Multiple Scattering in Solids

This book describes general techniques for solving linear partial differential equations by dividing space into regions to which the equations are independently applied and then assembling a global solution from the partial ones. It is intended for researchers and graduate students involved in calculations of the electronic structure of materials, but will also be of interest to workers in quantum chemistry, electron microscopy, acoustics, optics, and other fields.

Contenu
1 Introduction.- 1.1 Basic Characteristics of MST.- 1.2 Electronic Structure Calculations.- 1.3 The Aim of This Book.- References.- 2 Intuitive Approach to MST.- 2.1 Huygens' Principle and MST.- 2.2 Time-Independent Green Functions.- References.- 3 Single-Potential Scattering.- 3.1 Partial-Wave Analysis of Single Potential Scattering.- 3.2 General Considerations.- 3.3 Spherically Symmetric Potentials.- 3.4 Nonspherical Potentials.- 3.5 Wave Function in the Moon Region.- 3.6 Effect of the Potential in the Moon Region.- 3.7 Convergence of Basis Function Expansions ().- References.- 4 Formal Development of MST.- 4.1 Scattering Theory for a Single Potential.- 4.2 Two-Potential Scattering.- 4.3 The Equations of Multiple Scattering Theory.- 4.4 Representations.- 4.5 Muffin-Tin Potentials.- References.- 5 MST for Muffin-Tin Potentials.- 5.1 Multiple Scattering Series.- 5.2 The Green Function in MST.- 5.3 Impurities in MST.- 5.4 Coherent Potential Approximation.- 5.5 Screened MST.- 5.6 Alternative Derivation of MST.- 5.7 Korringa's Derivation.- 5.8 Relation to Muffin-Tin Orbital Theory.- 5.9 MST for E < 0.- 5.10 The Convergence Properties of MST ().- References.- 6 MST for Space-Filling Cells.- 6.1 Historical Development of Full-Cell MST.- 6.2 Derivations of MST for Space-Filling Cells.- 6.3 Full-Cell MST.- 6.4 The Green Function and Bloch Function.- 6.5 Variational Formalisms.- 6.6 Second Variational Derivation ().- 6.7 Construction of the Wave Function.- 6.8 The Closure of Internal Sums ().- 6.9 Numerical Results.- 6.10 Square Versus Rectangular Matrices ().- References.- 7 Augmented MST().- 7.1 General Comments.- 7.2 MST with a Truncated Basis Set: MT Potentials.- 7.3 General Potentials.- 7.4 Green Functions and the Lloyd Formula.- 7.5 Numerical Study of TwoMuffin-Tin Potentials.- 7.6 Convergence of Electronic Structure Calculations.- References.- 8 Relativistic Formalism.- 8.1 General Comments.- 8.2 Generalized Partial Waves.- 8.3 Generalized Structure Constants.- 8.4 Free-Particle Solutions.- 8.5 Relativistic Single-Site Scattering Theory.- 8.6 Relativistic Multiple Scattering Theory.- References.- 9 The Poisson Equation.- 9.1 General Comments.- 9.2 Multipole Moments.- 9.3 Comparison with the Schrödinger Equation.- 9.4 Convex Polyhedral Cells.- 9.5 Numerical Results for Convex Cells.- 9.6 Concave Cells.- 9.7 Direct Analogy with MST.- References.- A Time-Dependent Green Functions.- B Time-Independent Green Functions.- C Spherical Functions.- C.1 The Spherical Harmonics.- C.2 The Bessel, Neumann, and Hankel Functions.- C.3 Solutions of the Helmholtz Equation.- References.- D Displacements of Spherical Functions References D.- References.- E The Two-Dimensional Square Cell.- E.1 Numerical Results (*).- References.- F Formal Scattering Theory.- F.1 General Comments.- F.2 Initial Conditions and the Møller Operators.- F.3 The Møller Wave Operators.- F.4 The LippmannSchwinger Equation.- References.- G Irregular Solutions to the Schrödinger Equation.- H Displacement of Irregular Solutions.- K Conversion of Volume Integrals.- L Energy Derivatives.- M Convergence of the Secular Matrix.- N Summary of MST.- N.1 General Framework.- N.2 Single Potential.- N.3 Multiple Scattering Theory.