Roger Haydock

Vector continued fractions using a generalized inverse

A real vector space combined with an inverse (involution) for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse permits construction of vector analogues of the Jacobi continued fraction. These vector Jacobi fractions are related to vector and scalar-valued polynomial functions of the vectors, which satisfy recurrence relations similar to those of orthogonal polynomials. The vector Jacobi fraction has strong convergence properties which are demonstrated analytically, and illustrated numerically.

Estimation of surface charge densities for low‐energy atom diffraction

For purposes of comparing models of surface atomic structure with atom diffraction data, it is proposed that the surface of critical charge density for atom scattering be obtained from the superposition of atomic charge densities. The results of this method, which has the advantage of permitting fast calculation for many structures, including complex ones, are compared with the self‐consistent calculations of Hamann for GaAs(110) and Ni(110). The method is then applied to the Si(111) surface and used to illustrate the effects of atom shifts on corrugations of the surface of critical charge density.

Electronic structure and relative stability of the MgO (001) and (111) surfaces

The relative stability and electronic structure of the MgO (001), (111)‐Mg, and (111)‐O surfaces have been studied with the Harris–Foulkes total energy functional. The computed energies of the (001), (111)‐O, and (111)‐Mg surfaces of MgO are 2.64, 12.80, and 13.02 J m−2, respectively. Our calculations exhibit a qualitative distinction between the Mg‐terminated polar (111) surface, which has a nonzero density of one‐electron states at the Fermi level, and the other (001) and (111)‐O surfaces which retain the band gap properties of the bulk material. The projected local density of states for atoms in the bulk and at the various surfaces are presented, and used to explain the relative stability of different surfaces in terms of the properties of surface bonds.

The electronic structure of neutral and charged surface vacancy defects in periclase

Abstract An understanding of the electronic structure of defects on mineral surfaces is critical to the development of microscopic models of the chemistry which occurs at these surfaces in natural environments, and in catalytic applications. In this work, the formation energies and electronic structure of neutral and charged surface oxygen vacancy defects (Fs, F+s and F2+s centers) on the (001) cleavage face of periclase (MgO) are computed using a recently developed ab-initio electronic structure method. The position of the defect state in the fundamental energy gap of MgO is found to be in qualitative agreement with a model for optical absorption and emission, and energy-dependent electron energy-loss experiments, and is used to explain the formation energies of these defects.

Study of a mobility edge by a new perturbation theory

Abstract The recursion method is applied to the Anderson model of disorder on a Cayley tree, i.e. a Bethe lattice. For small disorder an infinite order perturbation theory is applied which determines the distributions of parameters of a tight-binding linear chain which is exactly equivalent to the original system. The linear chain possesses localized states and extended states separated in energy by mobility edges. Analysis of the linear chain gives the position of the mobility edges, the nature of the singularity at the mobility edge, and the spatial extent of the localized states. A universal property of the Gaussian distribution is demonstrated and comments are made about a Cauchy distribution.

The Block recursion library: accurate calculation of resolvent submatrices using the block recursion method

Abstract The Block Recursion Library, a collection of FORTRAN subroutines, calculates submatrices of the resolvent of a linear operator. The resolvent, in matrix theory, is a powerful tool for extracting information about solutions of linear systems. The routines use the block recursion method and achieve high accuracy for very large systems of coupled equations. This technique is a generalization of the scalar recursion method, an accurate technique for finding the local density of states. A sample program uses these routines to find the quantum mechanical transmittance of a randomly disordered two-dimensional cluster of atoms.

Densities of states, moments, and maximally broken time-reversal symmetry

Power moments, modified moments, and optimized moments are powerful tools for solving microscopic models of macroscopic systems; however the expansion of the density of states as a continued fraction does not converge to the macroscopic limit point-wise in energy with increasing numbers of moments. In this work the moment problem is further constrained by minimal lifetimes or maximal breaking of time-reversal symmetry, to yield approximate densities of states with point-wise macroscopic limits. This is applied numerically to models with one and two finite bands with various singularities, as well as to a model with infinite band-width, and the results are compared with the maximum entropy approximation where possible.

Accuracy of the recursion method and basis non-orthogonality

Abstract The utility of resolvent calculations is reviewed for condensed matter systems where quantum mechanical equations of motion can be expressed in terms of large sparse matrices. The recursion method for calculating both diagonal and off-diagonal elements of resolvents is described and compared with other methods. In many recursion calculations there is a dramatic loss of basis orthogonality due to rounding error. An example is described, and it is argued that orthogonality is restored in the recursion method by projection of the resolvent. This is applied to the calculation of the density of states and the quantum mechanical transmittance.

Asymptotic forms for the states of weakly disordered systems

Abstract The Schrodinger equation is solved asymptotically in separation for a two-dimensional and a three-dimensional system, each with weak disorder similar to the Anderson model. Distributions of the phase and logarithm of the amplitude of the wavefunctions are found to be Gaussian with means and widths which in two dimensions describe power-law localized states and exponentially localized states separated in energy by a mobility edge, and in three dimensions describe extended states and exponentially localized states also separated in energy by a mobility edge. The results do not obey a single-parameter scaling law.

Classical Localization of an Unbound Particle in a Two-Dimensional Periodic Potential and Surface Diffusion

In periodic, two-dimensional potentials a classical particle might be expected to escape from any finite region if it has enough energy to escape from a single cell. However, for a class of sinusoidal potentials in which the barriers between neighboring cells can be varied, numerical tridiagonalization of Liouvilles equation for the evolution of functions on phase space reveals a transition from localized to delocalized motion at a total energy significantly above that needed to escape from a single cell. It is argued that this purely elastic phenomenon increases the effective barrier for diffusion of atoms on crystalline surfaces and changes its temperature dependence at low temperatures when inelastic events are rare.