ELECTRON-PAIR LOGARITHMIC CONVEXITY AND INTERELECTRONIC MOMENTS IN ATOMS - APPLICATION TO HELIUM-LIKE IONS

The electron-pair function h (u) of a finite many-electron system is n ot monotonic, but the related quantity h (u)/u(alpha), alpha > 0, is n ot only monotonically decreasing from the origin but also convex for t he values alpha1 and alpha2, respectively, as has been recently found. Here, it is first argued that this quantity is also logarithmically c onvex for any alpha>alpha' with alpha'=max{-u2d2[Inh(u)]/du2}. Then th is property is used to obtain a general inequality which involves thre e interelectronic moments [u(t)]). Particular cases of this inequality involve relevant characteristics of the system such as the number of electrons and the total electron-electron repulsion energy. Second, th e logarithmic-convexity property of h (u) as well as the accuracy of t his inequality are investigated by the optimum 20-term Hylleraas-type wave functions for two-electron atoms with nuclear charge Z=1, 2, 3, 5 , and 10. It is found that (i) 14<alpha'<20 for these atomic cases (we remark that alpha' much greater than alpha2 much greater than alpha1) and (ii) the accuracy of the inequality which involves moments of con tiguous orders oscillates between 62.4% and 96.7% according to the spe cific He-like atom and the moments involved. Finally, the importance o f the logarithmic-convexity effects on the interelectronic moments rel ative to those coming from other monotonicity properties of h (u)/u(al pha) are analyzed in the same numerical Hylleraas framework.