Line-integral formulas for exchange and correlation potentials separately

Formal separate expressions for the exact exchange and correlation potentials ${\mathrm{v}}_{\mathrm{x}}$ and ${\mathrm{v}}_{\mathrm{c}}$ are extracted from the formal line-integral expression of Holas and March for the whole exact exchange-correlation potential. Relations for the components of ${\mathrm{v}}_{\mathrm{c}}$ are extracted for each order in the electron-electron repulsion coupling constant, through use of the coupling-constant expansion of Görling and Levy for the external potential. The resultant expressions for ${\mathrm{v}}_{\mathrm{c}}$ and ${\mathrm{v}}_{\mathrm{x}}$ are separately path independent. The difference between ${\mathrm{v}}_{\mathrm{x}}$ and the Harbola-Sahni approximation to it, ${\mathrm{v}}_{\mathrm{x}}^{\mathrm{HS}}$, is identified as arising from a first-order contribution to the kinetic-energy density tensor. It is shown that this small correction to ${\mathrm{v}}_{\mathrm{x}}^{\mathrm{HS}}$, which we express in terms of perturbation theory, would be precisely zero if the Kohn-Sham determinant were identical to the Hartree-Fock determinant for the same density. In other words, ${\mathrm{v}}_{\mathrm{x}}^{\mathrm{HS}}$ would equal ${\mathrm{v}}_{\mathrm{x}}$ if the optimized effective potential determinant were the same as the Hartree-Fock determinant. This same property is shared by the Slater potential.