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Frontiers and Challenges in Warm Dense Matter pp 25–60Cite as

Thermal Density Functional Theory in Context

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 96)

Abstract

This chapter introduces thermal density functional theory, starting from the ground-state theory and assuming a background in quantum mechanics and statistical mechanics. We review the foundations of density functional theory (DFT) by illustrating some of its key reformulations. The basics of DFT for thermal ensembles are explained in this context, as are tools useful for analysis and development of approximations. This review emphasizes thermal DFT’s strengths as a consistent and general framework.

Keywords

  • Density Functional Theory
  • Local Density Approximation
  • Degenerate Ground State
  • Thermal Ensemble
  • Single Slater Determinant

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Fig. 1
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Fig. 4

Notes

  1. 1.

    See Refs. [24] or [25] for quantum mechanical background that is useful for this chapter.

  2. 2.

    In this work, we discuss only spin-unpolarized electrons.

  3. 3.

    Here and in the remainder of the chapter, we restrict ourselves to square-integrable wavefunctions over the domain \({\mathbb{R}}^{3N}\).

  4. 4.

    For a more extended discussion of these topics, see Ref. [60].

  5. 5.

    Note that, we eventually choose to work in a system of units such that the Boltzmann constant is k B  = 1, that is, temperature is measured in energy units.

  6. 6.

    The interested reader may find the extension of the Hohenberg-Kohn theorem to the thermal framework in Mermin’s paper.

  7. 7.

    Uniform coordinate scaling may be considered as (very) careful dimensional analysis applied to density functionals. Dufty and Trickey analyze non-interacting functionals in this way in Ref. [15].

References

  1. N.R.C.C. on High Energy Density Plasma Physics Plasma Science Committee, Frontiers in High Energy Density Physics: The X-Games of Contemporary Science (The National Academies Press, Washington, D.C., 2003)

    Google Scholar 

  2. T.R. Mattsson, M.P. Desjarlais, Phys. Rev. Lett. 97, 017801 (2006)

    CrossRef  Google Scholar 

  3. F.R. Graziani, V.S. Batista, L.X. Benedict, J.I. Castor, H. Chen, S.N. Chen, C.A. Fichtl, J.N. Glosli, P.E. Grabowski, A.T. Graf, S.P. Hau-Riege, A.U. Hazi, S.A. Khairallah, L. Krauss, A.B. Langdon, R.A. London, A. Markmann, M.S. Murillo, D.F. Richards, H.A. Scott, R. Shepherd, L.G. Stanton, F.H. Streitz, M.P. Surh, J.C. Weisheit, H.D. Whitley, High Energy Density Phys. 8(1), 105 (2012)

    CrossRef  Google Scholar 

  4. K.Y. Sanbonmatsu, L.E. Thode, H.X. Vu, M.S. Murillo, J. Phys. IV France 10(PR5), Pr5 (2000)

    Google Scholar 

  5. S. Atzeni, J. Meyer-ter Vehn, The Physics of Inertial Fusion: Beam-Plasma Interaction, Hydrodynamics, Hot Dense Matter (Oxford University Press, New York, 2004)

    CrossRef  Google Scholar 

  6. M.D. Knudson, M.P. Desjarlais, Phys. Rev. Lett. 103, 225501 (2009)

    CrossRef  Google Scholar 

  7. A. Kietzmann, R. Redmer, M.P. Desjarlais, T.R. Mattsson, Phys. Rev. Lett. 101, 070401 (2008)

    CrossRef  Google Scholar 

  8. S. Root, R.J. Magyar, J.H. Carpenter, D.L. Hanson, T.R. Mattsson, Phys. Rev. Lett. 105(8), 085501 (2010)

    CrossRef  Google Scholar 

  9. K. Burke, J. Chem. Phys. 136, 150901 (2012)

    CrossRef  Google Scholar 

  10. P. Hohenberg, W. Kohn, Phys. Rev. 136(3B), B864 (1964)

    CrossRef  MathSciNet  Google Scholar 

  11. M. Levy, Proc. Natl. Acad. Sci. USA 76(12), 6062 (1979)

    CrossRef  Google Scholar 

  12. E.H. Lieb, Int. J. Quantum Chem. 24(3), 243 (1983)

    CrossRef  Google Scholar 

  13. N.D. Mermin, Phys. Rev. 137, A: 1441 (1965)

    Google Scholar 

  14. S. Pittalis, C.R. Proetto, A. Floris, A. Sanna, C. Bersier, K. Burke, E.K.U. Gross, Phys. Rev. Lett. 107, 163001 (2011)

    CrossRef  Google Scholar 

  15. J.W. Dufty, S.B. Trickey, Phys. Rev. B 84, 125118 (2011). Interested readers should note that Dufty and Trickey also have a paper concerning interacting functionals in preparation

    Google Scholar 

  16. L.H. Thomas, Math. Proc. Camb. Phil. Soc. 23(05), 542 (1927)

    CrossRef  MATH  Google Scholar 

  17. E. Fermi, Rend. Acc. Naz. Lincei 6, 602–607 (1927)

    Google Scholar 

  18. E. Fermi, Z. für Phys. A Hadrons Nucl. 48, 73 (1928)

    MATH  Google Scholar 

  19. V. Fock, Z. Phys. 61, 126 (1930)

    CrossRef  MATH  Google Scholar 

  20. D.R. Hartree, W. Hartree, Proc. R. Soc. Lond. Ser. A – Math. Phys. Sci. 150(869), 9 (1935)

    Google Scholar 

  21. W. Kohn, L.J. Sham, Phys. Rev. 140(4A), A1133 (1965)

    CrossRef  MathSciNet  Google Scholar 

  22. K. Burke. The ABC of DFT. http://dft.uci.edu/doc/g1.pdf (created April 10, 2007)

  23. K. Burke, L.O. Wagner, Int. J. Quant. Chem. 113, 96 (2013)

    CrossRef  Google Scholar 

  24. F. Schwabl, Quantum Mechanics (Springer, Berlin/Heidelberg/New York, 2007)

    Google Scholar 

  25. J.J. Sakurai, Modern Quantum Mechanics, Rev. Edn. (Addison Wesley, Reading, 1993)

    Google Scholar 

  26. E. Engel, R.M. Dreizler, Density Functional Theory: An Advanced Course (Springer, Heidelberg/Dordrecht/London/New York, 2011)

    CrossRef  Google Scholar 

  27. C.D. Sherrill, J. Chem. Phys. 132(11), 110902 (2010)

    CrossRef  Google Scholar 

  28. R.M. Dreizler, E.K.U. Gross, Density Functional Theory: An Approach to the Quantum Many-Body Problem (Springer, Berlin/New York, 1990)

    CrossRef  MATH  Google Scholar 

  29. W. Kohn, in Highlight of Condensed Matter Theory, ed. by F. Bassani, F. Fumi, M.P. Tosi (North-Holland, Amsterdam, 1985), p. 1

    Google Scholar 

  30. G.F. Giuliani, G. Vignale (eds.), Quantum Theory of the Electron Liquid (Cambridge University Press, Cambridge, 2008)

    Google Scholar 

  31. R. Dreizler, J. da Providência, N.A.T.O.S.A. Division (eds.), Density Functional Methods in Physics. NATO ASI B Series (Springer, Dordrecht, 1985)

    Google Scholar 

  32. H. Englisch, R. Englisch, Phys. A Stat. Mech. Appl. 121(1–2), 253 (1983)

    CrossRef  MathSciNet  Google Scholar 

  33. F.W. Averill, G.S. Painter, Phys. Rev. B 15, 2498 (1992)

    CrossRef  Google Scholar 

  34. S.G. Wang, W.H.E. Scharz, J. Chem. Phys. 105, 4641 (1996)

    CrossRef  Google Scholar 

  35. P.R.T. Schipper, O.V. Gritsenko, E.J. Baerends, Theor. Chem. Acc. 99, 4056 (1998)

    CrossRef  Google Scholar 

  36. P.R.T. Schipper, O.V. Gritsenko, E.J. Baerends, J. Chem. Phys. 111, 4056 (1999)

    CrossRef  Google Scholar 

  37. C.A. Ullrich, W. Kohn, Phys. Rev. Lett. 87, 093001 (2001)

    CrossRef  Google Scholar 

  38. M. Reed, B. Simon, I: Functional Analysis (Methods of Modern Mathematical Physics) (Academic, San Diego, 1981)

    Google Scholar 

  39. J. Harriman, Phys. Rev. A 24, 680 (1981)

    CrossRef  Google Scholar 

  40. R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989)

    Google Scholar 

  41. R. van Leeuwen, Adv. Q. Chem. 43, 24 (2003)

    Google Scholar 

  42. M. Levy, Phys. Rev. A 26, 1200 (1982)

    CrossRef  Google Scholar 

  43. S.V. Valone, J. Chem. Phys. 73, 1344 (1980)

    CrossRef  MathSciNet  Google Scholar 

  44. H. Englisch, R. Englisch, Phys. Stat. Solidi B 123, 711 (1984)

    CrossRef  Google Scholar 

  45. H. Englisch, R. Englisch, Phys. Stat. Solidi B 124, 373 (1984)

    CrossRef  Google Scholar 

  46. J.T. Chayes, L. Chayes, M.B. Ruskai, J. Stat. Phys 38, 497 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  47. J. Perdew, Phys. Rev. B 33, 8822 (1986)

    CrossRef  Google Scholar 

  48. A.D. Becke, Phys. Rev. A 38(6), 3098 (1988)

    CrossRef  Google Scholar 

  49. A.D. Becke, J. Chem. Phys. 98(7), 5648 (1993)

    CrossRef  Google Scholar 

  50. C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37(2), 785 (1988)

    CrossRef  Google Scholar 

  51. J.P. Perdew, K. Schmidt, in Density Functional Theory and Its Applications to Materials, ed. by V.E.V. Doren, K.V. Alsenoy, P. Geerlings (American Institute of Physics, Melville, 2001)

    Google Scholar 

  52. J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77(18), 3865 (1996); Ibid. 78, 1396(E) (1997)

    Google Scholar 

  53. J. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 80, 891 (1998)

    CrossRef  Google Scholar 

  54. V.V. Karasiev, R.S. Jones, S.B. Trickey, F.E. Harris, Phys. Rev. B 80, 245120 (2009)

    CrossRef  Google Scholar 

  55. V.V. Karasiev, R.S. Jones, S.B. Trickey, F.E. Harris, Phys. Rev. B 87, 239903 (2013)

    CrossRef  Google Scholar 

  56. V. Karasiev, S. Trickey, Comput. Phys. Commun. 183(12), 2519 (2012)

    CrossRef  MATH  Google Scholar 

  57. Y.A. Wang, E.A. Carter, in Theoretical Methods in Condensed Phase Chemistry, ed. by S.D. Schwartz (Kluwer, Dordrecht, 2000), chap. 5, p. 117

    Google Scholar 

  58. J.C. Snyder, M. Rupp, K. Hansen, K.R. Mueller, K. Burke, Phys. Rev. Lett. 108, 253002 (2012)

    CrossRef  Google Scholar 

  59. M. Levy, J. Perdew, Phys. Rev. A 32, 2010 (1985)

    CrossRef  Google Scholar 

  60. J.P. Perdew, S. Kurth, in A Primer in Density Functional Theory, ed. by C. Fiolhais, F. Nogueira, M.A.L. Marques (Springer, Berlin/Heidelberg, 2003), pp. 1–55

    Google Scholar 

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

    CrossRef  Google Scholar 

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

    CrossRef  Google Scholar 

  63. M. Ernzerhof, K. Burke, J.P. Perdew, in Recent Developments and Applications in Density Functional Theory, ed. by J.M. Seminario (Elsevier, Amsterdam, 1996)

    Google Scholar 

  64. R. Jones, O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989)

    CrossRef  Google Scholar 

  65. K. Burke, J. Perdew, D. Langreth, Phys. Rev. Lett. 73, 1283 (1994)

    CrossRef  Google Scholar 

  66. A.C. Cancio, C.Y. Fong, J.S. Nelson, Phys. Rev. A 62, 062507 (2000)

    CrossRef  Google Scholar 

  67. H. Eschrig, Phys. Rev. B 82, 205120 (2010)

    CrossRef  Google Scholar 

  68. A. Theophilou, J. Phys. C 12, 5419 (1979)

    CrossRef  Google Scholar 

  69. E. Gross, L. Oliveira, W. Kohn, Phys. Rev. A 37, 2809 (1988)

    CrossRef  MathSciNet  Google Scholar 

  70. L. Oliveira, E. Gross, W. Kohn, Phys. Rev. A 37, 2821 (1988)

    CrossRef  MathSciNet  Google Scholar 

  71. A. Nagy, Phys. Rev. A 57, 1672 (1998)

    CrossRef  Google Scholar 

  72. N.I. Gidopoulos, P.G. Papaconstantinou, E.K.U. Gross, Phys. Rev. Lett. 88, 033003 (2002)

    CrossRef  Google Scholar 

  73. F. Perrot, Phys. Rev. A 20, 586 (1979)

    CrossRef  MathSciNet  Google Scholar 

  74. F. Perrot, M.W.C. Dharma-wardana, Phys. Rev. A 30, 2619 (1984)

    CrossRef  Google Scholar 

  75. F. Perrot, M.W.C. Dharma-wardana, Phys. Rev. B 62(24), 16536 (2000)

    CrossRef  Google Scholar 

  76. R.G. Dandrea, N.W. Ashcroft, A.E. Carlsson, Phys. Rev. B 34(4), 2097 (1986)

    CrossRef  Google Scholar 

  77. J. Perdew, in Density Functional Method in Physics, ed. by R. Dreizler, J. da Providencia. NATO Advanced Study Institute, Series B: Physics, vol. 123 (Plenum Press, New York, 1985)

    Google Scholar 

  78. S. Kurth, J.P. Perdew, in Strongly Coupled Coulomb Systems, ed. by G.J. Kalman, J.M. Rommel, K. Blagoev (Plenum Press, New York, 1998)

    Google Scholar 

  79. J.P. Perdew, Int. J. Quantum Chem. 28(Suppl. 19), 497 (1985)

    Google Scholar 

  80. M. Greiner, P. Carrier, A. Görling, Phys. Rev. B 81, 155119 (2010)

    CrossRef  Google Scholar 

  81. V.V. Karasiev, T. Sjostrom, S.B. Trickey, Phys. Rev. B 86, 115101 (2012)

    CrossRef  Google Scholar 

  82. K.U. Plagemann, P. Sperling, R. Thiele, M.P. Desjarlais, C. Fortmann, T. Döppner, H.J. Lee, S.H. Glenzer, R. Redmer, New J. Phys. 14(5), 055020 (2012)

    CrossRef  Google Scholar 

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Acknowledgements

We would like to thank the Institute for Pure and Applied Mathematics for organization of Workshop IV: Computational Challenges in Warm Dense Matter and for hosting APJ during the Computational Methods in High Energy Density Physics long program. APJ thanks the U.S. Department of Energy (DE-FG02-97ER25308), SP and KB thank the National Science Foundation (CHE-1112442), and SP and EKUG thank European Community’s FP7, CRONOS project, Grant Agreement No. 280879.

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Authors and Affiliations

  1. Department of Chemistry, University of California, Irvine, CA, 92697, USA

    Aurora Pribram-Jones, Stefano Pittalis & Kieron Burke

  2. CNR–Istituto Nanoscienze, Centro S3, Via Campi 213a, I-41125, Modena, Italy

    Stefano Pittalis

  3. Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120, Halle, Germany

    E. K. U. Gross

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  1. Aurora Pribram-Jones
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  2. Stefano Pittalis
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Correspondence to Aurora Pribram-Jones .

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Editors and Affiliations

  1. Lawrence Livermore National Laboratory, Livermore, California, USA

    Frank Graziani

  2. Sandia National Laboratories, Albuquerque, New Mexico, USA

    Michael P. Desjarlais

  3. Institute of Physics, University of Rostock, Rostock, Germany

    Ronald Redmer

  4. Department of Physics, University of Florida, Gainesville, Florida, USA

    Samuel B. Trickey

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Pribram-Jones, A., Pittalis, S., Gross, E.K.U., Burke, K. (2014). Thermal Density Functional Theory in Context. In: Graziani, F., Desjarlais, M., Redmer, R., Trickey, S. (eds) Frontiers and Challenges in Warm Dense Matter. Lecture Notes in Computational Science and Engineering, vol 96. Springer, Cham. https://doi.org/10.1007/978-3-319-04912-0_2

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