Density Functional Methods for Plasmas and Liquid Metals

  • Chandre Dharma-wardana
  • François Perrot
Part of the NATO ASI Series book series (NSSB, volume 337)


Density functional theory (DFT)[1–3] has proved itself to be an effective first principles calculational method for the electronic and structural properties of a large variety of condensed matter systems. Most of these applications are to atomic, molecular, or solid systems where the ions merely provide a static “external potential” acting on the electronic system. DFT provides a means of reducing this many-electron problem to an effective “single-electron” problem couched in terms of a universal exchange correlation functional of the one-electron density. Given that the universe is mostly made up of ionized matter, i.e., plasmas, it is natural to turn to DFT to develop microscopic theories of matter in the plasma state. The main characteristics of the plasma state from our point of view are (i) the existence of electron populations in continuum as well as in bound states (ii) need to consider not just the electronic subsystem, but also the ionic subsystem, and (iii) finite temperature effects. Thus an Al-plasma at a temperature of 105K and at a density of 1/10 of the normal solid density contains continuum electrons, and not just Al3+, but also Al2+, Al1+, Al, and possibly even some Al-molecular species. Such a mixture is analogous to a molten alloy in many ways and could be much more complex because of partial electron-degeneracy effects and the complex nature of the continuum electron states. If the mass density of Al were maintained at the normal value and the temperature raised towards 106K the dominant species evolve from Alz+, z=3 to a Be-like “Aluminum” with a net charge of z=7, while the free electrons become classical. As another example consider a hydrogen plasma at sufficiently high temperatures and pressures; then it could well be a fully ionized system of electrons and protons. If the same hydrogen plasma were examined at a lower temperature of interest to planetary scientists, e.g., say, at T= 2eV, a pressure of 0.5 Mbar, and a mass density of around 0.2–0.5 g/cc, it would be found to contain e, H H+, H+ 2, H2, and H 2, interacting with each other, with most vibrational, rotational and electronic (bound- as well as continuum-) states occupied to form distributions consistent with the given temperature and chemical potentials. In other words, the electronic and ionic distributions are a function of each other and the pressure and temperature of the system.


Density Functional Theory Local Density Approximation Finite Temperature Hydrogen Plasma Pair Distribution Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    P. Hohenberg and W. Kohn, Phys. Rev B 136, 864 (1964).MathSciNetADSCrossRefGoogle Scholar
  2. W. Kohn and L.J. Sham, Phys. Rev. A. 140, 1133 (1965). See S. Luindqvist and N. March, ed., Theory of the Inhomogeneous Electron Gas (Plenum, New York 1983). R. Dreizler and E. Gross, Density Functional Theory, An approach to the Quantum Many-Body problem (Springer-Verlag, Berlin 1990).MathSciNetADSGoogle Scholar
  3. 2.
    N.D. Mermin, Phys. Rev. A 132, 1141 (1965).MathSciNetGoogle Scholar
  4. 3.
    M.W.C. Dharma-wardana and Francois Perrot, Phys. Rev. A 26, 2096 (1982).ADSCrossRefGoogle Scholar
  5. M.W.C. Dharma-wardana and Francois Perrot, Phys. Rev. 29, 1378 (1984). also, see the reviews in Strongly Coupled Plasma Physics, F.J. Rogers and H.E. DeWitt, Eds., (Plenum, New York 1987).CrossRefGoogle Scholar
  6. 4.
    M.W.C. Dharma-wardana and R. Taylor, J. Phys. C14, 629 (1981).ADSGoogle Scholar
  7. 5.
    F. Perrot and M.W.C. Dharma-wardana, Phys. Rev. A 30, 2619 (1984), also U. Gupta and A.K. Rajagopal, Phys. Rev A22, 2792.ADSCrossRefGoogle Scholar
  8. 6.
    F. Perrot, Y. Furutani and M.W.C. Dharma-wardana, Phys. Rev. A 41, 1096 (1990).Google Scholar
  9. 7.
    F. Grimaldi, A. Grimaldi-Lecourt and M.W.C. Dharma-wardana, Phys. Rev. A 32, 1063 (1985).CrossRefGoogle Scholar
  10. 8.
    F. Perrot and M.W.C. Dharma-wardana, Phys. Rev. A 36, 238 (1987).ADSCrossRefGoogle Scholar
  11. 9.
    D. Ofer, E. Nardi, and Y. Rosenfeld, Phys. Rev A 38, 5801, (1988).ADSCrossRefGoogle Scholar
  12. 10.
    F. Perrot and M.W.C. Dharma-wardana Equations of State for Astrophysics Eds. G. Chabrier and E. Schatzman (Cambridge University Press, U. K. 1994).Google Scholar
  13. 11.
    R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985).ADSCrossRefGoogle Scholar
  14. 12.
    W. Kohn and M. Luttinger Phys. Rev 118, 41 (1960).MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. 13.
    A.K. Fetter and J.D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, N.Y. 1971).Google Scholar
  16. 14.
    S. Vosko, L. Wilks, and M. Nusair, Can. J. Phys. 58, 388 (1980).MathSciNetCrossRefGoogle Scholar
  17. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 41, 566 (1980).ADSCrossRefGoogle Scholar
  18. A.H. MacDonald, M.W.C. Dharma-wardana and D.J.W. Geldart, J. Phys. F Iß, 1719 (1980).Google Scholar
  19. 15.
    D. G. Kanhere, P.V. Panat, R.K. Rajagopal, and J. Callaway, Phys. Rev A33, 490 (1986).ADSGoogle Scholar
  20. 16.
    R.G. Dundrea, N.W. Ashcroft, and A.E. Carlsson, Phys. Rev B34, 2097 (1986).ADSGoogle Scholar
  21. 17.
    S. Ichimaru, H. Iyetomi, and S. Tanaka, Phys. Rep. 149, 91 (1987).ADSCrossRefGoogle Scholar
  22. 18.
    R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1975).zbMATHGoogle Scholar
  23. J.-P. Hansen and I. MacDonald, Theory of Simple Liquids, (Academic, London 1986).Google Scholar
  24. 19.
    K.-C. Ng, J. Chem. Phys. 61, 2680 (1974).ADSCrossRefGoogle Scholar
  25. 20.
    F. Lado, Y. Rosenfeld and N. W. Ashcroft, Phys. Rev A20, 1028 (1976).Google Scholar
  26. 21.
    D.C. Langreth and J.P. Perdew, Phys. Rev. B15, 2884 (1977).ADSGoogle Scholar
  27. 22.
    J. Chihara Phys. Rev. A33, 2575 (1986), also J. Phys. C17, 1633 (1986).ADSGoogle Scholar
  28. 23.
    F. Perrot and M.W.C. Dharma-wardana, Phys. Rev A41, 3281 (1990).ADSGoogle Scholar
  29. 24.
    G.D. Mahan, Many-Particle Physics, Plenum, NY (1990).CrossRefGoogle Scholar
  30. 25.
    F. Perrot, A20, 586 (1979), for T=0K calculations see e.g., E. Zaremba et al J. Phys. F7, 1763 (1977).ADSCrossRefGoogle Scholar
  31. 26.
    J.P. Perdew and A. Zunger, Phys. Rev B23, 5048 (1981).ADSGoogle Scholar
  32. see also J. Dobson, J.Phys. C 4, 7877 (1992).Google Scholar
  33. 27.
    J. Rosenfeld(private communication), we have confirmed this independently.Google Scholar
  34. 28.
    M.W.C. Dharma-wardana and F. Perrot, Phys. Rev. A45, 5883 (1992).ADSGoogle Scholar
  35. 29.
    W. Kohn and C. Majumdar, Phys. Rev. A138, 1617 (1965).MathSciNetADSCrossRefGoogle Scholar
  36. 30.
    D. Saumon and G. Chabrier, Phys. Rev A46, 2084 (1992).ADSGoogle Scholar
  37. D.D.J. Stevensen and E.E. Salpeter, Astrophys. J. Suppl. Ser. 35, 221 (1977).ADSCrossRefGoogle Scholar
  38. E.B. Wigner and H.B. Huntington, J. Chem. Phys. 3,764 (1935).ADSCrossRefGoogle Scholar
  39. 31.
    Stauffer, Physics Reports, Scaling theory of percolation clusters. 54, 1 (1979).Google Scholar
  40. 32.
    e.g. See D.H. Reitze, H. Ahn and M.C. Downer, Phys. Rev. B 45, 2677 (1992).CrossRefGoogle Scholar
  41. 33.
    N.W. Ashcroft, Il Nuovo Cimento 12D, 597 (1990), also Z. Badirkhan et. al. ibid, 619.ADSCrossRefGoogle Scholar
  42. 34.
    M.W.C. Dharma-wardana and F. Perrot, Phys. Rev. Lett. 65, 76 (1990), Modern Physics Letters B. 5, 161 (1991).ADSCrossRefGoogle Scholar
  43. 35.
    P.S. Salmon, J. Phys. F. 18, 2345 (1988).ADSCrossRefGoogle Scholar
  44. 36.
    G. Galli, R. Martin, R. Car and M. Parrinello, Phys. Rev. Lett. 63, 988 (1989).ADSCrossRefGoogle Scholar
  45. I. Stich, R. Car and M. Parrinello, ibid,. 63, 2240 (1989).ADSCrossRefGoogle Scholar
  46. 37.
    F.A. Stillinger and T.A. Weber, Phys. Rev. B. 31, 5265 (1985).ADSCrossRefGoogle Scholar
  47. 38.
    F. Perrot, Phys. Rev. A42, 4871 (1990).ADSGoogle Scholar
  48. J.M. Ziman, Proc. R. Soc. London. 91, 701 (1976).Google Scholar
  49. 39.
    F. Nardin, J. Jaccucci, and M.W.C. Dharma-wardana, Phys. Rev A. 37, 1025 (1988).ADSCrossRefGoogle Scholar
  50. 40.
    A. Zangwill and P. Soven, Phys. Rev. A21 1561, (1980).ADSGoogle Scholar
  51. 41.
    T. Ando, Phys Rev B13, 3468 (1976).ADSGoogle Scholar
  52. T. Ando et al, Rev. Mod. Phys. 54, 437, (1982).ADSCrossRefGoogle Scholar
  53. 42.
    F. Grimaldi, A. Grimaldi-Lecourt, and M.W.C. Dharma-wardana, Phys. Rev. A32, 1063 (1985).ADSGoogle Scholar
  54. 43.
    F. Perrot and M.W.C. Dharma-wardana, Phys. Rev Lett. 71, 797 (1993).ADSCrossRefGoogle Scholar
  55. 44.
    B.K. Godwal A. Ng L. DaSilva Y.T. Lee and D.A. Liberman Phys. Rev. A 40 4521 1989.ADSCrossRefGoogle Scholar
  56. also T.A. Hall et. al. Phys. Rev. Lett. 60, 2034 (1988).ADSCrossRefGoogle Scholar
  57. 45.
    G.I. Kerley, Report No. LA-8833-M, Los Alamos (1981).Google Scholar
  58. 46.
    Y.T. Lee and R.M. More, Phys. Fluids 27, 1973 (1984).Google Scholar
  59. G.A. Rinker, Phys. Rev. B 31, 4207 (1985).ADSCrossRefGoogle Scholar
  60. 47.
    For a review of the Keldysh method see J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).ADSCrossRefGoogle Scholar
  61. Other methods: see D.N. Zubarev, Non-Equilibrium Statistical Thermodynamics, PJ. Shepherd Trs., (Consultants bureau, New York 1974).Google Scholar
  62. 48.
    The calculation for the equilibrium case has been given by G.D. Mahan, J. Phys. Chem. Solids. 31, 1477 (1970).ADSCrossRefGoogle Scholar
  63. 49.
    P. Cellier and Andrew Ng (Private communication).Google Scholar
  64. 50.
    H.M. Milchberg et al, Phys. Rev. Lett. 61, 2364 (1988).ADSCrossRefGoogle Scholar
  65. 51.
    M.W.C. Dharma-wardana and F. Perrot, Physics Letters A. 163, 223 (1992).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Chandre Dharma-wardana
    • 1
  • François Perrot
    • 2
  1. 1.National Research Council of CanadaOttawaCanada
  2. 2.Centre d’Etudes de Limeil-ValentonVilleneuve-St Georges CEDEXFrance

Personalised recommendations