Approximations for the interparticle interaction energy in an exactly solvable two-electron model atom

The capability of different ansatz kernels, denoted as K(r,r^'), in the calculation of the electron-electron interaction energy is investigated here for an exactly solvable two-electron model atom proposed by Moshinsky. The model atom is in the spin-compensated, paramagnetic ground state. The exact expression for the interaction energy in this state, derived by the diagonal of the second-order density matrix, is used as a rigorous background for comparison. It is found that the form of K_M(r,r^')=2(r)(r^')-^p(r,r^')^q(r^',r), expressed via the (r) density distributions and operator powers of the one-body density matrix (r,r^'), results in the exact value for the interparticle interaction energy of the two-electron model atom if and only if p=q=1/2. Approximate forms with p=q1/2 and with pq at (p+q)=1 give deviations from the exact expression. ©2010 The American Physical Society