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We calculate the non--adiabatic contributions to the free energy of metalsdue to the electron--phonon interaction at intermediate temperatures, $0\leqslant k_{B} T < \epsilon_{F}$ for four different nearly free electronmetals, Na, K, Al, and Pb. We calculate its value for $T = 0$ which has notbeen calculated before and we study its low--temperature behavior

Nonadiabatic contributions to the free energy from the electron-phonon interaction in Na, K, Al, and Pb