  The rate of energy transfer between electrons and phonons is investigated by
a first principles framework for electron temperatures up to $T_e=50000$ K
while considering the lattice at ground state. Two typical but differently
complex metals are investigated, namely Aluminium and Copper. In order to
reasonably take the electronic excitation effect into account, we adopt finite
temperature density functional theory and linear response to determine the
electron-temperature-dependent Eliashberg function and electron density of
states. Of the three branch-dependent electron-phonon coupling strengths, the
longitudinal acoustic mode plays a dominant role in the electron-phonon
coupling for Aluminium for all temperatures considered here, but for Copper it
only dominates above an electron temperature of $T_e=40000$ K. The second
moment of the Eliashberg function and the electron phonon coupling constant at
room temperature $T_e=315$ K show good agreement with other results. For
increasing electron temperatures, we show the limits of the $T=0$ approximation
for the Eliashberg function. Our present work provides a rich perspective on
the phonon dynamics and this will help to improve insight into the underlying
mechanism of energy flow in ultra-fast laser-metal interaction.
