The Recursion Method and Its Applications

1. Front Matter

   Pages I-VIII

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2. Introduction

   1. Front Matter

      Pages 1-1

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   2. Why Recur?

      • Volker Heine

      Pages 2-7

   3. The Recursive Solution of Schroedinger’s Equation

      • Roger Haydock

      Pages 8-20

3. Asymptotic Behaviour

   1. Front Matter

      Pages 21-21

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   2. Asymptotic Behaviour of Continued Fraction Coefficients Related to Singularities of the Weight Function

      • Alphonse Magnus

      Pages 22-45

   3. Band Gaps and Asymptotic Behaviour of Continued Fraction Coefficients

      • F. Ducastelle, P. Turchi, G. Tréglia

      Pages 46-51

   4. Computing Greenians: Quadrature and Termination

      • C. M. M. Nex

      Pages 52-60

   5. Application of Linear Prediction for Extrapolating Recursion Coefficients

      • G. Allan

      Pages 61-69

4. Related Methods

   1. Front Matter

      Pages 71-71

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   2. On a Generalized-Moments Method

      • J. P. Gaspard, Ph. Lambin

      Pages 72-83

   3. The Equation of Motion Method

      • A. MacKinnon

      Pages 84-90

   4. Use of Cyclic Matrices to Obtain Analytic Expressions for Crystals

      • Philippe Audit

      Pages 91-101

5. Solid State Applications

   1. Front Matter

      Pages 103-103

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   2. Continued Fractions and Perturbation Theory: Application to Tight Binding Systems

      • P. Turchi, F. Ducastelle

      Pages 104-119

   3. Response Functions and Interatomic Forces

      • M. W. Finnis, D. G. Pettifor

      Pages 120-131

   4. The Recursion Method with a Non-Orthogonal Basis

      • R. Jones

      Pages 132-137

6. Lanczos Method Applications

   1. Front Matter

      Pages 139-139

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   2. Hamiltonian Eigenvalues for Lattice Gauge Theories

      • A. C. Irving

      Pages 140-148

   3. The Lanczos Method in Lattice Gauge Theories

      • I. M. Barbour, N.-E. Behilil, P. E. Gibbs, G. Schierholz, M. Teper

      Pages 149-164

This volume reviews recent advances in the development and application of the recursion method in computational solid state physics and elsewhere. It comprises the invited papers which were presented at a two-day conference at Imperial College, London during September 1984. The recursion method is based on the Lanczos algorithm for the tridiago­ nalisation of matrices, but it is much more than a straightforward numerical technique. It is widely regarded as the most elegant framework for a variety of calculations into which one may incorporate physical insights and a num­ ber of technical devices. The standard reference is Volume 35 of Solid State Physics, which contains all the early ideas of Heine, Haydock and others, upon which the method was established. The present volume provides the first review of subsequent developments. It also indicates where problems remain, or opinions differ, in the interpretation of the mathematical details or choice of practical techniques in applications. The field is still very li­ vely and much remains to be done, as the summary chapter clearly demonstra­ tes. We are grateful to the S. E. R. C. 's Collaborative Computational Project No. 9 on the electronic structure of solids and the Institute of Physics's Solid State Sub-committee for their sponsorship of the conference. We thank Angus MacKinnon for his help in conference organisation and Jacyntha Crawley for secretarial assistance. December 1984 David G. Pettifor Denis L. Weaire v Contents Part I Introduction Why Recur? By V.

• Applications
• Hamiltonian
• Pet
• algorithms
• crystal
• development
• eigenvalue
• paper
• physics
• solid state physics
• solution
• structure

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