Article Contents
Article Contents

# Understanding Thomas-Fermi-Like approximations: Averaging over oscillating occupied orbitals

• The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, nature's glue". (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.
Mathematics Subject Classification: 41, 45, 46, 49, 81.

 Citation:

•  [1] A. Messiah, "Quantum Mechanics," Dover, 1999. [2] L. H. Thomas, The calculation of atomic fields, Proc. Cambridge Philos. Soc., 23 (1926), 542-548.doi: 10.1017/S0305004100011683. [3] E. Fermi, Un metodo statistico per la determinazione di alcune proprieta dell atomo, Rend. Accad. Naz. Licei, 6 (1927), 602-607. [4] J. A. Goldstein and G. R. Rieder, Some extensions of Thomas-Fermi theory, Lecture Notes in Mathematics, 1223 (1986), 110-121.doi: 10.1007/BFb0099187. [5] J. A. Goldstein and G. R. Rieder, Recent rigorous results in Thomas-Fermi theory, Lecture Notes in Mathematics, 1394 (1989), 68-82.doi: 10.1007/BFb0086753. [6] P. Benilan, J. A. Goldstein and G. R. Rieder, Nonlinear elliptic system arising in electron-density theory, Communications in Partial Differential Equations, 17 (1992), 2079-2092.doi: 10.1080/03605309208820914. [7] G. R. Rieder, J. A. Goldstein and N. Naima, A convexified energy functional for the Fermi-Amaldi correction, Discrete and Continuous Systems, 28 (2010), 41-65.doi: 10.3934/dcds.2010.28.41. [8] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev., 136 (1964), B864-B871.doi: 10.1103/PhysRev.136.B864. [9] W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation, Phys. Rev., 140 (1965), A11333-A1138.doi: 10.1103/PhysRev.140.A1133. [10] S. Kurth and J. P. Perdew, Role of the exchange-correlation energy: Nature's glue, Int. J. Quantum Chem., 77 (2000), 819-830.doi: 10.1002/(SICI)1097-461X(2000)77:5<814::AID-QUA3>3.0.CO;2-F. [11] J. P. Perdew, L. A. Constantin, E. Sagvolden and K. Burke, Relevance of the slowly-varying electron gas to atoms, molecules, and solids, Phys. Rev. Lett., 97 (2006), 223002, 4 pages.doi: 10.1103/PhysRevLett.97.223002. [12] J. Schwinger, Thomas-Fermi model: The leading correction, Phys. Rev. A, 22 (1980), 1827-1832; Thomas-Fermi model: The second correction, ibid., 24 (1981), 2353-2361.doi: 10.1103/PhysRevA.22.1827. [13] B. G. Englert and J. Schwinger, Statistical atom: Some quantum improvements, Phys. Rev. A, 29 (1984), 2339-2352; Semiclassical atom, ibid., 32 (1985), 26-35.doi: 10.1103/PhysRevA.29.2339. [14] E. H. Lieb, The stability of matter, Rev. Mod. Phys., 48 (1976), 553-569.doi: 10.1103/RevModPhys.48.553. [15] L. A. Constantin, J. C. Snyder, J. P. Perdew and K. Burke, Ionization potentials in the limit of large atomic number, J. Chem. Phys., 133 (2010), 241103, 4 pages.doi: 10.1063/1.3522767. [16] J. P. Perdew and S. Kurth, Density functionals for non-relativistic Coulomb systems in the new century, in "A Primer in Density Functional Theory" ( eds. C. Fiolhais, F. Nogueira and M. Marques), Lecture Notes in Physics, 620 (2003), 1-55.doi: 10.1007/3-540-37072-2_1. [17] J. P. Perdew and L. A. Constantin, Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy, Phys. Rev. B, 75 (2007), 155109, 9 pages.doi: 10.1103/PhysRevB.75.155109. [18] A. Cangi, D. Lee, P. Elliott, K. Burke and E. K. U. Gross, Electronic structure via potential functional approximations, Phys. Rev. Lett., 106 (2011), 236404, 4 pages.doi: 10.1103/PhysRevLett.106.236404. [19] D. C. Langreth and J. P. Perdew, The exchange-correlation energy of a metallic surface, Solid State Commun., 17 (1975), 1425-1429.doi: 10.1016/0038-1098(75)90618-3. [20] O. Gunnarsson and B. I. Lundqvist, Exchange and correlation in atoms, molecules, and solids, Phys. Rev. B, 13 (1976), 4274-4298.doi: 10.1016/0375-9601(76)90557-0. [21] D. C. Langreth and J. P. Perdew, Exchange-correlation energy of a metallic surface: Wavevector analysis, Phys. Rev. B, 15 (1977), 2884-2901.