Progress on New Approaches to Old Ideas: Orbital-Free Density Functionals

  • Valentin V. Karasiev
  • Debajit Chakraborty
  • S. B. Trickey
Chapter
Part of the Mathematical Physics Studies book series (MPST)

Abstract

State-of-the-art treatment of nuclei and electrons in materials uses ab initio molecular dynamics for nuclear motion driven by Born-Oppenheimer forces from the electrons. Almost universally, those forces are calculated from density functional theory in the Kohn–Sham form. The computational costs of the conventional KS implementation scale at least as the cube of the number of electrons. This is a formidable barrier to complex system simulations with long MD runs on department-scale machines, since the DFT force calculation dominates the per step cost. The difficulty arises from the explicit dependence of the non-interacting kinetic energy on the KS non-interacting orbitals. The cost scaling worsens with use of explicitly orbital-dependent exchange-correlation functionals. The alternative approach, use of DFT in its basic form, dates to Thomas-Fermi-Dirac theory. The challenge is to have sufficiently accurate orbital-free expressions for the KS kinetic energy and exchange-correlation functionals. We discuss progress on these tasks via constraint-based methods, with emphasis on developments since the Sept. 2010 “New Approaches to Many-Electron Theory” meeting.

Notes

Acknowledgments

We acknowledge, with thanks, the many contributions and influences especially of Travis Sjostrom and of our other primary collaborators, Jim Dufty, Frank Harris, Keith Runge, Alberto Vela, Jose Luis Gázquez, and Jorge Martín del Campo. Thanks also go to Emilio Orgaz, Victor Medel, Támas Gál, and J.C. Pacheco-Kato. Helpful conversations with P. Ayers, K. Burke, K. Capelle, E.A. Carter, L. Delle Site, D. Gárcia-Aldea, A. Görling, M. Levy, M.M. Odashima, J.P. Perdew, N. Rösch, A. Ruzsinszky, and T. Wesolowski are acknowledged with thanks. We thank Luigi Delle Site and Volker Bach for the opportunity to participate in this volume and for the opportunity provided to SBT to attend the NAMET meeting in 2010. This work was supported under U.S. Dept. of Energy BES grant DE-SC 0002139.

References

  1. 1.
    Hohenberg, P., Kohn, W.: Phys. Rev. 136, B864 (1964)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    Levy, M.: Proc. Natl. Acad. Sci. USA 76, 6062 (1979)ADSCrossRefGoogle Scholar
  3. 3.
    Lieb, E.H.: Int. J. Quantum Chem. 24, 243 (1983)CrossRefGoogle Scholar
  4. 4.
    Thomas, L.H.: Proc. Camb. Phil. Soc. 23, 542 (1927)ADSCrossRefMATHGoogle Scholar
  5. 5.
    Fermi, E.: Rend. Accad. Naz. Lincei 6, 602 (1927)Google Scholar
  6. 6.
    Fermi, E.: Z. Phys. 48, 73 (1928)ADSCrossRefMATHGoogle Scholar
  7. 7.
    Dirac, P.A.M.: Proc. Cambridge Phil. Soc. 26, 376 (1930)ADSCrossRefMATHGoogle Scholar
  8. 8.
    von Weizsäcker, C.F.: Z. Phys. 96, 431 (1935)ADSCrossRefMATHGoogle Scholar
  9. 9.
    Teller, E.: Rev. Mod. Phys. 34, 627 (1962)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Lieb, E.H.: Rev. Mod. Phys. 53, 603 (1981)ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kohn, W., Sham, L.J.: Phys. Rev. 140, A1133 (1965)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    Boettger, J.C., Trickey, S.B.: Phys. Rev. B 51, 15623(R) (1995)ADSCrossRefGoogle Scholar
  13. 13.
    Boettger, J.C., Trickey, S.B.: Phys. Rev. B 53, 3007 (1996)ADSCrossRefGoogle Scholar
  14. 14.
    Barnett, R.N., Landman, U.: Phys. Rev. B 48, 2081 (1993)ADSCrossRefGoogle Scholar
  15. 15.
    Marx, D., Hutter, J.: In: Grotendorst, J. (ed.) Modern Methods and Algorithms of Quantum Chemistry, J. von Neumann Institute for Computing, Jülich, NIC Series, vol. 1, p. 301 (2000) (refs. therein)Google Scholar
  16. 16.
    Tse, J.S.: Annu. Rev. Phys. Chem. 53, 249 (2002)ADSCrossRefGoogle Scholar
  17. 17.
    Marx, D., Hutter, J.: Ab Initio Molecular Dynamics: Basic Theory and Advanced Methods. Cambridge University Press, Cambridge (2009) (refs. therein)Google Scholar
  18. 18.
    Kühne, T.D.: arXiv:1201.5945
  19. 19.
    Parr, R.G., Yang, W.: Density Functional Theory of Atoms and Molecules. Oxford, New York (1989).Google Scholar
  20. 20.
    Dreizler, R.M., Gross, E.K.U.: Density Functional Theory. Springer, Berlin (1990)CrossRefMATHGoogle Scholar
  21. 21.
    Becke, A.D.: J. Chem. Phys. 98, 1372 (1993)ADSCrossRefGoogle Scholar
  22. 22.
    Stephens, P.J., Devlin, F.J., Chabalowski, C.F., Frisch, M.J.: J. Phys. Chem. 98, 11623 (1994)CrossRefGoogle Scholar
  23. 23.
    Görling, A., Levy, M.: J. Chem. Phys. 106, 2675 (1997)ADSCrossRefGoogle Scholar
  24. 24.
    Görling, A.: Phys. Rev. B 53, 7024 (1996), erratum, ibid. 59, 10370 (1999)Google Scholar
  25. 25.
    Ivanov, S., Hirata, S., Bartlett, R.J.: Phys. Rev. Lett. 83, 5455 (1999)ADSCrossRefGoogle Scholar
  26. 26.
    Perdew, J.P., Schmidt, K.: A.I.P. Conf. Proc. 577, 1 (2001)Google Scholar
  27. 27.
    Engel, E.: In: Fiolhais, C., Nogueira, F., Marques M.A.L. (eds.) A Primer in Density Functional Theory, p. 56. Springer, Berlin (2003) (refs. therein)Google Scholar
  28. 28.
    Heyd, J., Scuseria, G.E., Ernzerhof, M.: J. Chem. Phys. 118, 8207 (2003), erratum, ibid. 124, 219906 (2006)Google Scholar
  29. 29.
    Krukau, A.V., Vydrov, O.A., Izmaylov, A.F., Scuseria, G.E.: J. Chem. Phys. 125, 224106 (2006)ADSCrossRefGoogle Scholar
  30. 30.
    Heßelmann, A., Götz, A.W., Della Sala, F., Görling, A.: J. Chem. Phys. 127, 054102 (2007)ADSCrossRefGoogle Scholar
  31. 31.
    Kümmel, S., Kronik, L.: Rev. Mod. Phys. 80, 3 (2008) (refs. therein)Google Scholar
  32. 32.
    Karasiev, V.V., Ludeña, E.V.: Phys. Rev. A 65, 062510 (2002)ADSCrossRefGoogle Scholar
  33. 33.
    González, I., González, C., Karasiev, V.V., Ludeña, E.V., Hernández, A.J.: J. Chem. Phys. 118, 8161 (2003)ADSCrossRefGoogle Scholar
  34. 34.
    Karasiev, V.V.: J. Chem. Phys. 118, 8576 (2003)ADSCrossRefGoogle Scholar
  35. 35.
    Grabo, T., Kreibach, T., Kurth, S., Gross, E.K.U.. In: Anisimov, V.I. (ed.) Strong Coulomb Correlations in Electronic Structure: Beyond the Local Density Approximation, p. 203. Gordon and Breach, Tokyo (2000) (refs. therein)Google Scholar
  36. 36.
    Clerouin, J., Pollock, E.L., Zerah, G.: Phys. Rev. A 46, 5130 (1992)ADSCrossRefGoogle Scholar
  37. 37.
    Chacón, E., Alvarellos, J.E., Tarazona, P.: Phys. Rev. B 32, 7868 (1985)ADSCrossRefGoogle Scholar
  38. 38.
    Wang, L.W., Teter, M.P.: Phys. Rev. B 45, 13196 (1992)ADSCrossRefGoogle Scholar
  39. 39.
    Foley, M., Smargiassi, E., Madden, P.A.: J. Phys.: Condens. Matt. 6, 5231 (1993)Google Scholar
  40. 40.
    Pearson, M., Smargiassi, E., Madden, P.A.: J. Phys.: Condens. Matt. 5, 3221 (1993)Google Scholar
  41. 41.
    Smargiassi, E., Madden, P.A.: Phys. Rev. B 49, 5220 (1994)ADSCrossRefGoogle Scholar
  42. 42.
    Glossman, M.D., Bálbas, L.C., Rubio, A., Alonso, J.A.: Int. J. Quantum Chem. 49, 171 (1994)CrossRefGoogle Scholar
  43. 43.
    Karasiev, V.V., Sjostrom, T., Chakraborty, D., Dufty, J.W., Harris, F.E., Runge, K., Trickey, S.B.. In: Graziani, F., et al. (eds.) Frontiers and Challenges in Warm Dense Matter, p. 61, Springer, Heidelberg (2014).Google Scholar
  44. 44.
    Wesołowski, T.A.: Phys. Rev. A 77, 012504 (2008) (refs. therein)Google Scholar
  45. 45.
    Kaminski, J.W., Gusarov, S., Wesołowski, T.A., Kovalenko, A.: J. Phys. Chem. A 114, 6082 (2010) (refs. therein)Google Scholar
  46. 46.
    Perdew, J.P., Burke, K., Ernzerhof, M.: Phys. Rev. Lett. 77, 3865 (1996), erratum, ibid. 78, 1396 (1997)Google Scholar
  47. 47.
    Perdew, J.P., Zunger, A.: Phys. Rev. B 23, 5048 (1981)ADSCrossRefGoogle Scholar
  48. 48.
    Cancio, A.C., Wagner, C.E., Wood, S.A.: Int. Quant. Chem. 112, 3796 (2012)CrossRefGoogle Scholar
  49. 49.
    SBT is indebted to So Hirata for this observation (2005)Google Scholar
  50. 50.
    Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Phys. Rev. A 16, 1782 (1977); see Eq. (2.20). (also see Romera, E., Dehesa. J.S. Phys. Rev. A 50, 256 (1994))Google Scholar
  51. 51.
    Sears, S.B., Parr, R.G., Dinur, U.: Isr. J. Chem. 19, 165 (1980)CrossRefGoogle Scholar
  52. 52.
    Harriman, J.E.: J. Chem. Phys. 83, 6283 (1985)ADSCrossRefGoogle Scholar
  53. 53.
    Herring, C.: Phys. Rev. A 34, 2614 (1986)ADSCrossRefGoogle Scholar
  54. 54.
    Levy, M., Perdew, J.P., Sahni, V.: Phys. Rev. A 30, 2745 (1984)ADSCrossRefGoogle Scholar
  55. 55.
    Tal, Y., Bader, R.F.W.: Int. J. Quantum Chem. S12, 153 (1978)Google Scholar
  56. 56.
    Bartolotti, L.J., Acharya, P.K.: J. Chem. Phys. 77, 4576 (1982)ADSCrossRefGoogle Scholar
  57. 57.
    Harriman, J.E.. In: Erdahl R., Smith Jr. V.H. (eds.) Density Matrices and Density Functionals, p. 359. D. Reidel, Dordrecht (1987)Google Scholar
  58. 58.
    Levy, M., Ou-Yang, H.: Phys. Rev. A 38, 625 (1988)ADSCrossRefGoogle Scholar
  59. 59.
    Baltin, R.: J. Chem. Phys. 86, 947 (1987)ADSCrossRefGoogle Scholar
  60. 60.
    Wang, Y.A., Carter, E.A.: In: Schwartz, S.D. (ed.) Theoretical Methods in Condensed Phase Chemistry, Chap. 5, p. 117. Kluwer, New York (2000) (refs. therein)Google Scholar
  61. 61.
    Zhou, B.J., Lignères, V.L., Carter, E.A.: J. Chem. Phys. 122, 044103 (2005)ADSCrossRefGoogle Scholar
  62. 62.
    Huang, C., Carter, E.A.: Phys. Rev. B 81, 045206 (2010)ADSCrossRefGoogle Scholar
  63. 63.
    Xia, J., Carter, E.A.: Phys. Rev. B 86, 235109 (2012)ADSCrossRefGoogle Scholar
  64. 64.
    Hodak, M., Lu, W., Bernholc, J.: J. Chem. Phys. 128, 014101 (2008)ADSCrossRefGoogle Scholar
  65. 65.
    Blanc, X., Cances, E.: J. Chem. Phys. 122, 214106 (2005)ADSCrossRefGoogle Scholar
  66. 66.
    Zhao, Y., Truhlar, D.G.: J. Chem. Phys. 125, 194101 (2006)ADSCrossRefGoogle Scholar
  67. 67.
    Karasiev, V.V., Jones, R.S., Trickey, S.B., Harris, F.E.: Phys. Rev. B 80, 245120 (2009), erratum, ibid. Phys. Rev. B 87, 239902 (2013)Google Scholar
  68. 68.
    Karasiev, V.V., Trickey, S.B., Harris, F.E.: J. Comput. Aided Mat. Des. 13, 111 (2006)ADSCrossRefGoogle Scholar
  69. 69.
    Karasiev, V.V., Jones, R.S., Trickey, S.B., Harris, F.E.: In: Paz J.L., Hernández, A.J. (eds.) New Developments in Quantum Chemistry, p. 25. Transworld Research Network, Kerala (2009)Google Scholar
  70. 70.
    Trickey, S.B., Karasiev, V.V., Jones, R.S.: Int. J. Quantum Chem. 109, 2943 (2009)ADSCrossRefGoogle Scholar
  71. 71.
    Vela, A., Medel, V., Trickey, S.B.: J. Chem. Phys. 130, 244103 (2009)ADSCrossRefGoogle Scholar
  72. 72.
    Vela, A., Pacheco-Kato, J.C., Gázquez, J.L., del Campo, J.M., Trickey, S.B.: J. Chem. Phys. 136, 144115 (2012)ADSCrossRefGoogle Scholar
  73. 73.
    Perdew, J.P.: Phys. Lett. A 165, 79 (1992)ADSCrossRefGoogle Scholar
  74. 74.
    Lacks, D.J., Gordon, R.G.: J. Chem. Phys. 100, 4446 (1994)ADSCrossRefGoogle Scholar
  75. 75.
    DePristo, A.E., Kress, J.D.: Phys. Rev. A 35, 438 (1987)ADSCrossRefGoogle Scholar
  76. 76.
    Thakkar, A.J.: Phys. Rev. A 46, 6920 (1992)ADSCrossRefGoogle Scholar
  77. 77.
    Tran, F., Wesołowski, T.A.: Int. J. Quantum Chem. 89, 441 (2002)CrossRefGoogle Scholar
  78. 78.
    Karasiev, V.V., Trickey, S.B.: Comput. Phys. Commun. 183, 2519 (2012)ADSCrossRefMATHGoogle Scholar
  79. 79.
    Borgoo, A., Tozer, D.J.: J. Chem. Theory Comput. 9, 2250 (2013)Google Scholar
  80. 80.
    Karasiev, V.V., Chakraborty, D., Shukruto, O.A., Trickey, S.B.: Phys. Rev. B 88, 161108(R) (2013)Google Scholar
  81. 81.
    Hodges, C.H.: Can. J. Phys. 51, 1428 (1973)ADSCrossRefGoogle Scholar
  82. 82.
    Kato, T.: Commun. Pure Appl. Math. 10, 151 (1957)CrossRefMATHGoogle Scholar
  83. 83.
    del Campo, J.M., Gázquez, J.L., Alvarez-Mendez, R.J., Trickey, S.B., Vela, A.: In: Ghosh, S.K., Chattaraj, P.K. (eds.) Concepts and Methods in Modern Theoretical Chemistry, vol. 1, p. 295. CRC Press, Boca Raton (2013) (refs. therein)Google Scholar
  84. 84.
    del Campo, J.M., Gázquez, J.L., Trickey, S.B., Vela, A.: J. Chem. Phys. 136, 104108 (2012)ADSCrossRefGoogle Scholar
  85. 85.
    del Campo, J.M., Vela, A., Gázquez, J.L., Trickey, S.B.: Chem. Phys. Lett. 543, 179 (2012)ADSCrossRefGoogle Scholar
  86. 86.
    Perdew, J.P., Ruzsinszky, A., Csonka, G.I., Constantin, L.A., Sun, J.W.: Phys. Rev. Lett. 88, 026403 (2009), erratum, ibid. 103, 179902 (2011)Google Scholar
  87. 87.
    Lieb, E.H., Oxford, S.: Int. J. Quantum Chem. 19, 427 (1981)CrossRefGoogle Scholar
  88. 88.
    Lieb, E.H.: Phys. Lett. 70A, 444 (1979)ADSCrossRefMathSciNetGoogle Scholar
  89. 89.
    Perdew, J.P., Ruzsinszky, A., Csonka, G.I., Vydrov, O.A., Scuseria, G.E., Constantin, L.A., Zhou, X.L., Burke, K.: Phys. Rev. Lett. 100, 136406 (2008)ADSCrossRefGoogle Scholar
  90. 90.
    Haas, P., Tran, F., Blaha, P., Pedroza, L.S., da Silva, A.J.R., Odashima, M.M., Capelle, K.: Phys. Rev. B 81, 125136 (2010)ADSCrossRefGoogle Scholar
  91. 91.
    Chan, G.K-L., Cohen, A.J., Handy, N.C.: J. Chem. Phys. 114, 631 (2001)Google Scholar
  92. 92.
    Karasiev, V.V., Sjostrom, T., Trickey, S.B.: Phys. Rev. E 86, 056704 (2012)ADSCrossRefGoogle Scholar
  93. 93.
    Gonze, X., Amadon, B., Anglade, P.-M., Beuken, J.-M., Bottin, F., Boulanger, P., Bruneval, F., Caliste, D., Caracas, R., Cote, M., Deutsch, T., Genovese, L.: Ghosez, Ph, Giantomassi, M., Goedecker, S., Hamann, D.R., Hermet, P., Jollet, F., Jomard, G., Leroux, S., Mancini, M., Mazevet, S., Oliveira, M.J.T., Onida, G., Pouillon, Y., Rangel, T., Rignanese, G.-M., Sangalli, D., Shaltaf, R., Torrent, M., Verstraete, M.J., Zerah, G., Zwanziger, J.W. Comput. Phys. Commun. 180, 2582 (2009)ADSCrossRefGoogle Scholar
  94. 94.
    Gonze, X., Rignanese, G.-M., Verstraete, M., Beuken, J.-M., Pouillon, Y., Caracas, R., Jollet, F., Torrent, M., Zerah, G., Mikami, M.: Ghosez, Ph, Veithen, M., Raty, J.-Y., Olevano, V., Bruneval, F., Reining, L., Godby, R., Onida, G., Hamann, D.R., Allan, D.C.: Zeit. Kristallogr. 220, 558 (2005)Google Scholar
  95. 95.
    Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G.L., Cococcioni, M., Dabo, I., Dal Corso, A., de Gironcoli, S., Fabris, S., Fratesi, G., Gebauer, R., Gerstmann, U., Gougoussis, C., Kokalj, A., Lazzeri, M., Martin-Samos, L., Marzari, N., Mauri, F., Mazzarello, R., Paolini, S., Pasquarello, A., Paulatto, L., Sbraccia, C., Scandolo, S., Sclauzero, G., Seitsonen, A.P., Smogunov, A., Umari, P., Wentzcovitch, R.M.: J. Phys.: Condens. Matt. 21, 395502 (2009)Google Scholar
  96. 96.
    Kresse, G., Hafner, J.: Phys. Rev. B 47, 558 (1993)ADSCrossRefGoogle Scholar
  97. 97.
    Kresse, G., Hafner, J.: Phys. Rev. B 49, 14251 (1994)ADSCrossRefGoogle Scholar
  98. 98.
    Kresse, G., Furthmüller, J.: Comput. Mat. Sci. 6, 15 (1996)CrossRefGoogle Scholar
  99. 99.
    Kresse, G., Furthmüller, J.: Phys. Rev. B 54, 11169 (1996)ADSCrossRefGoogle Scholar
  100. 100.
    Heine, V., Abarenkov, I.: Phil. Mag. 9, 451 (1964)ADSCrossRefGoogle Scholar
  101. 101.
    Goodwin, L., Needs, R.J., Heine, V.: J. Phys.: Condens. Matt. 2, 351 (1990)Google Scholar
  102. 102.
    Karasiev, V.V., Sjostrom, T., Trickey, S.B.: Phys. Rev. B 86, 115101 (2012)ADSCrossRefGoogle Scholar
  103. 103.
    Geng, H.Y., Song, H.X., Li, J.F., Wu, Q.: J. Appl. Phys. 111, 063510 (2012)Google Scholar
  104. 104.
    Ho, G.S., Lignères, V.L., Carter, E.A.: Comput. Phys. Commun. 179, 839 (2008)ADSCrossRefGoogle Scholar
  105. 105.
    Hung, L., Huang, C., Shin, I., Ho, G.S., Lignères, V.L., Carter, E.A.: Comput. Phys. Commun. 181, 2208 (2010)ADSCrossRefMATHGoogle Scholar
  106. 106.
    Karasiev, V.V., Sjostrom, T., Trickey, S.B.: Finite-temperature orbital-free DFT molecular dynamics: coupling Profess and Quantum Espresso. Comput. Phys. Commun. (submitted)Google Scholar
  107. 107.
    Constantin, L.A., Fabiano, E., Laricchia, S., Della Sala, F.: Phys. Rev. Lett. 106, 186406 (2011)Google Scholar
  108. 108.
    Wang, B., Stott, M.J., von Barth, U.: Phys. Rev. A 63, 052501 (2001)ADSCrossRefGoogle Scholar
  109. 109.
    Trickey, S.B., Vela, A.: J. Mex. Chem. Soc. 57, 105 (2013)Google Scholar
  110. 110.
    Acharya, P.K., Bartolotti, L.J., Sears, S.B., Parr, R.G.: Proc. Natl. Acad. Sci. USA 77, 6978 (1980)ADSCrossRefGoogle Scholar
  111. 111.
    Yang, W., Parr, R.G., Lee, C.: Phys. Rev. A 34, 4586 (1986)ADSCrossRefGoogle Scholar
  112. 112.
    Gárcia-Aldea, D., Alvarellos, J.E.: J. Chem. Phys. 127, 144109 (2007)ADSCrossRefGoogle Scholar
  113. 113.
    Gárcia-Aldea, D., Alvarellos, J.E.: Phys. Chem. Chem. Phys. 14, 1756 (2012)CrossRefGoogle Scholar
  114. 114.
    Delle Site, L.: Eur. Phys. Lett. 86, 40004 (2009), erratum, ibid. 88, 19901 (2009)Google Scholar
  115. 115.
    Ghiringhelli, L.M., Hamilton, I.P., Delle Site, L.: J. Chem. Phys. 132, 014106 (2010)Google Scholar
  116. 116.
    Ghiringhelli, L.M., Delle Site, L.: Phys. Rev. B 77, 073104 (2008)Google Scholar
  117. 117.
    Trickey, S.B., Karasiev, V.V., Vela, A.: Phys. Rev. B 84, 075146 (2011)ADSCrossRefGoogle Scholar
  118. 118.
    Levy, M., Perdew, J.P.: Phys. Rev. A 32, 2010 (1985)ADSCrossRefGoogle Scholar
  119. 119.
    Ludeña, E.V., López-Boada, R.: Top. Curr. Chem. 180, 169 (1996)CrossRefGoogle Scholar
  120. 120.
    Stoitsov, M.V., Petkov, I.Zh.: Annals Phys. 185, 121 (1988)Google Scholar
  121. 121.
    Ludeña, E.V., Karasiev, V., López-Boada, R., Valderrama, E., Maldonado, J.: J. Chem. Phys. 20, 155 (1999)Google Scholar
  122. 122.
    Ludeña, E.V., Karasiev, V.V.: In: Sen, K.D. (ed.) Reviews of Modern Quantum Chemistry: A Celebration of the Contributions of Robert Parr, p. 612. World Scientific, Singapore (2002)Google Scholar
  123. 123.
    Wang, Y.A., Govind, N., Carter, E.A.: Phys. Rev. B 58, 13465 (1998), erratum, ibid. 64, 129901 (2001)Google Scholar
  124. 124.
    Wang, Y.A., Govind, N., Carter, E.A.: Phys. Rev. B 60, 16350 (1999), erratum, ibid. 64, 089903 (2001)Google Scholar
  125. 125.
    Perdew, J.P., Constantin, L.A.: Phys. Rev. B 75, 155109 (2007)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Valentin V. Karasiev
    • 1
  • Debajit Chakraborty
    • 1
  • S. B. Trickey
    • 1
  1. 1.Quantum Theory Project, Departments of Physics and of ChemistryUniversity of FloridaGainesvilleUSA

Personalised recommendations