THE ELECTRON-PAIR DENSITY OF ATOMIC SYSTEMS - RIGOROUS BOUNDS AND APPLICATION TO HELIUM

Firstly, the monotonicity properties of the electron-pair density I(u) of atomic systems are investigated. Let h(u) denote the spherically-averaged electron-pair density of an arbitrary N-electron system, which essentially coincides with I(u) in the case of atoms. It is found that the interelectronic function g(alpha)(u)=h(u)/u(alpha), alpha greater-than-or-equal-to 0, is not only monotonically decreasing from the origin for alpha greater-than-or-equal-to alpha1=max{uh'(u)/h(u)} but it also has the property of convexity for alpha greater-than-or-equal-to alpha2 the value of alpha2 is given in the text. Secondly, the Stieltjes technique is used to obtain rigorous, simple and compact inequalities which involve three interelectronic radial expectation values [u(k)]. These inequalities are universal in the sense that they are valid for both ground and excited states in the whole periodic table. Thirdly, for those systems with a unimodal h(u), i.e. having a single maximum at u=u(max), are found (i) upper bounds to u(max) in terms of any number of moments [u(k)] via the above-mentioned technique, and (ii) lower bounds to the maximal value h(max) = h(u(max)) by means of two arbitrary moments [u(k)] in a variational way. A particular case of the latter bound leads to a rigorous upper bound to the total electron-electron repulsion energy E(ee) of the system, namely E(ee) less-than-or-equal-to [9pi/8 N2(N-1)2h(max)]1/3. Finally, the electron-pair density of Helium is analysed in detail and the quality of the above mentioned inequalities is studied by means of the M-term Hylleraas-type wavefunctions, with M=1, 2, 3, 6, 10 and 20. We observe that in the 20-term case, which is shown to be very close to the exact one, alpha1 and alpha2 take the values 0.0414 and 0.2067, respectively. Moreover, in such a case we found that some of the above mentioned inequalities are very accurate.