We use first-principles density functional theory total energy and linearresponse phonon calculations to compute the Helmholtz and Gibbs free energy asa function of temperature, pressure, and cell volume in the flexiblemetal-organic framework material MIL-53(Cr) within the quasiharmonicapproximation. GGA and metaGGA calculations were performed, each includingempirical van der Waals (vdW) forces under the D2, D3, or D3(BJ)parameterizations. At all temperatures up to 500 K and pressures from -30 MPato 30 MPa, two minima in the free energy versus volume are found, correspondingto the narrow pore ($np$) and large pore ($lp$) structures. Critical positiveand negative pressures are identified, beyond which there is only one freeenergy minimum. While all results overestimated the stability of the $np$ phaserelative to the $lp$ phase, the best overall agreement with experiment is foundfor the metaGGA PBEsol+RTPSS+U+J approach with D3 or D3(BJ) vdW forces. Forthese 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$_4$)$_4$.
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