As a second model system of interest, we will consider the ionisation potential (IP) of a system of two electrons in a singlet configuration in four spin orbitals. We will enforce an exact fulfilment of Koopman's theorem (or specifically, its DFT version) in the dressed orbital basis. The exact IP of this system can be calculated as IP = E N-1 -E where the ground state energy of the N -1 electron system can is simply E N-1 = h 00 . In the dressed orbital basis, the HOMO energy is given as the following, where we again suppress the perturbative superscript: