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Final Report Summary - ULTRA BSE (Understanding elementary excitations in correlated materials: pushing the frontiers of thefirst-principles Bethe-Salpeter equation)

The ab initio solution of the Bethe-Salpeter equation (BSE) is nowadays the method of choice to obtain absorption spectra in bulk materials containing sp electrons.
The objective of ULTRA BSE was to overcome the limitations of its standard implementation that prevent its application to understanding electronic excitations in correlated materials, notably transition metal oxides containing localised d electrons.

The BSE implementation has been successfully extended to include the required features and used for studying neutral excitations at finite momentum transfer in prototypical correlated oxides.
Orbital dd excitations (including their dispersion and angular dependence) have been simulated within the parameter-free BSE framework. In the terminology of ab initio many-body perturbation theory (MBPT), they can be understood as localised Frenkel excitons with huge binding energies of several eV, in contrast to simple semiconductors where they are of the order of tens of meV. The BSE approach correctly reproduces this huge binding energy and confirms the expectation based on cluster models that double excitations are present in the spectra. The calculations have also put forward that important cancellation effects
exist between the band-gap opening due to the self-energy in the GW approximation (GWA) and compensating excitonic effects.
These achievements have also boosted a parallel activity on the exciton dispersion in layered compounds and 2D materials and the investigation of vertex corrections beyond the state-of-the-art random-phase approximation (RPA) used in the GWA,

The application to real material revealed that the real bottleneck of the computational scheme is the need to perform a self-consistent quasiparticle GW calculation (also hybrid functionals that contain a nonlocal Fock exchange term are still very expensive).
This represented the main motivation to devise an alternative strategy based on a novel simplified scheme, where the key quantity is a dynamical local effective potential instead of the self-energy as in the GWA.
The main outcome was the proposition of two shortcuts with respect to the standard method. The first one consists in the introduction of an auxiliary system that, in principle exactly, targets the excitation spectrum of the real system. The prototypical example for an auxiliary system is density functional theory, in which the auxiliary system is the Kohn-Sham system: it exactly reproduces the density of the real system via a real and static potential, the Kohn-Sham potential. However, the Kohn-Sham system does not correctly describe excited-state properties: an example is the famous band-gap problem. The proposed potential (named "spectral potential"), which is frequency dependent, yet local and real (so much cheaper than the GW self-energy), can be viewed as a dynamical generalisation of the Kohn-Sham potential that yields in principle the exact excitation spectrum.
The second shortcut was the idea of calculating this potential just once and forever in a model system, the homogeneous electron gas (HEG), and tabulating it. To study real materials, a “connector” was designed that prescribes the use of the HEG results for calculating electronic spectra. All the non-trivial interplay between electron interaction and inhomogeneity of the real system enters the form of the connector. Thus the great challenge of the approach is finding an accurate expression for the connector. The proposed approximation is based on local properties of the system, and was hence called the "dynamical local connector approximation”.
The first results of this completely new approach are very promising and can potentially have a very important impact for the calculation of electronic excitations.

An important part of the project was the development of a synergy with experimental colleagues working on the valence electron spectroscopy of materials.
The theoretical predictions have served as a guide for the realisation of photoemission, non-resonant Inelastic X-Ray Scattering (IXS), Resonant Inelastic X-Ray Scattering (RIXS) and electron energy loss spectroscopy (EELS) experiments.
The collaborations with experimental colleagues have been very productive and stimulating all along the duration of the project. This has been strongly facilitated by a formal association to the synchrotron SOLEIL

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