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 Fρⁿ{d}Dr, n∈{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^-1〉 and 〈r^-2〉. The limit of large dimensionality is also considered; in particular, it is found that T₀,D{≥}D²〈r^-2〉/2e² and K₀,D{≤}-2〈r^-1〉/{π}e, where e=2.718 28. In addition rigorous lower bounds to the exact kinetic energy of real (D=3) fermionic systems are given.