G. D. Mahan, Many-Particle Physics, 2000.

E. N. Economou, Green's Functions in Quantum Physics, 1992.

S. Hüfner, Photoelectron Spectroscopy: Principles and Applications. Advanced Texts in Physics, 2003.

L. Hedin, New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem, Physical Review, vol.139, issue.3A, pp.796-823, 1965.
DOI : 10.1103/PhysRev.139.A796

F. Aryasetiawan and O. Gunnarsson, method, Reports on Progress in Physics, vol.61, issue.3, p.237, 1998.
DOI : 10.1088/0034-4885/61/3/002

G. Wilfried, L. Aulbur, J. W. Jnsson, and . Wilkins, Quasiparticle calculations in solids, of Solid State Physics, pp.1-218, 1999.

P. Hohenberg and W. Kohn, Inhomogeneous Electron Gas, Physical Review, vol.136, issue.3B, pp.864-871, 1964.
DOI : 10.1103/PhysRev.136.B864

F. Aryasetiawan, L. Hedin, and K. Karlsson, Cumulant Expansion, Physical Review Letters, vol.77, issue.11, pp.2268-2271, 1996.
DOI : 10.1103/PhysRevLett.77.2268

M. Vos, A. Kheifets, S. Weigold, . Canney, . Holm et al., Determination of the energy-momentum densities of aluminium by electron momentum spectroscopy, Journal of Physics: Condensed Matter, vol.11, issue.18, p.113645, 1999.
DOI : 10.1088/0953-8984/11/18/302

M. Guzzo, G. Lani, F. Sottile, P. Romaniello, M. Gatti et al., Description of Multiple Satellites, Physical Review Letters, vol.107, issue.16, p.166401, 2011.
DOI : 10.1103/PhysRevLett.107.166401
URL : https://hal.archives-ouvertes.fr/hal-00659118

J. Lischner, D. Vigil-fowler, and S. G. Louie, Plus Cumulant Study, Physical Review Letters, vol.110, issue.14, p.146801, 2013.
DOI : 10.1103/PhysRevLett.110.146801

L. Hedin, B. I. Lundqvist, and S. Lundqvist, New structure in the single-particle spectrum of an electron gas, Solid State Communications, vol.5, issue.4, pp.237-239, 1967.
DOI : 10.1016/0038-1098(67)90264-5

B. I. Lundqvist, Single-particle spectrum of the degenerate electron gas, Physik der kondensierten Materie, vol.6, issue.3, pp.193-205, 1967.

C. Blomberg and B. Bergersen, Spurious Structure from Approximations to the Dyson Equation, Canadian Journal of Physics, vol.50, issue.19, pp.2286-2293, 1972.
DOI : 10.1139/p72-303

D. C. Langreth, Singularities in the X-Ray Spectra of Metals, Physical Review B, vol.1, issue.2, p.471, 1970.
DOI : 10.1103/PhysRevB.1.471

M. Guzzo, Dynamical correlation in solids: a perspective in photoelectron spectroscopy . Theses, Ecole Polytechnique X, 2012.
URL : https://hal.archives-ouvertes.fr/pastel-00784815

G. Lani, Towards a novel approach for the calculation of many-body Green's functions. Theses, Ecole Polytechnique X, 2011.

J. J. Kas, J. J. Rehr, and L. Reining, Cumulant expansion of the retarded one-electron Green function, Physical Review B, vol.90, issue.8, p.85112, 2014.
DOI : 10.1103/PhysRevB.90.085112
URL : https://hal.archives-ouvertes.fr/hal-01113971

J. Sky-zhou, J. J. Kas, L. Sponza, I. Reshetnyak, M. Guzzo et al., Dynamical effects in electron spectroscopy, The Journal of Chemical Physics, issue.18, pp.143-2015

M. Cardona and L. Ley, Photoemission in solids: General principles. Topics in applied physics, 1978.

A. Damascelli, Z. Hussain, and Z. Shen, Angle-resolved photoemission studies of the cuprate superconductors, Reviews of Modern Physics, vol.75, issue.2, pp.473-541, 2003.
DOI : 10.1103/RevModPhys.75.473

. Reinle-schmitt, Soft-x-ray arpes at the swiss light source: From 3d materials to buried interfaces and impurities, Synchrotron Radiation News, vol.27, issue.2, pp.31-40, 2014.

. Saiht, An experimental setup of angle-resolved photoemission spectroscopy ? Wikipedia, the free encyclopedia, 2009.

S. Huotari, J. A. Soininen, T. Pylkkänen, K. Hämäläinen, A. Issolah et al., Momentum Distribution and Renormalization Factor in Sodium and the Electron Gas, Physical Review Letters, vol.105, issue.8, p.86403, 2010.
DOI : 10.1103/PhysRevLett.105.086403
URL : https://hal.archives-ouvertes.fr/hal-00533335

H. Höchst, P. Steiner, and S. Hüfner, Xps investigation of simple metals, Zeitschrift fr Physik B Condensed Matter, pp.145-154, 1978.

P. Nozi-`-eresnozi-`-nozi-`-eres, C. T. De, and . Dominicis, Singularities in the X-Ray Absorption and Emission of Metals. III. One-Body Theory Exact Solution, Physical Review, vol.178, issue.3, pp.1097-1107, 1969.
DOI : 10.1103/PhysRev.178.1097

P. Minnhagen, Aspects on diagrammatic expansion for models related to a homogeneous electron gas, Journal of Physics C: Solid State Physics, vol.8, issue.10, p.1535, 1975.
DOI : 10.1088/0022-3719/8/10/010

L. Hedin, Effects of Recoil on Shake-Up Spectra in Metals, Physica Scripta, vol.21, issue.3-4, p.477, 1980.
DOI : 10.1088/0031-8949/21/3-4/039

F. Bechstedt, Electronic Relaxation Effects in Core Level Spectra of Solids, physica status solidi (b), vol.45, issue.1, pp.9-49, 1982.
DOI : 10.1002/pssb.2221120102

C. Almbladh and L. Hedin, Handbook on Synchrotron Radiation, chapter 8, 1983.

L. Hedin, approximation, Journal of Physics: Condensed Matter, vol.11, issue.42, p.489, 1999.
DOI : 10.1088/0953-8984/11/42/201

C. N. Berglund and W. E. Spicer, Photoemission Studies of Copper and Silver: Theory, Physical Review, vol.136, issue.4A, pp.1030-1044, 1964.
DOI : 10.1103/PhysRev.136.A1030

M. Guzzo, J. J. Kas, F. Sottile, M. G. Silly, F. Sirotti et al., Plasmon satellites in valence-band photoemission spectroscopy, The European Physical Journal B, vol.29, issue.9, p.2012
DOI : 10.1140/epjb/e2012-30267-y
URL : https://hal.archives-ouvertes.fr/hal-00781253

L. Hedin, J. Michiels, and J. Inglesfield, Transition from the adiabatic to the sudden limit in core-electron photoemission, Physical Review B, vol.58, issue.23, pp.15565-15582, 1998.
DOI : 10.1103/PhysRevB.58.15565

A. B. Migdal and V. M. Galitskii, Application of quantum field theory methods to the many body problem, JETP and ZhETF(Russian original), vol.7, p.96, 1958.

R. M. Dreizler and E. K. Gross, Density Functional Theory, 1990.
DOI : 10.1007/978-3-642-86105-5

R. Van-leeuwen, Density Functional Approach to the Many-Body Problem: Key Concepts and Exact Functionals, Advances in Quantum Chemistry, vol.43, pp.25-94, 2003.
DOI : 10.1016/S0065-3276(03)43002-5

K. Capelle, A bird's-eye view of density-functional theory, Brazilian Journal of Physics, vol.36, issue.4a, pp.1318-1343, 2006.
DOI : 10.1590/S0103-97332006000700035

X. Gonze, B. Amadon, P. Anglade, J. Beuken, F. Bottin et al., ABINIT: First-principles approach to material and nanosystem properties, YEARS OF CPC: A celebratory issue focused on quality software for high performance, grid and novel computing architectures, pp.1802582-2615, 2009.
DOI : 10.1016/j.cpc.2009.07.007
URL : http://orbi.ulg.ac.be/request-copy/2268/14506/44084/2009-CPC180-2582.pdf

W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Physical Review, vol.140, issue.4A, pp.1133-1138, 1965.
DOI : 10.1103/PhysRev.140.A1133

C. Almbladh and U. Barth, Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues, Physical Review B, vol.31, issue.6, pp.3231-3244, 1985.
DOI : 10.1103/PhysRevB.31.3231

D. M. Ceperley and B. J. Alder, Ground State of the Electron Gas by a Stochastic Method, Physical Review Letters, vol.45, issue.7, pp.566-569, 1980.
DOI : 10.1103/PhysRevLett.45.566

P. Steiner, H. Höchst, and S. Hüfner, XPS investigation of simple metals, Zeitschrift für Physik B Condensed Matter, pp.129-143, 1978.
DOI : 10.1007/BF01320978

I. Lyo and E. W. Plummer, Quasiparticle band structure of Na and simple metals, Physical Review Letters, vol.60, issue.15, pp.1558-1561, 1988.
DOI : 10.1103/PhysRevLett.60.1558

M. Van-schilfgaarde, T. Kotani, and S. Faleev, Theory, Physical Review Letters, vol.96, issue.22, p.226402, 2006.
DOI : 10.1103/PhysRevLett.96.226402

I. Mois, A. Aroyo, C. Kirov, J. M. Capillas, H. Perez-mato et al., Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups, Acta Crystallographica Section A, vol.62, issue.2, pp.115-128, 2006.

I. Mois, D. Aroyo, G. Orobengoa, E. S. De-la-flor, J. Tasci et al., Brillouin-zone database on the Bilbao Crystallographic Server, Acta Crystallographica Section A, vol.70, issue.2, pp.126-137, 2014.

J. P. Perdew, R. G. Parr, M. Levy, and J. L. Balduz, Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy, Physical Review Letters, vol.49, issue.23, pp.1691-1694, 1982.
DOI : 10.1103/PhysRevLett.49.1691

G. Strinati, Application of the Green???s functions method to the study of the optical properties of semiconductors, La Rivista del Nuovo Cimento, vol.49, issue.12, pp.111-86, 1978.
DOI : 10.1007/BF02725962

M. David, M. Ceperley-richard, L. Martin, and . Reining, Interacting electrons: theory and computational approaches, 2016.

D. Bohm and D. Pines, A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas, Physical Review, vol.92, issue.3, pp.609-625, 1953.
DOI : 10.1103/PhysRev.92.609

J. Lindhard, On the properties of a gas of charged particles, 1954.

N. W. Ashcroft and N. D. Mermin, Solid state physics, Science: Physics. Saunders College, 1976.

P. Minnhagen, Vertex correction calculations for an electron gas, Journal of Physics C: Solid State Physics, vol.7, issue.17, p.3013, 1974.
DOI : 10.1088/0022-3719/7/17/011

G. Stefanucci, Y. Pavlyukh, A. Uimonen, and R. Van-leeuwen, : Application to vertex corrections in the electron gas, Physical Review B, vol.90, issue.11, p.115134, 2014.
DOI : 10.1103/PhysRevB.90.115134

L. Stephen and . Adler, Quantum theory of the dielectric constant in real solids, Phys. Rev, vol.126, pp.413-420, 1962.

N. Wiser, Dielectric Constant with Local Field Effects Included, Physical Review, vol.129, issue.1, pp.62-69, 1963.
DOI : 10.1103/PhysRev.129.62

A. G. Marinopoulos, L. Reining, V. Olevano, A. Rubio, T. Pichler et al., Anisotropy and Interplane Interactions in the Dielectric Response of Graphite, Physical Review Letters, vol.89, issue.7, pp.266406-2002
DOI : 10.1103/PhysRevLett.89.076402

R. Hambach, C. Giorgetti, N. Hiraoka, Y. Q. Cai, F. Sottile et al., Anomalous Angular Dependence of the Dynamic Structure Factor near Bragg Reflections: Graphite, Physical Review Letters, vol.101, issue.26, p.266406, 2008.
DOI : 10.1103/PhysRevLett.101.266406
URL : https://hal.archives-ouvertes.fr/hal-00476164

C. Kramberger, R. Hambach, C. Giorgetti, M. H. Rümmeli, M. Knupfer et al., Linear plasmon dispersion in singlewall carbon nanotubes and the collective excitation spectrum of graphene, Phys. Rev. Lett, vol.100, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00347264

S. Waidmann, M. Knupfer, B. Arnold, J. Fink, A. Fleszar et al., Local-field effects and anisotropic plasmon dispersion in diamond, Physical Review B, vol.61, issue.15, pp.10149-10153, 2000.
DOI : 10.1103/PhysRevB.61.10149

H. Dröge, A. Fleszar, W. Hanke, M. Sing, M. Knupfer et al., Complex loss function of CdTe, Physical Review B, vol.59, issue.8, pp.5544-5550, 1999.
DOI : 10.1103/PhysRevB.59.5544

B. Bergersen, F. W. Kus, and C. Blomberg, Single Particle Green's Function in the Electron???Plasmon Approximation, Canadian Journal of Physics, vol.51, issue.1, pp.102-110, 1973.
DOI : 10.1139/p73-012

H. Jiang, R. I. Gomez-abal, P. Rinke, and M. Scheffler, approach, Physical Review B, vol.82, issue.4, p.45108, 2010.
DOI : 10.1103/PhysRevB.82.045108
URL : https://hal.archives-ouvertes.fr/hal-00919467

H. Jiang, Approach, The Journal of Physical Chemistry C, vol.116, issue.14, pp.7664-7671, 2012.
DOI : 10.1021/jp300079d
URL : https://hal.archives-ouvertes.fr/hal-00919467

B. Holm and U. Barth, self-energy of the electron gas, Physical Review B, vol.57, issue.4, pp.2108-2117, 1998.
DOI : 10.1103/PhysRevB.57.2108

M. Gatti and M. Guzzo, approximation and beyond, Physical Review B, vol.87, issue.15, p.155147, 2013.
DOI : 10.1103/PhysRevB.87.155147
URL : https://hal.archives-ouvertes.fr/hal-00931183

F. Bruneval and M. Gatti, Quasiparticle Self-Consistent GW Method for the Spectral Properties of Complex Materials, First Principles Approaches to Spectroscopic Properties of Complex Materials, pp.99-135, 2014.
DOI : 10.1007/128_2013_460
URL : https://hal.archives-ouvertes.fr/hal-01073577

T. J. Pollehn, R. W. Schindlmayr, and . Godby, approximation using Hubbard chains, Journal of Physics: Condensed Matter, vol.10, issue.6, pp.1273-1283, 1998.
DOI : 10.1088/0953-8984/10/6/011

M. Vos, A. S. Kheifets, E. Weigold, and F. Aryasetiawan, Electron correlation effects in the spectral momentum density of graphite, Physical Review B, vol.63, issue.3, p.33108, 2001.
DOI : 10.1103/PhysRevB.63.033108

O. Gunnarsson, V. Meden, and K. Schönhammer, Corrections to Migdal???s theorem for spectral functions: A cumulant treatment of the time-dependent Green???s function, Physical Review B, vol.50, issue.15, pp.10462-10473, 1994.
DOI : 10.1103/PhysRevB.50.10462

F. Caruso, H. Lambert, and F. Giustino, Band Structures of Plasmonic Polarons, Physical Review Letters, vol.114, issue.14, p.146404, 2015.
DOI : 10.1103/PhysRevLett.114.146404

M. Cazzaniga, and beyond approaches to quasiparticle properties in metals, Physical Review B, vol.86, issue.3, p.35120, 2012.
DOI : 10.1103/PhysRevB.86.035120

F. Leo and . Kadanoff, Quantum statistical mechanics green's function methods in equilibrium and nonequilibrium problems. Frontiers in physics, 1962.

G. Lani, P. Romaniello, and L. Reining, Approximations for many-body Green's functions: insights from the fundamental equations, New Journal of Physics, vol.14, issue.1, 2012.
DOI : 10.1088/1367-2630/14/1/013056

G. Stefanucci and R. Van-leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction, 2013.
DOI : 10.1017/CBO9781139023979

J. Rammer, Quantum Field Theory of Non-Equilibrium States, Cambridge Univ, 2007.
DOI : 10.1017/CBO9780511618956

D. C. Langreth, Linear and Nonlinear Response Theory with Applications, Linear and Nonlinear Electron Transport in Solids, pp.3-32, 1976.
DOI : 10.1007/978-1-4757-0875-2_1

B. I. Lundqvist, Single-particle spectrum of the degenerate electron gas, Physik der kondensierten Materie, vol.6, issue.3, pp.206-217

A. Georges and G. Kotliar, Hubbard model in infinite dimensions, Physical Review B, vol.45, issue.12, pp.6479-6483, 1992.
DOI : 10.1103/PhysRevB.45.6479

M. Jarrell, Hubbard model in infinite dimensions: A quantum Monte Carlo study, Physical Review Letters, vol.69, issue.1, pp.168-171, 1992.
DOI : 10.1103/PhysRevLett.69.168

A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Reviews of Modern Physics, vol.68, issue.1, pp.13-125, 1996.
DOI : 10.1103/RevModPhys.68.13

M. Casula, A. Rubtsov, and S. Biermann, Dynamical screening effects in correlated materials: Plasmon satellites and spectral weight transfers from a Green's function ansatz to extended dynamical mean field theory, Physical Review B, vol.85, issue.3, p.35115, 2012.
DOI : 10.1103/PhysRevB.85.035115

H. Ness, L. K. Dash, M. Stankovski, and R. W. Godby, gw approximations and vertex corrections on the keldysh time-loop contour: Application for model systems at equilibrium, Phys. Rev. B, vol.84, 2011.

A. Schindlmayr, approximation, Physical Review B, vol.56, issue.7, pp.3528-3531, 1997.
DOI : 10.1103/PhysRevB.56.3528

M. Guzzo, Sf repository. https://github.com/teoguso, 2016.

J. J. Kas, A. P. Sorini, M. P. Prange, L. W. Cambell, J. A. Soininen et al., Many-pole model of inelastic losses in x-ray absorption spectra, Physical Review B, vol.76, issue.19, 2007.
DOI : 10.1103/PhysRevB.76.195116

M. Cazzaniga, H. Weissker, S. Huotari, T. Pylkkänen, P. Salvestrini et al., Dynamical response function in sodium and aluminum from time-dependent density-functional theory, Physical Review B, vol.84, issue.7, p.75109, 2011.
DOI : 10.1103/PhysRevB.84.075109
URL : https://hal.archives-ouvertes.fr/hal-00696002

S. Huotari, M. Cazzaniga, H. Weissker, T. Pylkkänen, H. Müller et al., Dynamical response function in sodium studied by inelastic x-ray scattering spectroscopy, Physical Review B, vol.84, issue.7, p.75108, 2011.
DOI : 10.1103/PhysRevB.84.075108

F. Sottile, V. Olevano, and L. Reining, Parameter-Free Calculation of Response Functions in Time-Dependent Density-Functional Theory, Physical Review Letters, vol.91, issue.5, p.56402, 2003.
DOI : 10.1103/PhysRevLett.91.056402

R. , D. Sole, L. Reining, and R. W. Godby, GW ? approximation for electron selfenergies in semiconductors and insulators, Phys. Rev. B, vol.49, pp.8024-8028, 1994.

R. W. Wyckoff, Crystal structures, 1963.

N. Troullier and J. Martins, Efficient pseudopotentials for plane-wave calculations, Physical Review B, vol.43, issue.3, pp.1993-2006, 1991.
DOI : 10.1103/PhysRevB.43.1993

M. Gatti, I. V. Tokatly, and A. Rubio, Sodium: A Charge-Transfer Insulator at High Pressures, Physical Review Letters, vol.104, issue.21, p.216404, 2010.
DOI : 10.1103/PhysRevLett.104.216404
URL : http://arxiv.org/abs/1003.0540

B. Holm and F. Aryasetiawan, Total energy from the Galitskii-Migdal formula using realistic spectral functions, Physical Review B, vol.62, issue.8, pp.4858-4865, 2000.
DOI : 10.1103/PhysRevB.62.4858

T. Miyake, F. Aryasetiawan, H. Kino, and K. Terakura, Total energy from the many-body perturbation approach with a model spectral function: An application to simple metals, Physical Review B, vol.64, issue.23, p.233109, 2001.
DOI : 10.1103/PhysRevB.64.233109

N. Skkinen, Y. Peng, H. Appel, and R. Van-leeuwen, Many-body greens function theory for electron-phonon interactions: Ground state properties of the holstein dimer, The Journal of Chemical Physics, issue.23, pp.143-2015

B. Holm, Calculations, Physical Review Letters, vol.83, issue.4, pp.788-791, 1999.
DOI : 10.1103/PhysRevLett.83.788