Rigorous and approximate relations between expectation values of atoms

Abstract

The Z dependence of some expectation values of local operators, such as r^α and ρ^α−1, is derived within the Thomas–Fermi model for atoms. It is shown to have the general form F (α) =A (α) Z^β(α), where β (α) =aα+b. This equation leads to further relations among the expectation values and the total energy. The parameters A (α) and β (α) are also treated variationally to get the best agreement with the Hartree–Fock expectation values and it is shown that for a certain range of α (depending on the operator) the agreement is quite good. Further, some inequalities relating the aforementioned expectation values to ρ (0), the charge density at the nucleus, are developed. Finally, the N dependence of the Z⁻¹ expansion coefficients is considered and a conjecture by March and White [N.H. March and R.J. White, J. Phys. B 5, 466 (1972)] concerning their asymptotic behavior is proved for ε₁(N).