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The Thomas-Fermi theory of atoms, molecules and solids. (English) Zbl 0938.81568
The Thomas-Fermi (TF) model of the quantum theory of atoms, molecules and solids is put on a firm mathematical footing. A system of $$N$$ electrons moving about a number of fixed positive charges $$Z_i$$, where $$\sum Z_i=Z$$, is considered. If $$\rho(x)$$ denotes the electron charge density at any point $$x$$ and $$V(x)$$ the Coulomb potential due to positive charges at the point $$x$$, the energy $${\mathcal E}(\rho,V)$$ of the system can be written as a functional of $$\rho$$ only. For a suitable choice of units, $${\mathcal E}(\rho,V)=\tfrac 35\int\sigma(x)^{5/3} dx-\int V(x)\rho(x) dx+\tfrac 12\int\int(\rho(x)\rho(y)/|x-y|) dxdy$$. The Euler-Lagrange, equation for minimizing $${\mathcal E}(\rho,V)$$ with the subsidiary conditions $$\int\rho(x) dx=N$$ and $$\rho\geq 0$$ leads to the TF equation. It is shown that the TF equations have a unique solution for $$N\leq Z$$, that these solutions minimize the TF energy function and that no solution exists for $$N>Z$$. The chemical potential is monotone, strictly increasing and concave as a function of $$N$$. The Hamiltonian for the system of $$N$$ electrons and the nuclei gives rise to a quantum-mechanical operator that has the ground-state energy $$E_N{}^Q$$, taking into account the spins of the electrons and the Pauli principle. It is proved that, as the nuclear charges go to infinity, quantum mechanics and TF theory become identical, and the ratio of the ground-state energies tends to unity as $$N\rightarrow\infty$$. For the heavy atoms, far away from the nuclei, the TF density $$\rho(x)$$ is strictly positive for the neutral case and $$\rho(x)\rightarrow 27\pi^{-3}|x|^{-6}$$ as $$|x|\rightarrow\infty$$. For molecules one adds the intermolecular potential to the TF energy, and it is proved that theory does not lead to binding. The TF theory is also extended to solids, i.e., to a system of infinitely large, periodic molecules. The periodic TF equation is obtained and it is shown that $$\rho$$ exists and tends to a finite limit as the volume tends to infinity. The application of TF theory to solids under high pressures is justified. A TF theory of the screening of an impurity in a solid by the electrons in the solid is presented.

##### MSC:
 81V55 Molecular physics
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##### References:
  Auchmuty, J.F.G; Beals, R, Variational solutions of some nonlinear free boundary problems, Arch. rational mech. anal., 43, 255-271, (1971) · Zbl 0225.49013  Baker, E.B, The application of the Fermi-Thomas statistical model to the calculation of potential distribution in positive ions, Phys. rev., 36, 630-647, (1930) · JFM 56.1327.06  Balazs, N, Formation of stable molecules within the statistical theory of atoms, Phys. rev., 156, 42-47, (1967)  Balslev, E, Spectral theory of Schrödinger operators of many body systems with permutation and rotation symmetries, Ann. physics, 73, 49-107, (1972)  Baumgartner, B, The Thomas-Fermi-theory as result of a strong-coupling limit, (1975), University of Vienna, preprint  Bethe, H; Jackiw, R, Intermediate quantum mechanics, (1969), Benjamin New York  Bove, A; DaPrato, G; Fano, G, An existence proof for the Hartree-Fock time dependent problem with bounded two-body interaction, Comm. math. phys., 37, 183-192, (1974) · Zbl 0303.34046  Brillouin, L, La mécanique ondulatoire de Schrödinger; une méthode générale de résolution par approximations successives, C. R. acad. sci. Paris, 183, 24-26, (1926) · JFM 52.0967.05  Brownell, F.H; Clark, C.W, Asymptotic distribution of the eigenvalues of the lower part of the Schrödinger operator spectrum, J. math. mech., 10, 31-70, (1961) · Zbl 0148.09501  Chadam, J.M; Glassey, R.T, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. math. phys., 16, 1122-1130, (1975) · Zbl 0299.35084  (), 2306-2308  Clarkson, J, Uniformly convex spaces, Trans. amer. math. soc., 40, 396-414, (1936) · JFM 62.0460.04  Courant, R; Hilbert, D, ()  Dirac, P.A.M, The principles of quantum mechanics, (1930), Oxford Univ. Press Oxford · Zbl 0030.04801  Dirac, P.A.M, Note on exchange phenomena in the Thomas atom, (), 376-385 · JFM 56.0751.04  Dunford, N; Schwartz, J, Linear operators, (1957), Interscience New York, Part I  Fermi, E, Un metodo statistico per la determinazione di alcune priorietà dell’atome, Rend. accad. naz. lincei, 6, 602-607, (1927)  Feynman, R.P; Metropolis, N; Teller, E, Equations of state of elements based on generalized Fermi-Thomas theory, Phys. rev., 75, 1561-1573, (1949) · Zbl 0036.43007  Fock, V, Näherungsmethode zur Lösung des quantenmechanischen mehrkörperproblems, Z. physik, 61, 126-148, (1930) · JFM 56.1313.08  Fock, V, Bemerkung zum virialsatz, Z. physik, 63, 855-858, (1930) · JFM 56.1327.05  Fock, V, Über die Gültigkeit des virialsatzes in der Fermi-thomaschen theorie, Phys. Z. sowjetunion, 1, 747-755, (1932) · Zbl 0005.18502  Flügge, S, ()  Fonte, G; Mignani, R; Schiffrer, G, Solution of the Hartree-Fock equations, Comm. math. phys., 33, 293-304, (1973)  Gombás, P, Die statistische theorie des atoms und ihre anwendungen, (1949), Springer-Verlag Berlin · Zbl 0031.37703  Griffiths, R, A proof that the free energy of a spin system is extensive; appendix A, J. math. phys., 5, 1215-1222, (1964)  \scF. Guerra, L. Rosen, and B. Simon, Boundary conditions in the P(ø)2 Euclidean field theory, Ann. Inst. H. Poincaré, to appear.  Hartmann, P; Wintner, A, Partial differential equations and a theorem of Kneser, Rend. circ. mat. Palermo, 4, 237-255, (1955) · Zbl 0066.34501  Hartree, D, The wave mechanics of an atom with a non-Coulomb central field, part I, theory and methods, (), 89-132 · JFM 54.0966.05  Hertl, P; Lieb, E; Thirring, W, Lower bound to the energy of complex atoms, J. chem. phys., 62, 3355-3356, (1975)  Hertl, P; Narnhofer, H; Thirring, W, Thermodynamic functions for fermions with gravostatic and electrostatic interactions, Comm. math. phys., 28, 159-176, (1972)  Hertl, P; Thirring, W, Free energy of gravitating fermions, Comm. math. phys., 24, 22-36, (1971) · Zbl 0227.47034  Hille, E, On the Thomas-Fermi equation, (), 7-10 · Zbl 0179.12903  Hille, E, Methods in classical and functional analysis, (1972), Addison-Wesley Reading, Mass., · Zbl 0223.46001  Hylleraas, G, Über den grundterm der zweielektronenprobleme von H−, he, Li+, be++ usw., Z. für physik, 65, 209-225, (1930) · JFM 56.1311.05  Jeffreys, B; Jeffreys, H, Methods of mathematical physics, (1946), Cambridge Univ. Press Cambridge · Zbl 0037.31704  Jensen, H, Über die Gültigkeit des virialsatzes in der Thomas-fermischen theorie, Z. physik, 81, 611-624, (1933) · Zbl 0006.33003  Kato, T; Kato, T; Kato, T, On the convergence of the perturbation method, I, II, Progr. theoret. phys., Progr. theoret. phys., Progr. theoret. phys., 5, 207-212, (1950)  Kato, T, Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. amer. math. soc., 70, 195-211, (1951) · Zbl 0044.42701  Kato, T, On the eigenfunctions of many particle systems in quantum mechanics, Comm. pure appl. math., 10, 151-177, (1957) · Zbl 0077.20904  Kato, T, Perturbation theory for linear operators, (1966), Springer-Verlag Berlin · Zbl 0148.12601  Kato, T, Some mathematical problems in quantum mechanics, Suppl. progr. theoret. phys., 40, 3-19, (1967)  Kittel, C, Introduction to solid state physics, (), 112-114  Köthé, G, Topological vector spaces I, (1966), Springer-Verlag Berlin · Zbl 0137.31301  Kramers, H.A, Wellenmechanik and halbzählige quantisierung, Z. phys., 39, 828-840, (1926) · JFM 52.0969.04  Lieb, E; Lebowitz, J, The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei, Advances in math., 9, 316-398, (1972) · Zbl 1049.82501  Lenz, W, Über die anwendbarkeit der statistischen methode auf ionengitter, Z. phys., 77, 713-721, (1932) · JFM 58.0949.02  Lieb, E.H, Quantum mechanical extension of the Lebowitz-Penrose theorem on the van der Waals theory, J. math. phys., 7, 1016-1024, (1966)  Lieb́, E.H; Narnhofer, H, The thermodynamic limit for jellium, J. stat. phys., Errata, 14, No. 5, 291-310, (1975) · Zbl 0973.82500  Lieb, E.H; Simon, B, Thomas-Fermi theory revisited, Phys. rev. lett., 31, 681-683, (1973)  Lieb, E.H; Simon, B, On solutions to the Hartree-Fock problem for atoms and molecules, J. chem. phys., 61, 735-736, (1974)  \scE. H. Lieb and B. Simon, The Hartree-Fock Theory for Coulomb systems, Comm. Math. Phys., to appear.  Lieb, E.H; Thirring, W, Bound for the kinetic energy of fermions which proves the stability of matter, Phys. rev. lett., 33, 687-689, (1975), Errata 1116  Lieb, E.H; Thirring, W, Inequalities for the moments of the eigenvalues of the Schrödinger equation and their relation to Sobolev inequalities, () · Zbl 0342.35044  Lions, J.L, Lectures on elliptic partial differential equations, (1967), Tata Institute Bombay · Zbl 0253.35001  March, N.H, The Thomas-Fermi approximation in quantum mechanics, Advances in phys., 6, 1-98, (1957) · Zbl 0077.23101  Martin, A, Bound states in the strong coupling limit, Helv. phys. acta, 45, 140-148, (1972)  Maslov, V.P, Theory of perturbations and asymptotic methods, (1965), Izdat. Mosk. Gor. Univ.,, French trans. Dunod, Paris, 1972  McLeod, J.B, The distribution of eigenvalues for the hydrogen atom and similar cases, (), 139-158 · Zbl 0094.06103  Milne, E.A, The total energy of binding of a heavy atom, (), 794-799 · JFM 53.0868.05  Morrey, C.B, Multiple integrals in the calculus of variations, (1966), Springer-Verlag Berlin · Zbl 0142.38701  Narnhofer, H; Thirring, W, Convexity properties for Coulomb systems, Acta phys. austriaca, 41, 281-297, (1975)  von Neumann, J, Mathematische grundlagen der quantenmechanik, (1932), Springer-Verlag Berlin, English trans. Princeton Univ. Press, Princeton, N.J., 1955 · JFM 58.0929.06  Pauli, W; Villars, F, On the invariant regularization in relativistic quantum theory, Rev. modern phys., 21, 434-446, (1949) · Zbl 0037.12503  Protter, M.H; Weinberger, S.F, Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, N.J., · Zbl 0153.13602  Rebane, T.K; Rebane, T.K, Inequalities for nonrelativistic total energies and potential energy components of isoelectronic atoms, Opt. spektrosk. (USSR), Opt. spektrosk. (USSR), 34, 488-491, (1973), (English trans.  Reed, M; Simon, B, Methods of modern mathematical physics, vol. I: functional analysis, (1972), Academic Press New York  Reed, M; Simon, B, Methods of modern mathematical physics, vol. II: Fourier analysis, self-adjointness, (1975), Academic Press New York · Zbl 0308.47002  \scM. Reed and B. Simon, “Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators,” Academic Press, New York, in preparation. · Zbl 0401.47001  Reeken, M, General theorem on bifurcation and its application to the Hartree equation of the helium atom, J. math. phys., 11, 2505-2512, (1970)  Rellich, F; Rellich, F; Rellich, F; Rellich, F; Rellich, F, Störungstheorie der spektralzerlegung, I-V, Math. ann., Math. ann., Math. ann., Math. ann., Math. ann., 118, 462-484, (1942) · JFM 68.0243.02  Rijnierse, P.J, Algebraic solutions of the Thomas-Fermi equation for atoms, ()  Robinson, D, The thermodynamic pressure in quantum statistical mechanics, (1971), Springer-Verlag Berlin  Rudin, W, Fourier analysis on groups, (1962), Interscience New York · Zbl 0107.09603  Ruelle, D, Statistical mechanics-rigorous results, (1969), Benjamin New York · Zbl 0177.57301  Gustafson, K; Sather, D, A branching analysis of the Hartree equation, Rend. mat., 4, 723-734, (1971)  Schrödinger, E, Collected papers on wave mechanics, (1928), Blackie London · JFM 54.0963.02  Scott, J.M.C, The binding energy of the Thomas-Fermi atom, Philos. mag., 43, 859-867, (1952)  Sheldon, J.W, Use of the statistical field approximation in molecular physics, Phys. rev., 99, 1291-1301, (1955) · Zbl 0064.45203  Simon, B, On the infinitude vs. finiteness of the number of bound states of an N-body quantum system, I, Helv. phys. acta, 43, 607-630, (1970)  Simon, B, Quantum mechanics for Hamiltonians defined as quadratic forms, (1971), Princeton Univ. Press Princeton, N.J., · Zbl 0232.47053  Simon, B, Resonances in N-body quantum systems with dilatation analytic potentials and the foundations of time dependent perturbation theory, Ann. of math., 97, 247-274, (1973) · Zbl 0252.47009  Simon, B, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, I, (), 395-401 · Zbl 0285.35008  Simon, B, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, III, Trans. amer. math. soc., 208, 317-329, (1975) · Zbl 0305.35078  Slater, J.C, Note on Hartree’s method, Phys. rev., 35, 210-211, (1930)  Sommerfeld, A, Asymptotische integration der differentialgleichung des Thomas-fermischen atoms, Z. phys., 78, 283-308, (1932) · JFM 58.1353.03  Stenger, W; Weinstein, A, Method of intermediate problems for eigenvalues: theory and ramifications, (1972), Academic Press New York · Zbl 0291.49034  Stuart, C, Existence theory for the Hartree equation, Arch. rational mech. anal., 51, 60-69, (1973) · Zbl 0287.34032  Tamura, H, The asymptotic eigenvalue distribution for non-smooth elliptic operators, (), 19-22 · Zbl 0312.35058  Teller, E, On the stability of molecules in the Thomas-Fermi theory, Rev. modern phys., 34, 627-631, (1962) · Zbl 0111.46107  Thomas, L.H, The calculation of atomic fields, (), 542-548 · JFM 53.0868.04  Titchmarsh, E.C; Titchmarsh, E.C, Eigenfunction expansions, (1958), Oxford Univ. Press Oxford, Part II · Zbl 0097.27601  Tranin, R.F; Podurets, M.A; Simakov, G.V; Popov, L.V; Moiseev, B.N, An experimental verification of the Thomas-Fermi model for metals under high pressure, Zh. eksp. teor. fiz., Soviet physics JETP, 35, 550-552, (1972), English Trans.:  Weidmann, J, The virial theorem and its application to the spectral theory of Schrödinger operators, Bull. amer. math. soc., 73, 452-456, (1967) · Zbl 0156.23304  Wentzel, G, Eine verallgemeinerung der quantenbedingungen für die zwecke der wellenmechanik, Z. phys., 38, 518-529, (1926) · JFM 52.0969.03  Wolkowsky, J, Existence of solutions of the Hartree equation for N electrons, an application of the Schauder-Tychonoff theorem, Indiana univ. math. J., 22, 551-558, (1972)  Yosida, K, Functional analysis, (1965), Academic Press New York · Zbl 0126.11504  Zhislin, G.M, Discreteness of the spectrum of the Schrödinger operator for systems of many particles, Trudy moskov. mat. obšč., 9, 81-128, (1960)  \scE. H. Libe, Thomas-Fermi and Hartree-Fock theory, Proc. 1974 International Congress of Mathematicians, Vancouver.  Lieb, E.H, The stability of matter, Rev. mod. phys., 48, 553-569, (1976)  \scB. Simon, Classical boundary condition methods in modern mathematical physics, Advances in Math., to appear.
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