Modeling the flexibility of metal–organic
frameworks (MOFs)
requires the computation of mechanical properties from first principles,
e.g., for screening of materials in a database, for gaining insight
into structural transformations, and for force field development.
However, this paper shows that computations with periodic density
functional theory are challenged by the flexibility of these materials:
guidelines from experience with standard solid-state calculations
cannot be simply transferred to flexible porous frameworks. Our test
case, the MIL-47(V) material, has a large-pore and a narrow-pore shape.
The effect of Pulay stress (cf. Pulay forces) leads to drastic errors
for a simple structure optimization of the flexible MIL-47(V) material.
Pulay stress is an artificial stress that tends to lower the volume
and is caused by the finite size of the plane wave basis set. We have
investigated the importance of this Pulay stress, of symmetry breaking,
and of <i>k</i>-point sampling on (a) the structure optimization
and (b) mechanical properties such as elastic constants and bulk modulus,
of both the large-pore and narrow-pore structure of MIL-47(V). We
found that, in the structure optimization, Pulay effects should be
avoided by using a fitting procedure, in which an equation of state <i>E</i>(<i>V</i>) (EOS) is fit to a series of energy
versus volume points. Manual symmetry breaking could successfully
lower the energy of MIL-47(V) by distorting the vanadium–oxide
distances in the vanadyl chains and by rotating the benzene linkers.
For the mechanical properties, the curvature of the EOS curve was
compared with the Reuss bulk modulus, derived from the elastic tensor
in the harmonic approximation. Errors induced by anharmonicity, the
eggbox effect, and Pulay effects propagate into the Reuss modulus.
The strong coupling of the unit cell axes when the unit cell deforms
expresses itself in numerical instability of the Reuss modulus. For
a flexible material, it is therefore advisible to resort to the EOS
fit procedure