Showing papers on "Basis function published in 1975"

Møller–Plesset theory for atomic ground state energies

Abstract: Mo–Plesset theory, in which electron correlation energy is calculated by perturbation techniques, is used in second order to calculate energies of the ground states of atoms up to neon. The unrestricted Hartree–Fock (UHF) Hamiltonian is used as the unperturbed system and the technique is then described as unrestricted Mo–Plesset to second order (UMP2). Use of large Gaussian basis sets suggests that the limiting UMP2 energies with a complete basis of s, p, and d functions account for 75–84% of the correlation energy. Preliminary estimates of the contributions of basis functions with higher angular quantum numbers indicate that full UMP2 limits give even more accurate total energies.

Minimal Bases of Rational Vector Spaces, with Applications to Multivariable Linear Systems

Abstract: A minimal basis of a vector space V of n-tuples of rational functions is defined as a polynomial basis such that the sum of the degrees of the basis n-tuples is minimum. Conditions for a matrix G to represent a minimal basis are derived. By imposing additional conditions on G we arrive at a minimal basis for V that is unique. We show how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization. Transfer function matrices G solving equations of the form $PG = Q$ are also obtained by this method; applications to the problem of finding minimal order inverse systems are given. Previous applications to convolutional coding theory are noted. This range of applications suggests that minimal basis ideas will be useful throughout the theory of multivariable linear systems. A restatement of these ideas in the language of valuation theory is given in an Ap...

Use of energy derivative of the radial solution in an augmented plane wave method: application to copper

Abstract: By a suitable combination inside the muffin-tin sphere of a radial solution to the Schrodinger equation and its energy derivative, the dependence of the APW matrix elements on the energy E used to construct the basis functions can be greatly reduced. This has a number of advantages. The authors present an application of the method, previously suggested by Marcus (1967) and analysed by Andersen to the Chodorow copper potential and examine its precision and its convergence properties. It was found the method converges about equally well or somewhat more slowly than the standard APW method in number of basis functions. The eigenvalue error of the method is proportional to (E-E0)4. The error of the wavefunction is proportional to (E-Eo)2. The d states limit the range of mod E-E0 mod <0.1 Ryd for acceptable wavefunctions. The limit for non-d states is larger than 1 Ryd.

Rigorous method for computing photoabsorption cross sections from a basis-set expansion

Abstract: We present a rigorous technique, which should be applicable to both atoms and molecules, for calculating photoabsorption cross sections using square-integrable basis functions. The technique is based on the method of complex coordinates as developed by Nuttall and co-workers. In contrast to some other L2 methods, the calculations converge directly at real energies. The method is illustrated by application to the case of atomic hydrogen.

Three‐dimensional natural coordinate asymmetric top theory of reactions: Application to H + H2

Abstract: A particular partitioning of the Hamiltonian in natural collision coordinates is shown to lead to the use of hindered asymmetric top basis functions to represent all rotational motion (triangle tumbling plus internal bending) during reaction. These functions (along with perturbed Morse oscillator functions) are used as an adiabatic basis for expansion of the scattering wavefunction. The theory is discussed for both one and two reaction path potentials. The close coupled equations for the translational wavefunctions are then solved for the H + H2 reaction at total angular momentum J = 0. Wavefunction bifurcation and matching at the reactant–product boundary surface is considered in detail. Finally, numerical results (reaction probabilities, probability conservation, detailed balance, energy dependence of reactive S‐matrix elements, probability density, wavefunction real part, and flux) are presented and comparisons are made with other quantum mechanical, semiclassical, and statistical reaction studies.

An Introduction to Molecular Integral Evaluation

Abstract: The expectation value of the electronic energy of a n-electron system may be written as the ratio of two integrals, $$E = / $$ integration being taken over the 3n spatial and n spin coordinates of the electrons. Procedures for integration over the spin variables are not the concern of the present work, and we proceed under the assumption that the spin integrations of the energy expectation value have been completed. We are thus left with the evaluation of two integrals over the 3n spatial coordinates. A rather widely used procedure is to construct the trial form of the many electron wavefunction, Ψ, from a set of functions of the coordinates of one electron. We will denote the basis set of one electron functions by {Φ}. Notice that each basis function defines a set of n building blocks to be used in the construction of the wavefunction, since each function may be written in the coordinates of any of the n electrons.

Dipole properties of atoms and molecules in the random phase approximation

Abstract: A random phase approximation (RPA) calculation and a direct sum over states is used to calculate second−order optical properties and van der Waals coefficients. A basis set expansion technique is used and no continuumlike functions are included in the basis. However, unlike other methods we do not force the basis functions to satisfy any sum−rule constraints but rather the formalism (RPA) is such that the Thomas−Reiche−Kuhn sum rule is satisfied exactly. Central attention is paid to the dynamic polarizability from which most of the other properties are derived. Application is made to helium and molecular hydrogen. In addition to the polarizability and van der Waals coefficients, results are given for the molecular anisotropy of H2, Rayleigh scattering cross sections, and Verdet constants as a function of frequency. Agreement with experiment and other theories is good. Other energy weighted sum rules are calculated and compare very well with previous estimates. The practicality of our method suggests its applications to larger molecular systems and other properties.

Theory of J-matrix Green's functions with applications to atomic polarizability and phase-shift error bounds

Abstract: The recently introduced Jacobi or $J$-matrix techniques for quantum scattering are developed to include the construction of exact analytic matrix elements of regular and Coulomb partial-wave zeroth-order and full Green's functions. Very simple results obtain for the unperturbed Green's functions, while full Green's functions require a single diagonalization of an $N\ifmmode\times\else\texttimes\fi{}N$ Hamiltonian matrix, where $N$ is the number of basis functions coupled by the matrix truncated potential. In an application of the $J$-matrix Green's functions to the theory of atomic dynamic polarizabilities, the analytic result for hydrogen is derived, and it is shown how more general systems may be treated in a way which is superior to the usual $N$-term variational approach. In an application to error bounds for phase shifts, we show how the full Green's functions can be used to demonstrate the absence of false pseudoresonances in $J$-matrix scattering calculations, and bound the possible errors in computed phase shifts.

B-spline expansion bases for excited states and discretized scattering states

Abstract: High‐order B‐splines provide effective basis functions for constructing solutions to the Schrodinger equation. For high‐lying levels of the Morse potential septic splines yield an accuracy comparable to harmonic oscillators. They can also approximate hydrogenic wavefunctions; typically one‐third of the N eigenvalues of an N×N matrix are more accurate than 0.01%. Matrix eigenvectors can also represent scattering states. The approximation is uniformly valid spatially when basis functions terminate sharply.

Variational correction to Wigner R-matrix theory of scattering

Abstract: Three stages in the development of R-matrix scattering theory are discussed. The standard calculation using N basis functions is variationally stable in the N-dimensional basis space, yet results often converge slowly with increasing N. As an improvement the Hamiltonian is separated into two parts, a soluble H0 and a potential V. The Buttle correction (1967) is used to account for H0 exactly. This is equivalent to solving the approximate Hamiltonian H0+VN where VN is an N*N matrix approximation to V. The exact solution to the approximate Hamiltonian may be considered as a trial wavefunction for the full Hamiltonian H. The problem of obtaining scattering information from this trial wavefunction which is variationally stable against arbitrary variations, as opposed to variations in a finite trial space is discussed.

Limitations of the method of complex basis functions

Abstract: The method of complex basis functions proposed by Rescigno and Reinhardt is applied to the calculation of the amplitude in a model problem which can be treated analytically. It is found for an important class of potentials, including some of infinite range and also the square well, that the method does not provide a converging sequence of approximations. However, in some cases, approximations of relatively low order might be close to the correct result. The method is also applied to S-wave e-H elastic scattering above the ionization threshold, and spurious ''convergence'' to the wrong result is found. A procedure which might overcome the difficulties of the method is proposed.

Sturmian basis functions in the coupled state impact parameter method for H++H scattering

Abstract: When Sturmian basis functions are used in the standard coupled state impact parameter method for proton-hydrogen atom scattering, the correct boundary conditions cannot, in general, be satisfied. This means that for most choices of the initial and final states the approximate transition amplitude, as usually defined, does not exist. This difficulty is resolved and an approximate transition amplitude is defined which is a variational estimate of the exact transition amplitude for arbitrary initial and final states. In the Sturmian representation the exchange matrix elements can be calculated more directly by solving a set of coupled differential equations which is somewhat different from the set of equations given by Cheshire (1967). A procedure that does not require storage and interpolation of the matrix elements in order to exploit their time-symmetry is also described.

Generalization of the basis functions of the LCAO method for band-structure calculations

Abstract: A systematic discussion of various choices of the localized functions for constructing the bloch sums for LCAO calculations is presented. By generalizing the basis functions, improvement of efficiency and extension of the domain of applicability of the lcao method can be accomplished. The use of extended basis sets containing a series of single-gaussian bloch sums not only yields energy bands of accuracy comparable to those of the method of augmented plane waves, but also enables one to handle the highly excited states. For calculating the filled bands and lower conduction bands the basis functions formed by the optimized orbitals give considerably better results than those with the free-atom orbitals. To reduce the computational work, one can introduce the truncated orbitals by gradually damping the atomic wavefunctions beyond a certain distance, typically the mid-point to the second nearest neighbours. Specific calculations are given for the cases of lithium and diamond-type crystals.

The Correlated Basis Function Method and its Application to Liquid 3He, Nuclear Matter and Neutron Matter

Abstract: Abstract The method of correlated basis functions is studied and applied to the Fermi systems: liquid 3 He, nuclear matter and neutron matter. The reduced cluster integrals xijkl... and so the sub-normalization integrals Iijkl... are generalized to coinciding quantum numbers out of the set {i, j, k, I,...}. This generalization has an important consequence for the radial distribution function g (r) (and then for the liquid structure function) ; g(r) has no contributions of the order O (A-1). For 3 He the state-independent two-body correlation function g(r) is calculated from the Euler-Lagrange equation (in the limit of r → 0) for the unrenormalized two-body energy functional. For nuclear matter and neutron matter we adopt the three-parameter correlation function of Bäckman et al. Then the energy expectation values are calculated by including up to the three-body terms in the unrenormalized and renormalized version of the correlated basis functions method. The experimental ground-state energy and density of liquid s He can be well reproduced by the present method with the Lennard-Jones-(6 -12) potential. The same method is applied to the nuclear matter and neutron matter calculations with the OMY-potential. The results of the energy expectation values indicate a practical superiority of the unrenormalized cluster expansion method over the renormalized one.

Sparse matrix techniques applied to modal analysis of multi-section duct liners

Abstract: A simplified procedure is presented for analysis of ducts with discretely nonuniform properties. The analysis uses basis functions as the generalized coordinates. The duct eigenfunctions are approximated by finite series of these functions. The emphasis is on solution of the resulting large sparse set of linear equations. Characteristics of sparse matrix algorithms are outlined and some criteria for application are established. Analogies with structural methods are used to illustrate variations which can increase efficiency in generating values for design optimization routines. The effects of basis function selection, number of eigenfunctions and identification and ordering of equations on the sparsity and solution stability are included.

Method of correlated basis functions for the ground state of a one-dimensional many-particle boson system

Abstract: The method of correlated basis functions is used in the study of the ground-state properties of a one-dimensional system of bosons interacting through a repulsive $\ensuremath{\delta}$-function potential. Ground-state energies are determined numerically for intermediate values of the coupling constant in three steps: (1) a trial form of the radial distribution function generated by the Bijl-Dingle-Jastrow (BDJ) type of wave function is introduced to obtain approximate ground-state energies, (2) the paired-phonon analysis is applied to improve the BDJ description of the ground state, and (3) leading corrections to the improved approximate energies are evaluted from the second-order perturbation energy generated by the three-phonon vertex. The obtained energy values are found to agree closely with the exact results calculated by Lieb and Liniger.

The expectation value of the electronic energy of an-electron system may be written as the ratio of two integrals,

Abstract: integration being taken over the 3n spatial and n spin coordinates of the electrons. Procedures for integration over the spin variables are not the concern of the present work, and we proceed under the assumption that the spin integrations of the energy expectation value have been completed. We are thus left with the evaluation of two integrals over the 3n spatial coordinates. A rather widely used procedure is to construct the trial form of the many electron wavefunction, ~, from a set of functions of the coordinates of one electron. We will denote the basis set of one electron functions by {~}. Notice that each basis function defines a set of n building blocks to be used in the construction of the wavefunction, since each function may be written in the coordinates of any of the n electrons. It is now usual to find, irrespective of the exact rules used in the

Density and Phase Operator Approach

Abstract: The density and phase operator approach is used for deriving the ground-state energy in a weakly interacting many-boson system up to the perturbation energy of order 1/N; the inverse total number of particles. An expression exactly the same as that of the method of correlated basis functions is obtained by summing up several' perturbation terms .neglected in the previous theories of the collective variable approach. Some remarks are made on the other related theories, in particular, the current algebra approach of Dashen and Sharp.