  The temperature dependent effective potential (TDEP) method is generalized
beyond pair interactions. The second and third order force constants are
determined consistently from ab initio molecular dynamics simulations at finite
temperature. The reliability of the approach is demonstrated by calculations of
the Mode Gr\"uneisen parameters for Si. We show that the extension of TDEP to
higher order allows for an efficient calculation of the phonon life time, in Si
as well as in $\epsilon$-FeSi, a system that exhibits anomalous softening with
temperature.
