Stability of atoms and molecules in an ultrarelativistic Thomas-Fermi-Weizsäcker model

Abstract

Abstract. We consider the zero mass limit of a relativistic Thomas-FermiWeizsäcker model of atoms and molecules. We find bounds for the critical nuclear charges that insure stability.

A New Estimate on the Two-Dimensional Indirect Coulomb Energy

“…In the rest of this section we will give the proof of this theorem, which is similar to the proof of Theorem 1.2 in [1]. Notice that the upper limit z c (a, b) on z to insure stability is not sharp; in other words, there could still be values of z above our z c for which ξ(ρ) ≥ 0.…”

“…Then, as usual in this situation, the values of the coupling constant (i.e., the values of the nuclear charge) will be crucial to ensure stability of the system. Our main result in this section is the following stability theorem, which is the two dimensional analog of Theorem 1.2 in [1].…”

“…A key role in our proof of the Lieb-Oxford type bound in two dimensions will be played by a stability result on an auxiliary molecular system in two dimensions. This molecular system may be viewed as the two dimensional version of the zero mass limit of the relativistic Thomas-Fermi-Weizsäcker energy functional studied in [1] (which corresponds to the zero mass limit of the model introduced in [6,7,8]; the stability properties of the corresponding atomic system were studied also in [3]). Thus, let us consider the energy functional…”

Indirect Coulomb energy for two-dimensional atoms

“…The next lemma is a generalization of the analogous result introduced in [3] and used in the proof of Theorem 1.2 above (see [4]). This lemma is later needed to prove a Coulomb Uncertainty Principle.…”

Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

“…Relevant models include Thomas-Fermi-Dirac-von Weizsäcker (TFDW) models of Density Functional theory [5,19,21]; or Schrödinger-Poisson-Slater approximation to Hartree-Fock theory [9]. Nonquadratic (q = 2) Coulombic energies appear in a possible zero mass limit of the relativistic Thomas-Fermi-von Weizsacker (TFW) energy, see [7,8] where d = 3, s = 1, α = 2, q = 3; or [6, Section 2] where d = 2, s = 1, α = 1, q = 4. The fractional case s = 1/2 occurs in the ultra-relativistic models, cf.…”