Energy Density Functional Methods for Atomic Nuclei

PRELIMS.pdf
Preface
Reference
Editor biography
Nicolas Schunck
Contributors
Glossary
Acronyms
CH001.pdf
Chapter 1 Non-relativistic energy density functionals
1.1 Introduction
1.1.1 Definition of a configuration space
1.1.2 Microscopic approaches of nuclear systems
1.1.3 Vertical and horizontal philosophies
1.2 Energy density functional kernels
1.2.1 Skyrme functionals
1.2.2 Gogny functional
1.2.3 Phenomenological functionals
1.3 Pairing and Coulomb functionals
1.3.1 Pairing functionals
1.3.2 Treatment of the Coulomb potential
References
CH002.pdf
Chapter 2 Covariant energy density functionals
2.1 Relativistic description of quantum systems
2.1.1 Minkowski space–time
2.1.2 Lorentz and Poincaré groups
2.1.3 Relativistic wave equations
2.2 Symmetry properties of QCD
2.2.1 Symmetries of QCD at the classical level
2.2.2 Actual symmetries of QCD
2.3 Effective Lagrangians for nuclear systems
2.3.1 Effective Lagrangian à la Furnstahl, Serot and Tang
2.3.2 Effective Lagrangian à la Rho
2.4 Phenomenological Lagrangians
2.4.1 Non-linear Lagrangians
2.4.2 Density-dependent Lagrangians
2.4.3 Point-coupling Lagrangians
2.5 Derivation of the covariant energy density functional
2.5.1 Classical equations of motion
2.5.2 Classical Hamiltonian
2.5.3 Quantization of the relativistic classical Hamiltonian
2.5.4 Derivation of the relativistic energy density functional
2.6 Advantages of a relativistic description of nuclear systems
2.6.1 One-particle Dirac equation
2.6.2 Non-relativistic limit of the Dirac equation
2.6.3 Saturation of symmetric nuclear matter
2.6.4 Spin–orbit properties
2.6.5 Pseudospin symmetry
References
CH003.pdf
Chapter 3 Single-reference and multi-reference formulations
3.1 Single-reference implementation of nuclear energy density functionals
3.1.1 Spontaneous symmetry-breaking
3.1.2 Densities for a quasiparticle vacuum
3.1.3 HFB equation
3.1.4 Stability condition
3.1.5 Self-consistent symmetries
3.2 Multi-reference implementation of nuclear energy density functionals
3.2.1 General formalism
3.2.2 Calculation of overlaps and kernels
3.2.3 Projection of broken symmetries
3.2.4 MR-EDF calculations combining projection and GCM
References
CH004.pdf
Chapter 4 Time-dependent density functional theory
4.1 Time evolution equations
4.2 Role of pairing correlations in nuclear dynamics
4.3 Local DFT for superfluids
4.3.1 Pairing regularization of the anomalous density
4.3.2 Subtleties with regularization
4.4 Validation of the TDSLDA: the unitary Fermi gas
4.5 Symmetry-breaking
4.5.1 Translational motion
4.5.2 Rotation, pairing and parity symmetries
4.6 Time-dependent techniques
4.6.1 Accelerating/rotating frames
4.6.2 Chemical potentials
4.6.3 Neutron emission
4.6.4 Inclusion of pairing correlations within the BCS approximation
4.7 Selected examples
4.7.1 Relativistic Coulomb excitation of nuclei
4.7.2 Induced fission
4.7.3 Collisions of superfluid nuclei
References
CH005.pdf
Chapter 5 Small-amplitude collective motion
5.1 RPA with a Hamiltonian
5.1.1 Equations of motion method
5.1.2 Linear response
5.1.3 RPA response
5.1.4 Bethe–Salpeter equation for RPA response
5.2 RPA in density functional theory
5.2.1 Brief introduction to DFT for atoms and molecules
5.2.2 Time-dependent density functional theory
5.2.3 Adiabatic approximation
5.2.4 Significance of adiabatic approximation
5.3 Sum rules
5.3.1 Symmetries
5.3.2 Numerical implementations of RPA
5.3.3 Finite-amplitude method
5.4 Pairing correlations and QRPA formalism
5.4.1 QRPA through the equations-of-motion method
5.4.2 Implementation
5.4.3 HFB linear response and FAM generalization
5.5 Charge-changing QRPA
5.5.1 The equations-of-motion method
5.5.2 Finite-amplitude method
References
CH006.pdf
Chapter 6 Large-amplitude collective motion
6.1 Collective subspace
6.1.1 Classical Hamilton form of TDDFT
6.1.2 Basic concepts of a decoupled subspace
6.1.3 Decoupling conditions
6.2 Adiabatic time-dependent Hartree–Fock theory
6.2.1 Slow collective motion
6.2.2 Overview
6.2.3 Equation for the collective path
6.2.4 Problems with the equation of path
6.2.5 Conventional representation of ATDHF
6.3 Adiabatic self-consistent collective coordinate method
6.3.1 Covariant derivatives with Riemannian connection
6.3.2 Equations of collective subspace
6.3.3 Beyond point transformations
6.3.4 Anderson–Nambu–Goldstone modes: constants of motion
6.3.5 Arbitrariness of coordinate systems
6.3.6 Summary and practical implementation
6.3.7 Collective Hamiltonian and requantization
6.3.8 Derivation of the Bohr Hamiltonian
6.4 Gaussian overlap approximation of the GCM
6.4.1 The Gaussian overlap approximation
6.4.2 The collective Hamiltonian
6.4.3 Collective Hamiltonian with HFB generator states
6.4.4 Cranking approximation
6.4.5 Problems in the GCM+GOA approach
References
CH007.pdf
Chapter 7 Finite temperature
7.1 A reminder of statistical quantum mechanics
7.1.1 Basic concepts of thermodynamics
7.1.2 Partition functions
7.2 Finite-temperature Hartree–Fock theory
7.3 Finite-temperature Hartree–Fock–Bogoliubov theory
7.3.1 Derivation of the FT-HFB equation
7.3.2 Statistical operators in the HFB basis
7.3.3 Density matrix and pairing tensor
7.4 Finite-temperature RPA
7.4.1 The time-dependent density operator
7.4.2 The time-dependent density matrix
7.4.3 RPA equation at finite temperature
7.5 Beyond mean field
References
CH008.pdf
Chapter 8 Numerical implementations
8.1 Configuration space and basis expansions
8.1.1 The harmonic oscillator basis
8.1.2 Calculation of matrix elements
8.2 Lattice techniques
8.2.1 HFB equation in coordinate space
8.2.2 The simplest case of spherical symmetry and local EDFs
8.2.3 Broken symmetries and special lattices
8.2.4 Hybrid methods
8.3 The self-consistent loop
8.3.1 Non-linear eigenvalue problems
8.3.2 Gradient methods
8.4 Time-evolution algorithms
References
CH009.pdf
Chapter 9 Calibration of energy functionals
9.1 Parameters of energy functionals
9.1.1 From nucleon–nucleon potentials to energy functionals
9.1.2 Nuclear matter properties
9.1.3 Parameters of covariant functionals
9.2 Physical observables
9.3 Uncertainties of EDF parameters
9.3.1 Calibration of energy functionals
9.3.2 Bayesian inference techniques
9.4 Propagation of theoretical uncertainties
References