Advertisement

Thomas-fermi and related theories of atoms and molecules

  • Elliott H. Lieb

Abstract

This article is a summary of what is know rigorously about Thomas-Fermi (TF) theory with and without the Dirac and von Weizsacker corrections. It is also shown that TF theory agrees asymptotically, in a certain sense, with nonrelativistic quantum theory as the nuclear charge z tends to infiinity. The von Weizsacker correction is shown to correct certain undesirable features of TF theory and to yield a theory in much better agreement with what is believed (but as yet unproved) to be the structure of real atoms. Many open problems in the theory are presented.

Keywords

Related Theory Nuclear Charge Electron Repulsion Neutral Case Atomic Case 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, R. A., 1975, Sobolev SPaces (Academic, New York).MATHGoogle Scholar
  2. Balazs, N., 1967, “Formation of stable molecules within the statistical theory of atoms,” Phys. Rev. 156, 42–47.ADSCrossRefGoogle Scholar
  3. Baumgartner, B., 1976, “The Thomas—Fermi theory as result of a strong-coupling limit,” Commun. Math. Phys. 47, 215— 219.Google Scholar
  4. Baxter, J. R., 1980, “Inequalities for potentials of particle systems,” Ill. J. Math. 24, 645–652.MathSciNetMATHGoogle Scholar
  5. Benguria, R., 1979, “The von Weizsacker and exchange cor-rections in Thomas—Fermi theory,” Ph.D. thesis, Princeton University (unpublished).Google Scholar
  6. Benguria, R., 1981, “Dependence of the Thomas—Fermi En-ergy on the Nuclear Coordinates,” Commun. Math. Phys., to appear.Google Scholar
  7. Benguria, R., H. Brezis, and E. H. Lieb, 1981, “The Thomas— Fermi—von Weizsacker theory of atoms and molecules,” Commas. Math. Phys. 79, 167–180.Google Scholar
  8. Benguria, R., and E. H. Lieb, 1978a, “Many-body potentials in Thomas—Fermi theory,” Ann. of Phys. (N.Y.) 110, 34–45.MathSciNetADSCrossRefGoogle Scholar
  9. Benguria, R., and E. H. Lieb, 1978b, “The positivity of the pressure in Thomas—Fermi theory,” Common. Math. Phys. 63, 193–218, Errata 71, 94 (1980).ADSGoogle Scholar
  10. Berestycki, H., and P. L. Lions, 1980, “Existence of station-ary states in nonlinear scalar field equations,” in Bifurcation Phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht), 269— 292. • See also “Nonlinear Scalar field equations, Parts I and II,” Arch. Rat. Mech. Anal., 1981, to appear.Google Scholar
  11. Brezis, H., 1978, “Nonlinear problems related to the Thomas— Fermi equation,” in Contemporary Developments in Continu-um Mechanics and Partial Differential Equations edited by G. M. de la Penha, and L. A. Medeiros (North-Holland, Am-sterdam), 81–89.Google Scholar
  12. Brezis, H., 1980, “Some variational problems of the Thomas-Fermi type,” in Variational Inequalities and Complementarity Problems: Theory and Applications edited by R. W. Cottle, F. Giannessi, and J-L. Lions (Wiley, New York), 53–73.Google Scholar
  13. Brezis, H., and E. H. Lieb, 1979, “Long range atomic poten-tials in Thomas-Fermi theory,” Commun. Math. Phys. 65, 231–246.Google Scholar
  14. Brezis, H., and L. Veron, 1980, “Removable singularities of nonlinear elliptic equations,” Arch. Rat. Mech. Anal. 75, 1–6.Google Scholar
  15. Caffarelli, L. A., and A. Friedman, 1979, “The free boundary in the Thomas-Fermi atomic model,” J. Diff. Equ. 32, 335–356.Google Scholar
  16. Deift, P., W. Hunziker, B. Simon, and E. Vock, 1978, “ Point-wise bounds on eigenfunctions and wave packets in N-body quantum systems IV,” Commun. Math. Phys. 64, 1–34.Google Scholar
  17. Dirac, P. A. M., 1930, “Note on exchange phenomena in the Thomas-Fermi atom,” Proc. Cambridge Philos. Soc. 26, 376–385.Google Scholar
  18. Fermi, E., 1927. “Un metodo statistico per la determinazione di alcune priorieta dell’atome,” Rend. Accad. Naz. Lincei 6, 602–607.Google Scholar
  19. Firsov, 0. B., 1957, “Calculation of the interaction potential of atoms for small nuclear separations,” Zh. Eksper. i Teor. Fiz. 32, 1464. English transl. Soy. Phys.—JETP 5, 1192–1196 (1957)). See also Zh. Eksp. Teor. Fiz. 33, 696 (1957); 34, 447 (1958) [Soy. Phys.—JETP 6, 534–537 (1958); 7, 308–311 (1958)].Google Scholar
  20. Fock, V., 1932, “Uber die Giiltigkeit des Virialsatzes in der Fermi-Thomas’schen Theorie,” Phys. Z. Sowjetunion 1, 747–755.Google Scholar
  21. Gilbarg, D., and N. Trudinger, 1977, Elliptic Partial Differ-ential Equations of Second Order (Springer Verlag, Heidel-berg).Google Scholar
  22. Gombas, P., 1949, Die statistischen Theorie des Atomes und ihwe Anwenahingen (Springer Verlag, Berlin).Google Scholar
  23. Hille, E., 1969, “On the Thomas-Fermi equation,” Proc. Nat. Acad. Sc;. (USA) 62, 7–10.Google Scholar
  24. Hoffmann-Ostenhof, M., T. Hoffmann-Ostenhof, R. Ahlrichs, and J. Morgan, 1980, “On the exponential falloff of wave functions and electron densities,” Mathematical Problems in Theoretical Physics, Proceedings of the International Con-ference on Mathematical Physics held in Lausanne, Switzer-land, August 20–25, 1979 Springer Lectures Notes in Phys-ics, edited by K. Osterwalder (Springer-Verlag, Berlin, Heidelberg, New York, 1980), Vol. 116, 62–67.Google Scholar
  25. Hoffmann-Ostenhof, T., 1980, “A comparison theorem for differential inequalities with applications in quantum me-chanics,” J. Phys. A 13, 417–424.Google Scholar
  26. Jensen, H., 1933, “Uber die Giiltigkeit des Virialsatzes in der Thomas-Fermischen Theorie,” Z. Phys, 81, 611–624.Google Scholar
  27. Kato, T., 1957, “On the eigenfunctions of many-particle sys-tems in quantum mechanics,” Commun. Pure Appl. Math. 10, 151–171.Google Scholar
  28. Lee, C. E., C. L. Longmire, and M. N. Rosenbluth, 1974, “Thomas-Fermi calculation of potential between atoms,” Los Alamos Scientific Laboratory Report No. LA-5694-MS.Google Scholar
  29. Liberman, D. A., and E. H. Lieb, 1981, “Numerical calcula-tion of the Thomas-Fermi-von Weizsacker function for an in-finite atom without electron repulsion,” Los Alamos National Laboratory Report in preparation.Google Scholar
  30. Lieb, E. H., 1974, “Thomas-Fermi and Hartree-Fock theory,” in Proceedings of the International Congress of Mathematicians, Vancouver Vol. 2, 383–386.Google Scholar
  31. Lieb, E. H., 1976, “The stability of matter,” Rev. Mod. Phys. 48, 553–569.Google Scholar
  32. Lieb, E. H., 1977, “Existence and uniqueness of the minimiz-ing solution of Choquard’s nonlinear equation,” Stud. in Appl. Math. 57, 93–105.Google Scholar
  33. Lieb, E. H., 1979, “A lower bound for Coulomb energies,” Phys. Lett. A 70, 444–446.Google Scholar
  34. Lieb, E. H., 1981a, “A variational principle for many-fermion systems,” Phys. Rev. Lett. 46, 457–459; Erratum 47, 69 (1981).Google Scholar
  35. Lieb, E. H., 1981b, “Analysis of the Thomas-Fermi-von Weizsacker equation for an atom without electron repulsion,” in preparation.Google Scholar
  36. Lieb, E. H., and S. Oxford, 1981, “An improved lower bound on the indirect Coulomb energy,” Int. J. Quantum Chem. 19, 427–439.Google Scholar
  37. Lieb, E. H., and B. Simon, 1977, “The Thomas-Fermi theory of atoms, molecules and solids,” Adv. in Math. 23, 22–116.Google Scholar
  38. These results were first announced in “Thomas-Fermi theory revisited,” Phys. Rev. Lett. 31, 681–683 (1973). An outline of the proofs was given in Lieb, 1974.Google Scholar
  39. Lieb, E. H., and B. Simon, 1978, “Monotonicity of the elec-tronic contribution to the Born-Oppenheimer energy,” J. Phys. B 11, L537–542.Google Scholar
  40. Lieb, E. H., and W. Thirring, 1975, “Bound for the kinetic energy of fermions which proves the stability of matter,” Phys. Rev. Lett. 35, 687–689; Errata 35, 1116 (1975).Google Scholar
  41. Lieb, E. H., and W. Thirring, 1976, “A bound for the moments of the eigenvalues of the Schroedinger Hamiltonian and their relation to Sobolev inequalities,” in Studies in Mathematical Physics: Essays in Honor of Valentine Borgmann edited by E. H.Google Scholar
  42. Lieb, B. Simon, and A. S. Wightman (Princeton Uni-versity Press, Princeton), 269–303. March, N. H., 1957, “The Thomas-Fermi approximation in quantum mechanics,” Adv. in Phys. 6, 1–98.Google Scholar
  43. Mazur, S., 1933, “Uber konvexe Mengen in linearen normier-ten Riumen,” Studia Math. 4, 70–84. See p. 81.Google Scholar
  44. Morgan, J., III., 1978, “The asymptotic behavior of bound eigenfunctions of Hamiltonians of single variable systems,” J. Math. Phys. 19, 1658–1661.Google Scholar
  45. Morrey, C. B., Jr., 1966, Multiple integrals in the calculus of variations (Springer, New York). O’Connor, A. J., 1973, “Exponential decay of bound state wave functions,” Commun. Math. Phys. 32, 319–340.Google Scholar
  46. Reed, M., and B. Simon, 1978, Methods of Modern Mathemati-cal Physics (Academic, New York), Vol. 4. Ruskai, M. B., 1981, “Absence of discrete spectrum in highly negative ions,” Commun.Google Scholar
  47. Math. Phys. (to appear). Scott, J. M. C., 1952, “The binding energy of the Thomas-Fermi atom,” Philos. Mag. 43, 859–867.Google Scholar
  48. Sheldon, J. W., 1955, “Use of the statistical field assumption in molecular physics,” Phys. Rev. 99, 1291–1301.Google Scholar
  49. Simon, B., 1981, “Large time behavior of the Lt norm of Schroedinger semigroups.” J. Func. Anal. 40. 66–83.Google Scholar
  50. Stampacchia, G., 1965, Equations elliptiques du second ordre a coefficients discontinus (Presses de l’Universite, Montreal).Google Scholar
  51. Teller, E., 1962, “On the stability of molecules in the Thomas-Fermi theory,” Rev. Mod. Phys. 34. 627–631.Google Scholar
  52. Thirring, W., 1981, “A lower bound with the best possible con-stant for Coulomb Hamiltonians,” Commun. Math. Phys. 79, 1–7 (1981).Google Scholar
  53. Thomas, L. H., 1927, “The calculation of atomic fields,” Proc. Camb. Philos. Soc. 23, 542–548.Google Scholar
  54. Torrens, I. M., 1972, Interatomic Potentials (Academic, New York).Google Scholar
  55. Veron, L., 1979, “Solutions singulieeres d’equations elliptiques semilineaire,” C. R. Acad. Sci. Paris 288, 867–869.Google Scholar
  56. This is an announcement; details will appear in “Singular solutions of nonlinear elliptic equations,” J. Non-Lin. Anal., in press. von Weizsicker, C. F., 1935, “Zur Theorie der Kernmassen,” Z. Phys. 96, 431–458.Google Scholar
  57. Yonei, K., and Y. Tomishima, 1965, “On the Weizsacker cor-rection to the Thomas-Fermi theory of the atom,” Jour. Phys. Soc. Japan 20, 1051–1057.Google Scholar
  58. Yonei, K., 1971, “An extended Thomas-Fermi-Dirac theory for diatomic molecules,” Jour. Phys. Soc. Japan 31, 882–894.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations