Study of some interelectronic properties in helium-like atoms

By means of the optimumM-term Hylleraastype wavefunctions with 1≦M≦6 we study various interelectronic properties of the Helium-like atoms with nuclear chargeZ=1, 2, 3, 5 and 10. Leth(u) denote the spherically averaged electron-pair density of a finite many-electron system. Firstly we found that the intracule functionh(u)/u ^α of the above-mentioned atoms is (i) monotonically decreasing from the origin for α≥α₁ and (ii) convex for α≥α₂, where α₁ and α₂ are positive constants which depend onZ andM. Then we show that the electron-electron cusp condition, i.e. thath′(0)=h(0), may be extended in the sense that the inequalityh(u)?h′(u)≧0 is valid for anyu≥0. Thirdly, it is shown that the inequalities involving three interelectronic moments 〈u ⁿ 〉 recently found by the authors are, at times, of great quality. Finally the goodness of some bounds to the characteristics of the maximum ofh(u) and to the total interelectronic repulsion energy is discussed in detail.