Objectives:
The aim of the school was to give a comprehensive introduction to the theoretical and practical aspects of the electronic excitations  that are probed by experimental techniques such as optical  absorption, EELS and photoemission (direct or inverse). From a theoretical perspective, excitations and excited state properties are out of the reach of density-functional theory (DFT), which is a ground-state theory. Over the past three decades, alternative ab-initio theories and frameworks capable of describing electronic excitations and spectroscopy, have gained popularity including time-dependent density-functional theory (TDDFT) and many-body perturbation theory (MBPT) or Green’s function theory (GW approximation and Bethe-Salpeter equation BSE). In fact, computational solutions and codes have been developed to implement these theories and to provide tools to calculate excited state properties. The present school focused on these points, covering theoretical, practical, and also numerical aspects of TDDFT and MBPT, non-linear response and real-time spectroscopies. Additionally, the school provided valuable insights from an experimental perspective on spectroscopies and magnetic excitations, rarely covered in similar events. Finally, a large part of the school was devoted to getting familiar with the codes that implement such theories (ABINIT, 2Light, Lumen, DP, and EXC).

Organisers:
Francesco Sottile (Ecole Polytechnique, Palaiseau, France)
Valerio Olevano (CNRS Institut Neel, Grenoble, France)
Gian-Marco Rignanese (Université catholique de Louvain, Belgium)

Tutors:
Simo Huotari (University of Helsinki, Finland)
Valerie Veniard (Ecole Polytechnique and CNRS, Palaiseau, France)
Claudio Attaccalite (Institut Neel, CNRS, Grenoble, France)
Matteo Giantomassi (Université catholique de Louvain, Belgium)
Laura Urquiza (Ecole Polytechnique, Palaiseau, France)
Christoph Friedrich (Julich Research Centre, Germany)

Format:
Following the example of the previous edition, we have proposed a hybrid format, both online and in presence.
i) presentation of the theory and theoretical aspects of the implementation took place in the morning sessions; these sessions took place in the Aneesur Rahman room at CECAM HQ, but were also streamed over Zoom, for the online participants;
ii) the afternoon sessions were devoted to practical hands-on of the theory studied in the morning and were reserved to the onsite participants: DFT with ABINIT (Day 1), TDDFT with DP (Day 2), Non-linear response within TDDFT with 2Light and Lumen (Day 3), GW approximation to MBPT with Abinit (Day 4) and Bethe-Salpeter with EXC (Day 5). For the first time (in the Theoretical Spectroscopy Series of Lectures) we held the hands-on using the participants’ own laptop (computer room no longer available at Cecam).

Detailed description of the lectures:

• Introduction to spectroscopy
  This thorough introductory lecture was given by an expert in the field: Simo Huotari, an experimentalist from Finland with an outstanding record of activity in electronic excitations and spectroscopy spectra especially at synchrotron facilities. Several experimental techniques used to investigate the spectroscopic properties of matter are presented, ranging from scattering (EELS, IXS) to absorption (optical, XANES, EXAFS), from photoemission (and inverse-photoemission) to Auger spectroscopy. For all methods, a link has been given to the quantities that can be computed using the theoretical methods in the subsequent lectures, in particular the spectral function and the inverse dielectric function ε¹ or screening function.

• Density-Functional Theory (DFT)
  This lecture (presented by G.-M. Rignanese) covered the basics of DFT: formalism and implementation. A special care was taken to present the shortcomings of DFT regarding its use for the computation of band structures, and, on the other hand, its usefulness as a starting point for more elaborate theories. The topics covered were: the electronic N-body problem, functionals of the density, the Kohn-Sham approach, approximations and new functionals, and the band-gap problem. In addition, more technical concepts (related to the plane-wave approach to be followed in the hands-on) were explained: plane-wave basis set, Brillouin zone integration, pseudopotentials, relaxation via computing the forces, and iterative algorithms. And of course, the ABINIT code was presented. This year’s important addition was related to the high throughput and its importance to machine learning approaches. The lecture was followed by hands-on on Abinit, in the afternoon.
• Micro-Macro connection
  One-hour lecture (by F. Sottile) on the connection between measurable quantities (macroscopic) and calculated quantities (microscopic). This is a crucial concept that is often disregarded. We find that this micro-macro connection needs to be better explained in textbooks or articles, rather than given for granted, in particular for what concerns optical absorption.
• Time-Dependent Density Functional Theory (TDDFT)
  A review (by V. Olevano) of TDDFT and its fundamental assumptions, theorems, caveats and drawbacks has been presented. In particular, we have illustrated the linear-response TDDFT in an actual implementation which is in the frequency domain, reciprocal space and on a plane-wave basis, as implemented in the DP (Dielectric Properties, http://www.dp-code.org) code. This scheme is well suited to EELS and optical spectroscopy calculations, and particularly convenient for infinite periodic bulk solids, but also semi-infinite systems like surfaces, wires and tubes by the use of supercells. A critical analysis of all the classical approximations (RPA, Adiabatic LDA, with and without local-field effects) as well as the most recent ones (long-range contribution only, Nanonoquanta kernel, Bootstrap, etc.) has been presented, together with illustrating examples of spectra on prototype condensed matter systems, like bulk silicon, graphite, nanotubes, etc. This lecture was followed by a practical session on the use of DP code.

• Non-linear response: perturbative approach
  In this lecture (by V. Veniard) the perturbative approach for non-linear spectroscopy is presented, with a description of second- and third-order responses. In particular, the intrinsic difficulties of the inclusion of local fields and the exchange and correlation (xc) contribution beyond RPA (with terms like the third -or fourth-derivatives of the xc energy with respect to the density) were emphasized. Exercises with 2light in the afternoon to calculate the second harmonic generation of Silicon Carbide.Event website (with program, and list of participants): https://www.cecam.org/workshop-details/theoretical-spectroscopy-lectures-1195General Remarks:
  • This year’s event was particularly international: in addition to the 9 European nationalities, we welcomed 2 students from India, 1 from Morocco, 1 from Ethiopia, and 1 from the USA.
  • We had secured addition funding for this event from the GDR-REST and Psi-k. Futhermore, we obtained extra funding from the CECAM initiative for underrepresented and underprivileged groups within the CECAM Community (special thanks to Sara Bonella and Andrea Cavalli for this).
  • All participants attended the social dinner on Tuesday, and almost all attended the (exceptional) second social dinner that we managed to organize on Thursday.
  • For the first time, we decided to award a poster prize. Due to the exceptional quality of the posters, we chose to award two poster prizes ex-aequo on Friday. The two winners received a certificate and a 200 CHF award.
  • We extend our warm thanks to the organizing staff in Lausanne. The smooth running of this event was thanks to Aude Merola Failletaz, Bogdan Nichita, Nathalie Carminati, and Carole Albonico.
  • We hope that the feedback from the Survey compiled by the students is positive and we look forward to organizing another such event.

• Non-linear response: time evolution approach.
  Together with the perturbative approach, also the real time evolution approach was presented (by C. Attaccalite), in which all orders are automatically included. This permits both to tackle non-perturbative regimes (like high-harmonic generation) and to introduce important concepts (dephasing, Berry phase, etc.). Again, several points of view have been given: the theoretical explanation and derivation, as well as the implementation technical details. Exercises with the Lumen code in the afternoon.

• Many-Body Perturbation Theory
  In this (and next) lecture, presented by F. Sottile, we introduced the theoretical basics of many-body Green’s functions theory. Starting from the very basic concept of the Green’s function as a general strategy to solve a linear differential problem and later motivated by spectroscopic reasons (what is an electron? How do we measure the energy of an electron?), the lectures introduced the Hedin’s equations and, finally, the most popular approximations on the self-energy: Hartree-Fock, CohSex, and GW approximation. The latter is very successful in predicting the band gaps of solids. It improves significantly over the standard density-functional approaches for the electronic structure. The central quantities are the Green’s function G and the screened Coulomb interaction W. In this framework, the GW approximation appears naturally as a first-order approximation in the “small” quantity W. But, together with the formal derivation of the GW approximation, and in order to elucidate the physical content of the GW approximation, we show that the GW approximation is a natural improvement over the well-known Hartree-Fock method.

• The GW approximation in practice
  The second of this series of lectures (presented by M. Giantomassi) introducing MBPT is devoted to practical aspects: creation of the ground state with a correct pseudo-potential and the RPA screening file, convergence with respect to many parameters (bands, k-points, plane waves, G-vectors, self-consistent iterations, etc.), aspects of self-consistency (QPSCGW, self-consistent cohsex, only-energy self-consistency, etc.).

• Bethe-Salpeter Equation (BSE)
  This lecture (by L. Urquiza) presents the many-body approach for the description of the polarizability. Within the Green’s functions formalism, the linear response polarizability is given by the 2-particle Green’s function which obeys a Dyson-like equation, similar to the linear response TDDFT equation. The derivation of the Bethe-Salpeter equation as well as all the approximations involved in the (several) steps are illustrated in this lecture, before presenting the numerical aspects useful for the afternoon hands-on with the EXC code.

• Magnetic Response in MBPT
  This lecture (given by C. Friedrich) covers many aspects that are often neglected in all the other lectures: spin-polarized calculations, spin-orbit coupling, exchange splitting, and single-particle and collective magnetic excitations (Stoner and magnons). Besides the important connection to the relevant experiments, the link between magnetic and charge response functions in a unified and clear notation was particularly appreciated. The lecture was divided into two main parts: the theoretical (and implementation) details within the many-body approach via the GWT approximation, were followed by recent results on the electron-magnon interaction and band anomalies in iron.

Hands-on:
The practical exercises took place in the afternoon. Given the necessity for the participants to use their laptop, we devised a double strategy: i) on one side, we provided all the necessary tools (executables, doc, input files, reference results) for all those who wanted to use their laptops for all calculations; ii) on the other, we installed the codes (and all related and useful softwares and libraries, visualization tools) on the EPFL cluster, for a remote connection. The great majority chose the first strategy, while the latter was followed by those few without a Linux operative system. The exercises were very varied, ranging from the simple codes’ tutorials (available on the web pages of the codes) to more realistic calculations (converged results for a system of choice by the student).

Program

1.  Monday 21 March
   Welcome: objectives of the school (F.Sottile)
   Introduction on Spectroscopy (S.Huotari)
   Coffee Break
   Density Functional Theory (G.-M. Rignanese)
   Lunch break
   Hands-on with Abinit (tutors)
   Poster session and Aperitif
2.  Tuesday 22 March
   Micro-Macro Connection (F.Sottile)
   Time Dependent DFT (V. Olevano)
   Coffee Break
   Linear response and the DP code (V. Olevano)
   Lunch
   Hands-on with DP (tutors)
   Social Dinner
3.  Wednesday 23 March
   Non-linear approaches to TDDFT :: second and third order (V. Veniard)
   Coffee Break
   Spectroscopies in real-time (C. Attaccalite)
   Lunch
   Hands-on with 2Light and Lumen (tutors)
4. Thursday 24 March
   Many-Body Perturbation Theory (F. Sottile)
   Coffee Break
   GW in practice (M. Giantomassi)
   Lunch
   Hands-on with Abinit (tutors)
   2nd Social Dinner
5. Friday 21 March
   Bethe-Salpeter Equation (L. Urquiza)
   Coffee Break
   Magnetism in Spectroscopies (C. Friedrich)
   Lunch
   Hands-on with EXC (tutors)

Event website (with program, and list of participants): https://www.cecam.org/workshop-details/theoretical-spectroscopy-lectures-1195

General Remarks:

• This year’s event was particularly international: in addition to the 9 European nationalities, we welcomed 2 students from India, 1 from Morocco, 1 from Ethiopia, and 1 from the USA.
• We had secured additional funding for this event from the GDR-REST and Psi-k. Furthermore, we obtained extra funding from the CECAM initiative for underrepresented and underprivileged groups within the CECAM Community (special thanks to Sara Bonella and Andrea Cavalli for this).
• All participants attended the social dinner on Tuesday, and almost all attended the (exceptional) second social dinner that we managed to organize on Thursday.
• For the first time, we decided to award a poster prize. Due to the exceptional quality of the posters, we chose to award two poster prizes ex-aequo on Friday. The two winners received a certificate and a 200 CHF award.
• We extend our warm thanks to the organizing staff in Lausanne. The smooth running of this event was thanks to Aude Merola Failletaz, Bogdan Nichita, Nathalie Carminati, and Carole Albonico.
• We hope that the feedback from the Survey compiled by the students is positive and we look forward to organizing another such event.