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The present hands-on tutorial aims to provide the students with the basic aspects of DFT and TDDFT and with basic as well as more advanced aspects of the GW and BSE methods. Theoretical lectures on the foundations of the methods will be completed with technical lectures on numerical and computational aspects. A significant part of the school will be then dedicated to hands-on tutorials, where the students will be given the opportunity to carry out excited state calculations on several paradigmatic systems using the Yambo code, under the guide of the code developers who will be present as teachers at the school.

For the occasion of the school, moew extensions of the Yambo code will be released under the GPL licence and will form part of the school program, giving to the attendees the chance of learning very advanced methods, and providing them with critical tools to purse their research activity. Specifically, spin-polarized phenomena, surface spectroscopies and electron-phonon coupling.

The school will offer the students the chance to learn both state-of-the-art approaches (GW, TD-DFT and BSE) as well as cutting edge topics and applications (electron-phonon coupling, magneto-optical effects and surface spectroscopies). As objectives of the school, we expect that the students, after attending the school, will be able to:

- identify the method(s), among those taught, most appropriate for describing the system, the experiment they want to study and the physical effects they want to include
- set up the necessary calculations, starting from ground-state DFT;
- recognize the relevant numerical parameters of the calculations and perform convergence tests to establish their values;
- interpret the results output by the calculations, relating them to the experimental counterpart, and taking into account the limitations of the particular approach they are using.

The GW method addresses charged excitations in an electronic system and thus is appropriate for calculating photoelectron spectra, quasi-particle energies and life-times. Due to its ability to accurately describe band-gaps (errors ranging between 0.2-0.5 eV), it can be employed in technologically relevant area such as photovoltaics and microelectronics. The theoretical lectures will cover the relevant physics (photo-electron effect/spectroscopy) and key concepts (Green's function, quasi-particle, self-energy, Many-Body Perturbation-Theory) leading to Hedin's equations and the GW approximations. More advanced topics will also be discussed during the technical lecture and hands-on tutorial: the plasmon-pole approximation and the real-axis integration approach to the frequency dependence of the screening function, the random integration method to treat the singularity of the Coulomb potential, and the calculation of the quasi-particle life-times. TD-DFT and BSE are applied to calculations of neutral electronic excitations, and thus give access to vertical excitations and dynamical polarizabilities in molecules, as well as optical absorption and electron energy-loss spectra in extended systems. The difference between the two approaches is mainly in how the electron correlation is accounted for. In TD-DFT correlation is usually introduced through a mean-field approximation based on the homogeneous electron gas, while in BSE the approximation for correlation is derived from Many Body Perturbation Theory. During the first two days of the school, TD-DFT will be introduced within the linear response theory framework, and the BSE will be introduced starting from the Hedin equations. The session will be completed by a technical lecture on the computational and numerical aspects, such as the different implementations based either on the inversion of the response function or on the expansion in an electron-hole pair basis, efficient diagonalization algorithms and the Tamm-Dancoff approximation, scaling and feasibility of calculations. The dedicated hands-on tutorial will focus on how the approximation for electronic correlation influences the results of the optical spectra of electronic systems depending on their dimensionality.

The spin-polarized extension makes it possible to treat materials that may possess a magnetic ground state due to the presence of electrons in the localized d and f orbitals. The interest in those materials is growing rapidly, not only because of their possible applications in the field of spin electronics, but also following the observation of new physical effects. In particular, the Magneto-Optical Kerr Effect (MOKE), is at the basis of a powerful spectroscopic technique for probing the magnetic properties of materials. The MOKE is implemented in Yambo and will be the object of one of the advanced lectures.

Within surface science, optical and electronic spectroscopies are now commonly used alongside more traditional surface-sensitive techniques such as scanning tunneling microscope and low energy electron diffraction. In particular, reflectance anisotropy spectroscopy (RAS) and high resolution electron energy loss spectroscopy (HREELS) are used to characterize surface structure as well as complex processes like molecular adsorption, epitaxial growth and self-assembly. Nonetheless, these techniques are generally used in a phenomenological way, without a solid interpretation from first principles calculations. In this tutorial, the theoretical background and practical techniques for simulating these experiments will be described based on dedicated routines implemented into Yambo.

The electron-phonon coupling is another crucial ingredient in first-principles electronic structure that is however missing in the vast majority of calculations. Yambo has implemented the electron-phonon coupling within the Heine, Allen and Cardona approach. As there are only very few ab-initio calculations available in literature this part of the Yambo hands-on will provide the students with cutting-edge skills that will promote their critical understanding of the performance of any purely electronic calculation.