive integer, SD behaves like a conventional Euclidean vector space, but nonvector character otherwise occurs. These SD conform to informal usage of continuously variable D in several recent physical contexts, but surprisingly the number of mutually perpendicular lines in SD can exceed D. Integration rules for some classes of functions on SD are derived, and a generalized Laplacian operator is introduced. Rudiments are outlined for extension of Schrodinger wave mechanics and classical statistical mechanics to noninteger D. Finally, experimental measurement of D for the real world is discussed.

On differentiation of integrals.

Abstract: A technique is introduced to relate differentiation and covering properties of a basis. In particular, we find that the basis associated with a sparse set of directions differentiates integrals of functions locally in L2.

A Block–Stodola eigensolution technique for large algebraic systems with non‐symmetrical matrices

Abstract: A Block–Stodola eigensolution method is presented for large algebraic eigensystems of the form AU = λBU where A is real but non-symmetric. The steps in this method parallel those of a previous technique for the case when both A and B were real and symmetric. The essence of the technique is simultaneous iteration using a group of trial vectors instead of only one vector as is the case in the classical Stodola–Vianello iteration method. The problem is then transformed into a subspace where a direct solution of the reduced algebraic eigenvalue problem is sought. The main advantage is the significant reduction of computational effort in extracting a subset of eigenvalues and corresponding eigenvectors. Theorems from linear algebra serve to underlie the basis of the present technique. Complex eigendata that emerge during iteration can be handled without doubling the size of the problem. Higher order eigenvalue problems are reducible to first order form for which this technique is applicable. The treatment of the quadratic eigenvalue problem illustrates the details of this extension.

Uniform quality Gaussian basis sets

Abstract: A direct optimization technique was applied to determine uniformly balanced, optimum (4s2p), (6s3p), (8s4p), and (10s5p) Gaussian basis sets for the first row atoms. These bases formally correspond to 2G, 3G, 4G, and 5G function representations per symmetry type per shell. The basis sets are throroughly balanced and all satisfy a rather rigorous quality criterion in terms of the local properties of the energy hypersurface over the space of the orbital exponents.

On the Bartels--Golub decomposition for linear programming bases

Abstract: Many implementations of the simplex method require the row of the inverse of the basis matrixB corresponding to the pivot row at each iteration for updating either a pricing vector or the nonbasic reduced costs. In this note we show that if the Bartels--Golub algorithm [1, 2] or one of its variants is used to update theLU factorization ofB, then less computing is needed if one works with the factors of the updatedB than with those ofB. These results are discussed as they apply to the column selection algorithms recently proposed by Goldfarb and Reid [4, 5] and Harris [6].

Uniform quality constrained gaussian basis sets

Abstract: Uniformly balanced (6S3P), (7S3P), and (8S4P) gaussian basis sets with identical exponent sets for functions describing the 2s and 2p subshells have been obtained for the first row atoms. The basis sets have been determined using a direct optimization technique; they are thoroughly balanced and satisfy a rigorous quality criterion. These uniform quality constrained basis sets were designed for applications in ab initio programs of the type of the GAUSSIAN 70 program system, that may utilize the integration-time saving constraint α2s = α2p.

Expansion methods for Adams–Gilbert equations. I. Modified Adams–Gilbert equation and common and fluctuating basis sets

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.

Completeness and linear independence of basis sets used in quantum chemistry

Abstract: Infinite sets of functions in Hilbert space are characterized by their completeness properties and the extent of linear independence. Different measures of linear independence such as orthonormality, Gram's determinant, the special measure of linear independence, and the asymptotic dimension are related to each other and with the degrees of completeness such as overcompleteness, exact completeness, and incompleteness as far as possible.

A study of the convergence in iterative natural orbital procedures

Abstract: A series of five different Iterative Natural Orbital (INO) procedures are tested for the ground state of water and are compared on the basis of their respective convergence properties. The choice of configuration space employed in these methods is shown to be a key factor in determining the results of such calculations. If the CI space is generated by taking all single excitations with respect to a series of dominant or reference configurations, it is concluded that the practice of varying such generating species at each iteration is highly desirable. In general the choice of the configuration space is found to be much more important than the attainment of strict NO convergence, whereby experience indicates that inclusion of all singly and doubly excited configurations (or at least a select subset thereof) relative to a series of dominant configurations provides the most efficient means of approximating the true NOS of a given system within the general INO framework.

Some optimization problems for convolution systems over finite groups

Abstract: We define and study many-dimensional linear invariant discrete systems over finite groups. We consider the problem of optimum synthesis of such systems computing a given input/output pair. The optimum solution (or estimates for them) are obtained on the basis of two very simply computed criteria. Conditions are studied for the existence of an idempotent impulse function of a linear system over a group, computing a given input/output pair. The best approximation is found for many-dimensional linear invariant systems, defined on a finite interval of discrete time by systems over the given finite group.

Empirical eigenvector analysis of vector observations

Abstract: A method of deriving an empirical eigenvector or principal components representation of wind velocity measurements is given. The mathematical basis for generalizing the empirical eigenvector method to the treatment of vector data fields is stated briefly. The method presented can be employed in the analysis of regional wind velocity patterns. Applications of the analysis technique to other geophysical vector fields are also possible.

Rosetak Document 4: Rank Degeneracies and Least Square Problems

Abstract: In this paper we shall be concerned with the following problem. Let A be an m x n matrix with m being greater than or equal to n, and suppose that A is near (in a sense to be made precise later) a matrix B whose rank is less than n. Can one find a set of linearly independent columns of A that span a good approximation to the column space of B? The solution of this problem is important in a number of applications. In this paper we shall be chiefly interested in the case where the columns of A represent factors or carriers in a linear model which is to be fit to a vector of observations b. In some such applications, where the elements of A can be specified exactly (e.g. the analysis of variance), the presence of rank degeneracy in A can be dealt with by explicit mathematical formulas and causes no essential difficulties. In other applications, however, the presence of degeneracy is not at all obvious, and the failure to detect it can result in meaningless results or even the catastrophic failure of the numerical algorithms being used to solve the problem. The organization of this paper is the following. In the next section we shall give a precise definition of approximate degeneracy in terms of the singular value decomposition of A. In Section 3 we shall show that under certain conditions there is associated with A a subspace that is insensitive to how it is approximated by various choices of the columns of A, and in Section 4 we shall apply this result to the solution of the least squares problem. Sections 5, 6, and 7 will be concerned with algorithms for selecting a basis for the stable subspace from among the columns of A.

Surface Feature Interpolation on Two-Dimensional Time-Space Maps

Abstract: A procedure is described for interpolating surface features on a map based on a two-dimensional time-space configuration of points. Initially an objective procedure is used to give a best fit of a previously obtained time-space configuration of points to the physical configuration of the same points. The resultant set of vectors, each of which shows the residual displacement between a point in time space and physical space, is used as the basis of a fast, objective procedure for interpolating any other points or lines in the time space. This enables such things to be shown as the time-space distortion of an urban street network or of a square graticule from physical space. The process can also be reversed so that, for example, the solution points of a facility location problem solved in time space can be converted to their equivalent locations in physical space.

Application and Numerical Solution of Abel-Type Integral Equations.

Abstract: : This paper reviews the numerous applications in which the solution of equations like the Abel-type integral equation for discrete observational data (d sub i) is the basic step, compares, with respect to given discrete observational data (d sub i), the use of pseudo-analytic methods and the direct evaluation of its inversion formulas as a basis for solving this equation, proposes a specific algorithm based on these conclusions, and examines the consequences of the fact that, for the equation, linear functionals defined on its solution u(x) can be redefined as linear functionals on the data s(y). The justification for the latter is that, in applications involving separable first kind Abel-type integral equations, inferences are usually based on (linear) functionals defined on u(x), not on u(x) itself. This point is illustrated with an example from metallurgy.

Necessary conditions of the extremum for differential inclusions

Abstract: An optimal control problem is studied in which the plant is described by means of differential inclusions. Results are obtained, analogous to the maximum principle in the ordinary problem of optimal control, which are formulated on the basis of an investigation of the properties of multivalued mappings. In this the principal role is played by the concept of conjugate mapping, whose properties are investigated in detail. For the case of convex multivalued mappings necessary and sufficient conditions are obtained.

The effects of optimization and scaling of AO exponents on molecular properties

Abstract: The combined effects of direct optimization and various scaling techniques of Gaussian AO basis sets on the calculated molecular wavefunctions were analyzed for two examples, the hydrogen fluoride and ammonia molecules. An 8s4p Gaussian basis set for the fluorine atom which had been rigorously optimized for the total atomic energy by the conjugate gradient method was employed. For nitrogen, a similar atom optimized 9s5p basis set was employed. Three basis sets obtained in the course of the exponent optimization, the final optimized basis set, and basis sets derived from the optimum AO basis sets by various scalings of the exponents were employed in molecular calculations. The total energies and numerous one‐electron properties were analyzed. The convergence of molecular one‐electron properties to their limiting values is much slower than the energy convergence, and basis sets optimized for energy may be improved in terms of their property predictions by rigorous optimization. The simple scaling procedure ...

Rosetak Document 4: Rank Degeneracies and Least Square Problems

Abstract: In this paper we shall be concerned with the following problem. Let A be an m x n matrix with m being greater than or equal to n, and suppose that A is near (in a sense to be made precise later) a matrix B whose rank is less than n. Can one find a set of linearly independent columns of A that span a good approximation to the column space of B? The solution of this problem is important in a number of applications. In this paper we shall be chiefly interested in the case where the columns of A represent factors or carriers in a linear model which is to be fit to a vector of observations b. In some such applications, where the elements of A can be specified exactly (e.g. the analysis of variance), the presence of rank degeneracy in A can be dealt with by explicit mathematical formulas and causes no essential difficulties. In other applications, however, the presence of degeneracy is not at all obvious, and the failure to detect it can result in meaningless results or even the catastrophic failure of the numerical algorithms being used to solve the problem. The organization of this paper is the following. In the next section we shall give a precise definition of approximate degeneracy in terms of the singular value decomposition of A. In Section 3 we shall show that under certain conditions there is associated with A a subspace that is insensitive to how it is approximated by various choices of the columns of A, and in Section 4 we shall apply this result to the solution of the least squares problem. Sections 5, 6, and 7 will be concerned with algorithms for selecting a basis for the stable subspace from among the columns of A.

On two methods for discrete L p approximation

Abstract: A recent paper by Merle and Spath [3] gives some computational experience with two algorithms for linear discreteL p approximation. In this note, we establish a theoretical basis for some convergence properties observed by these authors.

Composition properties in uniform spaces

Abstract: In this paper we describe the algebraic operators derived from the inversionclosed and regular ring properties in terms of the metric-fine and measurable operators, respectively. In discussing the above relationships, the generalized and separable composition properties are also introduced. These concepts have roughly the same relationship to the sub-M-fine and locally sub-M-fine operators, respectively, as each of the above algebraic operators has to its respective uniform operator. This analogy is formalized by Theorems 2.3 and 2.4, which characterize the algebraic operators in terms of the least upper bound operation U, the real-valued operator c, and the respective uniform operators mentioned above. These results show (cf. 2.7) that the countable, separable, and generalized composition properties coincide for spaces generated by uniform real-valued functions and that the separable composition property is equivalent to the countable composition property and the real extension (RE) property for spaces with a basis of finite dimensional uniform covers. The analogy previously mentioned is further strengthened by Theorem 2.8, which shows that the sub-inversion-closed spaces are precisely the spaces closed under generalized composition. Finally, in Theorem 2.9 we show that the finite dimensional operator f is an isomorphism from the full subcategory of respectively, locally sub-M-fine spaces with a basis of point-finite uniform covers, sub-M-fine, or metric-fine spaces onto the full subcategory of spaces with a basis of finite dimensional uniform covers which are (i) closed under separable composition, generalized composition, or inversion, respectively, and (ii) are locally sub-M-fine in the following restricted sense: each cover two-dimensionally uniform on each member of some onedimensional Euclidean uniform cover is a uniform cover.

Uniform quality gaussian basis sets for organo-silicon compounds

Abstract: By applying the powerful direct optimization technique of conjugate gradients as adapted for the optimization of an open shell energy functional, a uniformly balanced (15s 10p) Gaussian basis set was obtained for the silicon atom. The quality of this basis set, as defined in terms of “exponent forces” or energy gradient |g|, is compatible with the quality of suitably chosen (10s 5p) carbon and (5s) hydrogen basis sets. Contractions better than double zeta were determined for all three bases of Si, C, and H. Using the primitive and contracted bases, ab initio SCF MO calculations were carried out on molecules of SiH4, CH4, and H2. Some of the computed results obtained for H2C = SiH2 are also included as an illustration for organo-silicon compounds.

The convergence to zero of Gaussian sequences

Abstract: Conditions for Gaussian sequences to converge to zero with unit probability are examined. A comparison theorem is proved, on the basis of which sufficient conditions are derived for the convergence to zero of Gaussian sequences, including, in particular, the previously known ones.

The role of double antisymmetrization

Abstract: With the basic equivalent transformation Pσ = ϵ(P)(Pr)−1 the dual bases of symmetric groups can be constructed. The importance of the double antisymmetrization is shown. The advantages of the use of the m basis as compared to the Wigner (matric) basis are discussed.