Solubility of the optimized-potential-method integral equation for finite systems

We provide a detailed analysis of the solubility of the optimized-potential-method (OPM) integral equation for the case of the orbital- and eigenvalue-dependent correlation energy functional E-c(MP2) obtained by second-order perturbation theory on the basis of the Kohn-Sham Hamiltonian. For this functional it was shown [Phys. Rev. Lett., 86, 2241 (2001)] that for free atoms no solution of the OPM equation can be found which satisfies the boundary condition v(c)(MP2)(r ->infinity)=0. On the other hand, there exists a proof that v(c)(MP2)(r) decays like 1/r(4) [J. Chem. Phys., 118, 9504 (2003)]. Here we resolve the obvious contradiction by demonstrating that (i) the OPM equation cannot be solved if continuum states are present, (ii) the OPM equation cannot be solved for a free atom if only a finite number of Rydberg states are included in E-c(MP2), and (iii) the OPM equation does allow a solution satisfying v(c)(MP2)(r ->infinity)=0 in the case of finite systems with a countable spectrum (exemplified by an atom in a spherical box), if the complete spectrum is taken into account in the OPM procedure.