The attraction between electrons in BCS theory is the result of an ‘overscreening’ of their Coulomb repulsion by vibrating ions. When the interaction between ion vibrations and electrons (i.e. the electron–phonon interaction) is strong, the electron Bloch states are affected even in the normal phase. Phonons are also affected by conduction electrons. In doped insulators (like high-temperature superconductors), ‘bare’ phonons are well defined in insulating parent compounds but a separation of electron and phonon degrees of freedom might be a problem in a metal. Here we have to start with the first-principle Hamiltonian describing conduction electrons and ions coupled by the Coulomb forces: