Can optimized effective potentials be determined uniquely?

Local (multiplicative) effective exchange potentials obtained from the line ar-combination-of-atomic-orbital (LCAO) optimized effective potential (OEP) method are frequently unrealistic in that they tend to exhibit wrong asymp totic behavior (although formally they should have the correct asymptotic b ehavior) and also assume unphysical rapid oscillations around the nuclei. W e give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of so lutions. We circumvent this problem by projecting the equation and the exch ange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The observed numerical problems are, therefore, primarily due to the slow convergence of the projected exchange potential with respect to the size of the expansion basis set for orbitals . Nonetheless, by making a judicious choice of the basis sets, we obtain ac curate exchange potentials for atoms and molecules from an LCAO OEP procedu re, which are significant improvements over local or gradient-corrected exc hange functionals or the Slater potential. The Krieger-Li-Iafrate scheme of fers better approximations to the OEP method. (C) 2001 American Institute o f Physics.