Subspace‐by‐subspace preconditioners for structured linear systems

We consider the iterative solution of symmetric positive‐definite linear systems whose coefficient matrix may be expressed as the outer product of low‐rank terms. We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly‐occuring case where the coefficient matrix is the linear sum of elements, some of which are of very low rank. Copyright © 1999 John Wiley & Sons, Ltd.     

Computational investigation of CO adsorbed on Au<Subscript>x</Subscript>, Ag<Subscript>x</Subscript> and (AuAg)<Subscript>x</Subscript> nanoclusters (x = 1 − 5, 147) and monometallic Au and Ag low-energy surfaces

Density functional theory calculations have been performed to investigate the use of CO as a probe molecule for the determination of the structure and composition of Au, Ag and AuAg nanoparticles. For very small nanoclusters (x = 1 ? 5), the CO vibrational frequencies can be directly correlated to CO adsorption strength, whereas larger 147-atom nanoparticles show a strong energetic preference for CO adsorption at a vertex position but the highest wavenumbers are for the bridge positions. We also studied CO adsorption on Au and Ag (100) and (111) surfaces, for a 1 monolayer coverage, which proves to be energetically favourable on atop only and bridge positions for Au (100) and atop for Ag (100); vibrational frequencies of the CO molecules red-shift to lower wavenumbers as a result of increased metal coordination. We conclude that CO vibrational frequencies cannot be solely relied upon in order to obtain accurate compositional analysis, but we do propose that elemental rearrangement in the core@shell nanoclusters, from Ag@Au (or Au@Ag) to an alloy, would result in a shift in the CO vibrational frequencies that indicate changes in the surface composition.     

Unitary group approach to the many-electron problem. III. Matrix elements of spin-dependent Hamiltonians

This is the final paper in a series of three directed toward the evaluation of spin-dependent Hamiltonians. In this paper we derive the reduced matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables a direct evaluation of the matrix elements of spin-dependent Hamiltonians in the spin-orbit basis. An alternative (indirect) method, which employs the use of U(2n) ↓ U(n) × U(2) subduction coefficients, is also discussed.     

Long-wavelength excitations in a Bose gas at zero temperature

The long-wavelength excitations in a simple model of a dilute Bose gas at zero temperature are investigated from a purely microscopic viewpoint. The role of the interaction and the effects of the condensate are emphasized in a dielectric formulation, in which the response functions are expressed in terms of regular functions that do not involve an isolated single-interaction line nor an isolated single-particle line. Local number conservation is incorporated into the formulation by the generalized Ward identities, which are used to express the regular functions involving the density in terms of regular functions involving the longitudinal current. A perturbation expansion is then developed for the regular functions, producing to a given order in the perturbation expansion an elementary excitation spectrum without a gap and simultaneously response functions that obey local number conservation and related sum rules.Explicit results to the first order beyond the Bogoliubov approximation in a simple one-parameter model are obtained for the elementary excitation spectrum ω_k, the dynamic structure function $\text{S}$(k, ω), the associated structure function $\text{S}$_m(k), and the one-particle spectral function $\text{A}$(k, ω), as functions of the wavevector k and frequency ω. These results display the sharing of the gapless spectrum ω_k by the various response functions and are used to confirm that the sum rules of interest are satisfied. It is shown that ω_k and some of the $\text{S}$_m(k) are not analytic functions of k in the long wavelength limit. The dynamic structure function $\text{S}$(k, ω) can be conveniently separated into three parts: a one-phonon term which exhausts the f sum rule, a backflow term, and a background term. The backflow contribution to the static structure function $\text{S}$₀(k) leads to the breakdown of the one-phonon Feynman relation at order k³. Both $\text{S}$(k, ω) and $\text{A}$(k, ω) display broad backgrounds because of two-phonon excitations. Simple arguments are given to indicate that some of the qualitative features found for various physical quantities in the first-order model calculation might also be found in superfluid helium.     

New crash procedures for large systems of linear constraints

Many algorithms for solving linearly constrained optimization problems maintain sets of basic variables. The calculation of the initial basis is of great importance as it determines to a large extent the amount of computation that will then be required to solve the problem. In this paper, we suggest a number of simple methods for obtaining an initial basis and perform tests to indicate how they perform on a variety of real-life problems.