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."

This book explains the observed trends in the bonding and structure of molecules and solids within the models of the electronic structure. Emphasis is placed throughout on recent theoretical developments that link structural stability to the local topology or connectivity of the lattice through the moments of the electronic density of states. The chemically-intuitive Tight Binding approximation provides a unified treatment of the covalent bond in small molecules and extended solids, while the physically-intuitive Nearly-Free Electron approximation provides a natural description of the metallic bonds in sp-valent metals. Unlike the conventional reciprocal-space formulation of band theory, this modern real-space approach allows an immediate understanding of the origin of structural trends within the periodic table for the elements and the AB structure map for binary compounds. Although this unique book is aimed primarily at postgraduates in physics, chemistry, and materials science, a chapter on basic quantum mechanical concepts is included for those readers with little or no basic knowledge of the subject.