Inhalt

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   Abstract
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   Contents
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   Introduction
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   The General Problem for N-atomic Molecules
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   General Solution Strategies
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   Objective of the Thesis
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   Structure and Organization of the thesis
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   Acknowledgements
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   Numerical Linear Algebra
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  Preliminaries
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   Eigensystems of General Matrices
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   Eigensystems of Hermitian Matrices
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   Basic Properties and Definitions
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   Variational Characterisations
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   Perturbation Analysis and Error Bounds
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  Technical Tools
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  Orthogonal and Unitary Matrices
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   Householder Reflections
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   Givens and Jacobi Rotations
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   QR Factorisation and Orthonormalisation of Vector Sets
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  Orthogonal Bases of Krylov Spaces
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   Arnoldi's Procedure
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   Lanczos Procedure
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   Singular Value Decomposition (SVD)
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   Kronecker Products
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  Methods for Computing Partial Eigensystems of Hermitian Matrices
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  Iterative Single Vector Methods
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   Power Method
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   Inverse iteration (INVIT)
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   Rayleigh Quotient Iteration (RQI)
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   Direct Methods
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   Reduction to Tridiagonal Form
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   Standard Approach
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   Two-stage Approach
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  Methods for the Symmetric Tridiagonal Eigenproblem
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   QR Algorithm
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   Bisection Method and Inverse Iteration
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   Divide-and-conquer Method
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   RRR Algorithm
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   Jacobi's Method
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  Assessment and Summary
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   Tridiagonalization Approaches
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   Tridiagonal Eigensolvers
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   Summary
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   Iterative Projection Methods
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   Information Extraction
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   Standard Extraction
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   Harmonic Extraction
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   Refined Extraction
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  Subspace Expansion
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   Lanczos Method
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   Implicitly Restarted Lanczos Method (IRLM)
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   Davidson's Method
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  The Jacobi-Davidson Method and its Variants
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  Motivation of the Algorithm
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   JOCC Method
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   Davidson's Method Revisited
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   The Basic Jacobi-Davidson Method for Computing one Eigenpair
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   Consistency of the Correction Equation
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  Relation to Other Methods
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   JD and RQI
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   JD and Davidson
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   JD and Olsen
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   JD and Lanczos
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   Solving the Correction Equation
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   Iterative Krylov Methods for Linear Systems
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   GMRES
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   MINRES
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   QMR and QMRS
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   Preconditioners
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   Preconditioning the Correction Equation
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   Convergence of the Jacobi-Davidson Method
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  The JDQR Variants for Computing Several Eigenpairs
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   Deflation
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   Restarts
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   Standard JDQR
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   Convergence of the JDQR Method
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   Preconditioning the Deflated Correction Equation
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  Variants of JDQR Using a Fixed Preconditioner
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   Preconditioned Jacobi-Davidson Correction Equation
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   Preconditioned Standard JDQR
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   Preconditioned Refined JDQR
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   Preconditioned Harmonic JDQR
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   Storage Requirements and Computational Costs
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  Summary and Guidelines for the Practical Use
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   Choice of the Parameters
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   Choice of the Krylov Solver
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   Choice of the Extraction Method
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   Related Approaches and Software Availability
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   Quantum Chemistry
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  Eigenvalue Problems in Theoretical Spectroscopy
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   Motivation and Introduction
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  Prerequisites from Functional Analysis
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   Linear Operators on Hilbert Spaces
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   Tensor Products of Hilbert Spaces and Operators
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   Schrödinger Equation for One-Particle Systems
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   Molecular Hamiltonian
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   Born-Oppenheimer Approximation
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   Nuclear Motion and Coordinate Systems
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   Variational Approach and Matrix Eigenvalue Problem
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   Product vs. Contracted Basis and Direct vs. Iterative Eigensolver
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   Product Basis
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   Contracted Basis
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   Dichotomies and General Approaches
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   General Framework for the Computation of Energy Levels
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  The Double Renner Effect for Triatomic Molecules
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   Breakdown of the Born-Oppenheimer Approximation
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   The Double Renner Hamiltonian
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   Choice of the Basis Set
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  Construction and Structure of the Hamiltonian Matrix
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   Hierarchies and Partitioning into Blocks
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   Block and Problem Sizes
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   Hamiltonian Matrix Blocks
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   Basic diagonal K-blocks
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   Diagonal and Off-Diagonal Perturbation K-blocks
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   Construction of the Hamiltonian Matrix
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   Application to the Problem
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  Eigensolvers for the Computation of Rovibronic Energy Levels
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   Matrix Properties and Specification of the Eigenproblem
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  Matrix-Vector Multiplication and Storage Scheme
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   Sparsity of the Off-Diagonal Hamiltonian Matrix Blocks
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   Sparsity and Compact Storage of the SO-Blocks
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   Sparsity and Compact Storage of the DK-Blocks
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  Storage Scheme for the Hamiltonian Matrix Blocks
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   Addressing the DIAG-blocks
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   Addressing the DK-blocks
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   Addressing the SO-blocks
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   Matrix-Vector Multiplication Exploiting Compact Storage
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   Contraction Scheme and Contracted Basis
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  Direct Solvers
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   Product Basis Calculation
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   Contracted Basis Calculation
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   JDQR Product Basis Calculation
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   Preconditioners for Exterior Eigenvalues
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   Specification and Properties
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   Numerical Results
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   Preconditioners for Interior Eigenvalues
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   Comparison with Other Methods
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   JDQR Contracted Basis Calculation
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   Parallelization
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   Summary and Outlook
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   Appendices and Surveys
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  Appendix
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   Conventions for the Usage of Fonts
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   Romanization of Russian Names
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   Mathematical Notation
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   Technical Details and Implementation Issues
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   Input Files for DR
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   Bibliography
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   List of Tables
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   List of Figures
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   List of Algorithms
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   List of Listings
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   List of Symbols
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  Index
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   A
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   B
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   C
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   D
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   E
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   F
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   G
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   H
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   I
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   J
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   K
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   L
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   M
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   N
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   O
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   P
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   Q
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   R
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   S
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   T
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   U
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   V
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   W
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   Z