Here we develop an alternative (but equivalent) diagrammatic method based on the variation approach, rooted in introducing infinitesimal external fields into the Hamiltonian. We first consider Hedin’s equations for interacting electrons and derive the famous Hedin equations for the self-energy, vertex, polarisation functions and screened interaction. Then we build a diagrammatic method for the electron–phonon interaction based on a phenomenological phonon Hamiltonian. This is followed by considering the Hamiltonian of electrons and bare nuclei; the corresponding equations for the self-energy, polarisation function, screened interactions and vertex function are derived using both the method of Maksimov and that of Hedin and Lundqvist. The relation to the perturbation approach of the previous chapter is discussed.