Orbital structure of the Kohn-Sham exchange potential and exchange kernel and the field-counteracting potential for molecules in an electric field

Exchange-only Kohn-Sham (KS) theory is developed based on a physically motivated common energy denominator approximation for the orbital Green's function G_iσ. An explicit expression for the exchange potential v_xσ in terms of the occupied KS orbitals is obtained via the analytical inverse of the resulting density response function χ_sσ, with v_xσ being subdivided into the Slater potential v_Sσ and the ``response'' potential v_respσ. The latter exhibits a characteristic orbital structure with ``diagonal'' contributions from the densities |ψ_iσ|² of the occupied KS orbitals as well as ``off-diagonal'' ones from the occupied-occupied orbital products ψ_iσψ_jσ*. An expression for the response part f_respσ of the exchange kernel is derived. It is established for the case of a symmetric molecular chain in an applied electric field that the kernel derived from the Krieger-Li-Iafrate potential with the Sharp-Horton approximation for G_iσ fails to produce the field-counteracting potential δv_respσ, which is lacking in local-density and generalized-gradient approximations but which is required to obtain realistic (hyper)polarizabilities. On the contrary, as is shown in the case of He₂, the present kernel f_respσ generates a field-counteracting potential δv_respσ⁽ᶠᶜ⁾. The field-counteracting exchange effect is seen to arise from the spatial dependence of the cross product ψgψᵤρ^-1 of the symmetric and antisymmetric orbitals which is coupled with an integral over itself times {δ}{ρ}. Similar ``self-coupling'' terms are indicated in the general case of a symmetric molecular chain.