Bounds to density-dependent quantities of D-dimensional many-particle systems in position and momentum spaces: Applications to atomic systems

Starting from any two finite expectation values, rigorous bounds to quantities of D-dimensional many-particle systems, depending on the single-particle density ρ(r) in the form F${\rho }^n$$d^D$r, n∈${\mathrm{openR}}^{+}$ (the set of positive real numbers), are explicitly given. Similar bounds are also valid in momentum space. The resulting expressions are used to rigorously bound the kinetic ($T_0$,D) and Dirac exchange ($K_0$,D) energies of fermionic systems in the plane-wave approximation (e.g., the Thomas-Fermi approach). As a numerical illustration, very accurate upper bounds to the Dirac exchange energy of atomic systems are found by means of 〈$r^{\mathrm{-}1}$〉 and 〈$r^{\mathrm{-}2}$〉. The limit of large dimensionality is also considered; in particular, it is found that $T_0$,D≥${\mathrm{D}}^2$〈${\mathrm{r}}^{\mathrm{-}2}$〉/2${\mathrm{e}}^2$ and $K_0$,D≤-2〈${\mathrm{r}}^{\mathrm{-}1}$〉/πe, where e=2.718 28. In addition rigorous lower bounds to the exact kinetic energy of real (D=3) fermionic systems are given.