Weizsäcker energy of many-electron systems

Rigorous lower bounds to the Weizsäcker energy ${\mathit{T}}_{\mathit{W}}$ of a many-fermion system are derived by means of two and three radial expectation values. Some of them are of variational nature and others are founded on classical integral inequalities and a theorem which allows us to extend universally the validity of the bounds to ${\mathit{T}}_{\mathit{W}}$ obtained for spherically symmetric densities. Also, rigorous and approximate upper bounds, in terms of a radial expectation value and the ionization potential, are encountered in the case of atomic systems by taking into account, at times, the properties of the monotonicity of the electron density. The role of the expectation value 〈${\mathit{r}}^{\mathrm{-}2}$〉 is highly remarkable in the determination of the bounds. The bounds found in this work allow us to correlate rigorously the Weizsäcker energy with numerous fundamental and/or experimentally measurable quantities of the system, such as, e.g., the number of constituents, the diamagnetic susceptibility, the diamagnetic-shielding correction, and the softness kernel in the density-functional theory. Finally, just for checking the quality of both lower and upper bounds, numerical comparisons employing Hartree-Fock atomic densities are done in the whole Periodic Table.