Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
Eric Cancès; Rachida Chakir; Yvon Maday
- Volume: 46, Issue: 2, page 341-388
- ISSN: 0764-583X
Access Full Article
topAbstract
topHow to cite
topReferences
top- [1] A. Anantharaman and E. Cancès, Existence of minimizers for Kohn-Sham models in quantum chemistry. Ann. Inst. Henri Poincaré26 (2009) 2425–2455. Zbl1186.81138MR2569902
- [2] R. Benguria, H. Brezis and E.H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Comm. Math. Phys.79 (1981) 167–180. Zbl0478.49035MR612246
- [3] X. Blanc and E. Cancès, Nonlinear instability of density-independent orbital-free kinetic energy functionals. J. Chem. Phys.122 (2005) 214–106.
- [4] M. Born and J.R. Oppenheimer, Zur quantentheorie der molekeln. Ann. Phys.84 (1927) 457–484. Zbl53.0845.04JFM53.0845.04
- [5] G. Bourdaud and M. Lanza de Cristoforis, Regularity of the symbolic calculus in Besov algebras. Stud. Math.184 (2008) 271–298. Zbl1139.46030MR2369144
- [6] E. Cancès, R. Chakir and Y. Maday, Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput.45 (2010) 90–117. Zbl1203.65237MR2679792
- [7] E. Cancès, R. Chakir, V. Ehrlacher and Y. Maday, in preparation.
- [8] E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational quantum chemistry: a primer, in Handbook of numerical analysis X. North-Holland, Amsterdam (2003) 3–270. Zbl1070.81534MR2008386
- [9] E. Cancès, C. Le Bris and Y. Maday, Méthodes mathématiques en chimie quantique. Springer (2006). MR2426947
- [10] E. Cancès, G. Stoltz, V.N. Staroverov, G.E. Scuseria and E.R. Davidson, Local exchange potentials for electronic structure calculations. MathematicS In Action2 (2009) 1–42. Zbl1177.47092MR2520849
- [11] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods: fundamentals in single domains. Springer (2006). Zbl1121.76001MR2223552
- [12] I. Catto, C. Le Bris and P.-L. Lions, Mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford University Press (1998). Zbl0938.81001MR1673212
- [13] H. Chen, X. Gong, L. He and A. Zhou, Convergence of adaptive finite element approximations for nonlinear eigenvalue problems. arXiv preprint, http://arxiv.org/pdf/1001.2344.
- [14] H. Chen, X. Gong and A. Zhou, Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model. Math. Methods Appl. Sci.33 (2010) 1723–1742. Zbl1194.35293MR2723492
- [15] R.M. Dreizler and E.K.U. Gross, Density functional theory. Springer (1990). Zbl0723.70002
- [16] A. Edelman, T.A. Arias and S.T. Smith, The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl.20 (1998) 303–353. Zbl0928.65050MR1646856
- [17] V. Gavini, J. Knap, K. Bhattacharya and M. Ortiz, Non-periodic finite-element formulation of orbital-free density functional theory. J. Mech. Phys. Solids55 (2007) 669–696. Zbl1162.74461MR2318929
- [18] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 3rd edition. Springer (1998). Zbl1042.35002
- [19] X. Gonze et al., ABINIT: first-principles approach to material and nanosystem properties. Computer Phys. Comm.180 (2009) 2582–2615.
- [20] P. Hohenberg and W. Kohn, Inhomogeneous electron gas. Phys. Rev. 136 (1964) B864–B871. MR180312
- [21] W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140 (1965) A1133–A1138. MR189732
- [22] B. Langwallner, C. Ortner and E. Süli, Existence and convergence results for the Galerkin approximation of an electronic density functional. Math. Mod. Methods Appl. Sci.20 (2010) 2237–2265. Zbl1208.82063MR2755499
- [23] C. Le Bris, Ph.D. thesis, École Polytechnique (1993).
- [24] W.A. Lester Jr. Ed., Recent advances in Quantum Monte Carlo methods. World Sientific (1997). Zbl1109.81008
- [25] W.A. Lester Jr., S.M. Rothstein and S. Tanaka Eds., Recent advances in Quantum Monte Carlo methods, Part II, World Sientific (2002).
- [26] M. Levy, Universal variational functionals of electron densities, first order density matrices, and natural spin-orbitals and solution of the V-representability problem. Proc. Natl. Acad. Sci. U.S.A.76 (1979) 6062–6065. MR554891
- [27] E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys.53 (1981) 603–641. Zbl1114.81336MR629207
- [28] E.H. Lieb, Density Functional for Coulomb systems. Int. J. Quant. Chem.24 (1983) 243–277.
- [29] Y. Maday and G. Turinici, Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations. Numer. Math.94 (2003) 739–770. Zbl1027.81043MR1990591
- [30] W. Sickel, Superposition of functions in Sobolev spaces of fractional order. A survey. Banach Center Publ. 27 (1992) 481–497. Zbl0792.47062MR1205849
- [31] P. Suryanarayana, V. Gavini, T. Blesgen, K. Bhattacharya and M. Ortiz, Non-periodic finite-element formulation of Kohn-Sham density functional theory. J. Mech. Phys. Solids58 (2010) 256–280. Zbl1193.81006MR2649224
- [32] N. Troullier and J.L. Martins, A straightforward method for generating soft transferable pseudopotentials. Solid State Commun.74 (1990) 613–616.
- [33] S. Valone, Consequences of extending 1matrix energy functionals from purestate representable to all ensemble representable 1 matrices. J. Chem. Phys.73 (1980) 1344–1349. MR580595
- [34] Y.A. Wang and E.A. Carter, Orbital-free kinetic energy density functional theory, in Theoretical methods in condensed phase chemistry, Progress in theoretical chemistry and physics 5. Kluwer (2000) 117–184.
- [35] A. Zhou, Finite dimensional approximations for the electronic ground state solution of a molecular system. Math. Methods Appl. Sci.30 (2007) 429–447. Zbl1119.35095MR2293570