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Linear response time-dependent density functional theory of the Hubbard dimer

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Abstract

The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. Many exact results, limits, and expansions about those limits are given in the Appendices.

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References

  1. E. Runge, E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984)

    Article  ADS  Google Scholar 

  2. C.A. Ullrich, Time-dependent density-functional theory: concepts and applications (Oxford University Press, Oxford, 2011)

  3. M.A. Marques, N.T. Maitra, F.M. Nogueira, E.K. Gross, A. Rubio, Eds., in Fundamentals of time-dependent density functional theory (Springer-Verlag, Berlin, Heidelberg, 2012), Vol. 837

  4. M. Casida, Recent advances in density functional methods, Part I (World Scientific, Singapore, 1995)

  5. M.E. Casida, in Recent developments and applications of modern density functional theory (Elsevier, Amsterdam, 1996), p. 391

  6. R.A.R. Bauernschmitt, Chem. Phys. Lett. 256, 454 (1996)

    Article  ADS  Google Scholar 

  7. T. Grabo, M. Petersilka, E. Gross, J. Mol. Struct. (Theochem) 501, 353 (2000)

    Article  Google Scholar 

  8. K. Yabana, T. Nakatsukasa, J.-I. Iwata, G. Bertsch, Phys. Status Solidi B 243, 1121 (2006)

    Article  ADS  Google Scholar 

  9. N.T. Maitra, J. Chem. Phys. 144, 220901 (2016)

    Article  ADS  Google Scholar 

  10. C. Adamo, D. Jacquemin, Chem. Soc. Rev. 42, 845 (2013)

    Article  Google Scholar 

  11. D. Jacquemin, V. Wathelet, E.A. Perpète, C. Adamo, J. Chem. Theory Comput. 5, 2420 (2009)

    Article  Google Scholar 

  12. P. Elliott, F. Furche, K. Burke, in Excited states from time-dependent density functional theory (Wiley, Hoboken, NJ, 2009), pp. 91–165

  13. M. Casida, M. Huix-Rotllant, Annu. Rev. Phys. Chem. 63, 287 (2012)

    Article  ADS  Google Scholar 

  14. M.E. Casida, M. Huix-Rotllant, in Density-Functional Methods for Excited States, edited by N. Ferré, M. Filatov, M. Huix-Rotllant (Springer International Publishing, Cham, 2016), pp. 1–60

  15. D. Tozer, R. Amos, N. Handy, B. Roos, L. Serrano-Andres, Mol. Phys. 97, 859 (1999)

    Article  ADS  Google Scholar 

  16. A. Dreuw, J.L. Weisman, M. Head-Gordon, J. Chem. Phys. 119, 2943 (2003)

    Article  ADS  Google Scholar 

  17. D. Tozer, J. Chem. Phys. 119, 12697 (2003)

    Article  ADS  Google Scholar 

  18. O. Gritsenko, E.J. Baerends, J. Chem. Phys. 121, 655 (2004)

    Article  ADS  Google Scholar 

  19. N.T. Maitra, J. Chem. Phys. 122, 234104 (2005)

    Article  ADS  Google Scholar 

  20. N.T. Maitra, J. Phys.: Condens. Matter 29, 423001 (2017)

    Google Scholar 

  21. Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, K. Hirao, J. Chem. Phys. 120, 8425 (2004)

    Article  ADS  Google Scholar 

  22. T. Stein, L. Kronik, R. Baer, J. Am. Chem. Soc. 131, 2818 (2009)

    Article  Google Scholar 

  23. R. Baer, E. Livshits, U. Salzner, Ann. Rev. Phys. Chem. 61, 85 (2010)

    Article  Google Scholar 

  24. L. Kronik, T. Stein, S. Refaely-Abramson, R. Baer, J. Chem. Theory Comput. 8, 1515 (2012)

    Article  Google Scholar 

  25. T. Körzdörfer, J.-L. Brédas, Acc. Chem. Res. 47, 3284 (2014)

    Article  Google Scholar 

  26. D.S.C. Jamorski, M.E. Casida, J. Chem. Phys. 104, 5134 (1996)

    Article  ADS  Google Scholar 

  27. D. Tozer, N. Handy, Phys. Chem. Chem. Phys. 2, 2117 (2000)

    Article  Google Scholar 

  28. N.T. Maitra, F. Zhang, R.J. Cave, K. Burke, J. Chem. Phys. 120, 5932 (2004)

    Article  ADS  Google Scholar 

  29. S. Tretiak, V. Chernyak, J. Chem. Phys. 119, 8809 (2003)

    Article  ADS  Google Scholar 

  30. P. Elliott, S. Goldson, C. Canahui, N.T. Maitra, Chem. Phys. 391, 110 (2011)

    Article  ADS  Google Scholar 

  31. R.J. Cave, F. Zhang, N.T. Maitra, K. Burke, Chem. Phys. Lett. 389, 39 (2004)

    Article  ADS  Google Scholar 

  32. G. Mazur, M. Makowski, R. Wldarczyk, Y. Aoki, Int. J. Quantum Chem. 111, 819 (2011)

    Article  Google Scholar 

  33. G. Mazur, R. Wlodarczyk, J. Comput. Chem. 30, 811 (2009)

    Article  Google Scholar 

  34. M. Huix-Rotllant, A. Ipatov, A. Rubio, M.E. Casida, Chem. Phys. 391, 120 (2011)

    Article  ADS  Google Scholar 

  35. J.P. Bergfield, Z.-F. Liu, K. Burke, C.A. Stafford, Phys. Rev. Lett. 108, 066801 (2012)

    Article  ADS  Google Scholar 

  36. G. Stefanucci, S. Kurth, Phys. Rev. Lett. 107, 216401 (2011)

    Article  ADS  Google Scholar 

  37. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994)

    Article  ADS  Google Scholar 

  38. P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

    Article  ADS  Google Scholar 

  39. P.W. Anderson, J. Phys. Conf. Ser. 449, 012001 (2013)

    Article  Google Scholar 

  40. R. Baer, J. Chem. Phys. 128, 044103 (2008)

    Article  ADS  Google Scholar 

  41. Y. Li, C.A. Ullrich, J. Chem. Phys. 129, 044105 (2008)

    Article  ADS  Google Scholar 

  42. C. Verdozzi, Phys. Rev. Lett. 101, 166401 (2008)

    Article  ADS  Google Scholar 

  43. S. Kurth, G. Stefanucci, E. Khosravi, C. Verdozzi, E.K.U. Gross, Phys. Rev. Lett. 104, 236801 (2010)

    Article  ADS  Google Scholar 

  44. I.V. Tokatly, Phys. Rev. B 83, 035127 (2011)

    Article  ADS  Google Scholar 

  45. J.I. Fuks, M. Farzanehpour, I.V. Tokatly, H. Appel, S. Kurth, A. Rubio, Phys. Rev. A 88, 062512 (2013)

    Article  ADS  Google Scholar 

  46. M. Farzanehpour, I.V. Tokatly, Phys. Rev. B 86, 125130 (2012)

    Article  ADS  Google Scholar 

  47. R. Requist, O. Pankratov, Phys. Rev. A 81, 042519 (2010)

    Article  ADS  Google Scholar 

  48. P. Schmitteckert, M. Dzierzawa, P. Schwab, Phys. Chem. Chem. Phys. 15, 5477 (2013)

    Article  Google Scholar 

  49. J.I. Fuks, N.T. Maitra, Phys. Chem. Chem. Phys. 16, 14504 (2014)

    Article  Google Scholar 

  50. J.I. Fuks, N.T. Maitra, Phys. Rev. A 89, 062502 (2014)

    Article  ADS  Google Scholar 

  51. N. Dittmann, J. Splettstoesser, N. Helbig, Phys. Rev. Lett. 120, 157701 (2018)

    Article  ADS  Google Scholar 

  52. S. Kurth, G. Stefanucci, arXiv:1803.03244 (2018)

  53. A. Kartsev, D. Karlsson, A. Privitera, C. Verdozzi, Sci. Rep. 3, 2570 (2013)

    Article  ADS  Google Scholar 

  54. D. Karlsson, C. Verdozzi, M.M. Odashima, K. Capelle, EPL 93, 23003 (2011)

    Article  ADS  Google Scholar 

  55. L. Mancini, J.D. Ramsden, M.J.P. Hodgson, R.W. Godby, Phys. Rev. B 89, 195114 (2014)

    Article  ADS  Google Scholar 

  56. V. Turkowski, T.S. Rahman, J. Phys.: Condens. Matter 26, 022201 (2014)

    Google Scholar 

  57. R. Requist, O. Pankratov, Phys. Rev. B 77, 235121 (2008)

    Article  ADS  Google Scholar 

  58. D.J. Carrascal, J. Ferrer, Phys. Rev. B 85, 045110 (2012)

    Article  ADS  Google Scholar 

  59. K. Capelle, V.L. Campo Jr., Phys. Rep. 528, 91 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  60. F. Aryasetiawan, O. Gunnarsson, Phys. Rev. B 66, 165119 (2002)

    Article  ADS  Google Scholar 

  61. D.J. Carrascal, J. Ferrer, J.C. Smith, K. Burke, J. Phys.: Condens. Matter 27, 393001 (2015)

    Google Scholar 

  62. M. Thiele, S. Kümmel, Phys. Rev. Lett. 112, 083001 (2014)

    Article  ADS  Google Scholar 

  63. M. Ruggenthaler, S.E.B. Nielsen, R. van Leeuwen, Phys. Rev. A 88, 022512 (2013)

    Article  ADS  Google Scholar 

  64. P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964)

    Article  ADS  Google Scholar 

  65. E. Gross, W. Kohn, Phys. Rev. Lett. 55, 2850 (1985)

    Article  ADS  Google Scholar 

  66. M.A. Marques, N.T. Maitra, F.M. Nogueira, E.K. Gross, A. Rubio (Eds.), in Fundamentals of time-dependent density functional theory (Springer-Verlag, Berlin, Heidelberg, 2012), Vol. 837, Chap. 1

  67. W. Thomas, Naturwissenschaften 13, 627 (1925)

    Article  ADS  Google Scholar 

  68. W. Kuhn, Z. Phys. 33, 408 (1925)

    Article  ADS  Google Scholar 

  69. F. Reiche, W. Thomas, Z. Phys. 34, 510 (1925)

    Article  ADS  Google Scholar 

  70. G. Mahan, Many-particle physics, 3rd edn. (Springer, New York, 2000)

  71. P.F. Maldague, Phys. Rev. B 16, 2437 (1977)

    Article  ADS  Google Scholar 

  72. D. Baeriswyl, C. Gros, T.M. Rice, Phys. Rev. B 35, 8391 (1987)

    Article  ADS  Google Scholar 

  73. W. Kohn, Phys. Rev. 133, A171 (1964)

    Article  ADS  Google Scholar 

  74. D. Langreth, J. Perdew, Solid State Commun. 17, 1425 (1975)

    Article  ADS  Google Scholar 

  75. O. Gunnarsson, B. Lundqvist, Phys. Rev. B 13, 4274 (1976)

    Article  ADS  Google Scholar 

  76. C. Li, R. Requist, E.K.U. Gross, J. Chem. Phys. 148, 084110 (2018)

    Article  ADS  Google Scholar 

  77. A.G.O.V. Gritsenko, S.J.A. van Gisbergen, E. Baerends, J. Chem. Phys. 113, 8478 (2000)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Universidad de Oviedo, 33007, Oviedo, Spain

    Diego J. Carrascal & Jaime Ferrer

  2. Nanomaterials and Nanotechnology Research Center, CSIC/Universidad de Oviedo, Oviedo, Spain

    Diego J. Carrascal & Jaime Ferrer

  3. Department of Physics, Hunter College, City University of New York, New York, NY, 1006, USA

    Neepa Maitra

  4. Department of Chemistry and of Physics, University of California, Irvine, CA, 92697, USA

    Kieron Burke

Authors
  1. Diego J. Carrascal
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  2. Jaime Ferrer
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  3. Neepa Maitra
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  4. Kieron Burke
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Correspondence to Jaime Ferrer.

Additional information

Contribution to the Topical Issue “Special issue in honor of Hardy Gross”, edited by C.A. Ullrich, F.M.S. Nogueira, A. Rubio, and M.A.L. Marques.

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Carrascal, D.J., Ferrer, J., Maitra, N. et al. Linear response time-dependent density functional theory of the Hubbard dimer. Eur. Phys. J. B 91, 142 (2018). https://doi.org/10.1140/epjb/e2018-90114-9

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