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Density functional theory [electronic resource] : modeling, mathematical analysis, computational methods and applications / Eric Cancs̈, Gero Friesecke, editors.

Cancs̈, Eric.; Friesecke, Gero.
2023
QD462.6.D45
Available Online
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Title
Density functional theory [electronic resource] : modeling, mathematical analysis, computational methods and applications / Eric Cancs̈, Gero Friesecke, editors.
ISBN
9783031223402 (electronic bk.)
3031223403 (electronic bk.)
303122339X
9783031223396
DOI
https://doi.org/10.1007/978-3-031-22340-2
Published
Cham : Springfer, 2023.
Language
English
Description
1 online resource.
Item Number
10.1007/978-3-031-22340-2 doi
Call Number
QD462.6.D45
Dewey Decimal Classification
541/.28
Summary
Density functional theory (DFT) provides the most widely used models for simulating molecules and materials based on the fundamental laws of quantum mechanics. It plays a central role in a huge spectrum of applications in chemistry, physics, and materials science. Quantum mechanics describes a system of $N$ interacting particles in the physical 3-dimensional space by a partial differential equation in $3N$ spatial variables. The standard numerical methods thus incur an exponential increase of computational effort with $N$, a phenomenon known as the curse of dimensionality; in practice these methods already fail beyond $N=2$. DFT overcomes this problem by 1) reformulating the $N$-body problem involving functions of $3N$ variables in terms of the density, a function of 3 variables, 2) approximating it by a pioneering hybrid approach which keeps important ab initio contributions and re-models the remainder in a data-driven way. This book intends to be an accessible, yet state-of-art text on DFT for graduate students and researchers in applied and computational mathematics, physics, chemistry, and materials science. It introduces and reviews the main models of DFT, covering their derivation and mathematical properties, numerical treatment, and applications. .
Access Note
Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed July 27, 2023).
Added Author
Cancs̈, Eric.
Friesecke, Gero.
Series
Mathematics and Molecular Modeling.
Available in Other Form
Print version: 9783031223396
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Intro
Preface
Reference
Prologue: Early Days of Modern DFT (1964-1979)
Contents
Contributors
List of Symbols
Parameters of the Electronic Problem
Electron Coordinates
Wavefunctions, Density Matrices, Densities, Orbitals
Functionals
Energy Levels, Eigenvalues, Energy Densities, Fermi-Dirac Function
Potentials and Kernels
Operators
Matrices
Discretization Parameters, Basis Functions, Discretized Models
1 Review of Approximations for the Exchange-Correlation Energy in Density-Functional Theory
1.1 Basics of Density-Functional Theory

1.1.1 The Many-Body Problem
1.1.2 The Universal Density Functional
1.1.3 The Kohn-Sham Scheme
1.1.3.1 Decomposition of the Universal Functional
1.1.3.2 The Kohn-Sham Equations
1.1.3.3 Extension to Spin Density-Functional Theory
1.1.4 The Generalized Kohn-Sham Scheme
1.2 Exact Expressions and Constraints for the Kohn-Sham Exchange and Correlation Functionals
1.2.1 The Exchange and Correlation Holes
1.2.2 The Adiabatic Connection
1.2.3 One-Orbital and One-Electron Spatial Regions
1.2.4 Coordinate Scaling
1.2.4.1 Uniform Coordinate Scaling

1.2.4.2 Non-uniform Coordinate Scaling
1.2.5 Atoms in the Limit of Large Nuclear Charge
1.2.6 Lieb-Oxford Lower Bound
1.3 Semilocal Approximations for the Exchange-CorrelationEnergy
1.3.1 The Local-Density Approximation
1.3.2 The Gradient-Expansion Approximation
1.3.3 Generalized-Gradient Approximations
1.3.4 Meta-Generalized-Gradient Approximations
1.4 Single-Determinant Hybrid Approximations
1.4.1 Hybrid Approximations
1.4.2 Range-Separated Hybrid Approximations
1.5 Multideterminant Hybrid Approximations
1.5.1 Double-Hybrid Approximations

1.5.2 Range-Separated Double-Hybrid Approximations
1.5.2.1 Range-Separated One-Parameter Double-Hybrid Approximations
1.5.2.2 Range-Separated Two-Parameter Double-Hybrid Approximations
1.6 Semiempirical Dispersion Corrections and Nonlocal van der Waals Density Functionals
1.6.1 Semiempirical Dispersion Corrections
1.6.2 Nonlocal van der Waals Density Functionals
1.7 Orbital-Dependent Exchange-Correlation Density Functionals
1.7.1 Exact Exchange
1.7.2 Second-Order Gr̲ling-Levy Perturbation Theory
1.7.3 Adiabatic-Connection Fluctuation-Dissipation Approach

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