  We develop ion-ion pair potentials for Al, Na and K for densities and
temperatures relevant to the warm-dense-matter (WDM) regime. Furthermore, we
emphasize non-equilibrium states where the ion temperature $T_i$ differs from
the electron temperature $T_e$. This work focuses mainly on ultra-fast
laser-metal interactions where the energy of the laser is almost exclusively
transferred to the electron sub-system over femtosecond time scales. This
results in a two-temperature system with $T_e>T_i$ and with the ions still at
the initial room temperature $T_i=T_r$. First-principles calculations, such as
density functional theory (DFT) or quantum Monte Carlo, are as yet not fully
feasible for WDM conditions due to lack of finite-$T$ features, e.g.
pseudopotentials, and extensive CPU time requirements. Simpler methods are
needed to study these highly complex systems. We propose to use two-temperature
pair potentials $U_{ii}(r, T_i,T_e)$ constructed from linear-response theory
using the non-linear electron density $n(\mathbf{r})$ obtained from finite-$T$
DFT with a single ion immersed in the appropriate electron fluid. We compute
equilibrium phonon spectra at $T_r$ which are found to be in very good
agreement with experiments. This gives credibility to our non-equilibrium
phonon dispersion relations which are important in determining thermophysical
properties, stability, energy-relaxation mechanisms and transport coefficients.
