Reliably Modeling the Mechanical Stability of Rigid and Flexible Metal–Organic Frameworks

ConspectusOver the past two decades, metal–organic frameworks (MOFs) have matured from interesting academic peculiarities toward a continuously expanding class of hybrid, nanoporous materials tuned for targeted technological applications such as gas storage and heterogeneous catalysis. These oft-times crystalline materials, composed of inorganic moieties interconnected by organic ligands, can be endowed with desired structural and chemical features by judiciously functionalizing or substituting these building blocks. As a result of this reticular synthesis, MOF research is situated at the intriguing intersection between chemistry and physics, and the building block approach could pave the way toward the construction of an almost infinite number of possible crystalline structures, provided that they exhibit stability under the desired operational conditions. However, this enormous potential is largely untapped to date, as MOFs have not yet found a major breakthrough in technological applications. One of the remaining challenges for this scale-up is the densification of MOF powders, which is generally achieved by subjecting the material to a pressurization step. However, application of an external pressure may substantially alter the chemical and physical properties of the material. A reliable theoretical guidance that can presynthetically identify the most stable materials could help overcome this technological challenge.In this Account, we describe the recent research the progress on computational characterization of the mechanical stability of MOFs. So far, three complementary approaches have been proposed, focusing on different aspects of mechanical stability: (i) the Born stability criteria, (ii) the anisotropy in mechanical moduli such as the Young and shear moduli, and (iii) the pressure-versus-volume equations of state. As these three methods are grounded in distinct computational approaches, it is expected that their accuracy and efficiency will vary. To date, however, it is unclear which set of properties are suited and reliable for a given application, as a comprehensive comparison for a broad variety of MOFs is absent, impeding the widespread use of these theoretical frameworks.Herein, we fill this gap by critically assessing the performance of the three computational models on a broad set of MOFs that are representative for current applications. These materials encompass the mechanically rigid UiO-66­(Zr) and MOF-5­(Zn) as well as the flexible MIL-47­(V) and MIL-53­(Al), which undergo pressure-induced phase transitions. It is observed that the Born stability criteria and pressure-versus-volume equations of state give complementary insight into the macroscopic and microscopic origins of instability, respectively. However, interpretation of the Born stability criteria becomes increasingly difficult when less symmetric materials are considered. Moreover, pressure fluctuations during the simulations hamper their accuracy for flexible materials. In contrast, the pressure-versus-volume equations of state are determined in a thermodynamic ensemble specifically targeted to mitigate the effects of these instantaneous fluctuations, yielding more accurate results. The critical Account presented here paves the way toward a solid computational framework for an extensive presynthetic screening of MOFs to select those that are mechanically stable and can be postsynthetically densified before their use in targeted applications

Radiation Products at 77 K in Trehalose Single Crystals: EMR and DFT Analysis

The radicals obtained in trehalose dihydrate single crystals after 77 K X-irradiation have been investigated at the same temperature using X-band electron paramagnetic resonance (EPR), electron nuclear double resonance (ENDOR), and ENDOR-induced EPR (EIE) techniques. Five proton hyperfine coupling tensors were unambiguously determined from the ENDOR measurements and assigned to three carbon-centered radical species (T1, T1*, and T2) based on the EIE spectra. EPR angular variations revealed the presence of four additional alkoxy radical species (T3 to T6) and allowed determination of their g tensors. Using periodic density functional theory (DFT) calculations, T1/T1*, T2, and T3 were identified as H-loss species centered at C4, C1′, and O2′, respectively. The T4 radical is proposed to have the unpaired electron at O4, but considerable discrepancies between experimental and calculated HFC values indicate it is not simply the (net) H-loss species. No suitable models were found for T5 and T6. These exhibit a markedly larger g anisotropy than T3 and T4, which were not reproduced by any of our DFT calculations

Radiation Products at 77 K in Trehalose Single Crystals: EMR and DFT Analysis

The radicals obtained in trehalose dihydrate single crystals after 77 K X-irradiation have been investigated at the same temperature using X-band electron paramagnetic resonance (EPR), electron nuclear double resonance (ENDOR), and ENDOR-induced EPR (EIE) techniques. Five proton hyperfine coupling tensors were unambiguously determined from the ENDOR measurements and assigned to three carbon-centered radical species (T1, T1*, and T2) based on the EIE spectra. EPR angular variations revealed the presence of four additional alkoxy radical species (T3 to T6) and allowed determination of their g tensors. Using periodic density functional theory (DFT) calculations, T1/T1*, T2, and T3 were identified as H-loss species centered at C4, C1′, and O2′, respectively. The T4 radical is proposed to have the unpaired electron at O4, but considerable discrepancies between experimental and calculated HFC values indicate it is not simply the (net) H-loss species. No suitable models were found for T5 and T6. These exhibit a markedly larger g anisotropy than T3 and T4, which were not reproduced by any of our DFT calculations

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

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

On the Thermodynamics of Framework Breathing: A Free Energy Model for Gas Adsorption in MIL-53

When adsorbing guest molecules, the porous metal–organic framework MIL-53­(Cr) may vary its cell parameters drastically while retaining its crystallinity. A first approach to the thermodynamic analysis of this “framework breathing” consists of comparing the osmotic potential in two distinct shapes only (large-pore and narrow-pore). In this paper, we propose a generic parametrized free energy model including three contributions: host free energy, guest–guest interactions, and host–guest interaction. Free energy landscapes may now be constructed scanning all shapes and any adsorbed amount of guest molecules. This allows us to determine which shapes are the most stable states for arbitrary combinations of experimental control parameters, such as the adsorbing gas chemical potential, the external pressure, and the temperature. The new model correctly reproduces the structural transitions along the CO₂ and CH₄ isotherms. Moreover, our model successfully explains the adsorption versus desorption hysteresis as a consequence of the creation, stabilization, destabilization, and disappearance of a second free energy minimum under the assumptions of a first-order phase transition and collective behavior. Our general thermodynamic description allows us to decouple the gas chemical potential μ and mechanical pressure <i>P</i> as two independent thermodynamic variables and predict the complete (μ, <i>P</i>) phase diagram for CO₂ adsorption in MIL-53­(Cr). The free energy model proposed here is an important step toward a general thermodynamics description of flexible metal–organic frameworks

Critical Analysis of the Accuracy of Models Predicting or Extracting Liquid Structure Information

This work aims at a critical assessment of properties predicting or extracting information on the density and structure of liquids. State-of-the-art NVT and NpT molecular dynamics (MD) simulations have been performed on five liquids: methanol, chloroform, acetonitrile, tetrahydrofuran, and ethanol. These simulations allow the computation of properties based on first principles, including the equilibrium density and radial distribution functions (RDFs), characterizing the liquid structure. Refinements have been incorporated in the MD simulations by taking into account basis set superposition errors (BSSE). An extended BSSE model for an instantaneous evaluation of the BSSE corrections has been proposed, and their impact on the liquid properties has been assessed. If available, the theoretical RDFs have been compared with the experimentally derived RDFs. For some liquids, significant discrepancies have been observed, and a profound but critical investigation is presented to unravel the origin of these deficiencies. This discussion is focused on tetrahydrofuran where the experiment reveals some prominent peaks completely missing in any MD simulation. Experiments providing information on liquid structure consist mainly of neutron diffraction measurements offering total structure factors as the primary observables. The splitting of these factors in reciprocal space into intra- and intermolecular contributions is extensively discussed, together with their sensitivity in reproducing correct RDFs in coordinate space

Assessment of Atomic Charge Models for Gas-Phase Computations on Polypeptides

The concept of the atomic charge is extensively used to model the electrostatic properties of proteins. Atomic charges are not only the basis for the electrostatic energy term in biomolecular force fields but are also derived from quantum mechanical computations on protein fragments to get more insight into their electronic structure. Unfortunately there are many atomic charge schemes which lead to significantly different results, and it is not trivial to determine which scheme is most suitable for biomolecular studies. Therefore, we present an extensive methodological benchmark using a selection of atomic charge schemes [Mulliken, natural, restrained electrostatic potential, Hirshfeld-I, electronegativity equalization method (EEM), and split-charge equilibration (SQE)] applied to two sets of penta-alanine conformers. Our analysis clearly shows that Hirshfeld-I charges offer the best compromise between transferability (robustness with respect to conformational changes) and the ability to reproduce electrostatic properties of the penta-alanine. The benchmark also considers two charge equilibration models (EEM and SQE), which both clearly fail to describe the locally charged moieties in the zwitterionic form of penta-alanine. This issue is analyzed in detail because charge equilibration models are computationally much more attractive than the Hirshfeld-I scheme. Based on the latter analysis, a straightforward extension of the SQE model is proposed, SQE+Q⁰, that is suitable to describe biological systems bearing many locally charged functional groups

Molecular Dynamics Kinetic Study on the Zeolite-Catalyzed Benzene Methylation in ZSM‑5

The methylation of arenes is a key step in the production of hydrocarbons from methanol over acidic zeolites. We performed ab initio static and molecular dynamics free energy simulations of benzene methylation in H-ZSM-5 to determine the factors that influence the reaction kinetics. Special emphasis is given to the effect of the surrounding methanol molecules on the methylation kinetics. It is found that for higher methanol loadings, methylation may also occur from a protonated methanol cluster, indicating that the exact location of the Brønsted acid site is not essential for the zeolite-catalyzed methylation reaction. However, methylations from a protonated methanol cluster exhibit higher free energy barriers than a methylation from a single methanol molecule. Finally, comparison with a pure methanol solvent reaction environment indicates that the main role of the zeolite during the methylation of benzene is to provide the acidic proton and to create a polar environment for the reaction. The metadynamics approach, which is specifically designed to sample rare events, allows exploring new reaction pathways, which take into account the flexibility of the framework and additional guest molecules in the pores and channels of the zeolite framework. This approach goes beyond the often applied static calculations to determine reaction kinetics

Solvent-Controlled Selective Transformation of 2-Bromomethyl-2-methylaziridines to Functionalized Aziridines and Azetidines

The reactivity of 2-bromomethyl-2-methylaziridines toward oxygen, sulfur, and carbon nucleophiles in different solvent systems was investigated. Remarkably, the choice of the solvent has a profound influence on the reaction outcome, enabling the selective formation of either functionalized aziridines in dimethylformamide (through direct bromide displacement) or azetidines in acetonitrile (through rearrangement via a bicyclic aziridinium intermediate). In addition, the experimentally observed solvent-dependent behavior of 2-bromomethyl-2-methylaziridines was further supported by means of DFT calculations

Dominant Stable Radicals in Irradiated Sucrose: <b>g</b> Tensors and Contribution to the Powder Electron Paramagnetic Resonance Spectrum

Ionizing radiation induces a composite, multiline electron paramagnetic resonance (EPR) spectrum in sucrose, that is stable at room temperature and whose intensity is indicative of the radiation dose. Recently, the three radicals which dominate this spectrum were identified and their proton hyperfine tensors were accurately determined. Understanding the powder EPR spectrum of irradiated sucrose, however, also requires an accurate knowledge of the <b>g</b> tensors of these radicals. We extracted these tensors from angular dependent electron nuclear double resonance-induced EPR measurements at 110 K and 34 GHz. Powder spectrum simulations using this completed set of spin Hamiltonian parameters are in good agreement with experimentally recorded spectra in a wide temperature and frequency range. However, as-yet nonidentified radicals also contribute to the EPR spectra of irradiated sucrose in a non-negligible way