This article is concerned with the numerical simulations of perfect crystals. We study the rate of convergence of the reduced Hartree−Fock (rHF) model in a supercell towards the periodic rHF model in the whole space. We prove that, whenever the crystal is an insulator or a semi-conductor, the supercell energy per unit cell converges exponentially fast towards the periodic rHF energy per unit cell, with respect to the size of the supercell.
Accepted:
DOI: 10.1051/m2an/2015084
Keywords: Reduced Hartree−Fock, supercell model, Riemann sums, analytic functions
@article{M2AN_2016__50_5_1403_0, author = {Gontier, David and Lahbabi, Salma}, title = {Convergence rates of supercell calculations in the reduced {Hartree\ensuremath{-}Fock} model}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1403--1424}, publisher = {EDP-Sciences}, volume = {50}, number = {5}, year = {2016}, doi = {10.1051/m2an/2015084}, zbl = {1356.35195}, mrnumber = {3554547}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015084/} }
TY - JOUR AU - Gontier, David AU - Lahbabi, Salma TI - Convergence rates of supercell calculations in the reduced Hartree−Fock model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1403 EP - 1424 VL - 50 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015084/ DO - 10.1051/m2an/2015084 LA - en ID - M2AN_2016__50_5_1403_0 ER -
%0 Journal Article %A Gontier, David %A Lahbabi, Salma %T Convergence rates of supercell calculations in the reduced Hartree−Fock model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1403-1424 %V 50 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015084/ %R 10.1051/m2an/2015084 %G en %F M2AN_2016__50_5_1403_0
Gontier, David; Lahbabi, Salma. Convergence rates of supercell calculations in the reduced Hartree−Fock model. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 5, pp. 1403-1424. doi : 10.1051/m2an/2015084. http://www.numdam.org/articles/10.1051/m2an/2015084/
A definition of the ground state energy for systems composed of infinitely many particles. Commun. Partial Differ. Eq. 28 (2003) 439–475. | DOI | MR | Zbl
, and ,Exponential localization of Wannier functions in insulators. Phys. Rev. Lett. 98 (2007) 046402. | DOI
, , , and ,E. Cancès, SCF algorithms for Hartree−Fock electronic calculations. In Mathematical Models and Methods for Ab Initio Quantum Chemistry, edited by M. Defranceschi and C. Le Bris. Springer (2000). | MR | Zbl
A new approach to the modeling of local defects in crystals: the reduced Hartree−Fock case. Commun. Math. Phys. 281 (2008) 129–177. | DOI | MR | Zbl
, and ,On the thermodynamic limit for Hartree−Fock type models. Ann. Inst. Henri Poincaré (C) 18 (2001) 687–760. | Numdam | MR | Zbl
, and ,Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and Zinc-blende structures. Phys. Rev. 141 (1966) 789–796. | DOI
and ,Analytical properties of -dimensional energy bands and Wannier functions. Phys. Rev. 135 (1964) A698–A707. | DOI | MR
,Energy bands and projection operators in a crystal: Analytic and asymptotic properties. Phys. Rev. 135 (1964) A685–A697. | DOI | MR
,T. Kato, Perturbation Theory for Linear Operators. Springer Science & Business Media (2012). | Zbl
Analytic properties of Bloch waves and Wannier functions. Phys. Rev. 115 (1959) 809–821. | DOI | MR | Zbl
,The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23 (1977) 22–116. | DOI | MR | Zbl
and ,Special points for Brillouin-zone integrations. Phys. Rev. B 13 (1976) 5188–5192. | DOI | MR
and ,Triviality of Bloch and Bloch–Dirac bundles. Ann. Henri Poincaré 8 (2007) 995–1011. | DOI | MR | Zbl
,M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press (1978). | MR
B. Simon, Trace Ideals and Their Applications. Mathematical Surveys and Monographs. American Mathematical Society (2005). | MR | Zbl
Cited by Sources: