Non--Adiabatic Contributions to the Free Energy from the Electron--Phonon Interaction for Na, K, Al, and Pb

We calculate the non--adiabatic contributions to the free energy of metals due to the electron--phonon interaction at intermediate temperatures, $0 \leqslant k_{B} T < \epsilon_{F}$ for four different nearly free electron metals, Na, K, Al, and Pb. We calculate its value for $T = 0$ which has not been calculated before and we study its low--temperature behavior.