Scientific Report for the
Theoretical Spectroscopy Lectures
March 21-25, 2022
CECAM-HQ-EPFL, Lausanne, Switzerland

Objectives:
The aim of the school was to give a deep introduction to the theoretical and practical aspects of the electronic excitations which are probed by experimental techniques such as optical absorption, EELS, and photoemission (direct or inverse). From the theory point of view, excitations and excited state properties are out of the reach of density-functional theory (DFT), which is a ground-state theory. In the last thirty years, other ab-initio theories and frameworks, which are able to describe electronic excitations and spectroscopy, have become more and more used: 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 in order 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. For the first time, this year we also covered theoretical aspects of magnetic excitations. Finally, a large part of the school was devoted to the codes implementing such theories (ABINIT, 2Light, DP, EXC).

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

Tutors:
Valérie Véniard (Ecole Polytechnique and CNRS, Palaiseau, France)
Matteo Gatti (Ecole Polytechnique and CNRS, Palaiseau, France)
Matteo Giantomassi (Université catholique de Louvain, Belgium)

External Speakers:
Claudio Attaccalite (Institut Neel, CNRS, Grenoble, France)
Simo Huotari (University of Helsinki, Finland)
Christoph Friedrich (Jülich Research Centre, Germany)

Format:
For the first time, we have proposed a hybrid format, both online and presential. i) presentation of the theory and theoretical aspects of the implementation took place in the morning sessions; these sessions took place in the Moser 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 (Day 3), GW approximation to MBPT with Abinit (Day 4) and Bethe-Salpeter with EXC (Day 5). They all took place in the Metropolis room at CECAM HQ.

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. An eminent role is covered by the inverse dielectric function $\varepsilon^1$, the screening function, which serves here as the main motivation and guideline for the whole school.
• Density-Functional Theory (DFT)
  This lecture (presented by G.-M. Rignanese) covered the basics of DFT: formalism and implementation. 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: the electronic N-body problem, functionals of the density, the Kohn-Sham approach, approximations, and new functionals, 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, iterative algorithms. And of course, the ABINIT code was presented. 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 the measurable quantities (macroscopic) and calculated quantities (microscopic). This is a crucial concept, often disregarded. We find that this micro-macro connection is never well explained in textbooks or articles, rather 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 frequency domain, reciprocal space, and on a plane-waves 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. Véniard) the perturbative approach for non-linear spectroscopy is presented, with the 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.
• Many-Body Perturbation Theory and GW approximation
  In these lectures (presented by M. Gatti and M. Giantomassi), we introduced the theoretical basics of many-body Green’s functions theory. Starting from the very basic concept of the Green’s function based on second quantization operators, and 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 has been shown to be 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. An important part of the lectures 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 F. Sottile) presents the many-body approach for the description of 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, similarly to the linear response TDDFT equation. The derivation of the Bethe-Salpeter equation as well 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.
• 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.). The time evolution technique is later also used to tackle the Bethe-Salpeter equation and go beyond BSE linear response. Again, several points of view have been given: the theoretical explanation and derivation, as well as the implementation technical details.
• Magnetic Response in MBPT
  This lecture (given by C. Friedrich) is a new entry in the school and covers many aspects that are often neglected in all the other lectures: spin-polarized calculations, spin-orbit coupling, exchange splitting, 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. We installed the codes (and all related and useful softwares and libraries, visualization tools) on the EPFL cluster, of which we had allocated 10 32-cores nodes for our school (Monday to Friday). 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), so reflecting the heterogeneous audience: some young students were actually trying a DFT code for the first time, some others were already proficient with DFT and wanted to do “real stuff” with TDDFT and GW, for instance. The possibility to run on a cluster (with many processors at our disposal) permitted this.

Program

• • Day 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)
    Social Dinner
  • Day 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)
  • Day 3 – Wednesday 23 March
    Non-linear approaches to TDDFT :: second and third-order (V. Veniard)
    Coffee Break
    Many-Body Perturbation Theory (M. Gatti)
    Lunch
    Hands-on with 2Light (tutors)
  • Day 4 – Thursday 24 March
    The GW approximation (M. Gatti)
    Coffee Break
    GWA :: practicalities and Abinit (M. Giantomassi)
    Lunch
    Hands-on with Abinit (tutors)
  • Day 5 Friday 21 March
    Bethe-Salpeter Equation (F. Sottile)
    Spectroscopies in real-time (C. Attaccalite)
    Coffee Break
    Magnetism in Spectroscopies (C. Friedrich)
    Lunch
    Hands-on with EXC (tutors)

Event website https://www.cecam.org/workshop-0-1573.html

Full program, with presentations https://www.cecam.org/workshop-5-1573.html

Full list of participants https://www.cecam.org/workshop-1-1573.html

General Remarks:
There were a lot of uncertainties related to the school in this post(?)-covid era. The event should have taken place in 2020 and was postponed twice, because we really wanted to have a presential event, not just an online one. However, we have, in these two years, cumulated many requests for participation. We have then decided to organize, for the first time, a hybrid event. We selected 30 participants for the onsite event and 57 other participants for the online sessions. In spite of our hesitations, everything worked out smoothly. The hybrid sessions, in the morning, were a success, thanks to a new audio/video system installed in the Moser room at CECAM HQ (even if this requires a little bit of adjusting from our side). The participation level has always been very high, and we have been glad to notice how the discussion time has always been stretched much over the allocated time, in particular for the most sensible lectures (MBPT and TDDFT).
The participation over Zoom was never very high. An average of 20 online students attended the morning sessions. However, we have to underline that: i) the lectures were recorded and put at students’ disposal on the very same day, on the CECAM website; ii) the time difference was disadvantageous for many participants, in particular those from the Americas. All in all, we are very happy with the outcome of the hybrid experiment and we will definitely consider this option in the future, but the presential part, especially for the hands-on where in many cases tutors handed the keyboards directly, is prioritary and we cannot stress enough the importance to have an event like this at CECAM HQ.

All participants attended the social dinner on Monday. We managed to organize an aperitif all together on Thursday night in the main common room.

Finally, we want to point out that the organization and the connection with Bogdan Nichita, Nathalie Carminati and particularly with Aude Failletaz were flawless (many thanks for all this).
We are looking forward to organising another event.