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(i sigma) An explicit expression for the exchange potential upsilon (x sigma) in terms of the occupied KS orbitals is obtained via the analytical inverse of the resulting density response function chi (s sigma) with upsilon (x sigma) being subdivided into the Slater potential upsilon (S sigma) and the response potential upsilon (res sigma). The latter exhibits a characteristic orbital structure with diagonal contributions from the densities \ psi (i sigma)\ (2) of the occupied KS orbitals as well as off-diagonal ones from the occupied-occupied orbital products psi (i sigma)psi*(j sigma). An expression for the response part f(resp sigma) 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(i sigma) fails to produce the field-counteracting potential delta upsilon (resp sigma) 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 He-2, the present kernel f(resp sigma) generates a field-counteracting potential delta upsilon ((fc))(resp sigma). The field-counteracting exchange effect is seen to arise from spatial dependence of the cross product psi (g)psi (u)rho (-1) of the symmetric and antisymmetric orbitals which is coupled with an integral over itself times delta rho. Similar self-coupling terms are indicated in the general case of a symmetric molecular chain.