Multiresolution analysis of electronic structure: semicardinal and wavelet bases

Abstract

This article reviews recent developments in multiresolution analysis which make it a powerful tool for the systematic treatment of the multiple length-scales inherent in the electronic structure of matter. Although the article focuses on electronic structure, the advances described are useful for non-linear problems in the physical sciences in general. The new language and notations introduced are well- suited for both formal manipulations and the development of computer software using higher-level languages such as C++. The discussion is self-contained, and all needed algorithms are specified explicitly in terms of simple operators and illustrated with straightforward diagrams which show the flow of data. Among the reviewed developments is the construction of_exact_ multiresolution representations from extremely limited samples of physical fields in real space. This new and profound result is the critical advance in finally allowing systematic, all electron calculations to compete in efficiency with state-of-the-art electronic structure calculations which depend for their celerity upon freezing the core electronic degrees of freedom. This review presents the theory of wavelets from a physical perspective, provides a unified and self-contained treatment of non-linear couplings and physical operators and introduces a modern framework for effective single-particle theories of quantum mechanics.

A "how-to from-scratch" book presently in press at Reviews of Modern Physics: 88 pages, 31 figures, 5 tables, 88 references. Significantly IMPROVED version, including (a) new diagrams illustrating algorithms; (b) careful proof-reading of equations and text; (c) expanded bibliography; (d) cosmetic changes including lists of figures and tables and a more reasonable font. Latest changes (Dec. 11, 1998): a more descriptive abstract, and minor lexicographical changes

References in corpus (5)

• Linear Scaling Solution of the Coulomb problem using wavelets
• Edge-driven transition in surface structure of nanoscale silicon
• Multiscale Computation with Interpolating Wavelets
• Paramagnetic structure for the soliton of the $30^{\circ }$ partial dislocation in silicon
• The Hopgrid algorithm: multilevel synthesis of multigrid and wavelet theory

Cited by in corpus (45)

• O(N) methods in electronic structure calculations
• Real-Space Mesh Techniques in Density Functional Theory
• Localized atomic basis set in the projector augmented wave method
• Simulating Physical Phenomena by Quantum Networks
• Daubechies wavelets as a basis set for density functional pseudopotential calculations
• Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework I: Total energy calculation
• Recent progress with large-scale ab initio calculations: the CONQUEST code
• A novel multigrid method for electronic structure calculations
• SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Extended systems
• SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters
• Fracture in Mode I using a Conserved Phase-Field Model
• Three real-space discretization techniques in electronic structure calculations
• Accelerating Atomic Orbital-based Electronic Structure Calculation via Pole Expansion and Selected Inversion
• Wavelet Scattering Regression of Quantum Chemical Energies
• A review on non-relativistic fully numerical electronic structure calculations on atoms and diatomic molecules
• From plane waves to local Gaussians for the simulation of correlated periodic systems
• Non-Equilibrium Electron Transport in Two-Dimensional Nano-Structures Modeled by Green's Functions and the Finite-Element Method
• Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties
• Symmetry-adapted real-space density functional theory for cylindrical geometries: application to large X (X=C, Si, Ge, Sn) nanotubes
• Multiscale Computation with Interpolating Wavelets
• An efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the Daubechies wavelet basis
• Comparison of variational real-space representations of the kinetic energy operator
• New ab initio approach for high pressure systems with application to a new high-pressure phase for boron: perturbative momentum-space potentials
• Multiresolution analysis for efficient, high precision all-electron density-functional calculations
• An efficient basis set representation for calculating electrons in molecules
• Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations
• Wavelet-Based Linear-Response Time-Dependent Density-Functional Theory
• First-principles density-functional calculations using localized spherical-wave basis sets
• GPU acceleration of local and semilocal density functional calculations in the SPARC electronic structure code
• Robust ab initio calculation of condensed matter: transparent convergence through semicardinal multiresolution analysis
• Wavelet treatment of the intra-chain correlation functions of homopolymers in dilute solutions
• Refinement trajectory and determination of eigenstates by a wavelet based adaptive method
• Chemical accuracy from small, system-adapted basis functions
• Wavelets for Density-Functional Theory and Post-Density-Functional-Theory Calculations
• SPARC: Simulation Package for Ab-initio Real-space Calculations
• MIKA: a multigrid-based program package for electronic structure calculations
• The wave packet propagation using wavelets
• Direct minimization on the complex Stiefel manifold in Kohn-Sham density functional theory for finite and extended systems
• Basis set convergence of Wilson basis functions for electronic structure
• Linear scaling solution of the all-electron Coulomb problem in solids
• An application of interpolating scaling functions to wave packet propagation
• Partition of unity finite element method for quantum mechanical materials calculations
• Solving One-Electron Systems in a Novel Gaussian-Sinc Mixed Basis Set
• Wavefunction optimization at the complete basis set limit with Multiwavelets and DMRG
• Universal Lattice Basis