On the leading energy correction for the statistical model of the atom: Interacting case

Abstract

Introducing the Hellmann-Weizsacker functional for large angular momenta and the orbitals of the Bohr atom for small angular momenta we obtain an upper bound on the quantum mechanical ground state energy of atoms that proves Scott's conjecture.

The ground-state energy of heavy atoms according to Brown and Ravenhall: absence of relativistic effects in leading order

“…By the analogon of Lieb's result [18,16] (see also Bach [1]) -which trivially transcribes from the Schrödinger setting to the present one -we have for all γ ∈ S N (21) E(c, N, Z) ≤ E HF (γ). (10), (11), (12), and (16)) turns the right hand side into the Thomas-Fermi kinetic energy modulo the positive error Z ∇g 2 R −2 (see Lieb [17, Formula (5.9)]), i.e., (24) tr…”

“…Lieb and Simon [20] proved that the leading behavior of the ground state energy is given by the Thomas-Fermi energy which decreases as Z 7/3 . The leading correction to this behavior, the so called Scott correction was established by Hughes [14,15] (lower bound), and Siedentop and Weikard [24,25,26,27,28] (lower and upper bound). In fact even the existence of the Z 5/3 -correction conjectured by Schwinger was proven (Fefferman and Seco [9,10,11,4,12,7,5,6,8]).…”

Asymptotic behavior of the ground state of large atoms

“…The first term was established by Lieb and Simon [13]. The proof of (19) has been given by Siedentop and Weikard [17,16] (see also Hughes [9] for the lower bound) for the neutral case and has been extended to general a by Bach [1].…”

Asymptotics for Two-Dimensional Atoms

“…By considering the hydrogen semiclassics we conjecture that the next term of E(λZ, Z) is of order Z 3/2 . In contrast, the ground state energy in three dimensions behaves as E(Z, Z) = c TF Z 7/3 + c S Z 2 + c DS Z 5/3 + o(Z 5/3 ), where the leading (Thomas-Fermi [28,6]) term was established in [15], the second (Scott [21]) term was proved in [7,23], and the third (Dirac-Schwinger [2,20]) term was shown in [5].…”