Improving gas adsorption modeling for MOFs by local calibration of Hubbard U parameters

While computational screening with density functional theory (DFT) is frequently employed for the screening of metal–organic frameworks (MOFs) for gas separation and storage, commonly applied generalized gradient approximations (GGAs) exhibit self-interaction errors, which hinder the predictions of adsorption energies. We investigate the Hubbard U parameter to augment DFT calculations for full periodic MOFs, targeting a more precise modeling of gas molecule–MOF interactions, specifically for N 2 , CO 2 , and O 2 . We introduce a calibration scheme for the U parameter, which is tailored for each MOF, by leveraging higher-level calculations on the secondary building unit (SBU) of the MOF. When applied to the full periodic MOF, the U parameter calibrated against hybrid HSE06 calculations of SBUs successfully reproduces hybrid-quality calculations of the adsorption energy of the periodic MOF. The mean absolute deviation of adsorption energies reduces from 0.13 eV for a standard GGA treatment to 0.06 eV with the calibrated U , demonstrating the utility of the calibration procedure when applied to the full MOF structure. Furthermore, attempting to use coupled cluster singles and doubles with perturbative triples calculations of isolated SBUs for this calibration procedure shows varying degrees of success in predicting the experimental heat of adsorption. It improves accuracy for N 2 adsorption for cases of overbinding, whereas its impact on CO 2 is minimal, and ambiguities in spin state assignment hinder consistent improvements of O 2 adsorption. Our findings emphasize the limitations of cluster models and advocate the use of full periodic MOF systems with a calibrated U parameter, providing a more comprehensive understanding of gas adsorption in MOFs.


I. INTRODUCTION
1][12] In designing MOFs for applications involving gases, both the absolute and relative strengths of gas binding are crucial for material selectivity. 13,14MOFs featuring open metal sites are particularly promising in this regard. 15The interaction between the metal and a gas molecule at these sites can be finely tuned through the chemical environment of the metal, enhancing selectivity for specific gases. 16,17To effectively design the selective interaction, quantum chemistry calculations are indispensable.They provide a deep understanding of the metal-gas molecule interactions, enabling the tailored design of MOFs with desired selectivity for certain gases. 180][21][22] DFT provides a practical balance between computational efficiency and accuracy, facilitating the calculation of large unit cells of MOFs. 235][26] Self-interaction errors can significantly impact the study of MOFs, leading to inaccuracies in reaction barriers, 27 adsorption energies, [28][29][30][31] and electronic properties. 32Moreover, as DFT operates as a single-reference method, approximating the many-electron wavefunction with a single Slater determinant, it incurs static correlation errors, which are particularly pronounced 4][35] There is a notable trade-off between self-interaction and static correlation errors, where methods that fix the delocalization error (e.g., incorporating an admixture of Hartree-Fock exchange) are known to be more prone to static correlation errors. 26,34,36,37or a more accurate analysis of gas molecule binding, post-DFT methods such as coupled cluster singles and doubles with perturbative triples [CCSD(T)] are necessary. 38While CCSD(T) remains a single-reference method and is subject to static correlation errors, it demonstrates sufficient accuracy for systems exhibiting moderate correlation (e.g., systems with low multi-reference character indicated by diagnostics). 39,40CCSD(T) is typically limited to molecular systems, and MOFs can be conveniently decomposed into molecular components, such as the secondary building unit (SBU) and organic linkers. 41Such a decomposition allows for the application of post-DFT methods to individual MOF components. 42While these approaches offer detailed insights into molecular interactions, they fall short in accurately representing the complete periodic structure of MOFs. 38,43,44Consequently, accurately calculating adsorption energies, while vital for understanding selectivity, remains challenging due to the difficulties in extending post-DFT methods to full periodic MOFs. 45o overcome this challenge, we investigate the application of the Hubbard U parameter to a GGA functional [here, Perdew-Burke-Ernzerhof (PBE)] in DFT calculations for full periodic MOFs, primarily focusing on correcting self-interaction errors.7][48][49] The U parameter has previously been adjusted to match experimental data for tuning oxidation energies, 50 electronic structure properties, 51,52 and catalytic activity. 53,54Leveraging the reticular structure of MOFs and the correlation between the multireference character of MOFs and their molecular counterparts, 55 we introduce a scheme that determines the U parameter by fitting the GGA+U gas molecule adsorption energy results of an SBU to the results of post-DFT calculations and applies the same U parameter to GGA+U (specifically, PBE+U) calculations of the full MOF.8][59][60][61] To validate a hypothesis that a U parameter that is effective for correcting SBU energetics will similarly apply to the MOF system, we initially benchmark against HSE06 calculation results.The screened hybrid HSE06 was selected because it is the highest-level method for solid-state systems that remains computationally feasible for MOFs that typically have large unit cells.Upon confirming that the U parameter capable of matching the HSE06 result for the SBU also increases agreement between PBE+U and HSE06 for the periodic MOF, we proceed to fit the U parameter to the CCSD(T) results of SBU.Finally, we compare the outcomes of PBE+U, employing the U parameter fitted to CCSD(T), with the experimental data.

II. METHODS
All DFT calculations of MOFs and DFT+U calculations of SBUs were performed using Quantum ESPRESSO version 7.0 62 with projector-augmented wave pseudopotentials 63 and an 80 Ry kinetic energy cutoff.An empirical dispersion correction of Grimme's DFT-D3 64 with Becke-Johnson damping 65 was applied.The geometry optimization and DFT-D3+U calculations of MOFs were performed with the PBE functional 66 on a 3×2×2 k-point mesh for M 2 (OH) 2 -BBTA and M 2 Cl 2 -BBTA and a 4×2×2 k-point mesh for M-DOBDC and M-DSBDC.The HSE06-D3 calculations 56 of MOFs were performed only at the Γ point due to the high computational cost of these calculations.The DFT-D3+U calculations of SBUs for Hubbard U parameter fitting were performed by adding 20 Å of vacuum in all directions and applying the Martyna-Tuckerman correction 67 for an isolated molecule.
The crystal structures of 24 MOFs were obtained by optimizing the atomic positions and the unit cell using the structures reported in Ref. 68 as the initial structure (Table S1 of the supplementary material).For calculations that included gases, one molecule was added per unit cell.The initial orientations of N 2 and O 2 molecules were based on Ref. 68, and those of CO 2 molecules were based on Ref. 69, which were determined through DFT calculations.Subsequently, the positions of both MOF atoms and the gas molecule were fully relaxed, with the cell parameters being kept constant.
SBU calculations using Gaussian type orbital basis sets were performed using ORCA version 5.0.1. 70The DFT calculations employed the def2-TZVP basis set. 71To ensure consistency with MOF calculations, Grimme's DFT-D3 correction with Becke-Johnson damping was applied as well.However, as ORCA does not have default Becke-Johnson parameters for the HSE06 functional, the parameters used by Quantum ESPRESSO were used, which are s 6 = 1, a 1 = 0.383, s 8 = 2.310, and a 2 = 5.685.We employed cc-pVDZ 72 and cc-pVTZ 73 for the CCSD(T) calculations and extrapolated these to approach the complete basis set limit following a recommended two-point formula. 74Given the large number of atoms in the SBUs ranging from 44 to 85, domain-based local pair natural orbital (DLPNO) CCSD(T) 75 with "normal" PNO thresholds was used to make CCSD(T) calculations feasible.
Although we employ two types of basis sets, plane wave basis sets and Gaussian type orbital basis sets, the choice of the basis set has a minimal impact on the adsorption energy (Fig. S1 of the supplementary material).A summary of the calculation details is provided in Table S2 of the supplementary material.
The heat of adsorption was obtained by adding 5/2RT to the calculated adsorption energy to account for the translational and rotational degrees of freedom of the gas molecule, where R is the gas constant and T is the temperature.The median temperature of the range used in the relevant experiment was employed (Table S3 of the supplementary material).The temperature variation during the experiment only influences the heat of adsorption by at most 3 kJ/mol.We neglect the zero-point vibrational (i.e., phonon) energy contribution to the adsorption energy, which is less than 5 kJ/mol and should not significantly impact the comparison with experimental data. 58The energy difference between the ferromagnetic and antiferromagnetic phases of the bare MOF's metal centers is also not expected to have a substantial effect on the results.For example, the difference is 4 kJ/mol for Co 2 (OH) 2 -BBTA 60 and 10 kJ/mol for Fe-DOBDC. 76he Journal of Chemical Physics ARTICLE pubs.aip.org/aip/jcp

III. SYSTEMS AND APPROACH FOR U PARAMETER CALIBRATION
We investigate a total of 24 MOFs that are derived from the combination of six 3d metals (Cr, Mn, Fe, Co, Ni, and Cu) and four MOF structures: M-DOBDC (DOBDC = 2,5dioxido-1,4-benzenedicarboxylate), M-DSBDC (DSBDC = 2,5disulfhydryl-1,4-benzenedicarboxylate), M 2 (OH) 2 -BBTA (BBTA = 1H,5H-benzo(1,2-d:4,5-d')bistriazole), and M 2 Cl 2 -BBTA (Fig. S2 of the supplementary material).8][59][60][61] These MOFs possess a hexagonal array of onedimensional channels that facilitate the diffusion of gas molecules within the pores. 77Their undercoordinated metal sites serve as adsorption sites that can be tailored by switching the metal to selectively capture specific types of molecules. 61,78,79These features make these MOFs particularly promising for gas storage, capture, and separation, where selectivity is a critical factor in device performance. 14he SBUs of these MOFs were identified by modifying the extraction scheme implemented in MOFSimplify. 41Each metal atom and all non-metal atoms up to two bonds away from it were first added to the SBU.Distinct from our previous approach, in the case where any non-metal atom is a member of a ring, all atoms within the ring are included as part of the SBU.Since the MOFs in our dataset have metal clusters that are infinitely extended in one dimension, they were truncated to a repeat of three metal atoms, and the adsorbate was aligned with the central metal atom (Fig. 1).To avoid dangling bonds or strongly negative charges, truncated bonds were capped with hydrogen.All metals were assumed to have oxidation states of +2 with metals in high-spin states and ferromagnetically coupled.The triplet spin of O 2 was assumed to be parallel with the spin of the metal atoms unless specified otherwise.The charge of the SBU was determined by summing the charge of the metal and the ligands.The full list of charge and magnetization states of MOFs and SBUs is provided in Table S4 of the supplementary material.
The U parameter is only applied to the 3d orbitals of metal atoms.The binding energy (E b ) of a gas molecule to an SBU is calculated using PBE-D3+U as follows: Note that the value of U applied to the bare MOF depends on the type of adsorbate molecule being considered.The U value is calibrated to minimize the difference in binding energy between the SBU calculated using HSE06-D3 or CCSD(T) and that calculated using PBE-D3+U.The range of U values considered is from 0 to 15 eV, with stepwise increments of 0.1 eV.
Our study focuses on the adsorption energy of N 2 , O 2 , and CO 2 .This enables us to compare the impact of U on the redox-dependent adsorption of N 2 , the additional spin effect of O 2 , and the physisorption characteristics of CO 2 .In particular, for CO 2 , we study only the end-on O-based adsorption of CO 2 and do not consider binding between the metal center and C that is common in chemisorption, following the geometry determined by DFT in Ref. 69.

A. Adsorption energies of small molecules on MOF SBUs
We first assess the sensitivity of the adsorption energy of the SBU to the calculation method by comparing the PBE-D3, HSE06-D3, and CCSD(T) results (Fig. 2).PBE-D3 and HSE06-D3 calculations were performed for the full dataset of 72 MOF-molecule systems.CCSD(T) calculations were limited to a subset of 19 MOF-molecule systems, chosen based on the availability of experimental data related to the heat of adsorption for direct benchmarking.In line with previous findings, 80 PBE-D3 consistently predicts a stronger binding affinity between gas molecules and SBUs compared to HSE06-D3 and CCSD(T).For the entire dataset, the mean absolute difference (MAD) in the adsorption energy calculated using PBE-D3 compared to HSE06-D3 is 0.14 eV.For the subset of systems with available experimental data, the adsorption energies calculated using PBE-D3 show a larger deviation from HSE06-D3 results, giving a MAD of 0.23 eV.The deviation is even larger when comparing PBE-D3 with CCSD(T) over this same set of 19 MOFs, where the MAD reaches 0.31 eV.While HSE06-D3 gives adsorption energy values that align more closely with CCSD(T) and are particularly accurate for CO 2 , it is not sufficient to provide an accurate representation of adsorption behavior of N 2 and O 2 (Fig. S3 of the supplementary material).For example, while HSE06-D3 overestimates the O 2 binding strength to Fe-DOBDC, yielding an adsorption energy of −0.39 eV, CCSD(T) predicts it to be repulsive (i.e., unbound) at 0.16 eV.Incorporating exact exchange in DFT calculations significantly alters the adsorption energy range for all the molecules on SBUs.The adsorption energy calculated using PBE-D3 ranges from −0.93 to −0.06 eV, while the adsorption energy calculated using HSE06-D3 is shifted upward ranging from −0.39 to 0.41 eV, meaning that some interactions are repulsive even when the DFT-D3 dispersion correction is applied.For the HSE06-D3 calculations, the adsorption energy of CO 2 exhibits the narrowest range, from −0.37 to −0.14 eV.N 2 displays a slightly wider range, spanning from −0.36 to 0.06 eV.In addition, O 2 shows the broadest range, varying from −0.39 to 0.41 eV, i.e., the same as the total range over all molecules.Molecules that show small ranges of binding energies tend to display better agreement between PBE-D3 and HSE06-D3 calculations.That is, CO 2 adsorption energies have a MAD of only 0.03 eV, the MAD for N 2 is higher at 0.13 eV, and O 2 demonstrates the largest deviation at 0.26 eV.It is worth noting that the adsorption of O 2 is particularly susceptible to self-interaction errors arising from a considerable charge transfer from the metal atom to the O 2 molecules, 81 leading to erroneous conclusions regarding the adsorption behavior. 82,83For example, PBE-D3 calculations suggest a strong binding of O 2 to Fe 2 Cl 2 -BBTA with an absorption energy of −0.48 eV, whereas HSE06-D3, which corrects for the impact of SIE, indicates a repulsive interaction, estimating an adsorption energy of 0.17 eV.

B. Correcting PBE-D3 energies with Hubbard U parameters calibrated on isolated SBUs
We explore two approaches to determine the U parameter for the dataset consisting of 72 MOF/adsorbate systems.First, taking into account the specific characteristics of the SBU composition, we individually fit the U parameter for each SBU/gas-molecule system.We refer to this customized U value as the SBU-specific U, or U SBU .Second, focusing solely on the chemistry between the metal atom and the gas molecule, we fit the U parameter for a group of SBUs that share the same metal/gas-molecule pair.In this case, we determine the U parameter by minimizing the MAD of each grouping.We refer to this calibrated U value as the metal-specific U, or U metal (Table S5 of the supplementary material).Even though we refer to this as the metal-specific U, it is also unique for each adsorbate (i.e., CO 2 , N 2 , or O 2 ).
By fitting the U parameter to each SBU/gas-molecule system, it is possible to align the adsorption energy of the molecule on the SBU model calculated with PBE-D3+U SBU to that of HSE06-D3 withins a maximum deviation of 0.2 eV for all systems under consideration.Deviations between PBE-D3+U SBU and HSE06-D3 persist primarily if the adsorption energy is insensitive to the U correction or if the effect of U is to tune the adsorption in the wrong direction with respect to HSE06-D3.The MAD of the adsorption energy of SBUs calculated with respect to HSE06-D3 decreases from 0.15 eV with PBE-D3 to 0.02 eV when the SBU-specific U is used in a PBE-D3+U calculation (Fig. 3).Even when fitting the U parameter for a group of SBUs that share the same metal/gas-molecule pair, we found that only two cases, N 2 and O 2 binding on Co 2 Cl 2 -BBTA, exhibit PBE-D3+U metal errors exceeding 0.2 eV.The MAD of the adsorption energy of SBUs calculated using PBE-D3+U metal with respect to HSE06-D3 remains at 0.04 eV, which is comparable to the previous scenario where the SBU-specific U is applied, with the benefit of making the approach more general (Fig. S4 of the supplementary material).The two sets of U parameters show a comparable performance even though the U SBU and U metal parameters that would be used for a MOF-adsorbate complex differ by 3.1 eV across the set of MOFs studied here.They differ because U metal is fit to reduce errors on average across all MOFs with that metal and adsorbate, whereas U SBU is fit to each SBU (Fig. S5 of the supplementary material).The small difference in the performance of the two U parameter fitting approaches can be attributed to the relatively low sensitivity of the adsorption energy to changes in the U parameter.In fact, the adsorption energy of 43 of 72 cases shows a deviation of less than 0.1 eV when the U parameter is varied from 0 to 15 eV (Fig. S6 of the supplementary material).
The value of the calibrated U parameters is influenced by the sensitivity of the adsorption energies to the U parameters and by the magnitude of the original deviation between the two functionals.Both the SBU-specific and metal-specific U parameters exhibit a distribution centered around 3-4 eV (Fig. 4).For SBU-specific U parameters in the most extreme cases, the calibrated U parameters reach 15 eV for nine out of 72 cases, and, in the opposite direction, the PBE-D3 results of another ten cases already agree well with HSE06-D3 without the need for the U parameter.The highest values of the U parameter are observed mainly for the cases of O 2 and CO 2 adsorption.A large U parameter is required when the adsorption energy demonstrates relatively low sensitivity to the U parameter, as is the case for CO 2 , or when there is a significant difference between PBE-D3 and HSE06-D3 results, as is the case for O 2 (Fig. 4).For example, the SBU-specific U parameter is set at 15 eV for both O 2 on Ni 2 (OH) 2 -BBTA and CO 2 on Fe 2 Cl 2 -BBTA.While the adsorption energy of O 2 on Ni 2 (OH) 2 -BBTA exhibits a notable variation of 0.26 eV when the U parameter is varied from 0 to 15 eV, the adsorption energy of CO 2 on Fe 2 Cl 2 -BBTA experiences a much smaller variation of 0.06 eV.To rationalize differences in needed U parameters, the sensitivity of the adsorption energy to the U parameter can be estimated by the difference in the PBE-D3 fractionality of the bare SBU and SBU/gas-molecule system, Δ Tr [n(1 − n)], which is directly proportional to the Hubbard energy correction term.Here, n is the occupation matrix of the orbitals where the U parameter is applied, i.e., the metal 3d orbitals.For each orbital, the term n(1 − n) takes on a value of 0 when the orbital is either completely filled or empty.Its value increases to a maximum of 0.25 when the orbital is half-filled, the regime in which the system is maximally impacted by corrections from the U parameter. 29,84,85The PBE-D3 fractionality analysis shows that, in general, the adsorption energy of CO 2 exhibits the least sensitivity, whereas the adsorption energy of O 2 is the most sensitive (Fig. 5).

C. Applicability of U values calibrated on isolated SBUs to periodic MOFs
We tested the effectiveness of the U parameter calibrated from the SBU by using it to perform PBE-D3+U calculations of the full periodic MOF structure.The MAD of PBE-D3+U calculations, when compared to HSE06-D3 results on the full periodic MOF, is found to be 0.06 eV for both the SBU-specific and metalspecific U parameters, exhibiting a substantial improvement over the MAD of 0.13 eV when using standard PBE-D3 (Fig. 6 and Fig. S7 of the supplementary material).Consequently, our findings confirm the hypothesis that a U value calibrated using a truncated SBU model will successfully correct the properties of the periodic MOF structure.PBE-D3+U calculations using an SBUcalibrated U value match adsorption energies from periodic hybrid calculations better than hybrid calculations on SBU models do (MAD = 0.06 vs 0.07 eV, Fig. S8 of the supplementary material).Therefore, the calibrated U parameter enables the GGA-level calculation on the periodic MOF to achieve similar accuracy to the hybrid-GGA-level calculation on the SBU, while also benefitting from the incorporation of the environment effects of incorporating periodicity.

D. Performance of the U parameter fitted to CCSD(T)
Given the promise of U parameters calibrated on SBU models to improve the energetics of full periodic MOFs, we investigated whether we could increase the fidelity with which we model the SBU.CCSD(T) is widely regarded as the gold standard in quantum chemistry, and thus, we selected it as our reference method for SBU calculations obtained at a higher level of theory.We confirmed the reliability of CCSD(T) calculations using the T 1 diagnostic, which estimates the multireference character through the single-excitation amplitude vector from CCSD calculations. 86Apart from one exception, all systems of interest were shown to have a T 1 value below 0.02 (Table S7 of the supplementary material).The cutoff value of 0.02 is generally accepted as indicating that static correlation errors are manageable for single-reference methods. 87,88The one system exhibiting a high T 1 value is Co 2 (OH) 2 -BBTA with an O 2 adsorbate, which will be discussed in detail below.Nevertheless, the ultimate test for theoretical calculations should be experimental reference data.As a more stringent test, we investigated the effectiveness of the U parameter fitted to SBU CCSD(T) calculation results, U CCSD(T) , in predicting the experimental heat of adsorption for the periodic MOF.This corresponds to the same approach as U SBU fitting where each adsorbate and MOF get their own U parameter, but, in this case, the tuning is carried out to match CCSD(T) results.For systems where experimental heat of adsorption data was available, we calculated the adsorption energy for N 2 , CO 2 , and O 2 using PBE-D3 and PBE-D3+U CCSD(T) (Tables S3 and S8 of the supplementary material).The performance of the U CCSD(T) parameter varies with different gas molecules as described in detail below.
For N 2 , PBE-D3 gives a mean absolute error (MAE) of 8.4 kJ/mol, which drops to 4.9 kJ/mol with the application of the U CCSD(T) correction (Fig. 7).In the cases of Mn-DOBDC, Ni-DOBDC, and Co 2 Cl 2 -BBTA, the heat of adsorption calculated using PBE-D3 is already small, within 3 kJ/mol of the experimentally measured value.Here, introducing the U CCSD(T) parameter does not greatly alter the results.For Co-DOBDC and Co 2 (OH) 2 -BBTA, PBE-D3 overbinds N 2 , giving errors of 10.7 and 16.4 kJ/mol, respectively.The U CCSD(T) parameter proves to be effective for these systems, decreasing the error to 4.1 kJ/mol for Co-DOBDC 2.0 kJ/mol for Co 2 (OH) 2 -BBTA.In contrast, with Cu-DOBDC and Fe-DOBDC, PBE-D3 exhibits underbinding for N 2 .Given that the U parameter tends to weaken the adsorption strength, 89 it amplifies the error for Cu-DODBC.While the heat of adsorption of Fe-DOBDC atypically gets stronger upon applying the U CCSD(T) parameter, the increase in heat of adsorption for PBE-D3+U is insufficient to achieve good agreement with the experimental heat of adsorption.For CO 2 , the interaction between the metal atom and the CO 2 molecule in these MOFs is primarily driven by the dispersion interaction, with minimal hybridization. 80Therefore, the U CCSD(T) parameter has little impact on the heat of adsorption, as discussed previously.In summary, while the U CCSD(T) correction is effective in improving the accuracy for N 2 in cases of PBE-D3 overbinding, it does not enhance the accuracy of CO 2 adsorption values, which are primarily due to physisorption processes.
O 2 adsorption presents additional challenges.There is an uncertainty associated with whether the O 2 molecule retains its spin when adsorbing and if that magnetic moment is coupled ferromagnetically (FM) or antiferromagnetically (AFM) to the metal center to which it binds.For all the studied MOFs, PBE-D3 exhibits a significant overbinding with AFM coupling, leading to a substantial MAE of 41.3 kJ/mol with respect to experiment.The U CCSD(T) parameter corrects much of this overbinding, reducing the MAE to 12.9 kJ/mol when choosing the coupling case with the lower energy for computing the heat of adsorption (Fig. 7 and Table S9 of the supplementary material).When the U CCSD(T) parameter is applied, FM coupling shows stronger adsorption than AFM coupling for Cu-DOBDC, Fe-DOBDC, Co 2 Cl 2 -BBTA, and Co 2 (OH) 2 -BBTA.However, the U CCSD(T) parameter overcompensates for two outlier systems, resulting in errors of 21.9 kJ/mol for Fe-DOBDC and 46.3 kJ/mol for Co 2 (OH) 2 -BBTA.The overcorrection is evident when examining Co 2 (OH) 2 -BBTA in detail.While the metal-O 2 distance of Co 2 (OH) 2 -BBTA is 1.9 Å when optimized using PBE-D3, the U CCSD(T) correction extends it to 2.9 Å.Consequently, O 2 loses the extra stabilization it had from the hydrogen bonding with The Journal of Chemical Physics the nearby hydroxy group, leading to even weaker adsorption of O 2 (Fig. S9 of the supplementary material).In stark contrast, experimental measurements place the metal-O 2 distance at 1.9 Å, aligning with the PBE-D3 results. 60he observed discrepancy between PBE-D3+U CCSD(T) and the experiment can likely be attributed to two primary factors.First, as previously mentioned, this system may be prone to significant static correlation errors, evidenced by the moderate T 1 value of 0.0202, suggesting that CCSD(T) results may not be sufficiently reliable.Second, our current U CCSD(T) is fitted only to the SBU with an FM spin configuration.The potential energy curve of the FM spin configuration calculated using CCSD(T) exhibits a notably flat and repulsive The Journal of Chemical Physics ARTICLE pubs.aip.org/aip/jcpenergy curve, contrary to the experimental observations (Fig. S10 of the supplementary material).It is important to note that while our current U CCSD(T) value is fitted only to the SBU with an FM spin configuration, the optimal U CCSD(T) value varies based on the spin configuration.Consequently, the CCSD(T) result of the FM spin configuration is not likely to estimate the U value needed to calculate the adsorption energy for the Co 2 (OH) 2 -BBTA-O 2 system with an AFM spin configuration.Systems with ambiguous spin coupling are, therefore, not good candidates for this calibration approach and rather require a comprehensive broken symmetry calculation to thoroughly understand their adsorption processes. 90verall, the quality of the CCSD(T) evaluation on SBUs in comparison with experimental adsorption energies on the full MOF suggests some limitations for gas adsorption (Fig. 7).Nevertheless, there is still an important role for the U CCSD(T) parameter in yielding more accurate results for the full MOF system.Optimizing the geometry of a periodic MOF using a GGA such as PBE-D3 and then employing CCSD(T) calculations on a cluster SBU model of the MOF, a typical approach in MOF studies, 38 can misrepresent the heat of adsorption.PBE-D3 tends to overbind the gas molecules, leading to an underestimation of the metal-molecule distances (Tables S10-S12 of the supplementary material). 80Especially for Co 2 (OH) 2 -BBTA and Co 2 Cl 2 -BBTA, the short metal-molecule distance of the PBE-D3-optimized structure causes the CCSD(T)evaluated single-point energy to indicate a positive (repulsive) N 2 adsorption energy.The potential energy curve for Co 2 (OH) 2 -BBTA interacting with N 2 exemplifies the efficacy of the U CCSD(T) parameter in correcting the overbinding tendency of PBE-D3.Indeed, the potential energy curve of the SBU calculated using PBE-D3+U CCSD(T) aligns closely with that calculated using CCSD(T) (Fig. S11 of the supplementary material).Still, potential energy curves from both PBE-D3 and PBE-D3+U CCSD(T) calculations reveal that relying solely on the SBU can underestimate the adsorption strength compared to the MOF.Therefore, a full periodic MOF model combined with the U CCSD(T) correction can offer a more accurate representation of the gas adsorption, especially when the CCSD(T) results are unambiguous regarding the spin of MOF metal nodes.

V. CONCLUSIONS
In summary, we examined the calibration of the Hubbard U parameter for use in MOFs that we achieve by tuning the properties of corresponding SBU cluster models.We applied this calibration approach to the prediction of adsorption energies on both the cluster and the full periodic MOF and demonstrated the utility of this approach for accurately calculating the adsorption energies of N 2 , CO 2 , and O 2 on 24 well-known MOF systems.We evaluated the sensitivity of calculated adsorption energies of isolated SBU cluster models across three different methods: the semi-local GGA PBE-D3, screened hybrid GGA HSE06-D3, and CCSD(T).While PBE-D3 generally predicted higher binding affinities than HSE06-D3 and CCSD(T), incorporating exact exchange through HSE06-D3 brought the adsorption energies closer to those obtained with CCSD(T).We verified the applicability of the U parameter calibrated against the SBU in improving the accuracy of the calculation of the full periodic MOF system by comparing PBE-D3+U and HSE06-D3 results.The U parameter was calibrated to align the PBE-D3+U adsorption energies with those calculated using HSE06-D3 for SBUs in two calibration approaches.In the first approach, a U SBU was tailored to each specific SBU/gas-molecule system, and, in the second approach, a U metal was obtained by fitting a single U value for the metal-gas chemistry across SBUs sharing the same metal/gas pair.Both U SBU and U metal successfully reduced the MAD of PBE-D3 compared to HSE06-D3.Applying these calibrated U parameters to PBE-D3+U SBU/metal calculations for the adsorption energy of full periodic MOF structures closely approximates the HSE06-D3 results as well.Our results confirmed that a calibrated U value for an SBU is aligned well with the U parameter needed for the full periodic MOF structure.
Furthermore, we investigated whether fitting the U parameter to calculations performed on isolated SBUs at the CCSD(T) level of theory, considered the gold standard, could help in predicting the experimental heat of adsorption.The calibrated parameter, termed U CCSD(T) , showed varying degrees of success.For N 2 adsorption, the PBE-D3+U CCSD(T) calculations exhibited improved accuracy over the standard PBE-D3 calculations in cases where PBE-D3 was prone to overbinding.However, the impact of U CCSD(T) was minimal for CO 2 adsorption, which is primarily driven by physisorption.O 2 adsorption presented additional challenges, particularly due to uncertainties in spin states upon adsorption.The U CCSD(T) parameter, while correcting the overbinding in PBE-D3 calculations for O 2 , leads to overcorrections in certain systems, highlighting the need to consider the full spin configuration.
Finally, our findings emphasized the necessity of using the full periodic MOF model for computing adsorption energies robustly.Although the cluster model cannot provide a quantitative estimation of gas molecule adsorption on MOFs, we, nevertheless, showed that it is useful for calibrating the U parameter.Thus, the proposed strategy provides a path to overcoming the computational limitations of full periodic MOFs, where the accuracy of calculations is typically limited to the GGA level, by calibrating methods at higher accuracy on the cluster model to predict the accurate adsorption in the full system.Our analysis of how the U parameter influences the adsorption energy will guide future researchers in determining the utility of the U parameter in their MOF studies.The deeper insights into the interactions between gas molecules and MOFs will facilitate the design of MOFs with improved selectivity for specific gases, addressing a critical need in applications requiring precise gas separation.We expect this approach to generalize well to other adsorbates and properties such as catalysis, thereby broadening the scope of MOF research and application.

SUPPLEMENTARY MATERIAL
The supplementary material contains the lattice parameters of studied MOFs; the adsorption energy of SBU calculated using two basis sets; the summary of calculation details; available experimental measurement data for the heat of adsorption; the structure of studied MOFs; the magnetic moment and charge of the extracted SBU; U parameters fitted to the HSE06-D3 calculation results of the SBU; the magnetization of the metal atom calculated using PBE-D3 and PBE-D3+U SBU ; the adsorption energy of SBUs calculated using The Journal of Chemical Physics

FIG. 1 .FIG. 2 .
FIG. 1. Flow chart for extracting an SBU from a MOF and example of SBU binding with N 2 extracted from Cr-DOBDC.The black border indicates the MOF unit cell.The atoms are colored as follows: Cr, cyan; C, gray; H, white; O, red; N, blue.

FIG. 3 .FIG. 4 .
FIG. 3. Adsorption energies of small molecules on MOF SBUs in eV calculated with (a) PBE-D3 and (b) PBE-D3+U using the SBU-specific U, with respect to the HSE06-D3 calculation results on the same SBU.PBE-D3 and PBE-D3+U calculations were performed using plane wave (PW) basis sets.The black parity lines are shown.

FIG. 6 .FIG. 7 .
FIG. 6.Adsorption energies of the small molecules on periodic MOFs in eV calculated with (a) PBE-D3 and (b) PBE-D3+U using the U parameters fitted to each SBU-gas molecule system, with respect to HSE06-D3 calculation results.The black parity lines are shown.
ARTICLE pubs.aip.org/aip/jcpHSE06-D3 and CCSD(T); the adsorption energy of SBUs calculated using PBE-D3, PBE-D3+U metal , and HSE06-D3; the SBU-specific U parameters and the metal-specific U parameters; the change in the adsorption energy of SBUs with the U parameter; the adsorption energy of MOFs calculated using HSE06-D3 and PBE-D3+U metal ; the adsorption energy of MOFs and SBUs calculated using HSE06-D3; T 1 values from CCSD(T) calculations of the SBUs; U parameters fitted to the CCSD(T) calculation results of the SBU; the O 2 adsorption energy of MOFs; the structure of O 2 adsorbed on Co 2 OH 2 -BBTA; the distance between the metal atom and the N 2 molecule; the distance between the metal atom and the CO 2 molecule; the distance between the metal atom and the O 2 molecule; the potential energy curve of Co 2 (OH) 2 -BBTA and N 2 ; the potential energy curve of Co 2 (OH) 2 -BBTA and O 2 (PDF) example input files; the initial and optimized structures of MOFs; and the structure of SBUs (ZIP).