Abstract
We show that the ground state energy \(E_\mathrm {M}(Z)\) of the Müller functional of (neutral) atoms of atomic number \(Z\) equals to the quantum mechanical ground state energy \(E_\mathrm {S}(Z)\) up order \(o(Z^{5/3})\), i.e., \( E_\mathrm {M}(Z)= E_\mathrm {S}(Z)+ o(Z^{5/3}). \)
©Heinz Siedentop. Based on a translation of J. Phys. A: Math. Theor. 42, 085201 (2009), doi:10.1088/1751-8113/42/8/085201.
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Notes
- 1.
Here and in the following \(\mathrm {C}\) denote a generic constant.
- 2.
Note the operator \(P\) introduced this way is no projection.
References
Bach, V.: Error bound for the Hartree–Fock energy of atoms and molecules. Comm. Math. Phys. 147, 527–548 (1992)
Bach, V.: Accuracy of mean field approximations for atoms and molecules. Comm. Math. Phys. 155, 295–310 (1993)
Cioslowski, J., Pernal, K.: Constraints upon natural spin orbital functionals imposed by properties of a homogeneous electron gas. J. Chem. Phys. 111(8), 3396–3400 (1999)
Fefferman, C., Seco, L.: Eigenfunctions and eigenvalues of ordinary differential operators. Adv. Math. 95(2), 145–305 (1992)
Fefferman, C., Seco, L.: The density of a one-dimensional potential. Adv. Math. 107(2), 187–364 (1994)
Fefferman, C., Seco, L.: The eigenvalue sum of a one-dimensional potential. Adv. Math. 108(2), 263–335 (1994)
Fefferman, C., Seco, L.: On the Dirac and Schwinger corrections to the ground-state energy of an atom. Adv. Math. 107(1), 1–188 (1994)
Fefferman, C., Seco, L.: The density in a three-dimensional radial potential. Adv. Math. 111(1), 88–161 (1995)
Fefferman, C.L., Seco, L.A.: An upper bound for the number of electrons in a large ion. Proc. Nat. Acad. Sci. USA 86, 3464–3465 (1989)
Fefferman, C.L., Seco, L.A.: On the energy of a large atom. Bull. AMS 23(2), 525–530 (1990)
Fefferman, C.L., Seco, L.A.: Aperiodicity of the Hamiltonian flow in the Thomas-Fermi potential. Rev. Math. Iberoamericana 9(3), 409–551 (1993)
Frank, R.L., Lieb, E.H., Seiringer, R., Siedentop, H.: Müller’s exchange-correlation energy in density-matrix-functional theory. Phys. Rev. A 76(5), 052517 (2007)
Frank, R.L., Lieb, E.H., Seiringer, R., Siedentop, H.: Müller’s exchange-correlation energy in density-matrix-functional theory. arXiv:0705.1587v3 (2009)
Friesecke, G.: On the infinitude of non-zero eigenvalues of the single-electron density matrix for atoms and molecules. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2029), 47–52 (2003)
Gilbert, T.L.: Hohenberg-Kohn theorem for nonlocal external potentials. Phys. Rev. B 12(6), 2111–2120 (1975)
Graf, G.M., Solovej, JP.: A correlation estimate with applications to quantum systems with Coulomb interactions. Rev. Math. Phys. 6(5A), 977–997 (1994) (Special issue dedicated to E.H. Lieb)
Hohenberg, P., Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 2(136), B864–B871 (1964)
Lee, D., Burke, K., Constantin, L.A., Perdew, J.P.: Exact condition on the Kohn–Sham kinetic energy, and modern parametrization of the Thomas-Fermi density. ArXiv.org http://arxiv.org/abs/0810.1992v1 (Oct 2008)
Lewin, M.: Quelques modèles non linéaires en mécanique quantique. PhD thesis, Paris Dauphine. http://tel.archives-ouvertes.fr/documents/archives0/00/00/63/06/index.html (2004)
Lieb, E.H.: A lower bound for Coulomb energies. Phys. Lett. 70A, 444–446 (1979)
Lieb, E.H..: Erratum: Variational principle for many-fermion systems [Phys. Rev. Lett. 46(7), 457–459 (1981), MR 81m:81083]. Phys. Rev. Lett. 47(1), 69 (1981)
Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53(4), 603–641 (1981)
Lieb, E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett. 46(7), 457–459 (1981)
Lieb, E.H., Oxford, S.: Improved lower bound on the indirect Coulomb energy. Intern. J. Quantum Chem. 19, 427–439 (1981)
Lieb, E.H., Simon, B.: On solutions of the Hartree–Fock problem for atoms and molecules. J. Chem. Phys. 61(2), 735–736 (1974)
Lieb, E.H., Simon, B.: The Hartree–Fock theory for Coulomb systems. Comm. Math. Phys. 53(3), 185–194 (1977)
Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977)
Müller, A.M.K.: Explicit approximate relation between reduced two- and one-particle density matrices. Phys. Lett. A 105(9), 446–452 (1984)
Schwinger, J.: Thomas-Fermi model: the second correction. Phys. Rev. A 24(5), 2353–2361 (1981)
Sharma, S., Dewhurst, J.K., Lathiotakis, N.N., Gross, E.K.U.: Reduced density matrix functional for many-electron systems. Phys. Rev. B (Condens. Matter Mater. Phys.) 78(20), 201103 (2008)
Siedentop, H.: Das asymptotische Verhalten der Grundzustandsenergie des Müllerfunktionals für schwere Atome. J. Phys. A 42(8), 085201, 9 (2009)
Thirring, W.: A lower bound with the best possible constant for Coulomb Hamiltonians. Comm. Math. Phys. 79, 1–7 (1981)
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Partial support by the SFB-TR 12 of the DFG is gratefully acknowledged.
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Siedentop, H. (2014). The Quantum Energy Agrees with the Müller Energy up to Third Order. In: Bach, V., Delle Site, L. (eds) Many-Electron Approaches in Physics, Chemistry and Mathematics. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-06379-9_11
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