  We use first-principles density functional theory total energy and linear
response phonon calculations to compute the Helmholtz and Gibbs free energy as
a function of temperature, pressure, and cell volume in the flexible
metal-organic framework material MIL-53(Cr) within the quasiharmonic
approximation. GGA and metaGGA calculations were performed, each including
empirical van der Waals (vdW) forces under the D2, D3, or D3(BJ)
parameterizations. At all temperatures up to 500 K and pressures from -30 MPa
to 30 MPa, two minima in the free energy versus volume are found, corresponding
to the narrow pore ($np$) and large pore ($lp$) structures. Critical positive
and negative pressures are identified, beyond which there is only one free
energy minimum. While all results overestimated the stability of the $np$ phase
relative to the $lp$ phase, the best overall agreement with experiment is found
for the metaGGA PBEsol+RTPSS+U+J approach with D3 or D3(BJ) vdW forces. For
these parameterizations, the calculated free energy barrier for the $np$-$lp$
transition is only 3 to 6 kJ per mole of Cr$_4$(OH)$_4$(C$_8$H$_4$O$_4$)$_4$.
