CHEMPHYSCHEM MINIREVIEWS DOI: 10.1002/cphc.201402291

Assessments of Semilocal Density Functionals and Corrections for Carbon Dioxide Adsorption on Metal– Organic Frameworks Hyunjun Ji, Joonho Park, Moses Cho, and Yousung Jung*[a] The significant amount of attention that has been directed toward metal–organic frameworks (MOFs) for a wide spectrum of applications can be attributed to their variety and tunability, which are precisely the aspects that computational modeling can offer by systematically exploring the chemical space. In this minireview, we describe density functional theory calculations for gas adsorption on MOFs, mainly focusing on the interaction of CO2 with MOF-74. The generalized gradient approximation (GGA) level of density functional studies seems

suited to treat MOFs, owing to the balance between its practical applicability and its useful accuracy, although this method is not without deficiencies such as the lack of nonlocal correlations and self-interactions. We review and analyze the effects of correction schemes to the GGA to amend the latter weaknesses, and the choice of exchange correlation functionals to treat MOFs for gas capture and separation. We also discuss a few topical questions that are currently missing in the present literature and that require further investigations.

1. Introduction Metal–organic frameworks (MOFs) have been receiving increasing attention as a new class of microporous materials. MOFs consist of metal–ligand complexes interconnected with organic linkers to form a porous structure. MOFs discriminate themselves from other microporous materials by their variety and tunability; an enormous number of MOFs with different pore structures and properties can be synthesized by controlling the metals and ligands. Some MOFs such as MOF-74, HKUST-1, and MIL-101 have open metal sites (OMSs), that is, coordinatively unsaturated metal sites that are exposed without any geometric hindrance, whereas the whole material remains as a solid. This structure gives rise to the possibility of various applications, such as gas capture and storage,[1, 2] separation,[3, 4] catalysis,[5] sensors,[6] drug delivery,[6] and lightweight magnets.[7, 8] One of the urgent applications of MOFs in addressing global climate changes is the capture of CO2 in flue gas to mitigate the increase in the concentration of CO2 in the atmosphere. The uptake capacity of Mg–MOF-74 for CO2, for example, reaches 6.0 mmol g1 at 15 kPa at room temperature,[9, 10] which is one of the highest CO2 uptake values reported to date. This large capacity is ascribed to the high density of the OMSs and the strong interaction between the metal ion of the MOF (i.e. OMS) and CO2, which makes OMS–MOFs applicable to lowpressure CO2 scrubbing in postcombustion flue gas at 100 kPa (15 % of CO2) and 50 8C. For these reasons, MOF-74 has been [a] H. Ji,+ J. Park,+ M. Cho, Prof. Y. Jung Graduate School of EEWS, KAIST Daejeon 305-701 (Republic of Korea) E-mail: [email protected] [+] These authors contributed equally to this work. Supporting Information for this article is available on the WWW under http://dx.doi.org/10.1002/cphc.201402291.

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studied extensively both experimentally and computationally. Figure 1 shows the structure of MOF-74 (or synonymously CPO-27), in which a one-dimensional (1D) hexagonal microchannel with a diameter of approximately 13  develops. The neutron diffraction snapshot in Figure 1 clearly shows that each CO2 molecule is bound to a metal ion exposed to the channel.

Figure 1. CO2 images from neutron powder diffraction were superimposed on the Mg–MOF-74 structure. Mg: blue, C: gray, O: red, and H: white. Reproduced with permission from Ref. [11]. Copyright 2005 American Chemical Society.

Calculation of the electronic structure of MOFs can be classified into two types: cluster calculations[12, 13] and periodic calculations. If the aim is to obtain highly accurate predictions or to reproduce experimental observations with great precision, one is forced to sacrifice the periodicity of the structure due to the ChemPhysChem 2014, 15, 3157 – 3165

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CHEMPHYSCHEM MINIREVIEWS high cost of ab initio and hybrid density functional calculations for periodic systems. In this case, a nonperiodic model is constructed so that the loss of representability can be compensated by the gain in accuracy. On the other hand, if one wants to retain the periodicity, density functional theory (DFT) calculations are performed with semilocal functionals, usually at the generalized gradient approximation (GGA) level. In this case, however, a certain degree of loss in accuracy is inevitable due to several known limitations of GGA functionals. With lengthened “error bars”, deviation in the calculated properties will be somewhat more dependent on the choice of the computation scheme, including the choice of GGA functional itself and corrections compensating for the limitation of the semilocal functionals, compared to the case in which more accurate methods are employed. However, it seems that there has been no thorough and systematic comparison of methods for MOF calculations, especially if gas adsorption is involved in metal–organic hybrid materials that are closer to atoms and molecules than traditional solids. The purpose of this paper is to provide a comparison between computational schemes for gas adsorption onto MOFs at the GGA level. In particular, we will set our focus mainly on, but not restricted to, CO2 adsorption onto MOF-74 due to the rich experimental and theoretical data to compare, not to mention its practical importance as one of the highest CO2capture materials. The criteria used to assess the accuracy will be mainly the gas binding energies and geometric parameters, as acquiring these quantities are usually the primary and most basic goals of electronic structure calculations, and the benchmark comparison of other properties is difficult to establish due to the limited availability of computational and/or experimental data. Physical interpretation and theoretical consideration will be provided if available, and further possible topics in this field will be briefly mentioned.

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Figure 2. Local density of states (LDOS) of metal (black) and CO2 (red) before (dashed line) and after (solid line) CO2 adsorption. Reproduced with permission from Ref. [14]. Copyright 2005 American Chemical Society.

interact with the formal positive charge of the open metal sites in MOF-74. Park et al. assessed this local electrostatic effect due to a positive charge developed on the OMS by calculating the energy of CO2 in the absence and presence of a point charge of the metal in its optimized distance by using the RIMP2/cc-pVTZ method.[14] Partial charges of open metals within the periodic MOF were obtained by using Bader population analysis. As shown in Figure 3, the overall trend of the

2. Various Gas Adsorption Phenomena in MOF-74 Before discussing various computation schemes, we briefly mention the binding nature of CO2 on MOF-74 systems,[11, 14] as these details must be the most influencing factors in choosing state-of-the-art, yet economical, calculation methods, which is the focus of the present paper. The dispersion interaction is clearly a major contribution, ranging approximately between 1/3 and 2/3 of the total interaction energy. For example, the relative affinity of CS2 versus CO2 was studied for a Hoffmann-type coordination polymer; theoretical analysis indicated that the larger polarizability of CS2 and hence its larger dispersion interaction is mainly responsible for the larger binding energy of CS2 relative to that of CO2.[15] However, other aspects of CO2 binding such as electrostatic and orbital interactions are, of course, also important. In the density of state (DOS) analysis of Figure 2,[14, 16] it is shown that after binding with the MOF, each molecular orbital of CO2 shifts to lower energy (see arrow) with some overlap between the DOSs of the metal ion and CO2. It has been speculated that the quadrupole moment of CO2 can favorably  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Figure 3. The CO2 binding energies (gray bar), electrostatic energies scaled by 0.4 (blue line), CO2–organic interaction energies (green line), and the charge transferred from the lone pair of CO2 to the empty d levels of the metal (red line). Reproduced with permission from Ref. [14]. Copyright 2005 American Chemical Society.

increased CO2 stability in the presence of the external point charge (blue line) shows a very good agreement with the actual total interaction energies of CO2 with various MOFs (bar graph). However, the electrostatic effect alone cannot explain the entire trend clearly (especially for Ti), and the authors noted that there are additional orbital interactions between CO2 and the empty d levels of the metals. Thus, mixing of the lone-pair orbitals of CO2 with the empty d levels occurs, which results in electron donation from the ligand to the metal as a very weak coordination bond. The higher CO2 affinity of Ti– and V–MOF-74 can thus be explained by the large extent of forward donation (Figure 3; red line). ChemPhysChem 2014, 15, 3157 – 3165

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CHEMPHYSCHEM MINIREVIEWS A similar orbital interaction can be even more crucial if it comes to unsaturated hydrocarbons with p bonds. These olefins make strong coordination bonds with metal ions through p–d orbital complexation, which was used as a key molecular recognition principle of olefin/paraffin separation in experiments and theory[4] by using MOF-74. Furthermore, the same orbital interaction was also shown to be able to separate 1butene from other butene isomers (i.e. separation of butene isomers that differ only in the position of the double bond) by additionally utilizing steric interactions in the gas-framework recognition (Figure 4).[17] In this case, 1-butene with a terminal

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Figure 5. The relative energies of FM (black) and AFM (red) coupling and their energy difference (blue) for bare Fe–MOF-74 as a function of FeFe distance. The bare MOF can undergo intrachain FM-to-AFM transition at an elongated FeFe distance of 3.12 . Reproduced with permission from Ref. [18]. Copyright 2005 American Chemical Society.

scription of electrostatic and orbital interaction, dispersion interaction, and spin polarization. Below, we discuss theoretical aspects of dispersion (Section 3), spin polarization (Section 4), and general accuracy of GGA for electrostatic and orbital interactions (Section 5).

3. Dispersion Correction

Figure 4. Schematic representation of the potential energy curves (solid line) for the adsorption of 1-butene and 2-butene. The green and purple zigzag lines between the gas and Fe indicate bonding and backbonding, respectively. Steric repulsion in 2-butene prevents the gas from approaching the open metal site to yield weak bonding only, unlike 1-butene with strong bonding and backbonding. Reproduced with permission from Ref. [17]. Copyright 2005 American Chemical Society.

double bond has the smallest steric interactions with the framework, and therefore, it can approach the metal binding sites more closely for stronger p complexation (bonding and backbonding possible) than 2-butene (only bonding possible without backbonding due to the large spatial separation). The effect of spin polarization is also non-negligible in some MOF systems, and definitely so for the present focus, MOF-74. If transition metals are constituents of MOF-74, intra- and interchain magnetic coupling can take place due to spin polarization. Ferromagnetic (FM) coupling through the 1D chain of Fe– MOF-74 was first reported by Bloch et al.,[3] but the most interesting magnetic behavior was the observation that olefin adsorption changed this intrachain FM coupling of the bare MOF to antiferromagnetic (AFM) coupling but paraffin adsorption did not show this behavior. Later, Park et al. revealed that the change in the FeFe distance induced by olefin adsorption was a primary determining factor.[18] Thus, upon varying the FeFe distance from 2.79 (bare MOF) to 3.31  (olefin-adsorbed MOF), the same magnetic transition from FM to AFM was reproduced at 3.12  even without the presence of gas molecules, that is, without additional p–d orbital interactions, as shown in Figure 5. As briefly discussed in this section, understanding various gas adsorption phenomena in MOF-74 requires a good de 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

As discussed above, noncovalent interactions are a major contribution in the adsorption of gases onto MOFs, which is a well-recognized weak point of semilocal density functional approximations. Indeed, the adsorption energies obtained by GGA are heavily underestimated, as shown in Figure 6. On the other hand, a stronger adsorption is observed for the local density approximation (LDA) functional; however, this is due to well-known fortuitous error cancelation and is not the result of correct physical description. In this regard, the “combination” of LDA and GGA functionals once suggested in a previous study[19] may not be theoretically justified. Thus, a nonlocal correlation correction is essential for compensation of the missing nonlocal part of GGA. Among them, the most popular is the semiempirical pairwise dispersion correction, such as Grimme’s D1,[20] D2,[21] and D3[22, 23] corrections. This class of approaches enhances the performance of DFT for noncovalent interactions such as hydrogen bonding and dispersion interactions to yield a mean absolute deviation (MAD) of < 0.40 kcal mol1 with several GGAs[24] for the S66 benchmark set,[25] for example. Another and perhaps the major advantage of these approaches lies in its cost effectiveness, as the computational burden is negligible relative to that of the density functional part. On the other hand, they are unable to capture the environmental effect of a change in the electronic density or orbital, though in the D3 correction such effects are partly considered by setting the van der Waals parameters coordination number as dependent. Other kinds of approaches are based on the adiabatic connection fluctuation–dissipation theorem: van der Waals density functionals, such as vdW-DF[26, 27] and vdW-DF2.[28] Such functionals are designed to compensate only for the nonlocal part of correlation energy that semilocal functionals are missing, and the local correlation part is approximated by LDA correlation energy. They, unlike semiempirical corrections, can describe the density dependence of nonlocal interactions by ChemPhysChem 2014, 15, 3157 – 3165

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Figure 6. Binding energies (BEs) of CO2 on M–MOF-74. The solid horizontal line represents the average of the experimental DH, and the gray shaded area corresponds to one standard deviation of the experimental values.

design, but in return the computational cost is higher than that for pure GGA. The computational cost of vdW-DF can be reduced in (R)VV10[29, 30] by a simpler nonlocal kernel, but this has not been tested for MOFs yet.  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Figure 6 shows the CO2 binding energies calculated by several GGAs, van der Waals functionals,[11, 14, 16, 31–35] and cluster second-order Møller–Plesset (MP2) calculation results,[32] as well as some experimental heats of adsorption.[9, 10, 32, 36–44] For the ChemPhysChem 2014, 15, 3157 – 3165

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CHEMPHYSCHEM MINIREVIEWS exact values of all the binding energies and their references, refer to the Supporting Information (SI). Given that experimental results fluctuate severely, we direct our discussion to the comparison of the calculated results. The binding energy is heavily dependent on the calculation method and not only on the choice of exchange functional, which shall be discussed later, but also on the correction method. For example, the binding energies obtained with Perdew–Burke–Ernzerhof (PBE)-D2 and PBE-vdW differ significantly, by about 18 kJ mol1. For van der Waals functionals, binding energies change with respect to the exchange functional, an effect also observed for the S22 set,[27] as the dispersion energies are calculated from electronic densities. On the other hand, the semiempirical correction methods are less dependent on the choice of functional according to the previous benchmark study[24] but could not be compared in this table for MOF-74 due to a lack of data. In Table 1, calculated characteristic structure parameters are compared. Though other experimental results exist, because the geometry difference can also arise depending on the occupation number of CO2 in the framework,[45] only directly comparable experimental data are shown. It is clearly shown that LDA severely underestimates the CO2–metal distance, which supports the argument that the seemingly plausible binding energy of LDA is a result of coincidence and not a physically correct description. Due to the underbinding nature of GGA, the CO2–metal distance obtained with PBE is slightly overestimated, whereas the overestimation decreases in the presence of dispersion interaction. In contrary to the case of the binding energies, the effect of the dispersion correction scheme on the geometric parameters seems less influential upon comparing the PBE-D2 and PBE-vdW results. There is another outstanding discrepancy in the OCO bond angle between theory (176– 1788) and experiment (160–1678) in Table 1. However, Wu et al. argued that this can be ascribed to the other degree of freedom of adsorbed CO2 rather than the OCO bending angle itself.[11] The five normal modes of adsorbed CO2 have energies less than 160 cm1, which means CO2 on the OMSs is very flexi-

www.chemphyschem.org ble. The presence of disorder in the orientation of CO2 causes the degree of the bond angle derived from refinement of the diffraction data to be much larger than the calculations suggests. Though there is a little amount of charge donation as mentioned above, given that CO2 is reversibly physisorbed on OMS, a large degree of bending in the zero-temperature ground state is very unlikely. The potential energy curve for the bond angle of adsorbed CO2 is almost similar to that of free CO2, which requires more than approximately 15 kJ mol1 for 208 bending, as shown by Wu et al[11] (Figure 7).

Figure 7. Potential energy surfaces for the OCO bending motion of free and adsorbed CO2 obtained with the LDA. Reproduced with permission from Ref. [11]. Copyright 2005 American Chemical Society.

4. Hubbard U Correction

Semilocal functionals suffer from a notorious self-interaction problem that is believed to be responsible for many deficiencies of current DFT such as charge transfer, underestimated barrier heights, and poor description of Rydberg excitations. The self-interaction problem is especially serious for localized 3d orbitals of transition metals in insulators, MOFs being examples. For example, the band gaps of apparently insulating transition-metal oxides are severely underestimated or even calculated to be zero. This can be effectively addressed by introducing the Hubbard U correction;[46] Table 1. Comparison of geometries for CO2-containing MOFs. MO denotes the distance () between the a metal atom is assumed to be metal atom and the oxygen molecule of the adsorbed CO2 molecule. ]OCO is the bond angle (8) of CO2. Exoxidized to a certain degree, perimental results are presented only if the occupancy was identical to the computational model. then the 3d orbitals are artificialMg–MOF-74 Ni–MOF-74 Co–MOF-74 ly localized. MO ]OCO MO ]OCO MO ]OCO Indeed, some previous MOF 2.18 176.1 2.27 177.6 LDA[a] calculations have been per[a] 2.37 178.2 2.47 178.9 2.67 179.1 PBE formed with the U correcPBE-D2[a] 2.31 177.0 2.36 178.1 2.59 178.3 tion,[34, 47–49] but the direct com2.29 175.9 PBE-D2[b] 2.29 175.9 PBE-vdW[b] parison between corrected and 2.29 177.4 2.33 178.3 2.51 178.4 optB86b-vdW[a] uncorrected cases has received [a] 2.29 177.4 2.34 178.3 2.52 178.4 optB88-vdW less attention. It has been [a] 2.33 177.9 2.43 178.7 2.62 178.8 optPBE-vdW argued that the Hubbard U cor2.39 178.4 2.62 179.0 2.81 179.1 revPBE-vdW[a] 2.34 177.9 2.47 178.6 2.64 178.7 rPW86-vdW[a] rection indeed affects band 2.28(3) 167(4) Exp.[c] gaps; however, the effect on in[d] 160 Exp. teraction energies is somewhat [a] Ref. [31]. [b] Ref. [33]. CO2 occupancy is 1/6 instead of 1 per Mg atom. [c] Ref. [45]. [d] Ref. [11]. The values smaller.[34] Table 2 shows a comin parentheses represent one standard deviation. parison of the MOF–CO2 binding

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Table 2. MOF–CO2 interaction energies calculated with the vdW-DF2 functional,[34] with and without Hubbard U correction. Experimental results are presented if available. U parameters are taken from Ref. [50]. MOF

vdW-DF2 [kJ mol1]

vdW-DF2 + U [kJ mol1]

Ueff [eV]

Exp. [kJ mol1]

Ti–MOF-74 V–MOF-74 Cr–MOF-74 Mn–MOF-74 Fe–MOF-74 Co–MOF-74 Ni–MOF-74 Cu–MOF-74

53.1 62.8 30.1 34.5 36.9 33.5 35.4 29.4

49.8 54.3 30.2 37.6 41.1 41.8 43.0 30.8

2.0 3.0 3.5 3.8 4.0 3.3 6.4 3.8

42.3  3.1[a] 34.3[b] 35.8  1.8[a] 39.1 1.3[a] 28.5  5.1[a]

[a] Average and standard deviation of the experimental data. [b] Ref. [32].

energies in the corresponding literature obtained with and without U correction. On the basis of these results, it was concluded in the article that the U correction can shift interaction energies by about 8.5 kJ mol1, but the overall trend remains unaffected. In another study,[4] instead of CO2, the adsorption energies of ethanes and ethenes onto Fe–MOF-74 were compared with the experimental results. For both the PBE-D2 and RPBE-D2 functionals, the Hubbard U correction deteriorates the R2 value of the calculated and experimental energies. In both studies, the Ueff parameters used were values optimized for the oxides;[50] thus, they are not expected to be optimal for metal–organic hybrid materials, as will be discussed below. However, for lattice parameters, the U correction has a clear effect on them. As shown in Table 3, the cell shrinks as the U correction is applied, which is a natural consequence of d orbital localization. In these cases, the shrinkage is so large that the lattice parameters are underestimated, which can be interpreted in such a way that the value of Ueff is too large for PBE without dispersion correction, as far as the lattice parameters are concerned. Although the effect of the U correction remains somewhat unclear to this point due to the lack of sufficient data to assess, it appears that the use of U-corrected methods for MOF calculations can still be justified theoretically. The Hubbard correction is, in principle, a self-interaction correction, and it should decrease delocalization error or nonlinearity of energy for a fractional particle number. If we recall the fact that the binding of CO2–MOF involves an electrostatic effect and orbital

interaction, correct charge distribution is crucially important in the quantitative MOF calculations. Even for dispersion interactions, it is well known that the quality of exchange functional largely affects the noncovalent interactions.[54] The importance of the U correction becomes evident if we are to focus on the magnetic properties of MOFs such as FM–AFM transition upon gas adsorption, as described in Section 2. The optimal choice of U parameter itself is an important issue. Though Ueff can be calculated ab initio for a given system, practically it is tuned for a target application. For example, if the calculations are performed for oxides mostly, the U parameter is fitted to reproduce the oxidation energies of the metal oxides.[50] Besides the advantage in cost (predetermined U values can be used instead of calculating them on the fly or before calculations), the reason behind the idea of fitting is twofold: First, the U correction works as an empirical scheme that corrects something more than mere self-interaction. Second, the degree of localization of the d/f orbitals can differ by target system, even if the formal oxidation state is identical. In such a manner, it is possible to tune the U parameter specifically for a given MOF system in a way that reproduces accurate geometric parameters or binding energies. However, it should be noted that the accuracy for a set of properties does not necessarily lead to accuracy for the other. In such cases, the transferability of the U parameter cannot be guaranteed. The optimal value of Ueff must change depending on the functional and correction schemes. We note that the Ueff parameters cited in Tables 2 and 3 were not optimized for MOF74. The validity of the U correction itself or a specific Ueff parameter can thus be thoroughly tested by comparing both the corrected and uncorrected GGA results with their counterparts, namely, energies obtained by using the hybrid version of the GGA functional. An example would be the comparison of PBED2, PBE-D2 + U, and PBE0-D2. However, to the best of our knowledge, such a study has not been performed for MOFs.

5. The Choice of Exchange Functional

The use of the PBE functional is predominant in the periodic calculation of MOFs, whereas its variants or other types of functionals are relatively niched. However, as mentioned above, accurate description of noncovalent interactions requires a proper exchange functional, and many properties of MOFs are connected directly or indirectly with the exchange functional. In this section, we review the performance of various GGA functionals including many PBE variants, the design principles of which will be briefly Table 3. Lattice parameters of the hexagonal unit cell of M–MOF-74 calculated with and without Hubbard U described below to aid in the [41, 51–53] eff For all transition metals, U was set to 3.0 eV. correction, along with experimental data. understanding of the benchmark Mn Fe Co Ni results. a [] c [] a [] c [] a [] c [] a [] c [] It was previously reported that 26.36 7.12 26.39 6.70 26.20 6.84 25.96 6.82 PBE[a] s(r), the reduced dimensionless 25.76 6.75 25.79 6.66 25.73 6.69 25.36 6.65 PBE + U[b] gradient of electron density n(r) 7.04[c] 26.10[d] 6.85[d] 26.11[e] 6.71[e] 25.79[f] 6.77[f] Exp. 26.23[c] [Eq. (1)] around the adsorbed [a] Ref. [16]. [b] Ref. [47]. The anisotropy is averaged out and converted into a hexagonal cell. [c] Ref. [51]. gas molecule in MOFs mostly [d] Ref. [52]. [e] Ref. [53]. [f] Ref. [41]. distributes in the s < 3 region,[33]

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which is in line with the fact that the region in which s lies between 0.5 and 2.5 is known to give rise to variation in the chemisorption properties.[55]

sðrÞ ¼

jrnðrÞj ð2ð3pÞ2 Þ1=3 nðrÞ4=3 Þ

ð1Þ

Thus, the enhancement factor in F(s) in the 0.5  s  3.0 region practically determines the performance, whereas the other region is rather unimportant for the chemical performance of the MOF systems. The PBE exchange enhancement factor[56] is constructed by restricting the functional form with physically correct conditions and limits. Its variants, on the other hand, release or adjust one or more of these conditions. For example, in the revPBE functional[57] the local Lieb–Oxford bound is considered unnecessary and a parameter is fitted for atomic and atomization energies. Later, in the RPBE functional[55] the local bound is restored, but the behavior in the 0.5  s  2.5 region is made similar to that in the revPBE functional by changing the function form. These two functionals have larger enhancement factors than PBE. On the other hand, in the PBEsol functional,[58] a slowly varying limit is satisfied in the second order and the enhancement factor is decreased. This functional is known to be more suitable for solids. An intermediate approach was introduced to connect small and large s regions to yield the PBEint functional.[59] The experimental values and computational results with various exchange functionals were directly compared in two recent studies,[31, 33] in which the van der Waals density functional was used as the correlation functional. In a study by Rana et al., it was shown that the magnitude of the van der Waals functional correlation energy heavily depends on the choice of the exchange functional, which is in line with the aforementioned effect of exchange on dispersion[54] and is also a consequence of density difference in the vdW energy calculation. Figure 6 contains the published results of extensive previous studies. On the basis of these results, no clear relationship between the enhancement factors and the binding energies is observed. However, as claimed by Rana et al.,[31] revPBE-vdW shows the highest accuracy for the binding energies. Table 1 shows the geometric parameters of the Mg–MOF74–CO2 complex, both calculated and experimentally measured. Care must be taken in designing a calculation model to be consistent with experimental pressure in terms of CO2 occupation, as that can affect the fairness of the comparison.[31, 33] Here, one CO2 molecule is adsorbed per Mg atom, and so a direct comparison between experiment and computation is possible for given experimental conditions. This table shows that, contrary to the binding energy case, a clear correlation exists between the enhancement factor and the MgO distance, whereas such a tendency is not revealed for bond angles. It turns out that the gas–metal distance is larger for an exchange functional with a large enhancement factor, which is also true for bond lengths and lattice parameters.[33] The highest accuracy can be obtained from the optBXX functionals, the enhancement factors of which are relatively small.  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Though MOFs are more analogous to atoms and molecules than traditional solids, the application of revPBE or RPBE is often considered. The latter PBE variants tend to overestimate lattice parameters or bond lengths, along with PBE. The enhancement factor is smaller for the PBE variants tuned for gradient expansion of the s!0 limit, such as PBEsol or PBEint. Whereas the PBEint functional has not been tested for MOFs yet, to the best of our knowledge the geometries obtained with PBEsol were in a good agreement with the experimental results, but the binding energies were overestimated.[60, 61] However, the accuracy in geometry prediction does not necessarily connect to accuracy in the binding-energy calculations. For example, the optBXX-vdW functionals constantly overestimate the binding energies, and the revPBE-vdW functional constantly overestimates bond lengths and MCO2 distances. For a given correlation functional and its corrections, it might be conceivable to “tune up” the exchange functional for more accurate prediction of the geometric parameters, especially if the lattice parameters or bond lengths of one of the target MOFs are known. If the enhancement factor is required to be even smaller than that for the optBXX functionals, SOGGA[62] and SOGGA11[63] may be reasonable choices. Of course, this conclusion can change depending on the correlation part of the functional, but we expect this tendency to remain unaltered regardless of the correlation functional.

6. Conclusions The nature and complexity of metal–organic frameworks (i.e. the coexistence of transition metals and organic functions in a periodic solid with a large unit cell and possessing weak interactions in host–guest chemistry) make it difficult for the GGA level of theory to be naively applied. Correction methods for amending the shortcomings of the GGA functionals, together with the different GGA functionals, create a diversity of computational schemes in the periodic calculations of MOFs, whereas in most of the existing studies PBE-D2 was used with reasonable success. In this paper, the performance of various calculation schemes for treating MOFs, in particular MOF-74, was reviewed, especially with the involvement of gas adsorption. Our conclusions can be listed as follows: 1) Semiempirical dispersion corrections such as D, D2, and D3 are a “safe” choice, as their dependency on the functional is smaller and cost effective, which is the reason why PBED2 (the “safest” choice) is predominantly used. However, higher accuracy can be expected from the van der Waals functional, but only with a proper choice of exchange functional. For screening calculations, for example, semiempirical corrections can be used effectively. If quantitatively accurate results (especially for geometries) are needed or the electronic structure is expected to be extraordinary, the van der Waals density functional is the method of choice. 2) The effectiveness of the Hubbard U correction for the interaction energies is unclear at this stage, due to a lack of data. However, the U correction still seems essential for a physically correct description of MOFs with spin polarizaChemPhysChem 2014, 15, 3157 – 3165

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CHEMPHYSCHEM MINIREVIEWS tion. Also, the calculated lattice parameters are clearly related to the introduction of the U correction, which shrinks the cell. Importantly, the U parameters should be optimized for accurate MOF calculations, with proper consideration of other correction methods. A systematic study of validity and performance of the U correction is essential, which can be tested by comparing GGA+U with its hybrid functional version. 3) The geometries of MOFs have a clear relationship with the enhancement factor of the exchange part, as far as van der Waals density functionals are concerned: A larger enhancement factor leads to a larger metal–gas distance. Among the van der Waals density functionals tested, the optBXXvdW recorded the highest accuracy for geometry prediction, and revPBE-vdW recorded the highest accuracy for binding-energy calculation. However, neither of these methods was successful in predicting both simultaneously in a quantitative manner. 4) The performance of a GGA scheme to treat MOF systems depends mainly on three components, namely, the dispersion correction scheme, the Hubbard U correction, and the exchange functional. These effects can be separated to some extent; however, the overall performance should be benchmarked thoroughly en masse. Although a specific property can be calculated accurately by tuning one or two of these components, none of the current GGA level of methods was capable of providing a balanced description of all properties, perhaps not surprisingly, due to the desired simplicity of GGA. For the development of a transferrable scheme with reasonable accuracy across different properties, all three components need to be considered collectively.

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Received: May 1, 2014 Revised: June 23, 2014 Published online on August 28, 2014

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Assessments of semilocal density functionals and corrections for carbon dioxide adsorption on metal-organic frameworks.

The significant amount of attention that has been directed toward metal-organic frameworks (MOFs) for a wide spectrum of applications can be attribute...
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