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Approximate energy functionals for one-body reduced density matrix functional theory from many-body perturbation theory

  • Klaas J. H. GiesbertzEmail author
  • Anna-Maija Uimonen
  • Robert van Leeuwen
Open Access
Regular Article
Part of the following topical collections:
  1. Topical issue: Special issue in honor of Hardy Gross

Abstract

We develop a systematic approach to construct energy functionals of the one-particle reduced density matrix (1RDM) for equilibrium systems at finite temperature. The starting point of our formulation is the grand potential Ω[G] regarded as variational functional of the Green’s function G based on diagrammatic many-body perturbation theory and for which we consider either the Klein or Luttinger–Ward form. By restricting the input Green’s function to be one-to-one related to a set on one-particle reduced density matrices (1RDM) this functional becomes a functional of the 1RDM. To establish the one-to-one mapping we use that, at any finite temperature and for a given 1RDM γ in a finite basis, there exists a non-interacting system with a spatially non-local potential v[γ] which reproduces the given 1RDM. The corresponding set of non-interacting Green’s functions defines the variational domain of the functional Ω. In the zero temperature limit we obtain an energy functional E[γ] which by minimisation yields an approximate ground state 1RDM and energy. As an application of the formalism we use the Klein and Luttinger–Ward functionals in the GW-approximation to compute the binding curve of a model hydrogen molecule using an extended Hubbard Hamiltonian. We compare further to the case in which we evaluate the functionals on a Hartree–Fock and a Kohn–Sham Green’s function. We find that the Luttinger–Ward version of the functionals performs the best and is able to reproduce energies close to the GW energy which corresponds to the stationary point.

Supplementary material

References

  1. 1.
    O.V. Gritsenko, K. Pernal, E.J. Baerends, J. Chem. Phys. 122, 204102 (2005) CrossRefGoogle Scholar
  2. 2.
    D.R. Rohr, K. Pernal, O.V. Gritsenko, E.J. Baerends, J. Chem. Phys. 129, 164105 (2008) CrossRefGoogle Scholar
  3. 3.
    M. Piris, X. Lopez, F. Ruipérez, J.M. Matxain, J.M. Ugalde, J. Chem. Phys. 134, 164102 (2011) CrossRefGoogle Scholar
  4. 4.
    F. Ruiperez, M. Piris, J.M. Ugalde, J.M. Matxain, Phys. Chem. Chem. Phys. 15, 2055 (2013) CrossRefGoogle Scholar
  5. 5.
    S. Sharma, J.K. Dewhurst, N.N. Lathiotakis, E.K.U. Gross, Phys. Rev. B 78, 201103(R) (2008) CrossRefGoogle Scholar
  6. 6.
    S. Sharma, J.K. Dewhurst, S. Shallcross, E.K.U. Gross, Phys. Rev. Lett. 110, 116403 (2013) CrossRefGoogle Scholar
  7. 7.
    E. Tölö, A. Harju, Phys. Rev. B 81, 075321 (2010) CrossRefGoogle Scholar
  8. 8.
    M. Buijse, Ph.D. thesis, Vrije Universiteit, De Boelelaan 1105, Amsterdam, The Netherlands, 1991, https://doi.org/theochem.chem.rug.nl/publications/PDF/ft217.pdf
  9. 9.
    M. Buijse, E.J. Baerends, Mol. Phys. 100, 401 (2002) CrossRefGoogle Scholar
  10. 10.
    Ł.M. Mentel, R. van Meer, O.V. Gritsenko, E.J. Baerends, J. Chem. Phys. 140, 214105 (2014) CrossRefGoogle Scholar
  11. 11.
    D.A. Mazziotti, Chem. Phys. Lett. 338, 323 (2001) CrossRefGoogle Scholar
  12. 12.
    M. Piris, P. Otto, Int. J. Quantum Chem. 94, 317 (2003) CrossRefGoogle Scholar
  13. 13.
    M. Piris, Int. J. Quantum Chem. 106, 1093 (2006) CrossRefGoogle Scholar
  14. 14.
    M. Piris, J.M. Matxain, X. Lopez, J.M. Ugalde, J. Chem. Phys. 131, 021102 (2009) CrossRefGoogle Scholar
  15. 15.
    M. Piris, J.M. Matxain, X. Lopez, J. Chem. Phys. 139, 234109 (2013) CrossRefGoogle Scholar
  16. 16.
    M. Piris, J. Chem. Phys. 141, 044107 (2014) CrossRefGoogle Scholar
  17. 17.
    M. Piris, Phys. Rev. Lett. 119, 063002 (2017) CrossRefGoogle Scholar
  18. 18.
    K. Pernal, K.J.H. Giesbertz, in Density-Functional Methods for Excited States, edited by N. Ferré, M. Filatov, M. Huix-Rotllant (Springer, Berlin, Heidelberg, 2015), Vol. 368 of Topics in Current Chemistry, Chap. 4, p. 125 Google Scholar
  19. 19.
    A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publiations, Inc., 2003) Google Scholar
  20. 20.
    G. Stefanucci, R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge Univeristy Press, New York, 2013) Google Scholar
  21. 21.
    F. Furche, Phys. Rev. B 64, 195120 (2001) CrossRefGoogle Scholar
  22. 22.
    N.E. Dahlen, U. von Barth, Phys. Rev. A 69, 195102 (2004) Google Scholar
  23. 23.
    N.E. Dahlen, R. van Leeuwen, U. von Barth, Phys. Rev. A 73, 012511 (2006) CrossRefGoogle Scholar
  24. 24.
    F. Furche, J. Chem. Phys. 129, 114105 (2008) CrossRefGoogle Scholar
  25. 25.
    M. Hellgren, U. von Barth, J. Chem. Phys. 132, 044101 (2010) CrossRefGoogle Scholar
  26. 26.
    A. Heßelmann, A. Görling, Mol. Phys. 108, 359 (2010) CrossRefGoogle Scholar
  27. 27.
    H. Eshuis, J.E. Bates, F. Furche, Theor. Chem. Acc. 131, 1084 (2012) CrossRefGoogle Scholar
  28. 28.
    J.E. Bates, F. Furche, J. Chem. Phys. 139, 171103 (2013) CrossRefGoogle Scholar
  29. 29.
    P. Bleiziffer, A. Heßelmann, A. Görling, J. Chem. Phys. 139, 084113 (2013) CrossRefGoogle Scholar
  30. 30.
    M. Hellgren, F. Caruso, D.R. Rohr, X. Ren, A. Rubio, M. Scheffler, P. Rinke, Phys. Rev. B 91, 165110 (2015) CrossRefGoogle Scholar
  31. 31.
    W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965) CrossRefGoogle Scholar
  32. 32.
    T.L. Gilbert, Phys. Rev. B 12, 2111 (1975) CrossRefGoogle Scholar
  33. 33.
    K. Pernal, Phys. Rev. Lett. 94, 233002 (2005) CrossRefGoogle Scholar
  34. 34.
    K.J.H. Giesbertz, E.J. Baerends, J. Chem. Phys. 132, 194108 (2010) CrossRefGoogle Scholar
  35. 35.
    G. Friesecke, Proc. R. Soc. London A 459, 47 (2003) CrossRefGoogle Scholar
  36. 36.
    K.J.H. Giesbertz, R. van Leeuwen, J. Chem. Phys. 139, 104109 (2013) CrossRefGoogle Scholar
  37. 37.
    K.J.H. Giesbertz, R. van Leeuwen, J. Chem. Phys. 139, 104110 (2013) CrossRefGoogle Scholar
  38. 38.
    K.J.H. Giesbertz, R. van Leeuwen, J. Chem. Phys. 140, 184108 (2014) CrossRefGoogle Scholar
  39. 39.
    K.J.H. Giesbertz, M. Ruggenthaler, https://doi.org/arXiv:1710.08805 (2017)
  40. 40.
    A. Görling, M. Levy, Phys. Rev. B 47, 13105 (1993) CrossRefGoogle Scholar
  41. 41.
    A. Görling, M. Levy, Phys. Rev. A 50, 196 (1994) CrossRefGoogle Scholar
  42. 42.
    T. Baldsiefen, Ph.D. thesis, Institut für Theoretische Physik Freie Universität Berlin, 2012 Google Scholar
  43. 43.
    T. Baldsiefen, A. Cangi, E.K.U. Gross, https://doi.org/arXiv:1208.4703 (2012)
  44. 44.
    P.E. Blöchl, T. Pruschke, M. Potthoff, Phys. Rev. B 88, 205139 (2013) CrossRefGoogle Scholar
  45. 45.
    C.O. Almbladh, U. von Barth, R. van Leeuwen, Int. J. Mod. Phys. B 13, 535 (1999) CrossRefGoogle Scholar
  46. 46.
    F. Aryasetiawan, T. Miyake, K. Terakura, Phys. Rev. Lett. 88, 166401 (2002) CrossRefGoogle Scholar
  47. 47.
    T. Miyake, F. Aryasetiawan, T. Kotani, M. van Schilfgaarde, M. Usuda, K. Terakura, Phys. Rev. B 66, 245103 (2002) CrossRefGoogle Scholar
  48. 48.
    J.M. Luttinger, J.C. Ward, Phys. Rev. 118, 1417 (1960) MathSciNetCrossRefGoogle Scholar
  49. 49.
    A. Klein, Phys. Rev. 121, 950 (1961) MathSciNetCrossRefGoogle Scholar
  50. 50.
    Y.M. Niquet, M. Fuchs, X. Gonze, Phys. Rev. A 68, 032507 (2003) CrossRefGoogle Scholar
  51. 51.
    F. Caruso, D.R. Rohr, M. Hellgren, X. Ren, P. Rinke, A. Rubio, M. Scheffler, Phys. Rev. Lett. 110, 146403 (2013) CrossRefGoogle Scholar
  52. 52.
    G. Baym, L.P. Kadanoff, Phys. Rev. 124, 287 (1961) MathSciNetCrossRefGoogle Scholar
  53. 53.
    G. Baym, Phys. Rev. 127, 1391 (1962) MathSciNetCrossRefGoogle Scholar
  54. 54.
    U. vonBarth, N.E. Dahlen, R. van Leeuwen, G. Stefanucci, Phys. Rev. B 72, 235109 (2005) CrossRefGoogle Scholar
  55. 55.
    A. Szabo, N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover Publications, Inc., N.Y., 1989) Google Scholar
  56. 56.
    P. Löwdin, J. Chem. Phys. 18, 365 (1950) CrossRefGoogle Scholar
  57. 57.
    W. Kołos, L. Wolniewicz, J. Chem. Phys. 43, 2429 (1965) CrossRefGoogle Scholar
  58. 58.
    W. Kołos, L. Wolniewicz, J. Chem. Phys. 45, 509 (1966) CrossRefGoogle Scholar
  59. 59.
    W. Kołos, L. Wolniewicz, J. Chem. Phys. 50, 3228 (1969) CrossRefGoogle Scholar
  60. 60.
    L. Wolniewicz, K. Dressler, J. Chem. Phys. 82, 3292 (1985) CrossRefGoogle Scholar
  61. 61.
    R. Requist, O. Pankratov, Phys. Rev. B 77, 235121 (2008) CrossRefGoogle Scholar
  62. 62.
    M. Fuchs, Y.M. Niquet, X. Gonze, K. Burke, J. Chem. Phys. 122, 094116 (2005) CrossRefGoogle Scholar
  63. 63.
    N. Rosen, Phys. Rev. 38, 2099 (1931) CrossRefGoogle Scholar
  64. 64.
    J.O. Hirschfelder, J.W. Linnett, J. Chem. Phys. 18, 130 (1950) CrossRefGoogle Scholar
  65. 65.
    R.S. Mulliken, J. Chim. Phys. 46, 497 (1949) CrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://doi.org/creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Theoretical Chemistry, Faculty of Exact Sciences, VU UniversityHV AmsterdamThe Netherlands
  2. 2.Department of PhysicsNanoscience Center, University of JyväskyläJyväskyläFinland

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