B. Annexe, Evaluation de l'énereie totale en PAW

, Annexe C Expressions de l'opérateur Hamiltonien dans le formalisme PA\M Dans cette section, nous expliciterons l'évaluation des opérateurs recouvrement et Hamiltonien effectif tels qu'il

, 1-Opérateur de recouvrement La matrice de recouvrement dans la représentation AE est donnée par les éléments matricielles de l'opérateur unité. Dans cette représentation les ondes planes forment une base orthonormale

P. Dans-la-représentation, cette même base n'est plus orthogonale ô:r+ t lin,) [(on

. Àr-'àz-par-conséquent,

, Cette dernière relation s'interprète comme la conséquence de la relaxation de la condition de conservation de norme

P. Hohenberg and W. Kohn, Phys. Rev, vol.136, issue.8864, 1964.

W. Kohn and L. J. Sham, Phys. Rev', vol.140, p.41133, 1965.

J. P. Perdew, What do the Kohn-Sham orbital energies mean?" in Density-Functional Methods in Physics, NATO ASI Series B: Physics, p.265, 1985.

C. Almbladh and U. Barth, Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues, Physical Review B, vol.29, issue.6, p.3231, 1985.
DOI : 10.1103/PhysRevA.29.2322

D. M. Ceperley and B. J. Alder, Ground State of the Electron Gas by a Stochastic Method, Physical Review Letters, vol.34, issue.7, p.566, 1980.
DOI : 10.1039/tf9383400678

J. P. Perdew and A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Physical Review B, vol.1, issue.10, p.5048, 1981.
DOI : 10.1103/PhysRevB.1.1044

U. Barth, Proceedings of the workshop "Method,s of electronic structure calculations, p.2

O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B, vol.3, p.4274, 1976.

D. C. Langreth and J. Perdew, Phys. Rev. B, vol.2, p.5469, 1980.

D. C. Langreth and M. J. , Beyond the local-density approximation in calculations of ground-state electronic properties, Physical Review B, vol.10, issue.4, p.1809, 1983.
DOI : 10.1103/PhysRevB.10.2221

D. C. Langreth and M. J. , Erratum: Beyond the local-density approximation in calculations of ground-state electronic properties, Physical Review B, vol.29, issue.4, p.2310, 1984.
DOI : 10.1103/PhysRevB.29.2310.2

C. D. Hu and D. C. Langreth, A Spin Dependent Version of the Langreth-Mehl Exchange-Correlation Functional, Physica Scripta, vol.32, issue.4, p.391, 1985.
DOI : 10.1088/0031-8949/32/4/024

J. P. Perdew, Accurate Density Functional for the Energy: Real-Space Cutoff of the Gradient Expansion for the Exchange Hole, Physical Review Letters, vol.53, issue.16, p.1665, 1985.
DOI : 10.1103/PhysRevLett.53.2087

, J. P. Perdew, Physica B, vol.721, 1991.

J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson et al., Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation, Physical Review B, vol.20, issue.11, p.6671, 1992.
DOI : 10.1103/PhysRevB.20.3136

URL : http://estudogeral.sib.uc.pt/jspui/bitstream/10316/2535/1/Doutor%20Fiolhais%201.pdf

M. Gunnarsson, B. I. Jonsson, and . Lundqvist, Descriptions of exchange and correlation effects in inhomogeneous electron systems, Physical Review B, vol.10, issue.8, p.3136, 1979.
DOI : 10.1103/PhysRevB.10.3052

J. C. Slater and J. Wood, Int. J. Quant. Chem, 1971.

C. Almbladh and U. Barth, Phys. Rev. B, vol.33, p.3307, 1976.

J. Janak, in density-functional theory, Physical Review B, vol.184, issue.12, p.7165, 1978.
DOI : 10.1103/PhysRev.184.672

L. J. Sham and M. Schlûter, Phys. Rev. Lett, vol.5, p.1888, 1983.

L. J. Sham and M. Schlûter, Density-functional theory of the band gap, Physical Review B, vol.51, issue.2, p.3883, 1985.
DOI : 10.1002/pssb.2220900208

J. P. Bibliographie-l22l, M. Perdew, and . Levy, Phys. Rev. Lett, vol.5, p.1884, 1983.

R. W. Godby, M. Schlûter, L. , R. W. Godby, M. Schlûter et al., J. Sham, Phys. Rev. B Phys. Rev. Lett. J. Sham, Phys. Rev. B, vol.37, issue.36, pp.10159-24156497, 1986.

R. Car and M. Parrinello, Phys. Rev. B, vol.55, p.247, 1985.

S. Kirkpatrick, C. D. Gelatt, J. , and M. P. Vecchi, Science, vol.22, p.67, 1983.

L. Verlet, Phys. Rev, vol.59, p.98, 1967.

P. E. Blôchl, Projector augmented-wave method, Physical Review B, vol.44, issue.24, p.17953, 1994.
DOI : 10.1103/PhysRevB.44.13063

N. F. Mott, Proc. Royal Soc. London A 42, p.416, 1949.

J. Hubbard, Proc. Royal Soc. London A276, 1963.

J. C. Slater, Magnetic Effects and the Hartree-Fock Equation, Physical Review, vol.71, issue.4, p.538, 1951.
DOI : 10.1007/BF01341708

P. Dufek, P. Blaha, V. Sliwko, and K. Schwarz, Generalized-gradient-approximation description of band splittings in transition-metal oxides and fluorides, Physical Review B, vol.2, issue.15, p.10170, 1994.
DOI : 10.1088/0953-8984/2/24/003

K. Terakura, T. Oguchi, A. R. Williams, and J. Kûbler, Band theory of insulating transition-metal monoxides: Band-structure calculations, Physical Review B, vol.153, issue.8, p.4734, 1984.
DOI : 10.1103/PhysRev.153.673

A. Svane and O. Gunnarsson, Transition-metal oxides in the self-interaction???corrected density-functional formalism, Physical Review Letters, vol.153, issue.9, p.1148, 1990.
DOI : 10.1088/0953-8984/2/17/008

Z. Szolek and W. M. Temmerman, Phys. Rev B, vol.4, p.4029, 1993.

M. Arai and T. Fujiwara, Phys. Rev. B, vol.5, p.1477, 1995.

V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B, vol.44, pp.9-12, 1991.

V. Anisimov, L. V. Solovyev, M. A. Korotin, M. T. Czyzyk, and G. A. Sawatzky, Density-functional theory and NiO photoemission spectra, Physical Review B, vol.67, issue.23, p.16929, 1993.
DOI : 10.1103/PhysRevLett.67.1035

V. I. Anisimov and O. Gunnarsson, Phys. Rev. B, vol.43, issue.7570, 1991.

V. I. Anisimov, P. Kuiper, and J. Nordgren, First-principles calculation of NiO valence spectra in the impurity-Anderson-model approximation, Physical Review B, vol.43, issue.12, p.8257, 1994.
DOI : 10.1016/0304-8853(86)90187-3

W. E. Pickett, S. C. Erwin, and E. C. Ethridge, method for a local-orbital basis, Physical Review B, vol.46, issue.3, p.1201, 1998.
DOI : 10.1103/PhysRevB.46.3771

M. R. Castell, P. L. Wincott, N. G. Condon, G. Muggelberg, S. L. Thornton et al., Atomic-resolution STM of a system with strongly correlated electrons:NiO(001) surface structure and defect sites, Physical Review B, vol.5, issue.57, p.7859, 1997.
DOI : 10.1103/PhysRevB.5.290

S. L. Dudarev, A. I. Liechtenstein, M. R. Castell, G. A. Briggs, and A. P. Sutton, Surface states on NiO (100) and the origin of the contrast reversal in atomically resolved scanning tunneling microscope images, Physical Review B, vol.77, issue.57, p.4900, 1997.
DOI : 10.1103/PhysRevLett.77.1322

I. V. Solovyev, P. H. Dederichs, and V. I. Anisimov, Phys. Rev. B, vol.43, p.16861, 1994.

J. C. Slater, Quantum Theory of Molecules and Solids, 1974.

Î. S. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Electron-energy-loss spectra and the structural stability of nickel oxide:??????An LSDA+U study, Physical Review B, vol.50, issue.3, p.1505, 1998.
DOI : 10.1103/PhysRevB.50.5041

A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Density-functional theory and strong interactions: Orbital ordering in Mott-Hubbard insulators, Physical Review B, vol.71, issue.8, p.5467, 1995.
DOI : 10.1103/PhysRevLett.71.3214

A. B. Shick, A. I. Liechtenstein, and W. E. Pickett, Implementation of the LDA+U method using the full-potential linearized augmented plane-wave basis, Physical Review B, vol.57, issue.15, pp.10763-155, 1999.
DOI : 10.1103/PhysRevB.57.6884

F. M. De-groot, J. C. Fuggle, B. T. Thole, and G. A. Sawatzky, transition-metal compounds: An atomic multiplet description including the crystal field, Physical Review B, vol.40, issue.9, p.5459
DOI : 10.1103/PhysRevB.40.12542

N. Bulut, D. J. Scalapino, and S. R. White, Comparison of Monte Carlo and diagrammatic calculations for the two-dimensional Hubbard model, Physical Review B, vol.39, issue.189, p.2742, 1993.
DOI : 10.1103/PhysRevB.39.11663

M. Alouani and J. M. Wills,

P. Wei and Z. Q. Qi, correlation correction, Physical Review B, vol.46, issue.16, p.10864, 1994.
DOI : 10.1103/PhysRevB.46.12165

J. Hugel and M. Kamal, J. Phys. Condens. Matter, vol.9647, 1997.

W. L. Roth, Magnetic Structures of MnO, FeO, CoO, and NiO, Physical Review, vol.12, issue.6, p.1333, 1958.
DOI : 10.1051/jphysrad:01951001208076500

W. L. Roth, Phys. Rev, vol.7, issue.1, p.772, 1958.

W. L. Roth, J. Appl. Phys, vol.3, 1960.

T. Yamada and J. , Phys. Soc. Jpn, vol.31664, 1966.

M. Grimsditch, L. B. Mcneil, and D. J. Lockwood, Phys. Rev. B, vol.58, p.74462, 1998.

S. Hiifner, P. Steiner, I. Sander, F. Reinert, H. Schnitt et al., Phys.B, vol.86, p.207, 1992.

G. A. Sawatzky and J. W. Allen, Phys. Rev. Lett, vol.53, p.2239, 1984.

S. Hiifner, Solid State Commun, 1985.

S. Hûfner, J. Osterwaldr, T. Riesterer, and F. Hullinger, Solid State Commun, pp.52-793, 1984.

Z. Shen, C. K. Shih, W. E. Jepsen, I. Spicer, J. W. Lindau et al., Aspects of the correlation effects, antiferromagnetic order, and translational symmetry of the electronic structure of NiO and CoO, Physical Review Letters, vol.41, issue.20, p.2442
DOI : 10.1103/PhysRevB.41.6399

Z. Shen, J. W. Allen, P. A. Lindberg, D. S. Dessau, B. O. Wells et al., Phys. Rev. B, vol.42, pp.8-15, 1990.

Z. Shen, R. S. List, D. S. Dessau, B. O. Wells, A. J. Jepsen et al., , p.443604, 1991.

H. Kuhlenbeck, Phys. Rev. B, vol.43, 1969.

O. Jepsen and O. K. Andersen, The electronic structure of h.c.p. Ytterbium, Solid State Communications, vol.9, issue.20, p.1763, 1971.
DOI : 10.1016/0038-1098(71)90313-9

G. Lehmann and M. Taut, On the Numerical Calculation of the Density of States and Related Properties, Physica Status Solidi (b), vol.3, issue.2, p.469, 1972.
DOI : 10.1103/PhysRevB.5.1276

H. A. Alperin, J. Phys Soc. Jpn, vol.12, 1962.

A. K. Cheetham and D. A. Hope, Physical Review B, vol.12, issue.11, p.6964, 1983.
DOI : 10.1107/S0365110X59000871

V. Fernandez, C. Vettier, F. De-bergevin, C. Giles, and W. Neubeck, Observation of orbital moment in NiO, Physical Review B, vol.140, issue.144, p.7870
DOI : 10.1016/0304-8853(94)01378-0

R. W. Wyckoff, , 1963.

F. Aryasetiawan and . Gunnarsson, Approximation, Physical Review Letters, vol.39, issue.16, p.3221, 1995.
DOI : 10.1103/PhysRevB.39.1708

S. Massida, A. Continenza, M. Posternak, and A. Baldereschi, Quasiparticle energy bands of transition-metal oxides within a model GW scheme, Physical Review B, vol.50, issue.20, p.13494
DOI : 10.1103/PhysRevB.50.5041

F. Manghi, C. Calandra, and S. Ossicini, Quasiparticle Band Structure of NiO: The Mott-Hubbard Picture Regained, Physical Review Letters, vol.43, issue.23, p.3129, 1994.
DOI : 10.1103/PhysRevB.43.3671

S. L. Adler, Quantum Theory of the Dielectric Constant in Real Solids, Physical Review, vol.99, issue.2, p.413, 1962.
DOI : 10.1103/PhysRev.99.1128

N. Wiser, Phys. Rev, vol.29, issue.62, 1963.

H. Ehrenreich and M. L. Cohen, Self-Consistent Field Approach to the Many-Electron Problem, Physical Review, vol.34, issue.4, p.786, 1959.
DOI : 10.1139/p56-154

M. S. Hybertsen and S. G. Louie, static dielectric matrices from the density-functional approach. I. Formulation and application to semiconductors and insulators, Physical Review B, vol.27, issue.11, p.5585, 1987.
DOI : 10.1103/PhysRevB.27.3735

R. Powell and W. E. Spicer, Phys. Rev. B, vol.2, p.2185, 1970.

D. Cappus, C. Xu, D. Ehrlich, B. Dillman, C. A. Ventrice-jr et al., Hydroxyl groups on oxide surfaces: NiO(100), NiO(111) and Cr2O3(111), Chemical Physics, vol.177, issue.2, p.533, 1993.
DOI : 10.1016/0301-0104(93)80031-4

A. Fujimori, F. Minami, and S. Sugano, Multielectron satellites and spin polarization in photoemission from Ni compounds, Physical Review B, vol.317, issue.34, p.5225, 1984.
DOI : 10.1002/zaac.19623170502

A. Fujimori and F. Minami, Valence-band photoemission and optical absorption in nickel compounds, Physical Review B, vol.28, issue.2, p.957, 1984.
DOI : 10.1103/PhysRevB.28.5347

L. Messick, W. C. Walker, and R. Glosser, Direct and Temperature-Modulated Reflectance Spectra of MnO, CoO, and NiO, Physical Review B, vol.7, issue.10, p.3941, 1972.
DOI : 10.1103/PhysRevB.5.290

H. Kanda, M. Yoshiya, F. Oba, K. Ogasawara, and H. Adachi, -edge electron-energy-loss near-edge structure of NiO, Physical Review B, vol.54, issue.15, p.9693
DOI : 10.1103/PhysRevLett.54.2725

H. K. Bowen, D. Adler, and B. H. Auker, Electrical and optical properties of FeO, Journal of Solid State Chemistry, vol.12, issue.3-4, p.355, 1975.
DOI : 10.1016/0022-4596(75)90340-0

S. G. , Louie travail en ce moment sur les effets excitoniques sur le spectre optique de NiO (communication privée)

V. I. Anisimov, M. A. Korotin, J. Zaaren, and O. K. Andersen, from first principles, Physical Review Letters, vol.59, issue.3, p.345
DOI : 10.1143/JPSJ.59.3053

A. Liechtenstein, V. P. Antropov, and B. N. Harmon, Phys. Rev. B, vol.49, p.770, 1994.

J. Bouchet, B. Siberchicot, F. Jollet, and A. Pasturel, J. Phys.: Condens. Matter, vol.2, p.7723, 2000.

]. M. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, total-energy calculations: molecular dynamics and conjugate gradients, Reviews of Modern Physics, vol.20, issue.4, p.1045, 1992.
DOI : 10.1103/PhysRevB.20.4082

P. W. Tasker and J. Phys, Cr Solid State Phys, p.4977, 1979.

A. M. Stoneham, Defects and their structure in Non, 1976.

F. Bertaut, , 1958.

J. Bardeen, Phys. Rev. Lett, vol.6, issue.57, 1961.

R. M. Feenstra, J. A. Stroscio, and A. P. Fein, Tunneling spectroscopy of the Si(111)2 ?? 1 surface, Surface Science, vol.181, issue.1-2, p.295, 1987.
DOI : 10.1016/0039-6028(87)90170-1

R. Lacmann, Colloq. Int. CNRS, vol.152, issue.195, 1965.

D. Wolf, Phys. Rev. Lett. Phys. Rev, vol.68, issue.52, p.5323, 1992.

, 1C

C. Noguera, Physics and Chemistry of orides surfaces, 1996.

C. , V. Jr, . Th, H. Bertrams, A. Hannemann et al., Phys. Rev. B, vol.49, p.5773, 1994.

F. Rohr, K. Wirth, J. Libuda, D. Cappus, M. Bâumer et al., Surf. Sci. Lett, pp.315-977, 1994.

]. A. Barbier, G. Renaud, and A. Stierle, Surf. Sci, pp.4-6, 1998.

A. , Phys. Rev. Lett, vol.84, p.2897, 2000.

O. L. Warren and P. A. Thiel, J. Chem. Phys, vol.00, p.659, 1994.

G. T. Tyuliev and K. L. Kostov, XPS/HREELS study of NiO films grown on Ni(111), Physical Review B, vol.394, issue.153, p.2900, 1999.
DOI : 10.1016/S0039-6028(97)00787-5

N. Kitakatsu, V. Maurice, and P. Marcus, Surf. Sci, vol.41121, issue.5, 1998.

F. P. Netzer and M. Prutton, LEED and electron spectroscopic observations on NiO (100), Journal of Physics C: Solid State Physics, vol.8, issue.15, p.2401, 1975.
DOI : 10.1088/0022-3719/8/15/014

M. R. Welton-cook and M. Prutton, J. Phys. C, vol.3, p.3993, 1980.

P. Cantini, R. Tatarek, and G. P. Felcher, Phys. Rev. B, vol.9, p.1161, 1979.

J. P. Taennies, G. Witte, A. Shikin, and K. H. Rieder, J. Electron Spectrosc. Relat. Phenom, vol.64, issue.65, p.677, 1993.

S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets of Transition-Metal lons in Crystals, 1970.

A. Freitag, V. Staemmler, D. Cappus, C. A. Ventrice, K. Shamery et al., Chem. Phys. Lett, vol.2, issue.0, p.10, 1993.

A. Gorschlûter and H. Merz, excitations in NiO(100) and CoO(100), Physical Review B, vol.62, issue.57, p.17293, 1994.
DOI : 10.1103/PhysRevLett.62.478

G. J. Janssen and W. C. Nieuwpoort, Band gap in NiO: A cluster study, Physical Review B, vol.54, issue.57, p.3449, 1988.
DOI : 10.1016/0304-8853(86)90187-3

K. Okada and A. Kotani, Interatomic and Intra-Atomic Configuration Interactions in Core-Level X-Ray Photoemission Spectra of Late Transition-Metal Compounds, Journal of the Physical Society of Japan, vol.61, issue.12, p.4619, 1992.
DOI : 10.1143/JPSJ.61.4619

J. Zaanen, G. A. Sawatzky, and J. W. Allen, Band gaps and electronic structure of transition-metal compounds, Physical Review Letters, vol.29, issue.4, p.418, 1985.
DOI : 10.1103/PhysRevB.2.3918

URL : https://openaccess.leidenuniv.nl/bitstream/handle/1887/5216/8_400_108.pdf?sequence=1

M. F. Crommie, C. P. Lutz, D. M. Eigler, and B. J. Heller, WAVES ON A METAL SURFACE AND QUANTUM CORRALS, Surface Review and Letters, vol.02, issue.01, p.127, 1995.
DOI : 10.1142/S0218625X95000121

C. G. Kinniburgh and J. A. Walker, Surf. Sci', p.631274, 1977.