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

Temperature dependent effective third order interatomic force constants from first principles