Mechanical Properties from Periodic Plane Wave Quantum Mechanical Codes: The Challenge of the Flexible Nanoporous MIL-47(V) Framework

Abstract

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 in 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 transfered 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 k-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, where an equation of state E(V) (EOS) is fit to a series of energy versus volume points. Manual symmetry breaking could successfully lower MIL-47(V)'s energy 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.

Mechanical Properties in Metal–Organic Frameworks: Emerging Opportunities and Challenges for Device Functionality and Technological Applications

“…The stress–strain approach is implemented in DFT codes such as VASP whereas the energy‐strain method can be implemented in the CRYSTAL code . Given the shallow potential energy surface of some MOFs, extreme care should be taken in determining the appropriate deformation magnitudes for such calculations. These deformation gradients must be large enough to overcome the numerical noise in the energy calculations but also be small enough to stay within the valid elastic regime.…”

“…When comparing elastic constant values across the literature, it should also be noted that its components depend upon the choice of coordinate system and lattice vectors. Given their well‐studied experimental flexibility, the ab initio zero kelvin elastic tensor has been computed for a large number of Materials Institute Lavoisier (MIL) materials …”

Pillared-layered metal–organic frameworks for mechanical energy storage applications

“…The energy prole as a function of the volume was constructed by xed-volume relaxations in which the atomic positions and cell shape were optimized. 15,71 We also calculated the dynamical matrix at the G-point for the stable openpore phase of DMOF-1(Zn). More technical details on the DFT simulations can be found in the ESI.…”

“…1 †). An equation of state is tted following the procedure of Vanpoucke et al 71 and the resulting properties are tabulated in Table 1. The Zn variant has a slightly larger unit cell volume and it is less rigid, as reected in the equilibrium volume V 0 and bulk modulus B 0 , respectively.…”

“…85,86 We now focus on the intrinsic exibility of the empty DMOF-1 framework. This is rst achieved by constructing the energy prole as a function of the volume using DFT calculations at 0 K. 15,25,71 These DFT E(V) proles are shown in the top le panel of Fig. 3 for DMOF-1(Zn) and DMOF-1(Cu), together with the indication of various states as introduced earlier.…”