Differential inequality techniques are applied in the derivation of the following upper bound for the one-electron density $\rho \mathrm{}\left(x\right)\mathrm{}$ of atomic or molecular systems: $\sqrt{\rho \mathrm{}\left(x\right)\mathrm{}}\le C{\mathrm{}(1+r)\mathrm{}}^{\frac{\mathrm{}(Z-n+1\mathrm{})\mathrm{}}{\sqrt{2\epsilon }-1\mathrm{}}}{e}^{-\sqrt{2\epsilon }r}$ where $r=\left|x\right|\mathrm{}$, $\epsilon$ is the first ionization potential, $n$ the number of electrons, and $Z$ the nuclear charge (or the sum of nuclear charges in the molecular case). Related bounds on the decay of the $k$-electron density and the wave function itself are also given. These bounds improve upon previous results [TH-O, MH-O, RA Phys. Rev. A 18, 328 (1978)]. For the ground state of a two-electron atom (ion) we report a lower bound to $\rho \mathrm{}\left(x\right)\mathrm{}$ which exhibits the same functional form as the upper bound. Finally, for this case we give a lower bound to the wave function itself which shows essentially the same decay as the corresponding upper bound.

• Received 3 June 1980

DOI:https://doi.org/10.1103/PhysRevA.23.2106