A. Anantharaman and E. Cancès, Existence of minimizers for Kohn???Sham models in quantum chemistry, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol.26, issue.6, pp.2425-2455, 2009.
DOI : 10.1016/j.anihpc.2009.06.003

O. [. Aryasetiawan and . Gunnarsson, method, Reports on Progress in Physics, vol.61, issue.3, pp.237-312, 1998.
DOI : 10.1088/0034-4885/61/3/002

L. [. Aulbur, J. W. Jönsson, and . Wilkins, Quasiparticle calculations in solids, Solid State Phys, pp.1-218, 2000.

X. Blase, C. Attaccalite, and V. Olevano, calculations for fullerenes, porphyrins, phtalocyanine, and other molecules of interest for organic photovoltaic applications, Physical Review B, vol.83, issue.11, p.115103, 2011.
DOI : 10.1103/PhysRevB.83.115103
URL : https://hal.archives-ouvertes.fr/hal-00626224

V. Bach and L. Delle-site, Many-electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View, Mathematical Physics Studies, 2014.
DOI : 10.1007/978-3-319-06379-9

F. Bruneval and M. Gatti, Quasiparticle Self-Consistent GW Method for the Spectral Properties of Complex Materials, Top. Curr. Chem, vol.347, pp.99-135, 2014.
DOI : 10.1007/128_2013_460
URL : https://hal.archives-ouvertes.fr/hal-01073577

O. [. Blöchl, O. K. Jepsen, and . Andersen, Improved tetrahedron method for Brillouin-zone integrations, Physical Review B, vol.49, issue.23, pp.16223-16233, 1994.
DOI : 10.1103/PhysRevB.49.16223

X. Blanc, C. L. Bris, and P. , A Definition of the Ground State Energy for Systems Composed of Infinitely Many Particles, Communications in Partial Differential Equations, vol.35, issue.1-2, pp.439-475, 2003.
DOI : 10.1081/PDE-120019389

. Bpc-+-07-]-c, C. Brouder, M. Panati, C. Calandra, N. Mourougane et al., Exponential localization of Wannier functions in insulators, Phys. Rev. Lett, issue.4, pp.98-046402, 2007.

I. W. Bulik, G. Scalmani, M. J. Frisch, and G. E. Scuseria, Noncollinear density functional theory having proper invariance and local torque properties, Physical Review B, vol.87, issue.3, p.35117, 2013.
DOI : 10.1103/PhysRevB.87.035117

]. E. Can00 and . Cancès, SCF algorithms for Hartree-Fock electronic calculations, Mathematical Models and Methods for Ab Initio Quantum Chemistry, 2000.

M. L. Cohen and T. K. Bergstresser, Band Structures and Pseudopotential Form Factors for Fourteen Semiconductors of the Diamond and Zinc-blende Structures, Physical Review, vol.141, issue.2, pp.789-796, 1966.
DOI : 10.1103/PhysRev.141.789

A. [. Cancès, M. Deleurence, and . Lewin, A new approach to the modeling of local defects in crystals: the reduced Hartree-Fock case, Commun. Math. Phys, pp.281-129, 2008.

. Cdm-+-14-]-e, G. Cancès, M. Dusson, B. Maday, M. Stamm et al., A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, C. R. Math, vol.352, issue.11, pp.941-946, 2014.

. Ceg-+-15-]-e, V. Cancès, D. Ehrlacher, A. Gontier, D. Levitt et al., Fast numerical methods for Brillouin-zone integration, 2015.

[. Cancès, D. Gontier, and G. Stoltz, A mathematical analysis of the GW0 method for computing electronic excited energies of molecules, submitted, 2015.

I. Catto, C. L. Bris, and P. Lions, Limite thermodynamique pour des modèles de type Thomas-Fermi, C. R. Acad. Sci. Paris, vol.322, issue.4, pp.357-364, 1996.

]. A. Col63 and . Coleman, Structure of fermion density matrices, Rev. Mod. Phys, vol.35, issue.3, pp.668-686, 1963.

J. [. Cole and . Perdew, Calculated electron affinities of the elements, Physical Review A, vol.25, issue.3, pp.1265-1271, 1982.
DOI : 10.1103/PhysRevA.25.1265

F. Caruso, P. Rinke, X. Ren, A. Rubio, and M. Scheffler, approach, Physical Review B, vol.86, issue.8, p.81102, 2012.
DOI : 10.1103/PhysRevB.86.081102
URL : https://hal.archives-ouvertes.fr/hal-00861894

J. and D. Cloizeaux, -Dimensional Energy Bands and Wannier Functions, Physical Review, vol.135, issue.3A, pp.698-707, 1964.
DOI : 10.1103/PhysRev.135.A698
URL : https://hal.archives-ouvertes.fr/jpa-00209672

]. A. Del08 and . Deleurence, Modélisation mathématique et simulation numérique de la structure électronique de cristaux en présence des défauts ponctuels, 2008.

C. [. Derezi?ski and . Gérard, Scattering Theory of Classical and Quantum N-Particle Systems, Texts and Monographs in Physics Quantum mechanics of many-electron systems, Proc. R. Soc. Lond. A 123, pp.714-733, 1929.

E. Engel and R. M. Dreizler, Density Functional Theory: an advanced course, 2011.
DOI : 10.1007/978-3-642-14090-7

B. [. Engel and . Farid, Generalized plasmon-pole model and plasmon band structures of crystals, Physical Review B, vol.47, issue.23, pp.15931-15934, 1993.
DOI : 10.1103/PhysRevB.47.15931

B. Farid, Ground and low-lying excited states of interacting electron systems; a survey and some critical analyses, Electron Correlation in the Solid State, 1999.

]. C. Fef85 and . Fefferman, The thermodynamic limit for a crystal, Commun. Math. Phys, vol.98, issue.3, pp.289-311, 1985.

M. [. Fournais, T. Hoffmann-ostenhof, T. Hoffmann-ostenhof, and . Østergaard-sørensen, The Electron Density is Smooth Away from the Nuclei, Communications in Mathematical Physics, vol.7, issue.3, pp.401-415, 2002.
DOI : 10.1007/s002200200668

E. [. Frank, R. Lieb, H. Seiringer, and . Siedentop, M??ller???s exchange-correlation energy in density-matrix-functional theory, Physical Review A, vol.76, issue.5, p.52517, 2007.
DOI : 10.1103/PhysRevA.76.052517

]. T. Gil75 and . Gilbert, Hohenberg-Kohn theorem for nonlocal external potentials, Phys. Rev. B, vol.502, issue.6, pp.2111-2120, 1975.

D. Gontier and S. Lahbabi, Convergence rates of supercell calculations in the reduced Hartree???Fock model, ESAIM: Mathematical Modelling and Numerical Analysis, vol.50, issue.5, 2015.
DOI : 10.1051/m2an/2015084
URL : https://hal.archives-ouvertes.fr/hal-01238623

A. [. Galitskii, Application of quantum field theory methods to the many body problem, Sov. Phys. JETP, vol.139, 1958.

R. Godby and R. Needs, Metal-insulator transition in Kohn-Sham theory and quasiparticle theory, Physical Review Letters, vol.62, issue.10, pp.1169-1172, 1989.
DOI : 10.1103/PhysRevLett.62.1169

]. D. Gon13 and . Gontier, N-representability in noncollinear spin-polarized densityfunctional theory, Phys. Rev. Lett, vol.111, p.153001, 2013.

L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, 2004.
DOI : 10.1007/978-0-387-09432-8

N. [. Horsfield and . Ashcroft, The Fermi surface and pseudopotentials of aluminium, Journal of Physics: Condensed Matter, vol.5, issue.23, pp.3925-3936, 1993.
DOI : 10.1088/0953-8984/5/23/018

J. E. Harriman, Orthonormal orbitals for the representation of an arbitrary density, Physical Review A, vol.24, issue.2, pp.680-682, 1981.
DOI : 10.1103/PhysRevA.24.680

L. Hedin, New Method for Calculating the One-Particle Green's Function with Application to the Electron-Gas Problem, A796. [HJO14], 1965.
DOI : 10.1103/PhysRev.139.A796

T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Electronic-Structure Theory, 2014.
DOI : 10.1002/9781119019572

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

L. Hedin and S. Lundqvist, Effects of electron-electron and electron-phonon interactions on the one-electron states of solids, Solid State Phys, pp.1-181, 1970.

M. S. Hybertsen and S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies, Physical Review B, vol.34, issue.8, pp.5390-5413, 1986.
DOI : 10.1103/PhysRevB.34.5390

C. Hainzl, M. Lewin, and J. P. Solovej, The thermodynamic limit of quantum Coulomb systems Part I. General theory, Advances in Mathematics, vol.221, issue.2, pp.454-487, 2009.
DOI : 10.1016/j.aim.2008.12.010
URL : https://hal.archives-ouvertes.fr/hal-00334555

T. [. Hoffmann-ostenhof and . Hoffmann-ostenhof, "Schr??dinger inequalities" and asymptotic behavior of the electron density of atoms and molecules, Physical Review A, vol.16, issue.5, pp.1782-1785, 1977.
DOI : 10.1103/PhysRevA.16.1782

J. [. Hobby and . Rice, A moment problem in L 1 approximation, Proc. Amer, pp.665-670, 1965.

W. Hunziker, On the spectra of Schrödinger multiparticle Hamiltonians, Helv. Phys. Acta, vol.39, pp.452-462, 1966.

]. D. Joh74 and . Johnson, Local field effects and the dielectric response matrix of insulators: A model, Phys. Rev. B, vol.9, issue.10, pp.4475-4484, 1974.

T. Kato, Perturbation Theory for Linear Operators, 2012.

P. Koval, D. Foerster, and D. , Sánchez-Portal, Fully self-consistent GW and quasiparticle self-consistent GW for molecules, Phys. Rev. B, vol.81, p.85103, 2010.

J. Kübler, K. H. Höck, J. Sticht, and A. R. Williams, Density functional theory of non-collinear magnetism, Journal of Physics F: Metal Physics, vol.18, issue.3, p.469, 1988.
DOI : 10.1088/0305-4608/18/3/018

A. Klein, Perturbation Theory for an Infinite Medium of Fermions. II, Physical Review, vol.121, issue.4, pp.950-956, 1961.
DOI : 10.1103/PhysRev.121.950

]. W. Koh59 and . Kohn, Analytic properties of Bloch waves and Wannier functions, Phys. Rev, vol.115, pp.809-821, 1959.

L. [. Kohn and . 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

[. Bris, Quelques problèmes mathématiques en chimie quantique moléculaire, 1993.

]. M. Lev79 and . Levy, Universal variational functionals of electron densities, firstorder density matrices, and natural spin-orbitals and solution of the vrepresentability problem, Proc. Natl. Acad. Sci. U.S.A, vol.76, issue.12, pp.6062-6065, 1979.

]. E. Lie83 and . Lieb, Density functionals for Coulomb systems, Int. J. Quantum Chem, vol.24, issue.3, pp.243-277, 1983.

]. P. Lio84 and . Lions, The concentration-compactness principle in the calculus of variations . The locally compact case, part 1, Ann. Inst. H. Poincaré (C), vol.2, issue.1, pp.109-145, 1984.

O. [. Lieb and . Lazarev, A smooth, complex generalization of the Hobby- Rice theorem, Math. Jour, vol.62, pp.1133-1141, 2013.

B. [. Lieb and . Simon, The Thomas-Fermi theory of atoms, molecules and solids, Advances in Mathematics, vol.23, issue.1, pp.22-116, 1977.
DOI : 10.1016/0001-8708(77)90108-6

R. [. Lieb and . Schrader, Current densities in density-functional theory, Physical Review A, vol.88, issue.3, p.32516, 2013.
DOI : 10.1103/PhysRevA.88.032516

J. [. Luttinger and . Ward, Ground-State Energy of a Many-Fermion System. II, Physical Review, vol.118, issue.5, pp.1417-1427, 1960.
DOI : 10.1103/PhysRev.118.1417

M. A. Marques, N. T. Maitra, F. M. Nogueira, E. K. Gross, and A. Rubio, Fundamentals of time-dependent density functional theory, 2012.
DOI : 10.1007/978-3-642-23518-4

H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Physical Review B, vol.13, issue.12, pp.5188-5192, 1976.
DOI : 10.1103/PhysRevB.13.5188

M. A. Marques, C. A. Ullrich, F. M. Nogueira, A. Rubio, K. Burke et al., Time-dependent density functional theory, Lecture Notes in Physics, vol.706, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00438357

]. H. Nus72 and . Nussenzveig, Causality and Dispersion Relations, Mathematics in Science and Engineering, vol.95, 1972.

G. Onida, L. Reining, and A. Rubio, Electronic excitations: density-functional versus many-body Green???s-function approaches, Reviews of Modern Physics, vol.74, issue.2, pp.601-659, 2002.
DOI : 10.1103/RevModPhys.74.601

G. Panati, Triviality of Bloch and Bloch???Dirac Bundles, Annales Henri Poincar??, vol.8, issue.5, pp.995-1011, 2007.
DOI : 10.1007/s00023-007-0326-8

G. S. Pau, Reduced-basis method for band structure calculations, Physical Review E, vol.76, issue.4, p.46704, 2007.
DOI : 10.1103/PhysRevE.76.046704

K. [. Perdew, M. Burke, and . Ernzerhof, Generalized Gradient Approximation Made Simple, Physical Review Letters, vol.77, issue.18, pp.3865-3868, 1996.
DOI : 10.1103/PhysRevLett.77.3865

A. Pinkus, A simple proof of the Hobby-Rice theorem, Proc. Amer, pp.82-84, 1976.
DOI : 10.1090/S0002-9939-1976-0425470-0

J. P. Perdew and Y. Wang, Accurate and simple analytic representation of the electron-gas correlation energy, Physical Review B, vol.45, issue.23, pp.13244-13249, 1992.
DOI : 10.1103/PhysRevB.45.13244

J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Physical Review B, vol.23, issue.10, pp.5048-5079, 1981.
DOI : 10.1103/PhysRevB.23.5048

R. [. Rojas, R. J. Godby, and . Needs, Calculations of Self-Energies and Dielectric Response Functions of Solids, Physical Review Letters, vol.74, issue.10, pp.1827-1830, 1995.
DOI : 10.1103/PhysRevLett.74.1827

C. Roostgaard, K. W. Jacobsen, and K. S. Thygesen, Fully self-consistent GW calculations for molecules, Physical Review B, vol.81, issue.8, p.85103, 2010.
DOI : 10.1103/PhysRevB.81.085103

B. [. Reed and . Simon, Methods of Modern Mathematical Physics, Analysis of Operators, vol.IV, 1978.

. M. Rsw-+-99-]-m, L. Rieger, I. D. Steinbeck, H. N. White, R. W. Rojas et al., The GW space-time method for the self-energy of large systems, Comput. Phys. Commun, vol.117, issue.3, pp.211-228, 1999.

]. V. Rut13 and . Rutherfoord, On the Lazarev?Lieb extension of the Hobby?Rice theorem, Adv. Math, vol.244, pp.16-22, 2013.

S. Sharma, J. K. Dewhurst, C. Ambrosch-draxl, S. Kurth, N. Helbig et al., First-Principles Approach to Noncollinear Magnetism: Towards Spin Dynamics, Physical Review Letters, vol.98, issue.19, p.98, 2007.
DOI : 10.1103/PhysRevLett.98.196405

]. A. Sdvl06, N. E. Stan, R. Dahlen, and . Van-leeuwen, Fully self-consistent GW calculations for atoms and molecules, Europhys. Lett, vol.76, pp.298-304, 2006.

]. B. Sim05 and . Simon, Trace Ideals and Their Applications, Mathematical Surveys and Monographs, 2005.

]. J. Sla51 and . Slater, A simplification of the Hartree-Fock method, Phys. Rev, vol.81, pp.385-390, 1951.

J. Ph, Solovej, Proof of the ionization conjecture in a reduced Hartree-Fock model, Invent. Math, vol.104, issue.1, pp.291-311, 1991.

E. Seiler and B. Simon, Bounds in the Yukawa2 quantum field theory: Upper bound on the pressure, Hamiltonian bound and linear lower bound, Communications in Mathematical Physics, vol.93, issue.2, pp.99-114, 1975.
DOI : 10.1007/BF01629241

F. [. Schuch and . Verstraete, Computational complexity of interacting electrons and fundamental limitations of density functional??theory, Nature Physics, vol.8, issue.10, pp.1-8, 2009.
DOI : 10.1103/PhysRevA.78.012352

J. G. Taylor, Dispersion relations and Schwartz's distributions, Annals of Physics, vol.5, issue.4, pp.391-398, 1958.
DOI : 10.1016/0003-4916(58)90008-3

]. E. Tit48 and . Titchmarsh, Introduction to the Theory of Fourier Integrals, 1948.

S. [. Tellgren, T. Kvaal, and . Helgaker, -representability for prescribed density and paramagnetic current density, Physical Review A, vol.89, issue.1, p.12515, 2014.
DOI : 10.1103/PhysRevA.89.012515
URL : https://hal.archives-ouvertes.fr/hal-01082875

P. [. Taut, H. Machon, and . Eschrig, -representability of the exact solutions of the Schr??dinger equation for a two-electron quantum dot in a homogeneous magnetic field, Physical Review A, vol.80, issue.2, p.22517, 2009.
DOI : 10.1103/PhysRevA.80.022517

]. S. Val80 and . Valone, Consequences of extending 1-matrix energy functionals from pure-state representable to all ensemble representable 1 matrices A local exchange-correlation potential for the spin polarized case. I, J. Chem. Phys. J. Phys. C, vol.73, issue.5 13, pp.1629-1642, 1344.

]. U. Barth and B. Holm, within the random-phase approximation, Physical Review B, vol.54, issue.12, pp.8411-88, 1996.
DOI : 10.1103/PhysRevB.54.8411

W. Der-linden and P. Horsch, Precise quasiparticle energies and Hartree-Fock bands of semiconductors and insulators, Physical Review B, vol.37, issue.14, pp.8351-8362, 1988.
DOI : 10.1103/PhysRevB.37.8351

G. Vignale, Density-functional theory in strong magnetic fields, Physical Review Letters, vol.59, issue.20, pp.2360-2363, 1987.
DOI : 10.1103/PhysRevLett.59.2360

M. [. Vignale and . Rasolt, Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields, Physical Review B, vol.37, issue.18, pp.10685-10696, 1988.
DOI : 10.1103/PhysRevB.37.10685

L. [. Vosko, M. Will, and . Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis, Canadian Journal of Physics, vol.58, issue.8, pp.58-1200, 1980.
DOI : 10.1139/p80-159

H. Yserentant, Regularity and Approximability of Electronic Wave Functions, Lecture Notes in Mathematics, vol.2000, 2010.
DOI : 10.1007/978-3-642-12248-4

G. M. Zhislin, A study of the spectrum of the Schrödinger operator for a system of several particles, Trudy Moskov. Mat. Obsc, vol.9, pp.81-120, 1960.