where Ω qv are the renormalised frequencies, the last term in the second sum corresponds to a unitary transformation of harmonic frequencies, and α qν = 2Ωqv [1 + 2n(Ω qv )] where n(ω) is the Bose-Einstein distribution. The first sum corresponds to a simple rearrangement of Eq. ( ). Within TDEP, a slightly different approach is taken, where the free energy reads