Since this equation is also a direct consequence of the general second order energy equation (Eq. 17), the proof of existence for an Ĥ(0) , which regenerates the exact correlation energy at second order within this generalised orbital space, is identical to the proof presented in appendix A. For this reason, the orbital generalisation does not change the absence of a variational theorem for Ĥ(0) , but it also does not affect the fact that an exact energy Ĥ(0) is fully defined by the ground state density.