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

Abstract

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 ${E}_{\mathrm{c}}^{\mathrm{MP}2}$ 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 ${v}_{\mathrm{c}}^{\mathrm{MP}2}(r\ensuremath{\rightarrow}\ensuremath{\infty})=0$. On the other hand, there exists a proof that ${v}_{\mathrm{c}}^{\mathrm{MP}2}(r)$ 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 ${E}_{\mathrm{c}}^{\mathrm{MP}2}$, and (iii) the OPM equation does allow a solution satisfying ${v}_{\mathrm{c}}^{\mathrm{MP}2}(r\ensuremath{\rightarrow}\ensuremath{\infty})=0$ 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.