Second-order Hartree–Fock conditional electron densities of atoms

The second-order conditional electron density c(2)N(r1|r2) associated with the electron-pair radial density D(2)N(r1, r2) is studied in the Hartree–Fock theory of atoms with N ( ⩾ 2) electrons. In particular, it is shown that when r2 → ∞, 2c(2)N(r1|r2) becomes DN − 1(r1) which is the single-electron radial density of the (N–1)-electron system arising from the removal of the outermost electron from the original system under the frozen orbital approximation. Namely, we can extract the electron density dh(r) of the outermost subshell h from the total electron-pair density D(2)N(r1, r2). Conditional expectations of multiplicative radial operators have the same properties as above. These results apply not only to the spherically averaged densities but also to the densities with angular variables though in some cases we have to specify angular values of the outgoing electron. A numerical illustration is given for the Kr atom.