Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 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 : 10.1051/m2an/2011038
Classification : 65N25, 65N35, 65T99, 35P30, 35Q40, 81Q05
Mots clés : 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 },
     pages = {341--388},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {2},
     year = {2012},
     doi = {10.1051/m2an/2011038},
     mrnumber = {2855646},
     zbl = {1278.82003},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2011038/}
}
TY  - JOUR
AU  - Cancès, Eric
AU  - Chakir, Rachida
AU  - Maday, Yvon
TI  - Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 341
EP  - 388
VL  - 46
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2011038/
DO  - 10.1051/m2an/2011038
LA  - en
ID  - M2AN_2012__46_2_341_0
ER  - 
%0 Journal Article
%A Cancès, Eric
%A Chakir, Rachida
%A Maday, Yvon
%T Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 341-388
%V 46
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2011038/
%R 10.1051/m2an/2011038
%G en
%F 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 , Tome 46 (2012) no. 2, pp. 341-388. doi : 10.1051/m2an/2011038. http://archive.numdam.org/articles/10.1051/m2an/2011038/

[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 | Zbl

[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 | Zbl

[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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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

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

[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 | Zbl

[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

[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

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

[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 | Zbl

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

[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 | Zbl

[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

[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 | Zbl

Cité par Sources :