Universal nature of different methods of obtaining the exact Kohn-Sham exchange-correlation potential for a given density

Finding the external potential or the Kohn-Sham potential for a given ground state density is a fundamental inverse problem in density functional theory. Furthermore, it is important in generating the exact exchange-correlation potential for a density which can then serve as a benchmark for testingthe accuracy of an exchange-correlation energy functional. Over the years different methods have been proposed to do the density-to-potential inversion and all of these appear to be disjoint. In this paper we show that these methods are connected to one general algorithm that utilizes two forms of the equation for the density; these are the Kohn-Sham equation in terms of orbitals and the Euler equations in terms of the density. We derive the condition for the convergence of the general method and show that all the other methods considered by us satisfy this condition. We demonstrate the method and its flexibility by obtaining the Kohn-Sham potential for some spherical systems in a variety of ways.