Abstract
We prove the existence of solutions of the reduced Hartree–Fock equations at finite temperature for a periodic crystal with a small defect, and show the total screening of the defect charge by the electrons. We also show the convergence of the damped self-consistent field iteration using Kerker preconditioning to remove charge sloshing. As a crucial step of the proof, we define and study the properties of the dielectric operator.
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Notes
This reasoning also holds true for more complex methods. The Jacobian of the system has a condition number proportional to \(L^{2}\), and therefore we expect simple methods to require a number of iterations proportional to \(L^{2}\), and Krylov-type methods such as Anderson acceleration to require a number of iterations proportional to L [23, 26].
Note that if other occupation functions are used, the contour may need to be modified. For instance, Gaussian smearing [5] decays exponentially only if \(b < 1\). Our technique is less general than that of [20] based on the Helffer–Sjöstrand formula, which does not require any analyticity in \(f_{\varepsilon _{F}}\).
References
Adler , S.L.: Quantum theory of the dielectric constant in real solids. Phys. Rev. 126(2), 413, 1962
Benzi, M., Boito, P., Razouk, N.: Decay properties of spectral projectors with applications to electronic structure. SIAM Rev. 55(1), 3–64, 2013
Cancès , É., Deleurence , A., Lewin , M.: A new approach to the modeling of local defects in crystals: the reduced Hartree–Fock case. Commun. Math. Phys. 281(1), 129–177, 2008
Cancès , É., Ehrlacher , V.: Local defects are always neutral in the Thomas–Fermi–von Weiszäcker theory of crystals. Arch. Ration. Mech. Anal. 202(3), 933–973, 2011
Cancès, É., Ehrlacher, V., Gontier, D., Levitt, A., Lombardi, D.: Numerical quadrature in the Brillouin zone for periodic Schrodinger operators, 2018. arXiv preprint arXiv:1805.07144
Cancès, É., Le Bris, C.: On the convergence of SCF algorithms for the Hartree–Fock equations. ESAIM: Math. Model. Numer. Anal. 34(4), 749–774, 2000
Cancès , É., Lewin , M.: The dielectric permittivity of crystals in the reduced Hartree–Fock approximation. Arch. Ration. Mech. Anal. 197(1), 139–177, 2010
Cancès, É., Stoltz, G.: A mathematical formulation of the random phase approximation for crystals. Annales de l’Institut Henri Poincare (C) Non Linear Anal. 29(6), 887–925, 2012
Cancès, E., Kemlin, G., Levitt, A.: Convergence analysis of direct minimization and self-consistent iterations, 2020. arXiv preprint arXiv:2004.09088
Catto, I., Le Bris, C., Lions, P.-L.: On the thermodynamic limit for Hartree–Fock type models. Annales de l’Institut Henri Poincaré (C) Non Linear Anal. 18(6), 687–760, 2001
Chen, H., Lu, J., Ortner, C.: Thermodynamic limit of crystal defects with finite temperature tight binding. Arch. Ration. Mech. Anal. 230(2), 701–733, 2018
Frank , R., Lewin , M., Lieb , E., Seiringer , R.: A positive density analogue of the Lieb–Thirring inequality. Duke Math. J. 162(3), 435–495, 2013
Gontier , D., Lahbabi , S.: Supercell calculations in the reduced Hartree–Fock model for crystals with local defects. Appl. Math. Res. Express 2017(1), 1–64, 2016
Holmes , R.B.: A formula for the spectral radius of an operator. Am. Math. Mon. 75(2), 163–166, 1968
Kerker , G.P.: Efficient iteration scheme for self-consistent pseudopotential calculations. Phys. Rev. B 23(6), 3082, 1981
Lechleiter , A.: The Floquet–Bloch transform and scattering from locally perturbed periodic surfaces. J. Math. Anal. Appl. 446(1), 605–627, 2017
Levitt, A.: Convergence of gradient-based algorithms for the Hartree–Fock equations. ESAIM: Math. Model. Numer. Anal. 46(6), 1321–1336, 2012
Lieb , E., Simon , B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23(1), 22–116, 1977
Nazar , F., Ortner , C.: Locality of the Thomas–Fermi–von Weizsäcker Equations. Arch. Ration. Mech. Anal. 224(3), 817–870, 2017
Nier , F.: A variational formulation of Schrödinger–Poisson systems in dimension d \(\le \) 3. Commun. Partial Differ. Equ. 18(7–8), 1125–1147, 1993
Prodan , E., Garcia , S.R., Putinar , M.: Norm estimates of complex symmetric operators applied to quantum systems. J. Phys. A: Math. Gen. 39(2), 389, 2005
Reed, M., Simon, B.: Analysis of operators. In: Reed, M., Simon, B. (eds.) Methods of Modern Mathematical Physics, vol. IV. Academic Press, New York, NY 1978
Saad , Y.: Iterative Methods for Sparse Linear Systems, vol. 82. SIAM, Philadelphia 2003
Simon, B.: Trace Ideals and Their Applications. American Mathematical Society, Providence 2010
Solovej , J.-P.: Proof of the ionization conjecture in a reduced Hartree–Fock model. Invent. Math. 104(1), 291–311, 1991
Walker , H., Ni , P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735, 2011
Wiser , N.: Dielectric constant with local field effects included. Phys. Rev. 129(1), 62, 1963
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Stimulating discussions with Eric Cancès, Thierry Deutsch and Domenico Monaco are gratefully acknowledged.
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Levitt, A. Screening in the Finite-Temperature Reduced Hartree–Fock Model. Arch Rational Mech Anal 238, 901–927 (2020). https://doi.org/10.1007/s00205-020-01560-0
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DOI: https://doi.org/10.1007/s00205-020-01560-0