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The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint

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Numerical Analysis of Multiscale Computations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 82))

Abstract

The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.

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References

  1. Adler, S.L.: Quantum theory of the dielectric constant in real solids. Phys. Rev. 126, 413–420 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ambrosio, L., Friesecke, G. Giannoulis, J.: Passage from quantum to classical molecular dynamics in the presence of Coulomb interactions. Commun. Part. Diff. Eq. 35, 1490–1515 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L., Figalli, A., Friesecke, G. Giannoulis, J., Paul, T.: Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data. To appear in Comm. Pure Appl. Math., (2011)

    Google Scholar 

  4. Anantharaman, A., Cancès, É.: Existence of minimizers for Kohn-Sham models in quantum chemistry. Ann. I. H. Poincaré-An 26, 2425–2455 (2009)

    MATH  Google Scholar 

  5. Arnold, A.: Self-consistent relaxation-time models in quantum mechanics, Commun. Part. Diff. Eq., 21(3-4), 473–506 (1996)

    Article  MATH  Google Scholar 

  6. Baroni, S., Resta, R.: Ab initio calculation of the macroscopic dielectric constant in silicon. Phys. Rev. B 33, 7017–7021 (1986)

    Article  Google Scholar 

  7. 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, 129–177 (2008)

    Article  MATH  Google Scholar 

  8. Cancès, É., Deleurence, A., Lewin, M.: Non-perturbative embedding of local defects in crystalline materials. J. Phys.: Condens. Mat. 20, 294213 (2008)

    Google Scholar 

  9. Cancès, É., Lewin, M.: The dielectric permittivity of crystals in the reduced Hartree-Fock approximation. Arch. Ration. Mech. Anal. 197, 139–177 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cancès, É., Stoltz, G.: in preparation

    Google Scholar 

  11. Catto, I, Le Bris, C., Lions, P.-L.: On the thermodynamic limit for Hartree-Fock type models. Ann. I. H. Poincaré-An 18, 687–760 (2001)

    Google Scholar 

  12. Dautray, R. and Lions, J.-L. Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Evolution Problems I. Springer-Verlag Berlin (1992)

    Book  MATH  Google Scholar 

  13. Dreizler, R., Gross, E.K.U.: Density functional theory. Springer Verlag, Berlin (1990)

    MATH  Google Scholar 

  14. Engel, G. E., Farid, B.: Calculation of the dielectric properties of semiconductors. Phys. Rev. B 46, 15812–15827 (1992)

    Article  Google Scholar 

  15. Frank, R.L., Lieb, E.H., Seiringer, R., Siedentop, H.: Müllers exchange-correlation energy in density-matrix-functional theory. Phys. Rev. A 76, 052517 (2007)

    Article  Google Scholar 

  16. Gajdoš, M., Hummer, K., Kresse, G., Furthmüller, J., Bechstedt, F.: Linear optical properties in the projector-augmented wave methodology. Phys. Rev. B 73, 045112 (2006)

    Article  Google Scholar 

  17. Gravejat, P., Lewin, M, Séré, É.: Ground state and charge renormalization in a nonlinear model of relativistic atoms. Commun. Math. Phys. 286, 179–215 (2009)

    Google Scholar 

  18. Hainzl, C., Lewin, M., Séré, É.: Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation. Commun. Math. Phys. 257, 515–562 (2005)

    Article  MATH  Google Scholar 

  19. Hainzl, C., Lewin, M., Séré, E., Solovej, J.P.: A minimization method for relativistic electrons in a mean-field approximation of quantum electrodynamics. Phys. Rev. A 76, 052104 (2007)

    Article  Google Scholar 

  20. Hainzl, C., Lewin, M., Solovej, J.P.: The mean-field approximation in Quantum Electrodynamics: the no-photon case. Commun. Pur. Appl. Math. 60(4), 546–596 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136, B864-B871 (1964)

    MathSciNet  Google Scholar 

  22. Hybertsen, M.S., Louie, S.G.: Ab initio static dielectric matrices from the density-functional approach. I. Formulation and application to semiconductors and insulators. Phys. Rev. B 35, 5585–5601 (1987)

    Google Scholar 

  23. Hybertsen, M.S., Louie, S.G.:Ab initio static dielectric matrices from the density-functional approach. II. Calculation of the screening response in diamond, Si, Ge, and LiCl. Phys. Rev. B 35, 5602–5610 (1987)

    Google Scholar 

  24. Kohn, W., Sham L. J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133 (1965)

    Article  MathSciNet  Google Scholar 

  25. Kunc, K., Tosatti, E.: Direct evaluation of the inverse dielectric matrix in semiconductors. Phys. Rev. B 29, 7045–7047 (1984)

    Article  Google Scholar 

  26. Lieb E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett. 46, 457–459 (1981)

    Article  MathSciNet  Google Scholar 

  27. Lieb, E.H.: Density Functional for Coulomb systems. Int. J. Quantum Chem. 24, 243–277 (1983)

    Article  Google Scholar 

  28. Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences. Springer, New York (1983)

    Google Scholar 

  29. Pick R. M., Cohen, M.H., Martin R. M.: Microscopic theory of force constants in the adiabatic approximation. Phys. Rev. B 1, 910–920 (1970)

    Article  Google Scholar 

  30. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)

    Google Scholar 

  31. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)

    MATH  Google Scholar 

  32. Resta, R., Baldereschi, A.: Dielectric matrices and local fields in polar semiconductors. Phys. Rev. B 23, 6615–6624 (1981)

    Article  Google Scholar 

  33. Simon, B.: Trace ideals and their applications, vol. 35 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1979)

    Google Scholar 

  34. Solovej, J.P.: Proof of the ionization conjecture in a reduced Hartree-Fock model. Invent. Math. 104, 291–311 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thomas, L.E.: Time dependent approach to scattering from impurities in a crystal. Commun. Math. Phys. 33, 335–343 (1973)

    Article  Google Scholar 

  36. Wiser, N.: Dielectric constant with local field effects included. Phys. Rev. 129, 62–69 (1963)

    Article  MATH  Google Scholar 

  37. Zhislin, G.M., Sigalov, A.G.: The spectrum of the energy operator for atoms with fixed nuclei on subspaces corresponding to irreducible representations of the group of permutations. Izv. Akad. Nauk SSSR Ser. Mat. 29, 835–860 (1965)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. CERMICS, Project-team Micmac, INRIA-Ecole des Ponts, Université Paris-Est, 6 & 8 avenue Blaise Pascal, 77455, Marne-la-Vallée Cedex 2, France

    Éric Cancès & Gabriel Stoltz

  2. CNRS & Laboratoire de Mathématiques UMR 8088, Université de Cergy-Pontoise, 95300, Cergy-Pontoise, France

    Mathieu Lewin

Authors
  1. Éric Cancès
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  2. Mathieu Lewin
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  3. Gabriel Stoltz
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Corresponding author

Correspondence to Éric Cancès .

Editor information

Editors and Affiliations

  1. , Department of Mathematics, University of Texas, University Station C 1200 1, Austin, 78712-0257, Texas, USA

    Björn Engquist

  2. Dept. Numerical Analysis &, Computer Science (NADA), Royal Institute of Technology (KTH), Lindstedtsvägen 3, Stockholm, 100 44, Sweden

    Olof Runborg

  3. , Department of Mathematics, University of Texas at Austin, University Station C1200 1, Austin, 78712-0257, Texas, USA

    Yen-Hsi R. Tsai

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Cancès, É., Lewin, M., Stoltz, G. (2012). The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint. In: Engquist, B., Runborg, O., Tsai, YH. (eds) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21943-6_5

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