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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 a higher order allows for an efficient calculation of the phonon life time, in Si as well as in -FeSi; a system that exhibits anomalous softening with temperature.
Hellman et al. (2013) studied this question.