Atomic systems with a completely monotonic electron density

For a many-particle system with a spherically averaged single-particle density {ρ}(r) of a kth-order monotonic nature (i.e., a function whose successive derivatives up to that of order k alternate in sign), rigorous inequalities that involve the central values of {ρ}(r) and its derivatives as well as the radial expectation values are derived. These inequalities become optimal in the completely monotonic case, that is for k{→}{∞}. Then, it is argued that for atomic systems the electron density is completely monotone to quite a good approximation except in hydrogen, where it is rigorous. In this approximation, the corresponding atomic inequalities produce the following: (i) the famous tight upper bound to the electron density at the nucleus {ρ}(0){≤}(Z/2{π})〈r^-2〉 obtained by Hoffmann-Ostenhof, Hoffmann-Ostenhof, and Thirring [J. Phys. B 11, L571 (1978)] by assuming an infinite nuclear mass, (ii) the lower bound {ρ}(0){≥}〈r^-2{〉}²/(4π〈{r}^{{{-}}1}$〉), (iii) lower bounds to the values of any kth-order derivative of the electron density at the nucleus, and (iv) some inequalities involving two and/or three radial expectation values. These bounds and/or inequalities improve all the corresponding ones known at present. Finally, for completeness, the accuracy of these atomic bounds and/or inequalities are analyzed in the framework of the Hartree-Fock approximation.