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.

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.- 2.1.1 Informal Discussion: Point Scatterers.- 2.1.2 Formal Presentation.- 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.- 3.3.1 Free-Particle Solutions.- 3.3.2 The Radial Equation for Central Potentials..- 3.3.3 The Scattering Amplitude.- 3.3.4 Normalization of the Scattering Wave Function.- 3.3.5 Integral Expressions for the Phase Shifts.- 3.4 Nonspherical Potentials.- 3.4.1 Alternative Forms of the Solution.- 3.4.2 Direct Determination of thet-Matrix(*).- 3.5 Wave Function in the Moon Region.- 3.5.1 Displaced-Center Approach: Convex Cells.- 3.5.2 Displaced-Cell Approach: Convex Cells.- 3.5.3 Numerical Example: Convergence for Square Cell.- 3.5.4 Displaced-Cell Approach: Concave Cells (*).- 3.6 Effect of the Potential in the Moon Region.- 3.7 Convergence of Basis Function Expansions (*).- 3.7.1 First Justification.- 3.7.2 Second Justification.- References.- 4 Formal Development of MST.- 4.1 Scattering Theory for a Single Potential.- 4.1.1 The S-Matrix and the t-Matrix.- 4.1.2 t-Matrices and Green Functions.- 4.2 Two-Potential Scattering.- 4.2.1 An Integral Equation for thet-Matrix.- 4.3 The Equations of Multiple Scattering Theory.- 4.3.1 The Wave Functions of Multiple Scattering Theory.- 4.4 Representations.- 4.4.1 The Coordinate Representation.- 4.4.2 The Angular-Momentum Representation.- 4.4.3 Representability of the Green Function and the Wave Function.- 4.4.4 Example of Representability.- 4.4.5 The Representability Theorem.- 4.5 Muffin-Tin Potentials.- References.- 5 MST for Muffin-Tin Potentials.- 5.1 Multiple Scattering Series.- 5.1.1 The Angular-Momentum Representation.- 5.1.2 Electronic Structure of a Periodic Solid.- 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.- 5.9.1 The Two-Scatterer Problem in Three Dimensions.- 5.9.2 Arbitrary Number of MT Potentials.- 5.9.3 Convergence and Accuracy of MST (*).- 5.10 The Convergence Properties of MST (*).- 5.10.1 Energy Convergence.- 5.10.2 Convergence of the Wave Function.- 5.10.3 Convergence of Single-Center Expansion of the Wave Function.- 5.10.4 Summary.- 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.- 6.3.1 Outgoing-Wave Boundary Conditions.- 6.3.2 Empty-Lattice Test.- 6.3.3 Note on Convergence.- 6.3.4 Full-Potential Wave Functions.- 6.4 The Green Function and Bloch Function.- 6.4.1 The Green Function.- 6.4.2 Alternative Expressions for the Green Function.- 6.4.3 Bloch Functions for Periodic, Space-Filling Cells.- 6.5 Variational Formalisms.- 6.5.1 Variational Derivation of MST.- 6.5.2 A Variational Principle for MST.- 6.5.3 First Variational Derivation of MST.- 6.6 Second Variational Derivation (*).- 6.6.1 The Secular Equation for Nonspherical MT Potentials.- 6.6.2 Space-Filling Cells of Convex Shape.- 6.6.3 Displaced-Cell Approach: Convex Cells (*).- 6.6.4 Displaced-Cell Approach: Concave Cells(*).- 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.- 7.4.1 Green Functions.- 7.4.2 The Lloyd Formula.- 7.4.3 Single Scatterer.- 7.4.4 A Collection of Scatterers.- 7.4.5 First Derivation.- 7.4.6 Second Derivation.- 7.4.7 The Effects of Truncation.- 7.5 Numerical Study of Two Muffin-Tin Potentials.- 7.6 Convergence of Electronic Structure Calculations.- References.- 8 Relativistic Formalism.- 8.1 General Comments.- 8.2 Generalized Partial Waves.- 8.2.1 The Free-Particle Propagator in the Presence of Spin.- 8.2.2 The (?, ?) or ? Representation.- 8.3 Generalized Structure Constants.- 8.4 Free-Particle Solutions.- 8.4.1 Free-Particle Solution of the Dirac Equation.- 8.4.2 The Free-Particle Propagator.- 8.5 Relativistic Single-Site Scattering Theory.- 8.5.1 Spherically Symmetric Potentials.- 8.5.2 Spin-Orbit Coupling.- 8.5.3 Scalar Relativistic Expressions.- 8.5.4 Generally Shaped, Scalar Potentials.- 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.- 9.4.1 Mathematical Preliminaries.- 9.4.2 Non-MT, Space-Filling Cells of Convex Shape.- 9.5 Numerical Results for Convex Cells.- 9.6 Concave Cells.- 9.6.1 Analytic Continuation.- 9.7 Direct Analogy with MST.- 9.7.1 Single Cell Charges.- 9.7.2 Multiple Scattering Solutions.- 9.7.3 Space-Filling Charges of Arbitrary Shape.- 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 Lippmann—Schwinger 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.