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Density-Matrix Functional Theory and the High-Density Electron Gas

  • P. Ziesche
Chapter

Abstract

Often the ground state (GS) of the unpolarized uniform electron gas (EG) serves as a good model to describe the influence of the interelectron Coulomb repulsion on the energies and reduced densities 1. Essential kinematic features of this quantum-many-body system are described by the electron density ρ, the momentum distribution n(k), and the dimensionless pair distribution functions, or pair densities (PD), g and g pairs with parallel and antiparallel spins, respectively. n(k) satisfies 0 ≤ n(k) ≤ 1 (n(k) is non-idempotent), and the g’s are probabilities, thus g , g ≥ 0. ρ fixes the dimensionless density parameter r s from (4πr s 3 /3)a B 3 = 1/ρ and the Fermi wave number k F from k F 3 = 3π 2 ρ. The PD’s g and g are each referred to as either exchange or Fermi hole, and correlation or Coulomb hole, respectively. They are functions of the interelectron distance r 12 = |r 1r 2|. All the quantities of the EG depend parametrically on the density parameter r s . The Coulomb repulsion between each electron pair causes the phenomenon denoted electron correlation 2–9, which shows up in the correlation ‘tails’ of n(k), which means n(k < 1) ⪅ 1 for holes and n(k > 1) ⪆ 0 for particles, where k is measured in units of k F. Naturally related to these tails is z F < 1, the quasiparticle weight or reduced jump discontinuity of n(k) at k = 1, which causes the oscillatory long-range behavior of g and g for r 12 → ∞. Short-range (or dynamical) correlations show up in the curvature of g for zero-interelectron distance. We refer to this situation as the electrons being “in contact” with or “on-top” of each other. They also appear in the on-top value of g , which determines simultaneously both the on-top slope of g from the coalescing cusp theorem 10, and the asymptotic behavior of n(k) for k → ∞. The spin-traced PD g = (g + g )/2 and its parallel and antiparallel ‘components’ have been repeatedly studied over the years 11–35 This is especially true for the on-top value of the Coulomb hole g 25–27, 34, 35 and for the on-top curvature of the Fermi hole g 17, 28, 34, 35 which are short-range correlation properties. Examples of studies of the Fermi and Coulomb holes in molecules are given in Ref. 36.

Keywords

Momentum Distribution Coulomb Repulsion Virial Theorem Pair Density Reduce Density Matrice 
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References

  1. [1]
    With the ‘poor’ jellium model, pure electronic correlations are studied. Contrarily, in all real molecules, clusters, and solids the many-electron problem is intimately combined with the multi-centre problem. To explain or predict properties of molecules, clusters, and solids these two problems have to be solved simultaneously. For an overview of recent developments in solid state theory cf. the book Theoretical Materials Science. Tracing the Electronic Origins of Materials Behavior (MRS, Warrendale, 2000) by A. Gonis with the Preface by P. Ziesche. For the application of the jellium model to metal surfaces cf. A. Kiejna and K. F. Wojciechowski, Metal Surface Electron Physics, Pergamon, Kidlington (1996).Google Scholar
  2. [2]
    E.R. Davidson, Reduced Density Matrices in Quantum Chemistry, Academic, New York (1976).Google Scholar
  3. [3]
    P. Ziesche and G. Lehmann, ed., Ergebnisse in der Elektronentheorie der Metalle, Akademie-and Springer-Verlag, Berlin (1983).Google Scholar
  4. [4]
    R. Erdahl and V. H. Smith, Jr., ed., Density Matrices and Density Functionals, Reidel, Dortrecht (1987).Google Scholar
  5. [5]
    E. K. U. Gross and E. Runge, Vielteilchentheorie, Teubner, Stuttgart (1986); E. K. U. Gross, E. Runge, and O. Heinonen, Many-Particle Theory, Hilger, Bristol (1991).Google Scholar
  6. [6]
    P. Fulde, Electron Correlations in Molecules and Solids, Springer, Berlin (1991, 3rd enlarged ed. 1995 ).Google Scholar
  7. [7]
    W. Kutzelnigg and P. von Herigonte, Adv. Quant. Chem. 36: 185 (1999).Google Scholar
  8. [8]
    A. J. Coleman and V. I. Yukalov, Reduced Density Matrices,Springer, Berlin (2000). 320 Google Scholar
  9. [9]
    J. Cioslowski, ed., Many-Electron Densities and Reduced Density Matrices, Kluwer/Plenum, New York (2000).Google Scholar
  10. [10]
    T. Kato, Commun. Pure Appl. Math. 10:51(1957)Google Scholar
  11. J. H. Smith, Jr., Chem. Phys. Lett. 9:365(1971)Google Scholar
  12. A. J. Thakkar, V. H. Smith, Jr., Chem. Phys. Lett. 42:476(1976)Google Scholar
  13. W. Klopper and W. Kutzelnigg, J. Chem Phys. 94: 2020 (1991)Google Scholar
  14. J. Cioslowski, B. B. Stefanov, A. Tan, and C. J. Umrigar, J. Chem. Phys. 103: 6093 (1995)Google Scholar
  15. H. F. King, Theor. Chico. Acta 94: 345 (1996).Google Scholar
  16. [11]
    P. Noziéres and D. Pines, Phys. Rev. 111: 442 (1958).Google Scholar
  17. [12]
    D. J. W. Geldart, Can. J. Phys. 45: 3139 (1967).Google Scholar
  18. [13]
    P. Vashishta and K. S. Singwi, Phys. Rev. B 6: 875 (1972).Google Scholar
  19. [14]
    A. Holas, P. K. Aravind, and K. S. Singwi, Phys. Rev. B 20: 4912 (1979).Google Scholar
  20. [15]
    P. K. Aravind, A. Holas, and K. S. Singwi, Phys. Rev. B 25: 561 (1982).Google Scholar
  21. [16]
    J. T. Devreese, F. Brosens, and L. F. Lemmens, Phys. Rev. B 21: 1349, 1363 (1980)Google Scholar
  22. H. Nachtegaele, F. Brosens, and J. T. Devreese, Phys. Rev. B 28: 6064 (1983).Google Scholar
  23. [17]
    J. C. Kimball, Phys. Rev. A 7: 1648 (1973).Google Scholar
  24. [18]
    J. C. Kimball, Phys. Rev. B 14: 2371 (1976).Google Scholar
  25. [19]
    A. K. Rajagopal, J. C. Kimball, and M. Banerjee, Phys. Rev. B 18: 2339 (1978).Google Scholar
  26. [20]
    V. Contini, G. Mazzone, and F. Sacchetti, Phys. Rev. B 33: 712 (1986).Google Scholar
  27. [21]
    D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45: 566 (1980).Google Scholar
  28. [22]
    W. E. Pickett and J. Q. Broughton, Phys. Rev. B 48: 14859 (1993).Google Scholar
  29. [23]
    C. F. Richardson and N. W. Ashcroft, Phys. Rev. B 50: 8170 (1994)Google Scholar
  30. M. Lein, E. K. U. Gross, and J. P. Perdew, Phys. Rev. B 61: 13431 (2000).Google Scholar
  31. [24]
    G. Ortiz, M. Harris, and P. Ballone, Phys. Rev. Lett. 82: 5317 (1999).Google Scholar
  32. [25]
    H. Yasuhara, Solid State Commun. 11: 1481 (1972).Google Scholar
  33. [26]
    A. W. Overhauser, Can. J. Phys. 73: 683 (1995).Google Scholar
  34. [27]
    K. Burke, J. P. Perdew, and M. Ernzerhof, J. Chem. Phys. 109: 3760 (1998).Google Scholar
  35. [28]
    J. F. Dobson, J. Chem. Phys. 94:4328(1991); J. Phys.: Cond. Matt. 4: 7877 (1992).Google Scholar
  36. [29]
    J. P. Perdew and Y. Wang, Phys. Rev. B 46: 12947 (1992)Google Scholar
  37. J. P. Perdew and Y. Wang, Phys. Rev. B 56: 7018 (1997).Google Scholar
  38. [30]
    V. A. Rassolov, J. A. Pople, and M. A. Ratner, Phys. Rev. B 59: 15625 (1999).Google Scholar
  39. [31]
    P. Gori-Giorgi, F. Sacchetti, and G. B. Bachelet, Phys. Rev. B 61: 7353 (2000).Google Scholar
  40. [32]
    K. Schmidt, S. Kurth, J. Tao, and J. P. Perdew, Phys. Rev. B 62: 2227 (2000).Google Scholar
  41. [33]
    V. A. Rassolov, J. A. Pople, and M. A. Ratner, Phys. Rev. B 62: 2232 (2000).Google Scholar
  42. [34]
    P. Gori-Giorgi and J. P. Perdew, Phys. Rev. B 64: 155102 (2001).Google Scholar
  43. [35]
    P. Ziesche, Int. J. Quantum Chem., submitted.Google Scholar
  44. [36]
    E. J. Baerends and O. Gritsenko, J. Phys. Chem. A 101: 5383 (1997);Google Scholar
  45. M. A. Buijse, electron correlation - Fermi and Coulomb holes, dynamical and nondynamical correlation, Thesis, Amsterdam (1991); cf. alsoGoogle Scholar
  46. M. Slamet and V. Sahni, Phys. Rev. A 51: 2815 (1995)Google Scholar
  47. J. Wang and V. H. Smith, Jr., Int. J. Quantum Chem. 56:509(1995) for excited states in moleculesGoogle Scholar
  48. J. Wang, A. N. Tripathi, and V. H. Smith, Jr., J. Chem. Phys. 97:9188(1992) for intra-and extracule PDs; cf. also the articles byGoogle Scholar
  49. E. Valdemoro, by J. Cioslowski and by T. Koga in Ref. 9.Google Scholar
  50. [37]
    P. Ziesche, Pair Densities, Particle Number Fluctuations, and a Generalized Density Functional Theory, in: A. Gonis, N. Kioussis, M. Ciftan (ed.), Electron Correlations and Materials Properties, Kluwer/Plenum, New York (1999), p. 361. Note that after Eq. (3.15) therein the Coulomb hole (C-hole) normalization C is erroneously considered as a measure for the correlation strength. Correlation indeed creates the Coulomb hole, but its normalization C is always zero irrespective of the correlation strength. Note also that correlation widens the Fermi hole (X-hole), namely its on-top or short-range region, while further away from an electron the probability of finding another electron is enhanced (correlation reshovels the pair density from the nearby region of an electron to the region of its 1st pair-density shell, where g(x) has its 1st maximum), such that the Fermi-hole normalization X remains unchanged. So, C = 0 and X = N.Google Scholar
  51. [38]
    P. Ziesche, J. Tao, M. Seidl, and J. P. Perdew, Int. J. Quant. Chem. 77: 819 (2000).Google Scholar
  52. [39]
    R. A. Römer and P. Ziesche, Phys. Rev. B 62: 15279 (2000)Google Scholar
  53. R. A. Römer and P. Ziesche, J. Phys. A: Math. Gen. 34:1485(2001) and references cited therein.Google Scholar
  54. [40]
    P. Ziesche, J. Mol. Struc. (Theochem.) 527:35(2000) and references cited therein.Google Scholar
  55. [41]
    P. Ziesche, Solid State Commun. 82: 597 (1992).Google Scholar
  56. [42]
    P. Ziesche, Cumulant Expansions of Reduced Densities, Reduced Density Matrices, and Green’s Functions, in: Ref. 9, p. 33.Google Scholar
  57. [43]
    K. Kladko, P. Fulde, and D. A. Garanin, Europhys. Lett. 46: 425 (1999).Google Scholar
  58. [44]
    W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 110: 2800 (1999).Google Scholar
  59. [45]
    The relation between the concept of cumulants and the quantum chemical coupled-cluster method is described in Ref. 6. Note the formal similarity of the coupled-cluster wave function W = eTWYo and the exponential relation between the generating functional of the reduced density matrices and the generating functional of the cumulant matrices.Google Scholar
  60. [46] W. Macke and P. Ziesche, Acta Phys. Hung. 17:215(1964)
    with references therein to the original papers by Gell-Mann and Low (1951), Goldstone (1957), Hubbard (1957/58), Sucher (1957), Rodberg (1958), Bogolyubov and Shirkov (1957) and Bloch (1958)Google Scholar
  61. P. Ziesche, Commun. Math. Phys. 5: 191 (1967).Google Scholar
  62. [47]
    N. H. March, Phys. Rev. 110: 604 (1958).Google Scholar
  63. [48]
    W. Macke and P. Ziesche, Ann. Physik (Leipzig) 13: 26 (1964).Google Scholar
  64. [49]
    W. Macke, Z. Naturforsch. A 5: 192 (1950).Google Scholar
  65. [50]
    M. Gell-Mann and K. Brueckner, Phys. Rev. 106: 364 (1957).MathSciNetGoogle Scholar
  66. [51]
    L. Onsager, L. Mittag, and M. J. Stephen, Ann. Physik (Leipzig) 18: 71 (1966).Google Scholar
  67. [52]
    G. G. Hoffman, Phys. Rev. B 45: 8730 (1992).Google Scholar
  68. [53]
    E. Daniel and S. H. Vosko, Phys. Rev. 120: 2041 (1960).MathSciNetGoogle Scholar
  69. [54]
    I. O. Kulik, Z. Eksp. Thor. Fiz. 40:1343(1961) [Soy. Phys. JETP 13:946(1961)].Google Scholar
  70. [55]
    D. F. DuBois, Ann. Phys. (N.Y.) 7:174(1959). 8: 24 (1959).Google Scholar
  71. [56]
    W. J. Carr, Jr., and A. A. Maradudin, Phys. Rev. 133: A371 (1964).Google Scholar
  72. [57]
    T. Endo, M. Horiuchi, Y. Takada, and H. Yasuhara, Phys. Rev. B 59: 7367 (1999).Google Scholar
  73. [58]
    L. Hedin, Phys. Rev. 139: A796 (1965).Google Scholar
  74. [59]
    P. Ziesche, Ann. Physik (Leipzig) 21: 80 (1968).Google Scholar
  75. [60]
    J. C. Kimball, J. Phys. A 8: 1513 (1975).Google Scholar
  76. [61]
    H. Yasuhara and Y. Kawazoe, Physica A 85: 416 (1976).Google Scholar
  77. [62]
    Y. Takada and H. Yasuhara, Phys. Rev. B 44: 7879 (1991). 322Google Scholar
  78. [63]
    D. L. Freeman, Phys. Rev. B 15: 5512 (1977)Google Scholar
  79. R. F. Bishop, K.H. Lührmann, Phys. Rev. B 18: 3757 (1978)Google Scholar
  80. F. Bishop, K.H. Lührmann, Phys. Rev. B 26: 5523 (1982).Google Scholar
  81. [64]
    G. Ortiz and P. Ballone, Phys. Rev. B 50: 1391 (1994)Google Scholar
  82. G. Ortiz and P. Ballone, Phys. Rev. B 56: 9970 (1997).Google Scholar
  83. [65]
    T. L. Gilbert, Phys. Rev. B 12: 2111 (1975).Google Scholar
  84. [66]
    M. Levy, Proc. Natl. Acad. Sci. 76: 6062 (1979)Google Scholar
  85. M. Levy, Correlation Energy Functionals of One-Matrices and Hartree-Fock Densities, in: Ref. 4, p. 479.Google Scholar
  86. [67]
    G. Zumbach and K. Maschke, J. Chem. Phys. 82: 5604 (1985).Google Scholar
  87. [68]
    S. Goedecker and C. J. Umrigar, Phys. Rev. Lett. 81: 866 (1998).Google Scholar
  88. [69]
    S. Goedecker and C. J. Umrigar, Natural Orbital Functional Theory, in: Ref. 9, p. 165.Google Scholar
  89. [70]
    J. Cioslowski and R. Lopez-Boada, J. Chem. Phys. 109: 4156 (1998).Google Scholar
  90. [71]
    J. Cioslowski and R. Lopez-Boada, Chem. Phys. Lett. 307: 445 (1999).Google Scholar
  91. [72]
    J. Cioslowski and K. Pernal, J. Chem. Phys. 111: 3396 (1999).Google Scholar
  92. [73]
    J. Cioslowski and K. Pernal, Phys. Rev. A 61: 34503 (2000).Google Scholar
  93. [74]
    A. Bolas, Phys. Rev. A 59: 396 (1999).Google Scholar
  94. [75]
    G. Csanyi and T. A. Arias, Phys. Rev. B 61: 7348 (2000).Google Scholar
  95. [76]
    B. Barbiellini, J. Phys. Chem. Solids 61: 341 (2000).Google Scholar
  96. [77]
    J. Cioslowski, P. Ziesche, and K. Pernal, Phys. Rev. B 63:205105(2001). Therein, in Eq. (27) the erroneous constant 57r-1 must be replaced by (57r)-1, consequently in Eq. (28) the factor 1011 must be replaced by 2(57)-1, in Eq. (32) 57–1 by (5v)-1, and in Eq. (45) 57r/3 by 7r/15, so the correct value of the constant C should read -0.663788. In Eq. (64) replace IF by kF and in Ref. 12 replace L. M. Hag by L. Mittag. In footnote 14 read small-rs.Google Scholar
  97. [78]
    J. Cioslowski, P. Ziesche, and K. Pernal, J. Chem. Phys., in press.Google Scholar
  98. [79]
    C. Valdemoro, Adv. Quant. Chem. 31:37(1999); in: Progress in Theoretical Chemistry, S. Wilson and A. Hernandez-Laguna, ed., Kluwer, Dordrecht (1993) and references cited therein.Google Scholar
  99. [80]
    K. Yasuda and H. Nakatsuji, Phys. Rev. A 56: 2648 (1997);Google Scholar
  100. K. Yasuda, Phys. Rev. A 59: 4133 (1999);Google Scholar
  101. M. Ehara, Chem. Phys. Lett. 305:483(1999) and references cited therein.Google Scholar
  102. [81]
    K. Yasuda, Phys. Rev. A 63: 32517 (2001).Google Scholar
  103. [82]
    D. Mazziotti, Phys. Rev. A 60:4396(1999) and references cited therein.Google Scholar
  104. [83]
    R. J. Bartlett, Coupled Cluster Theory: An Overview of Recent Developments, in: D. R. Yarkony, ed., Modern Electronic Structure Theory, World Scientific, Singapore (1995)Google Scholar
  105. [84]
    P. Fulde, H. Stoll, and K. Kladko, Chem. Phys. Lett. 299:481(1999) and refs. cited therein.Google Scholar
  106. [85]
    M. B. Ruskai, J. Math. Phys. 11:3218(1970). Generally, in the more mathematically oriented community studying the properties of reduced density matrices, a series of theorems has been derived 2, 4, 8 which merit to be used and exploited more in quantum chemistry and solid state theory.Google Scholar
  107. [86]
    In Ref. 40 the factors 1/2 are erroneously incorporated in the definition of gir(x) and grl(x) such that therein g⇈(00) = g⇅(oo) = 1/2 appears instead of 1.Google Scholar
  108. [87]
    The chemical potential ζ is not to be confused with the electrochemical potential µ = ζ − D, where D is the dipole barrier of an semi-infinite EG, the work function of which is determined by µ. Note the relation ζN = E + pΩ with p, the pressure. In the equilibrium statistics it is known as Duhem-Gibbs relation. With the virial theorem 2T + V = 3pΩ it holds ζN = 5/3T + 4/3 V, or per particle, ζ= 5/3t + 4/3v. This follows also from the operator identity cf. Eq. (3.22).Google Scholar

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© Springer Science+Business Media New York 2002

Authors and Affiliations

  • P. Ziesche
    • 1
  1. 1.Max Planck Institute for the Physics of Complex SystemsDresdenGermany

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