Recent progress in low-order density matrix theory of inhomogeneous electron liquids by exact solution of two- and four-electron model atoms

Three aspects of low-order density matrix (DM) theory will be reviewed, following some brief comments on analogies with, and differences from, density functional theory (DFT). First, the local energy equation, involving first-to-third order DMs, will be set out, and applied exactly to model spin-compensated two-electron atoms. Explicit relations known between low-order DMs for harmonic confinement and arbitrary interparticle interactions will be reported for such model atoms. Secondly, the March-Young proposal for use variationally, satisfying $N$-representability, will be set out for spin-free systems such as a four-electron model in the quintet state. Thirdly, the equation of motion for the correlated 1DM is summarized, and brief comments made on its application to (a) the He atom, and (b) crystalline Si. Finally, a model two-electron atom with Coulomb confinement plus an $s$-wave Coulomb repulsion modified by a $\delta$-function radial correlation is reported as an exactly solvable example.