The nuclear energy density functional formalism

The present document focuses on the theoretical foundations of the nuclear energy density functional (EDF) method. As such, it does not aim at reviewing the status of the field, at covering all possible ramifications of the approach or at presenting recent achievements and applications. The objective is to provide a modern account of the nuclear EDF formalism that is at variance with traditional presentations that rely, at one point or another, on a {\it Hamiltonian-based} picture. The latter is not general enough to encompass what the nuclear EDF method represents as of today. Specifically, the traditional Hamiltonian-based picture does not allow one to grasp the difficulties associated with the fact that currently available parametrizations of the energy kernel $E[g',g]$ at play in the method do not derive from a genuine Hamilton operator, would the latter be effective. The method is formulated from the outset through the most general multi-reference, i.e. beyond mean-field, implementation such that the single-reference, i.e. "mean-field", derives as a particular case. As such, a key point of the presentation provided here is to demonstrate that the multi-reference EDF method can indeed be formulated in a {\it mathematically} meaningful fashion even if $E[g',g]$ does {\it not} derive from a genuine Hamilton operator. In particular, the restoration of symmetries can be entirely formulated without making {\it any} reference to a projected state, i.e. within a genuine EDF framework. However, and as is illustrated in the present document, a mathematically meaningful formulation does not guarantee that the formalism is sound from a {\it physical} standpoint. The price at which the latter can be enforced as well in the future is eventually alluded to.Comment: 64 pages, 8 figures, submitted to Euroschool Lecture Notes in Physics Vol.IV, Christoph Scheidenberger and Marek Pfutzner editor

Effect of Fock exchange on the electronic structure and magnetic coupling in NiO

The effect of Fock exchange on the periodic description of the geometrical structure, elastic constants, and electronic and magnetic properties of NiO is analyzed. Hybrid density functionals which combine a portion of "exact" Fock exchange with conventional local density approximation (LDA) or generalized gradient approximation (GGA) functionals remedy a number of serious inconsistencies with the traditional LDA or GGA descriptions of this prototypical "Mott" insulator. For example, the hybrid B3LYP functional (which mixes ∼20% Fock exchange with GGA functionals) introduces a significant insulating gap and yields antiferromagnetic Heisenberg coupling constants between Ni sites (J 2 ) in semiquantitative agreement with experiment. Closer inspection shows that while the B3LYP orbital band gap is in excellent agreement with experiment, the magnitude of the antiferromagnetic coupling is overestimated by slightly more than 50%. This has led us to examine a simplified model which combines Fock exchange with the LDA exchange and correlation functionals. This combination allows us to study the magnitude and nature of the band gap, the magnitude of the unpaired spin densities in the different magnetic phases, and the two most important magnetic coupling constants as a function of the fraction of Fock exchange included. It is concluded that ∼35% Fock exchange gives a reasonably balanced description of all properties, including structural parameters, magnetic form factors, the antiferromagnetic Ni-Ni exchange constant, and the character and magnitude of the band gap

Antiferromagnetic exchange interactions from hybrid density functional theory

A hybrid theory which combines the full nonlocal ¿exact¿ exchange interaction with the local spin-density approximation of density-functional theory is shown to lead to marked improvement in the description of antiferromagnetically coupled systems. Semiquantitative agreement with experiment is found for the magnitude of the coupling constant in La2CuO4, KNiF3, and K2NiF4. The magnitude of the unpaired spin population on the metal site is in excellent agreement with experiment for La2CuO4