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The Bethe-Salpeter equation has been extensively employed to compute the two-body electron-hole propagator and its poles which correspond to the neutral excitation energies of the system. Through a different time-ordering, the two-body Green's function can also describe the propagation of two electrons or two holes. The corresponding poles are the double ionization potentials and double electron affinities of the system. In this work, a Bethe-Salpeter equation for the two-body particle-particle propagator is derived within the linear-response formalism using a pairing field and anomalous propagators. This framework allows us to compute kernels corresponding to different self-energy approximations ($GW$, $T$-matrix, and second-Born) as in the usual electron-hole case. The performance of these various kernels is gauged for singlet and triplet valence double ionization potentials using a set of 23 small molecules. The description of double core hole states is also analyzed.

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In a recent letter [Phys. Rev. Lett. 131, 216401] we presented the multichannel Dyson equation (MCDE) in which two or more many-body Green's functions are coupled. In this work we will give further details of the MCDE approach. In particular we will discuss: 1) the derivation of the MCDE and the definition of the space in which it is to be solved; 2) the rationale of the approximation to the multichannel self-energy; 3) a diagrammatic analysis of the MCDE; 4) the recasting of the MCDE on an eigenvalue problem with an effective Hamiltonian that can be solved using standard numerical techniques. This work mainly focuses on the coupling between the one-body Green's function and the three-body Green's function to describe photoemission spectra, but the MCDE method can be generalized to the coupling of other many-body Green's functions and to other spectroscopies.

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Hedin's equations provide an elegant route to compute the exact one-body Green's function (or propagator) via the self-consistent iteration of a set of non-linear equations. Its first-order approximation, known as $GW$, corresponds to a resummation of ring diagrams and has shown to be extremely successful in physics and chemistry. Systematic improvement is possible, although challenging, via the introduction of vertex corrections. Considering anomalous propagators and an external pairing potential, we derive a new self-consistent set of closed equations equivalent to the famous Hedin equations but having as a first-order approximation the particle-particle (pp) $T$-matrix approximation where one performs a resummation of the ladder diagrams. This pp version of Hedin's equations offers a way to go systematically beyond the $T$-matrix approximation by accounting for low-order pp vertex corrections.

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The Bethe–Salpeter equation (BSE) is the key equation in many-body perturbation theory based on Green's functions to access response properties. Within the GW approximation to the exchange-correlation kernel, the BSE has been successfully applied to several finite and infinite systems. However, it also shows some failures, such as underestimated triplet excitation energies, lack of double excitations, ground-state energy instabilities in the dissociation limit, etc. In this work, we study the performance of the BSE within the GW approximation as well as the T-matrix approximation for the excitation energies of the exactly solvable asymmetric Hubbard dimer. This model allows one to study various correlation regimes by varying the on-site Coulomb interaction U as well as the degree of the asymmetry of the system by varying the difference of potential Δv between the two sites. We show that, overall, the GW approximation gives more accurate excitation energies than GT over a wide range of U and Δv. However, the strongly correlated (i.e., large U) regime still remains a challenge.

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The subject of the thesis focuses on new approximations studied in a formalism based on a perturbation theory allowing to describe the electronic properties of many-body systems in an approximate way. We excite a system with a small disturbance, by sending light on it or by applying a weak electric field to it, for example and the system "responds" to the disturbance, in the framework of linear response, which means that the response of the system is proportional to the disturbance. The goal is to determine what we call the neutral excitations or bound states of the system, and more particularly the single excitations. These correspond to the transitions from the ground state to an excited state. To do this, we describe in a simplified way the interactions of the particles of a many-body system using an effective interaction that we average over the whole system. The objective of such an approach is to be able to study a system without having to use the exact formalism which consists in diagonalizing the N-body Hamiltonian, which is not possible for systems with more than two particles.

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Sujets

Electronic properties of metal clusters and organic molecules Numbers 3360+q Molecules Metal cluster Mean-field Electron correlation High intensity lasers Dissipative effects Approximation GW Fonction de Green Aggregates Corrélations Nuclear Monte-Carlo Neutronic Greens function methods Molecular irradiation Instability Ionization mechanisms Clusters Coulomb presssure 3620Kd Density Functional Theory Méchanismes d'ionisation Environment Electronic emission Neutron Induced Activation Hierarchical model Inverse bremsstrahlung collisions Collision frequency Activation neutronique Correction d'auto-interaction Agregats MBPT Méthode multiréférence Nickel oxide Electronic properties of sodium and carbon clusters Dynamique moléculaire Semiclassic Oxyde de nickel Fission Atom laser Interactions de photons avec des systèmes libres Landau damping Effets dissipatifs Matel clusters Irradiation moléculaire GW approximation Chaos Corrélation forte Photon interactions with free systems Electronic excitation Relaxation 3640Cg Energy spectrum FOS Physical sciences Agrégats Au-delà du champ moyen Time-dependent density-functional theory Deposition Dynamics Ar environment 3115ee Damping Electron-surface collision Collisional time-dependent Hartree-Fock Méthodes des fonctions de Green Théorie de la fonctionnelle de la densité Diffusion Neutronique Optical response Hierarchical method Modèle de Hubbard Extended time-dependent Hartree-Fock Embedded metal cluster Metal clusters Density-functional theory Electric field Deposition dynamics Angle-resolved photoelectron spectroscopy Matrice densité Electron emission Instabilité Photo-Electron Spectrum Nucléaire Champ-moyen Nanoplasma Explosion coulombienne Molecular dynamics Coulomb explosion Photo-electron distributions TDDFT Hubbard model Laser CAO Multirefence methods Green's function Lasers intenses Dissipation Corrélations dynamiques

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