  Recent experiments [Schaeffer 2015] have shown that lithium presents an
extremely anomalous isotope effect in the 15-25 GPa pressure range. In this
article we have calculated the anharmonic phonon dispersion of $\mathrm{^7Li}$
and $\mathrm{^6Li}$ under pressure, their superconducting transition
temperatures, and the associated isotope effect. We have found a huge
anharmonic renormalization of a transverse acoustic soft mode along $\Gamma$K
in the fcc phase, the expected structure at the pressure range of interest. In
fact, the anharmonic correction dynamically stabilizes the fcc phase above 25
GPa. However, we have not found any anomalous scaling of the superconducting
temperature with the isotopic mass. Additionally, we have also analyzed whether
the two lithium isotopes adopting different structures could explain the
observed anomalous behavior. According to our enthalpy calculations including
zero-point motion and anharmonicity it would not be possible in a stable
regime.
