Exact density matrix for a two-electron model atom and approximate proposals for realistic two-electron systems

Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-1/2 particles interacting via harmonic forces and moving in a harmonic-oscillator potential. Here, the exact ground-state density ρ(r) is related to the (also analytically known) Hartree-Fock density ρ_HF(r). The generalization to the off-diagonal matrix γ(r,r^′) is then effected, this being related to the idempotent γ_HF(r,r^′)/2. This exact information on this “model atom” prompts us to propose an approximate form of γ(r,r^′) for the He-like ions, the H₂ molecule and, in general, all two-electron systems. γ(r,r^′) is constructed solely from the exact ρ(r) and its Hartree-Fock counterpart. Some detailed treatment of the two-electron Hookean atom with spring constant k=1/4 (atomic units) is also presented.