## Error bound for the Hartree-Fock energy of atoms and molecules.(English)Zbl 0771.46038

The ground state energy $$E_ Q$$ of the Hamiltonian $H_ N(\underline {Z},\underline {R})=\sum_{i=1}^ N \left(-\Delta_ i-\sum_{j=1} ^ K {{Z_ j} \over {| x_ j-R_ j|}}\right)+ \sum_{1\leq i<j}^ N {1\over {| x_ i-x_ j|}}$ where $$\underline {Z}=[Z_ 1,\dots,Z_ K],\;\underline {R}=[R_ 1,\dots,R_ k]$$, is investigated. The error of the Hartree-Fock energy $$E_{HF}$$ is estimated when $$Z\to \infty$$, $$N\approx Z$$, $$\underline{Z}/Z$$ fixed, $$\min| R_ i-R_ j| \geq CZ^{-2/3+\varepsilon}$$ $$(Z=\sum_{j=1}^ K Z_ j)$$. For any $$0<\delta< 2/21$$ there exists a $$C_ \delta>0$$ such that $$| E_ Q-E_{HF}|\leq C_ \delta Z^{5/3-\delta}$$.

### MSC:

 46N50 Applications of functional analysis in quantum physics 81V45 Atomic physics 81V55 Molecular physics

### Keywords:

ground state energy; Hartree-Fock energy
Full Text:

### References:

 [1] Coleman, A.J.: Structure of fermion density matrices. Rev. Mod. Phys.35(3), 668–689 (1963) [2] Dirac, P.A.M.: Note on exchange phenomena in the Thomas-Fermi atom. Proc. Cambridge Philos. Soc.26, 376–385 (1931) · JFM 56.0751.04 [3] Fefferman, C.L., de la Llave, R.: Relativistic stability of matter-I. Revista Matematica Iberoamericana,2(1, 2), 119–161 (1986) · Zbl 0602.58015 [4] Fefferman, C.L., Seco, L.A.: The ground-state energy of a large atom. Bull. A.M.S. (1990) · Zbl 0722.35072 [5] 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) [6] Hughes, W.: An Atomic Energy Lower Bound that Gives Scott’s Correction. PhD thesis, Princeton, Department of Mathematics, 1986 [7] Ivrii, V.Ja., Sigal, I.M.: Asymptotics of the ground state energies of large Coulomb systems. Annals Math. (to appear) · Zbl 0789.35135 [8] Lieb, E.H.: A lower bound for Coulomb energies. Phys. Lett.70A, 444–446 (1979) [9] Lieb, E.H.: On characteristic exponents in turbulence. Commun. Math. Phys.92, 473–480 (1984) · Zbl 0598.76054 [10] Lieb, E.H.: The stability of matter. Rev. Mod. Phys.48, 653–669 (1976) [11] Lieb, E.H.: Themas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys.53, 603–604 (1981) · Zbl 1114.81336 [12] Lieb, E.H.: Variational principle for many-fermion systems. Phys. Rev. Lett.46(7), 457–459 (1981) [13] Lieb, E.H., Oxford, S.: An improved lower bound on the indirect Coulomb energy. Int. J. Quantum Chem.19, 427–439 (1981) [14] Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23, 22–116 (1977) · Zbl 0938.81568 [15] Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In Lieb, E.H., Simon, B., Wightman, A.S. (eds.), Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton, NJ: Princeton University Press 1976 · Zbl 0342.35044 [16] Schwinger, J.: Thomas-Fermi model: The second correction. Phys. Rev. A24(5), 2353–2361 (1981) [17] Siedentop, H.K.H., Weikard, R.: A new phase space localization technique with application to the sum of negative eigenvalues of Schrödinger operators. Ann. Scient. Ec. Norm. Sup.24, 215–225 (1991) · Zbl 0762.47022 [18] Siedentop, H.K.H., Weikard, R.: On the leading energy correction for the statistical model of the atom: Interacting case. Commun. Math. Phys.112, 471–490 (1987) · Zbl 0920.35120 [19] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton, New Jersey: Princeton University Press, 2 edition, 1971 · Zbl 0232.42007 [20] Thirring, W.: Lehrbuch der Mathematischen Physik 4: Quantenmechanik großer Systeme. Wien, New York: Springer, 1 edition, 1980 · Zbl 0453.46054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.