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Jacobi Davidson type methods for computing rovibronic energy levels of triatomic molecules / von Peter Langer. 2008
Content
Abstract
Contents
Introduction
The General Problem for N-atomic Molecules
General Solution Strategies
Objective of the Thesis
Structure and Organization of the thesis
Acknowledgements
Numerical Linear Algebra
Preliminaries
Eigensystems of General Matrices
Eigensystems of Hermitian Matrices
Basic Properties and Definitions
Variational Characterisations
Perturbation Analysis and Error Bounds
Technical Tools
Orthogonal and Unitary Matrices
Householder Reflections
Givens and Jacobi Rotations
QR Factorisation and Orthonormalisation of Vector Sets
Orthogonal Bases of Krylov Spaces
Arnoldi's Procedure
Lanczos Procedure
Singular Value Decomposition (SVD)
Kronecker Products
Methods for Computing Partial Eigensystems of Hermitian Matrices
Iterative Single Vector Methods
Power Method
Inverse iteration (INVIT)
Rayleigh Quotient Iteration (RQI)
Direct Methods
Reduction to Tridiagonal Form
Standard Approach
Two-stage Approach
Methods for the Symmetric Tridiagonal Eigenproblem
QR Algorithm
Bisection Method and Inverse Iteration
Divide-and-conquer Method
RRR Algorithm
Jacobi's Method
Assessment and Summary
Tridiagonalization Approaches
Tridiagonal Eigensolvers
Summary
Iterative Projection Methods
Information Extraction
Standard Extraction
Harmonic Extraction
Refined Extraction
Subspace Expansion
Lanczos Method
Implicitly Restarted Lanczos Method (IRLM)
Davidson's Method
The Jacobi-Davidson Method and its Variants
Motivation of the Algorithm
JOCC Method
Davidson's Method Revisited
The Basic Jacobi-Davidson Method for Computing one Eigenpair
Consistency of the Correction Equation
Relation to Other Methods
JD and RQI
JD and Davidson
JD and Olsen
JD and Lanczos
Solving the Correction Equation
Iterative Krylov Methods for Linear Systems
GMRES
MINRES
QMR and QMRS
Preconditioners
Preconditioning the Correction Equation
Convergence of the Jacobi-Davidson Method
The JDQR Variants for Computing Several Eigenpairs
Deflation
Restarts
Standard JDQR
Convergence of the JDQR Method
Preconditioning the Deflated Correction Equation
Variants of JDQR Using a Fixed Preconditioner
Preconditioned Jacobi-Davidson Correction Equation
Preconditioned Standard JDQR
Preconditioned Refined JDQR
Preconditioned Harmonic JDQR
Storage Requirements and Computational Costs
Summary and Guidelines for the Practical Use
Choice of the Parameters
Choice of the Krylov Solver
Choice of the Extraction Method
Related Approaches and Software Availability
Quantum Chemistry
Eigenvalue Problems in Theoretical Spectroscopy
Motivation and Introduction
Prerequisites from Functional Analysis
Linear Operators on Hilbert Spaces
Tensor Products of Hilbert Spaces and Operators
Schrödinger Equation for One-Particle Systems
Molecular Hamiltonian
Born-Oppenheimer Approximation
Nuclear Motion and Coordinate Systems
Variational Approach and Matrix Eigenvalue Problem
Product vs. Contracted Basis and Direct vs. Iterative Eigensolver
Product Basis
Contracted Basis
Dichotomies and General Approaches
General Framework for the Computation of Energy Levels
The Double Renner Effect for Triatomic Molecules
Breakdown of the Born-Oppenheimer Approximation
The Double Renner Hamiltonian
Choice of the Basis Set
Construction and Structure of the Hamiltonian Matrix
Hierarchies and Partitioning into Blocks
Block and Problem Sizes
Hamiltonian Matrix Blocks
Basic diagonal K-blocks
Diagonal and Off-Diagonal Perturbation K-blocks
Construction of the Hamiltonian Matrix
Application to the Problem
Eigensolvers for the Computation of Rovibronic Energy Levels
Matrix Properties and Specification of the Eigenproblem
Matrix-Vector Multiplication and Storage Scheme
Sparsity of the Off-Diagonal Hamiltonian Matrix Blocks
Sparsity and Compact Storage of the SO-Blocks
Sparsity and Compact Storage of the DK-Blocks
Storage Scheme for the Hamiltonian Matrix Blocks
Addressing the DIAG-blocks
Addressing the DK-blocks
Addressing the SO-blocks
Matrix-Vector Multiplication Exploiting Compact Storage
Contraction Scheme and Contracted Basis
Direct Solvers
Product Basis Calculation
Contracted Basis Calculation
JDQR Product Basis Calculation
Preconditioners for Exterior Eigenvalues
Specification and Properties
Numerical Results
Preconditioners for Interior Eigenvalues
Comparison with Other Methods
JDQR Contracted Basis Calculation
Parallelization
Summary and Outlook
Appendices and Surveys
Appendix
Conventions for the Usage of Fonts
Romanization of Russian Names
Mathematical Notation
Technical Details and Implementation Issues
Input Files for DR
Bibliography
List of Tables
List of Figures
List of Algorithms
List of Listings
List of Symbols
Index
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