Density-matrix theory for the ground state of spin-compensated harmonically confined two-electron model atoms with general interparticle repulsion

For model two-electron atoms with harmonic confinement, the correlated first-order density matrix can be expressed in terms of the relative motion wave function ΨR(r). Here we demonstrate that the probability density P(r) associated with this wave function is directly related to the x-ray scattering factor f(G). This latter quantity, in turn, is determined by the ground-state electron density n(r). The Euler-Lagrange equation of the resulting density-matrix theory is thereby shown to take the form of a third-order integro-differential equation for n(r) in which the probability density P(r)=ΨR2(r) also appears. For two specific choices of the interaction between the two fermions under consideration, the above integro-differential equation derived here is shown to lead back to known linear homogeneous differential equations for the electron density. Finally, it is emphasized that specific equations summarized here will apply directly to theoretical study of the nonrelativistic groundstate electron density n(r,Z) in the He-like ions with atomic number Z