Study of the discontinuity of the exchange-correlation potential in an   exactly soluble case
  It was found by Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. {\bf 49}, 1691 (1982)] and by Sham and Schl\"uter [Phys. Rev. Lett. {\bf 51}, 1884 (1983)] that the exact Kohn-Sham exchange-correlation potential of an open system may jump discontinuosly as the particle number crosses an integer, with important physical consequences. Recently, Sagvolden and Perdew [Phys. Rev. A {\bf 77}, 012517 (2008)] have analyzed the discontinuity of the exchange-correlation potential as the particle number crosses one, with an illustration that uses a model density for the H$^-$ ion. In this work, we extend their analysis to the case in which the external potential is the simple harmonic confinement, choosing spring-constant values for which the two-electron hamiltonian has an analytic solution. This way, we can obtain the exact, analytic, exchange and correlation potentials for particle number fluctuating between zero and two, illustrating the discontinuity as the particle number crosses one without introducing any model or approximation. We also discuss exchange and correlation separately. 
Paola Gori-Giorgi and Andreas Savin