A proposed family of variationally correlated first-order density matrices for spin-polarized three-electron model atoms

In early work of March and Young (Phil Mag 4:384, 1959), it was pointed out for spin-free fermions that a first-order density matrix (1DM) for $N-1$ particles could be constructed from a 2DM ( $Gamma $ ) for $N$ fermions divided by the diagonal of the 1DM, the density $n(mathbf{r}_1)$ , as $2Gamma (mathbf{r}_1,mathbf{r}^{prime }_2;mathbf{r}_1,mathbf{r}_2)/n(mathbf{r}_1)$ for any arbitrary fixed $mathbf{r}_1$ . Here, we thereby set up a family of variationally valid 1DMS constructed via the above proposal, from an exact 2DM we have recently obtained for four electrons in a quintet state without confining potential, but with pairwise interparticle interactions which are harmonic. As an indication of the utility of this proposal, we apply it first to the two-electron (but spin-compensated) Moshinsky atom, for which the exact 1DM can be calculated. Then the 1DM is found for spin-polarized three-electron model atoms. The equation of motion of this correlated 1DM is exhibited and discussed, together with the correlated kinetic energy density, which is shown explicitly to be determined by the electron density.