Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, p. 341-388

In this article, we provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. We prove that, for any local minimizer Φ 0 of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of Φ 0 for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.

DOI : https://doi.org/10.1051/m2an/2011038
Classification:  65N25,  65N35,  65T99,  35P30,  35Q40,  81Q05
Keywords: electronic structure calculation, density functional theory, Thomas-Fermi-von Weizsäcker model, Kohn-Sham model, nonlinear eigenvalue problem, spectral methods
@article{M2AN_2012__46_2_341_0,
     author = {Canc\`es, Eric and Chakir, Rachida and Maday, Yvon},
     title = {Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {2},
     year = {2012},
     pages = {341-388},
     doi = {10.1051/m2an/2011038},
     zbl = {1278.82003},
     mrnumber = {2855646},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_2_341_0}
}
Cancès, Eric; Chakir, Rachida; Maday, Yvon. Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 2, pp. 341-388. doi : 10.1051/m2an/2011038. http://www.numdam.org/item/M2AN_2012__46_2_341_0/

[1] A. Anantharaman and E. Cancès, Existence of minimizers for Kohn-Sham models in quantum chemistry. Ann. Inst. Henri Poincaré 26 (2009) 2425-2455. | Numdam | MR 2569902 | Zbl 1186.81138

[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. | MR 612246 | Zbl 0478.49035

[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. | JFM 53.0845.04

[5] G. Bourdaud and M. Lanza De Cristoforis, Regularity of the symbolic calculus in Besov algebras. Stud. Math. 184 (2008) 271-298. | MR 2369144 | Zbl 1139.46030

[6] E. Cancès, R. Chakir and Y. Maday, Numerical analysis of nonlinear eigenvalue problems. J. Sci. Comput. 45 (2010) 90-117. | MR 2679792 | Zbl 1203.65237

[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. | MR 2008386 | Zbl 1070.81534

[9] E. Cancès, C. Le Bris and Y. Maday, Méthodes mathématiques en chimie quantique. Springer (2006). | MR 2426947

[10] E. Cancès, G. Stoltz, V.N. Staroverov, G.E. Scuseria and E.R. Davidson, Local exchange potentials for electronic structure calculations. MathematicS In Action 2 (2009) 1-42. | MR 2520849 | Zbl 1177.47092

[11] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods: fundamentals in single domains. Springer (2006). | MR 2223552 | Zbl 1121.76001

[12] I. Catto, C. Le Bris and P.-L. Lions, Mathematical theory of thermodynamic limits: Thomas-Fermi type models. Oxford University Press (1998). | MR 1673212 | Zbl 0938.81001

[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. | MR 2723492 | Zbl 1194.35293

[15] R.M. Dreizler and E.K.U. Gross, Density functional theory. Springer (1990). | Zbl 0723.70002

[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. | MR 1646856 | Zbl 0928.65050

[17] V. Gavini, J. Knap, K. Bhattacharya and M. Ortiz, Non-periodic finite-element formulation of orbital-free density functional theory. J. Mech. Phys. Solids 55 (2007) 669-696. | MR 2318929 | Zbl 1162.74461

[18] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, 3rd edition. Springer (1998). | Zbl 1042.35002

[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. | MR 180312

[21] W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects. Phys. Rev. 140 (1965) A1133-A1138. | MR 189732

[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. | MR 2755499 | Zbl 1208.82063

[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). | Zbl 1109.81008

[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. | MR 554891

[27] E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53 (1981) 603-641. | MR 629207 | Zbl 1114.81336

[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. | MR 1990591 | Zbl 1027.81043

[30] W. Sickel, Superposition of functions in Sobolev spaces of fractional order. A survey. Banach Center Publ. 27 (1992) 481-497. | MR 1205849 | Zbl 0792.47062

[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. Solids 58 (2010) 256-280. | MR 2649224 | Zbl 1193.81006

[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. | MR 580595

[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. | MR 2293570 | Zbl 1119.35095