A tale of two electrons: Correlation at high density

We review our recent progress in the determination of the high-density correlation energy E c in two-electron systems. Several two-electron systems are considered, such as the well known helium-like ions (helium), and the Hooke’s law atom (hookium). We also present results regarding two electrons on the surface of a sphere (spherium), and two electrons trapped in a spherical box (ballium). We also show that, in the large-dimension limit, the high-density correlation energy of two opposite-spin electrons interacting via a Coulomb potential is given by E c ∼ - 1 / ( 8 D 2 ) for any radial external potential V ( r ) , where D is the dimensionality of the space. This result explains the similarity of E c in the previous two-electron systems for D = 3 .