The self-consistent phonon equations were solved on an 8⇥8⇥8 reciprocal space grid , and three di↵erent additional bubble corrections termed QP[0], QP[S] and QP-NL were included besides the first order SC1 theory only based on the loop diagram . Anharmonic free energies were calculated including anharmonic correction from both loop and bubble diagram according to the method of Oba et al. .