Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks Density Functional Theory Studies of Catalytic Sites in Metal-Organic Frameworks

Theoretical methods have become indispensable tools in many fields of chemistry and materials research. Metal-organic frameworks (MOFs) are porous materials; they have been intensively developed due to their diverse properties suitable for a wide range of applications. Theoretical approaches have thus been frequently employed toward the design and characterization of MOFs. We focus here in particular on theoretical studies of single-site catalytic reactions that occur inside the cavities of MOFs. The density func tional method (DFT) has been the main approach used for such studies. We briefly review the uses of DFT to examine the catalytic reactions in MOFs. We note that DFT methods are versatile and can be made to work for different purposes such as, e.g., force-field development for molecular simulations. We shall, however, cover this field only very succinctly to put it into context with our main topic. functionalized groups on different linkers as using the anion. The concluded that the of the d-orbital was significantly by the The 3d orbital energy of Fe correlated with the electron-donating strength of the functional groups on the linkers. The results reveal that linkers with ▬ NH 2 groups reduce the enthalpic barrier for the most endothermic step in the oxidation pathway. This illustrates the use of to understand intensive The activ might by selecting which might useful for upgrading certain hydrocarbon


Introduction
Experiments in chemistry aim primarily at understanding, or insight. These insights lead in turn to "innovation" and new technologies [1]. However, in many instances, such an insight cannot be obtained purely from current experiments. The required fundamental theory has been available for some time. The more recent development of computational methodologies and the dramatic increase in computer power made computational methods an additional and competitive tool to describe chemical phenomena [2]. This field is broadly known as "modeling." In "molecular modeling," the premise is that the interatomic and intermolecular interactions can be known [3]. DFT has become the dominant tool for the purpose of determining such interactions [4]. In particular, in solid-state chemistry and physics, it has become increasingly popular since appropriate functionals and basis sets [5] were designed to provide a viable balance between the reliability of the numerical results and the computational costs.
Metal-organic frameworks (MOFs), also known as porous coordination polymers (PCPs) [6][7][8], are a class of hybrid organic-inorganic materials with high surface area, a permanent porosity with large internal pore volume and tunable pore sizes. They have a wide range of potential applications such as gas storage [9] and separation [10], catalysis [11], and many others [12][13][14]. MOFs are constructed by interlinking metal ions or, more generally, metalcontaining units with organic moieties (carboxylates, azolates, imidazolates, pyridyl, etc.) through coordination bonds, thus creating crystalline frameworks. Due to the variety of the structural and chemical elements, some MOFs show unexpected characteristics, which sometimes cannot be fully assessed experimentally. Theoretical approaches have been intensively employed in this particular case to investigate the systems at the atomic level. Beyond the analysis of observed phenomena, the prediction of unknown attributes has been attempted.
In this book-chapter, we review and discuss the current status and challenges of DFT studies of MOFs. We focus in particular on aspects relevant to catalysis at the metal centers found in these structures.

DFT calculations on MOFs
One of the main issues of DFT calculations is that a compromise between the dependability of the results and computational costs must be found, especially for large and complex systems [15]. A primary issue is how to select a functional suitable for the investigated system [16]. Different functionals, based on different approximations, may give different (even opposite) results. It is thus important to choose the functionals wisely. No functional devised up to now has proven to be suitable for all systems. Therefore, the validation of any selected functional is crucial.
MOFs are crystalline materials with extended frameworks. This needs, in some instances, to be taken into account when, for catalytic applications of MOFs, DFT is used to investigate the reaction mechanism. The reaction enthalpy can be obtained approximately from the energy at 0 K by including a zero-point energy correction calculated from (possibly scaled) vibrational frequencies.
To compute the entropy, harmonic or anharmonic low-frequency modes can be used. The temperature-dependent free and activation energies can be calculated by adding thermal enthalpic and entropic effects. The so-obtained free energies of reaction are used to calculate temperaturedependent equilibrium constants. Note that the kinetics of a reaction can also be investigated, e.g., by transition state theory (TST) [17]. However, this is beyond the scope of this chapter.
Since the properties of MOFs depend not only on their chemical composition but also their structures, atomistic structural characterization is essentials. Generally, the structure of a newly synthesized MOF is characterized by single-crystal X-ray diffraction (SC-XRD). If the quality of the single crystals is not sufficient, the structure can be inferred from powder diffraction patterns [18] which often remains incomplete because of the complicated molecular system, large unit cell, and low symmetry. Molecular modeling methods must thus complement these experiments [19].
The periodic structure of an MOF can be studied by DFT: properties such as cell parameters and elastic properties can be obtained from periodic calculations. However, such calculations consume large amounts of computational resources due to the typically large unit cell of MOFs. Simple functionals such as local-density approximation (LDA) or generalized gradient approximations (GGA) are, however, often sufficient in this case [20]. Studying catalytic effects requires more advanced "hybrid" functionals, e.g., BP86 [21], B3-LYP [22], or M06 series [23].
Molecular mechanics (MM) is a much cheaper computational method to study large systems. The reliability of these calculations depends on the quality of the force fields used to describe the interactions between the atoms in the investigated system. Some high-accuracy force fields for MOFs, such as BTW-FF [24], MOF-FF [25], Quick-FF [26], and others [27,28], were parametrized from DFT calculations. They were successfully used to determine and predict the periodic structures [29,30], as well as to examine phase transformations [31][32][33], guest diffusion in the pores and other mechanical properties such as bulk moduli [34], elastic constants [35] and mechanical stability [36]. Note that MM calculations cannot describe electronic structures. Bond breaking and forming cannot be accounted for in such calculations. Thus, it is not possible to study reaction mechanisms using conventional MM methods [37].
In summary, for structurally well-defined and small enough systems, DFT calculations are a cogent approach especially in the field of localized catalytic reactions. We aim here to outline the suitability of DFT techniques to study catalysis in MOFs materials through examples from the recent literature.

Catalytic MOFs
MOFs have well-defined crystal structures with high concentrations of metal centers organized at topical distances, and large pore volumes between these centers. Even though MOFs are less able to withstand high temperatures compared to some other catalytic porous materials, their undeniable qualities such as large internal surface areas and uniform pore and cavity sizes make them attractive for various applications [38]. The pore size/shape in the framework can be tuned for selectivity for a particular reaction. In contrast, the small pore windows of other nanoporous materials, such as zeolites, commonly limit the catalytic transformation of large molecules [39].
In principle, various types of active sites can be incorporated into MOFs. Coordinatively unsaturated metal sites (CUSs) and functional groups on the organic linkers (usually acid/ base sites) are the main catalytic sites. In addition, even though the MOF framework itself does not contain active sites, the catalytic reactivity can be enhanced by (a) postsynthetic functionalized modification [40] and (b) encapsulation of catalytic species [41]. The pore volume of MOFs is able to accommodate organic molecules, inorganic nanoparticles, metal complexes, and enzymes to conduct catalytic reactions. Here, we focus only catalytic reactions taking place on one CUS.
It is because of these functionalities [39] that MOFs were originally proposed for applications in catalysis. From this point of view, one tends to focus on the design and synthesis of MOFs with large pore sizes. The purpose is to allow the reactant to diffuse easily into the framework reaching the catalytic centers. Suitable synthetic techniques make it possible to choose the linkers and the metal-containing nodes to construct MOF suitable for particular reaction. Such a welldesigned MOF catalyst should be highly reactive, selective, and stable. In order to approach this goal, some understanding of the reaction mechanisms is essential. In many cases, the reactants and resulting products tend to be well localized, for a sufficiently long time, in the framework due to host-guest interaction. The catalytic site can thus often be separated from the framework and investigated independently, making it easier to follow the reaction mechanisms [42].
We focus in this brief overview, mostly through a series of examples, on such local analyses of the activity of single-site catalysts. Selected examples of work based on cluster, or truncated, models of reactions catalyzed by metal centers, metal substitutions, and deposited metal complexes are succinctly reviewed.

Catalytically reactive metal nodes
As mentioned above, MOFs are built up from metal ions or metal-containing clusters as the inorganic building node and organic ligands as linkers. In some MOFs, coordinatively unsaturated metal sites (CUSs) have a catalytic effect. The sites are spatially well separated and, ideally, structurally identical. Thus, to investigate the reaction mechanism with DFT methods, truncated structures can often be used to represent the entire MOFs.

Reactivity of coordinatively unsaturated metal sites
MOFs with CUSs provide identical active metal sites, which are spatially isolated from each another. The sites are structurally well-characterized, without coupling to their neighbors. In typical dense heterogeneous catalyst, the reaction takes place on the outer surface [43]. One way to enhance the catalytic effect is to increase the accessible surface area. In MOFs, reactions occur not only on the outer surface, easily accessible active sites inside the framework also contribute. (Figure 1a) was used. The reaction mechanism started with the N 2 O decomposition over Fe to generate an active Fe-oxo moiety, requiring an activation energy of 23.7 kcal/mol. An ethylenoxy intermediate, and eventually, ethylene oxide, were obtained as final products rather than acetaldehyde (Figure 1b).

Maihom et al. investigated the epoxidations of ethylene over Fe
An investigation of the epoxidation of propylene using Cu 3 (BTC) 2 and Fe 3 (BTC) 2 was recently carried out by the same group [45]. As mentioned, truncated clusters, [Cu 2 (H 2 BTC) 4 ] and [Fe 2 (H 2 BTC) 4 ], are suitable for these calculations, carried out at the M06-L/6-31G(d,p) level. As expected, the calculations showed that Fe 3 (BTC) 2 is more active than Cu 3 (BTC) 2 due to a larger charge transfer from the CUS to the oxidant O 2 . It was found that the production of propylene oxide is favored over that of carbonylic products (propanal and acetone). Propanal and acetone were formed on the Fe-MOF cluster via the formation of a C▬O bond. Then, the propyleneoxy intermediates and acetone are formed via a 1,2-hydride shift.
DFT calculations were performed on truncated models cut from the periodic geometries (Figure 2) optimized at the PBE-D2 level. The catalytic activities of M 2 (DOBDC) systems with respect to the Knoevenagel condensation and Michael additions were examined. The calculations at the B3LYP level with a 6-31 g(d) basis showed that Ni-MOF-74 is the most active catalyst for both reactions. In addition, it was found that the phenolate groups coordinated with the CUSs substantially increase the catalytic performance. The phenolate oxygen proved to be a stronger base than the carboxylate oxygen, resulting in more acidic CUSs, enhancing the catalytic activity.
Llabrés i Xamena et al. [50] demonstrated that Cu(2-pymo) 2 and Co(PhIM) 2 (2-pymo and PhIM are 2-hydroxypyrimidinolat and phenylimidazolate, respectively) promote the aerobic oxidation reaction converting tetralin hydrocarbon to ketone and alcohol derivatives. The tetralin was first oxidized to hydroperoxides under oxygen condition and then decomposed to ketone and alcohol under the influence of the Lewis acid sites Cu(II) and Co(II). Ryan et al. [51] studied the mechanism of the hydroperoxide decomposition over three different complexes possessing copper or cobalt CUSs, i.e., Co(imidazolate) 4 , Cu 2 (2-hydroxypyrimidinolat) 4 , and Cu 2 (acetate) 4 , as illustrated in Figure 3.
From DFT calculations (B3LYP/TZVP/6-31G(d,p)), the access of hydroperoxide to the CUSs of Cu 2 (2-hydroxypyrimidinolat) 4 and of Co(imidazolate) 4 was found to be blocked by this bulky ligand. As a consequence, no decomposition of hydroperoxide was observed for the two systems. On the other hand, for Cu 2 (acetate) 4 , i.e., without the bulky ligands, the decomposition should occur. However, the calculations revealed that Cu 2 (acetate) 4 was not active for the decomposition of tetralin. The energy barrier of the O▬O bond cleavage over Cu 2 (acetate) 4 (35.6-36.8 kcal/mol) was almost identical with that of the cleavage in the gas phase without catalyst (36.0-36.7 kcal/mol). The external surface was then considered as the active sites.
A complex consisting of Cu coordinated by three organic linkers and one water molecule was modeled representing the edge or outer surface of the framework (Figure 4a). The energy barrier of the hydroperoxide decomposition on the new model complex decreased to 25.3 kcal/mol, which was considerably lower than the value of the gas-phase reaction without MOFs as shown in Figure 4b.
Vanadium-based MIL-47(V), [VO(BDC)] (BDC = benzene-1,4-dicarboxylate), is an active catalyst for the liquid-phase cyclohexene oxidation-reaction using tert-butyl hydroperoxide (TBHP) as an oxidant [52]. MIL-47(V) is built up from linear V-(μ 2 -O)-V chains interconnected  by terephthalate linkers to form a 3D framework possessing 1D rhombic channels. Each V(IV)-center in this chain coordinates to six oxygen atoms-four from carboxylate and two from μ 2 O. Typically, MIL-47(V) does not provide CUSs. However, CUSs can be created from structural defects obtained by partially breaking the V▬O COO bonds and from the hydrolysis of the V▬O COO bonds.
The main products of the cyclohexene oxidation reaction are cyclohexene oxide, cyclohexane-1,2-diol, tert-butyl-2-cyclohexenyl-1-peroxide, and 2-cyclohexen-1-one with a conversion of about 80%. It was found that the catalytic performance of MIL-47(V) was comparable with that of typical homogeneous catalyst, VO(acac) 2 . Note that no leaching of V(IV) was observed when the oxidant was dissolved in the solution. The structures of the MIL-47(V) were maintained intact until the end of reaction. Leus and Vandichel et al. [52] proposed a reaction pathway from DFT calculations on finite clusters of MIL-47(V) (Figure 5).
The results were in good agreement with EPR and NMR measurement that are interpreted in terms of the existence of (at least) two different catalytic pathways, described as the "direct" and "radical" pathways [52]. The first step of both pathways was the formation of a vanadium hydroperoxide species from TBHP. For the direct pathway, cyclohexene was directly converted to cyclohexene oxide, while the oxidation state of vanadium (V(IV)) did not change during the reaction. In the case of the radical pathway, V(IV) was oxidized to V(V), which further reacted with the oxidant TBHP to give an active species of the cyclohexene epoxidation. Finally, the catalyst was regenerated by interacting with TBHP again. At the B3LYP-D3 and 311 + g(3df,2p) level of theory, the reaction free energy for epoxidation reaction was 37 kJ/mol for the more favorable radical route.
Moreover, Matthias et al. [53] studied the cyclohexene oxidation using MIL-47(V)-functionalized linkers (with the functional groups ▬OH, ▬F, ▬Cl, ▬Br, ▬CH 3 , and ▬NH 2 ) by means of experimental and theoretical works. Different catalytic conversions were observed depending on the functionalization of the linkers. DFT calculations on truncated clusters of MIL-47(V) and its derivatives were performed as in the abovementioned example for the radical pathway. The results showed that, compared to the parent MIL-47(V), the catalytic efficiency significantly increased due to the functional groups. Among these, the ▬OH-functionalized structure showed the largest improvement (lowest activation energy), which was in good agreement with the experimental results. This is because of a strong hydrogen bond between the ▬OH and the alkylperoxo group possibly stabilizing the transition state.

Reactivity of substituted metal centers as single-site catalysts
Partial substitution of metal ions while maintaining the structural integrity and the porosity is a promising strategy enabling some otherwise inactive MOFs to show a catalytic activity [54]. The metal substitution usually involves the cleavage and regeneration of coordination bonds between metal ions and organic ligands. The atomic-level understanding of this process is difficult, making room for molecular modeling. Here, we show examples of theoretical studies on the catalytic activity of metal-substituted MOFs.
Xiao et al. [55] reported the oxidation reaction of ethane to ethanol over magnesium-diluted  this magnesium-diluted MOFs using N 2 O as oxidant was proposed to proceed via a two-step consecutive reaction (Figure 6): the formation of the Fe(IV)-oxo unit via N 2 O decomposition and the hydroxylation of ethane to ethanol over the Fe(IV)-oxo complex. It is found that the first step is the rate-determining step with an activation energy of 82 kJ/mol. Compared to the uncatalyzed reaction, where nitrous oxide directly oxidizes the ethane to ethanol, it was found that the reaction requires an activation barrier of 280 kJ/mol. This suggests that the Fe(IV)-oxo species is indeed an active center for the oxidation of ethane to ethanol.
In the meantime, Hirrao et al. [57] also carried out combined quantum and molecular mechanics (  and 4-pyridyl functionalized groups on different linkers such as using methanol, pyridine, formate, benzoate anion, and the imidazole anion. The authors concluded that the population of the d-orbital was significantly influenced by the coordination ligand field. The 3d orbital energy of Fe correlated with the electron-donating strength of the functional groups on the linkers. The results reveal that linkers with ▬NH 2 groups reduce the enthalpic barrier for the most endothermic step in the ethane oxidation pathway. This study illustrates the use of simple models to understand complicated and computationally intensive systems. The activity of CUSs thus might be optimized by selecting suitable ligand environments, which might be useful for upgrading certain hydrocarbon process. The design of Fe(IV)-oxo complexes in MOFs was further reported by Impeng et al. [61].
Ortuño et al. [68] reported on a computational screening of the first row of divalent transition metals (i.e., Fe(II), Co(II), Ni(II), Cu(II), and Zn(II)) supported on NU-1000 (Figure 8) for acceptorless alcohol dehydrogenation. The author proposed the reaction taking place via a three-step reaction mechanism, composed of (i) a proton transfer, (ii) β-hydride elimination, and (iii) H▬H bond formation. The Fe(II), Co(II), and Ni(II) complexes, consistent with weakfield oxide ligands, had high-spin ground electronic states as quintet, quartet, and triplet, respectively. The Cu(II) and Zn(II) species were predicted to have doublet and singlet ground states, respectively. It was found that the Co(II) and Ni(II) supported NU-1000 were the two most promising catalysts for the acceptorless alcohol dehydrogenation with an activation free energy of 28.5 and 26.5 kcal/mol, respectively.
Later, the same group also studied Ni(II) and Co(II) deposited on Zr-NU-1000 as a catalyst for ethylene dimerization, which converts ethylene to 1-butene and 2-butene [69]. For the structure optimization, the DFT level of theory (M06-L) and def2-TZVPP basis set were employed, whereas complete active space self-consistent field (CASSCF) and second-order perturbation theory (CASPT2) were used for the electronic structure characterization of the reactive species. The NU-1000 models were used as the same as in the previous example (Figure 8). The grafted Ni and Co were terminated with active ▬OH and ▬OH 2 groups.
The Cossee-Arlman mechanism was found as the energetically preferable pathway. Ethylene insertion into the existing metal-ethyl bond was the rate-determining step. Concerning the spin state of the two catalysts, the Co(II) species can have doublet or quartet states and the Ni(II) can have singlet or triplet states. Based on these calculations, Ni(II)-modified NU-1000 (with an activation energy of 15.6 kcal/mol) presents a greater reactivity than the Co(II)modified system (with an activation energy of 24.1 kcal/mol) for the ethylene insertion into the metal-carbon bond. The energetics of ethylene dimerization for the Ni(II) and Co(II) complexes are shown in Figure 9. From the most relevant CASSCF molecular orbitals (MOs), the 3d orbitals of the low-spin Ni(II) complex hybridized with the 2p orbitals of the active carbon atom because of the empty fifth orbital. In contrast to the high-spin Co(II) complex, the degree of hybridization in the Ni-case is less able to stabilize the TS structure due to the half-filled 3d orbital, leading to a lower catalytic activity.  The oxidative dehydrogenation mechanism of propane on an active Co(II)-oxygen cluster anchored on the inorganic node of Zr-NU-1000 [70] was also explored by DFT with the M06-L functional and def2-TZVPP. A highly selective propane dehydrogenation at low temperature (230°C) has been found in experiments. Li et al. [70] found a computed propene formation in line with experimental results. DFT cluster calculations allowed to study the reaction and showed the influence on the reaction of adding Co(II) cations as a secondary metal. The computed enthalpies for the equilibrium structures are shown in Figure 10.
In more detail, the reactive Co(III)▬O˙ moiety was generated by the catalyst regeneration reaction with O 2 having a strong interaction with the propane molecules, promotes the propane dehydrogenation, and forms isopropyl at the Co(III) active site. The abstraction of the terminal C-H bond by the Co(III)▬O˙ intermediate is the rate-determining step, yielding Figure 10. Computed enthalpies, ΔH 503K (kcal/mol), for propane oxidative dehydrogenation. Reproduced from Ref. [70] with permission of the American Chemical Society, copyright 2017. Figure 11. Calculated enthalpy, ΔH 298.15K (kcal/mol), for ethylene hydrogenation to ethane and dimerization to 1-butene catalyzed by supported metal complexes initially present as Rh(C 2 H 4 ) 2 on NU-1000, UiO-67, or DAY zeolites. Reproduced from Ref. [71] with permission of the American Chemical Society, copyright 2017. Co(II)▬OH and isopropyl radicals, which are converted into propene. The energy required for the H abstraction producing propene (15.8 kcal/mol) was significantly lower than that for the isopropyl migration (23.1 kcal/mol), the propene formation path is thus preferable.
Gates et al. [71] showed that Rh(I) and Ir(I) diethylene complexes can be anchored on the Zr-NU-1000, Hf-NU-1000, and Zr-UiO-67 to catalyze ethylene hydrogenation and dimerization. They also used DAY zeolite as supporting material to compare the catalytic performance. DFT calculations with M06-L and def2-TZVPP were employed. The optimized geometries were in good agreement with the data analyzed from IR spectra and X-ray absorption fine structure (EXAFS) results. The ethylene insertion to the ethyl▬Rh bond (rate-determining step) significantly depended on the nature of the supporting MOF. The catalytic activity was compared between Zr-and Hf-nodes and DAY zeolites. The calculated enthalpies for the ethylene hydrogenation and dimerization were shown in Figure 11. The calculations reveal a lower energy barrier of the DAY zeolite as supporting material than Zr-NU-1000, Hf-NU-1000, and Zr-UiO-67 for the hydrogenation and dimerization. This is opposite to the experiments which show a better catalytic activity of the DAY zeolite than the MOFs materials. The authors proposed that this might be because of the spillover effect rather than a direct electronic effect on the Rh(I) complex.

Outlook
We focus in this brief overview on theoretical studies of the catalytic activity of single-site catalytic MOFs. DFT-based approaches can provide a detailed, if only local, description of the chemistry occurring in various environments encountered in such MOFs. This not only helps in the interpretation of experiment data; the abilities of DFT calculations are a crucial characterization method toward a deeper molecular perspective. This technique is now powerful and reliable enough to predict and guide the synthesis of materials.
There are, however, several problems which need to be overcome: I. To find a link between a predicted structure and its syntheses still a challenge.

II.
Sometime nonlocal phenomena such as a stimulus-responsive behavior of the catalytic site itself must be taken into account. Many works [ref] attempt to simulate larger models in order to describe such systems. This seems to require improved methodologies to limit the computational requirements.
III. The functionals: Since there is no "universal functional" DFT must be always used with a lot of caution. A lot of experience has been accumulated on the suitability of various approximations to investigate MOFs. Yet new, and costly, validations are still required whenever a "new" system is to be studied.

IV.
Last but not least, in some cases, a local view as described in this chapter may not be sufficient to catch the main features of a particular process. The overall outcome of the process under study may depend on many factors, in the worst case, none of which dominant. Various "local views" that may have been obtained could be combined as building blocks (e.g., by understanding the molecular motions between "sites"). This remains largely a task for theoretical improvements in the future.