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.

Inhalt

I Introduction.- Why Recur?.- The Recursive Solution of Schroedinger's Equation.- II Asymptotic Behaviour.- Asymptotic Behaviour of Continued Fraction Coefficients Related to Singularities of the Weight Function.- Band Gaps and Asymptotic Behaviour of Continued Fraction Coefficients.- Computing Greenians: Quadrature and Termination.- Application of Linear Prediction for Extrapolating Recursion Coefficients.- III Related Methods.- On a Generalized-Moments Method.- The Equation of Motion Method.- Use of Cyclic Matrices to Obtain Analytic Expressions for Crystals.- IV Solid State Applications.- Continued Fractions and Perturbation Theory: Application to Tight Binding Systems.- Response Functions and Interatomic Forces.- The Recursion Method with a Non-Orthogonal Basis.- V Lanczos Method Applications.- Hamiltonian Eigenvalues for Lattice Gauge Theories.- The Lanczos Method in Lattice Gauge Theories.- A Dedicated Lanczos Computer for Nuclear Structure Calculations.- VI Conference Summary.- Conference Summary.- Index of Contributors.