Multiple Scattering in Solids

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.