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# Density functional theory and self-consistent methods

Density functional theory (DFT) is the method of choice for large-scale electronic structure calculations in condensed matter physics, quantum chemistry and even materials science or molecular biology. This popularity results from the fact that DFT calculations are comparatively simple to implement, yet are often very accurate with a computational cost that makes them the ideal choice for systems with large numbers of electrons. For the same reasons, energy density functional (EDF) approaches play a central role in nuclear theory since they offer the only computationally feasible many-body framework capable of describing nuclei across the mass table. EDF approaches to nuclear structure are analogous to electronic DFT in that they map the computationally prohibitive many-body problem onto an effective one-body problem. Besides the obvious computational simplifications, this so-called Kohn-Sham framework provides a mean-field-like description in terms of intrinsic shape and single-particle degrees of freedom that lends itself to simple and physically intuitive interpretations. The price to pay for these simplifications is that the unknown EDF must be approximated with phenomenological functionals (e.g., Skyrme, Gogny, etc.), typically expressed as local powers and gradients of ground state nucleon densities and currents with empirical couplings adjusted to data. As a consequence of this phenomenological nature, it is difficult to systematically improve the performance of nuclear EDFs or provide reliable error estimates of theoretical predictions away from stability. In this course, students will learn, through various computational projects and hands-on exercises, the fundamental concepts and methods of modern density functional theory. Most of the course will be focused on nuclear structure applications, with short excursions in the theory of nuclear forces and electronic structure theory.

The goals of the course are threefold: (i) gain an in-depth understanding of the basic concepts, mathematical methods, and computational techniques used to solve the quantum many-body problem within the framework of density functional theory, (ii) learn about the current developments, challenges, and research opportunities in the specific context of nuclear structure, (iii) conduct and manage a small computational nuclear physics project. The skills acquired in this course will enable participants to quickly and efficiently write their own codes using state-of-the-art methods of scientific computing. The experience gained will also facilitate the student's insertion in research groups that already maintain a DFT software base. The course will thus give the participants the necessary proficiency to tackle a broad spectrum of research problems in atomic and nuclear physics. Special emphasis will be given on communicational aspects of modern scientific research via small oral presentations and the writing of a scientific report, which will be graded.