Direct and exchange contributions to inner and outer radii in many-electron atoms

Within the Hartree-Fock framework, the spinless two-electron density function Gamma(r(1), r(2)) consists of direct Gamma(di)(r(1), r(2)) and exchange Gamma(ex)(r(1), r(2)) parts. Accordingly, the inner < r(<)> and outer < r(>)> radii in many-electron systems are rigorously separated into the direct and exchange contributions, i.e., < r(<)> = < r(<)>(di) + < r(<)>(ex) and < r(>)> =< r(>)>(di) + < r(>)>(ex). It is generally shown that < r(<)>(di) + < r(>)>(di) = 2 < r > and < r(<)>(ex) + < r(>)>(ex) = 0, where < r > is the usual average radius of an electron. Numerical examinations of the direct and exchange contributions for the 102 atoms from He to Lr in their ground states find that the electron exchange works to decrease < r(<)> and increase < r(>)>. However, the exchange parts are very small and the direct parts essentially govern the inner and outer radii.