Ab Initio Prediction of Adsorption Isotherms for Small Molecules in Metal–Organic Frameworks: The Effect of Lateral Interactions for Methane/CPO-27-Mg

A hybrid method that combines density functional theory for periodic structures with wave function-based electron correlation methods for finite-size models of adsorption sites is employed to calculate energies for adsorption of CH₄ onto different sites in the metal–organic framework (MOF) CPO-27-Mg (Mg-MOF-74) with chemical accuracy. The adsorption energies for the Mg²⁺, linker, second layer sites are −27.8, −18.3, and −15.1 kJ/mol. Adsorbate–adsorbate interactions increase the average CH₄ adsorption energy by about 10% (2.4 kJ/mol). The free rotor-harmonic oscillator-ideal gas model is applied to calculate free energies/equilibrium constants for adsorption on the individual sites. This information is used in a multisite Langmuir model, augmented with a Bragg–Williams model for lateral interactions, to calculate adsorption isotherms. This ab initio approach yields the contributions of the individual sites to the final isotherms and also of the lateral interactions that contribute about 15% to the maximum excess adsorption capacity. Isotherms are calculated for both absolute amounts, for calculation of isosteric heats of adsorption as function of coverage, and excess amounts, for comparison with measured isotherms. Agreement with observed excess isotherms is reached if the experimentally determined limited accessibility of adsorption sites (78%) is taken into account

Dual-Site Model for <i>Ab Initio</i> Calculations of Gibbs Free Energies and Enthalpies of Adsorption: Methane in Zeolite Mobile Five (H-MFI)

Quantum chemical hybrid MP2:PBE+D2 calculations in combination with molecular statistics are employed to calculate enthalpies and Gibbs free energies of adsorption for CH4 at Brønsted acid sites [bridging Si–O(H)–Al groups] and silica wall sites (Si–O–Si) of the proton form of zeolite MFI (H-ZSM-5) and its purely siliceous analogue Silicalite-1. A Langmuir model is adopted to calculate the amounts of CH4 adsorbed at each type of site from the Gibbs free energies. The combination of these results according to the ratio of silica wall sites and Brønsted acid sites in the sample yields adsorption isotherms for zeolites with different Si/Al ratios. The zero-coverage isosteric heats of adsorption, calculated as thermal averages of the adsorption enthalpies of the individual sites, vary between 20.2 kJ/mol for the pore wall site and 29.2 kJ/mol for the acid site and agree well within ±1 kJ/mol with experimental results

Ab Initio Adsorption Isotherms for Molecules with Lateral Interactions: CO₂ in Metal–Organic Frameworks

Adsorption of carbon dioxide in the metal–organic framework CPO-27-Mg (Mg-MOF-74) is examined. We use accurate quantum chemical ab initio methods (wave function-type electron correlation methods for cluster models combined with density functional theory for periodic systems) to calculate gas–surface site and gas–gas interactions. At 298 K, the “zero-coverage” enthalpy and Gibbs free energy of CO₂ adsorption on Mg²⁺ sites are −46 and −9 kJ/mol, respectively; for linker sites these values are −30 and +5 kJ/mol, respectively. For full monolayer coverage lateral interactions from nearby molecules contribute −6 and −5 kJ/mol to the adsorption enthalpy for CO₂ at Mg²⁺ and linker sites, respectively. The predicted heats of adsorption and free energies of adsorption agree within 2.6 and 0.8 kJ/mol, respectively, with experimental values well within chemical accuracy limits (4.2 kJ/mol). We use two different ways of calculating isotherms from equilibrium constants for individual sites and interaction energies: (i) a Langmuir model, augmented with the mean-field (MF) approximation for lateral interactions, and (ii) grand canonical Monte Carlo (GCMC) simulations on a lattice of sites, which agree very well with each other. We use GCMC data to examine how different isotherm models (Langmuir, dual-site Langmuir, Sips, Toth, and mean-field) fit them. We conclude that the MF model yields the best fit over a wide pressure range with physically meaningful parameters, i.e., adsorption constants for individual sites and lateral interaction energies

Ab Initio Prediction of Adsorption Isotherms for Gas Mixtures by Grand Canonical Monte Carlo Simulations on a Lattice of Sites

Gibbs free energies of adsorption on individual sites and the lateral (adsorbate–adsorbate) interaction energies are obtained from quantum chemical ab initio methods and molecular statistics. They define a Grand Canonical Monte Carlo (GCMC) Hamiltonian for simulations of gas mixtures on a lattice of adsorption sites. Coadsorption of CO₂ and CH₄ at Mg²⁺ sites in the pores of the metal–organic framework CPO-27-Mg (Mg-MOF-74) is studied as an example. Simulations with different approximations as made in widely used coadsorption models such as the ideal adsorbed solution theory (IAST) show their limitations in describing adsorption selectivities for binary mixtures

Heats of Adsorption of CO and CO₂ in Metal–Organic Frameworks: Quantum Mechanical Study of CPO-27-M (M = Mg, Ni, Zn)

Density functional theory is applied with a hybrid functional to which a parametrized damped 1/<i>r</i>⁶ term has been added to account for dispersion (B3LYP+D*). This method is used with periodic boundary conditions to get the structures of the adsorption complexes. Dispersion has a substantial share on the calculated adsorption energies (46–77%). For these structures, adsorption energies are also calculated with a hybrid high-level (MP2 with complete basis set extrapolation):low level (B3LYP+D*) method. The MP2 calculations are performed on cluster models. Comparison is made with experimental heats of adsorption. B3LYP+D* underestimates heats of adsorption by about 5 kJ/mol, whereas hybrid MP2:B3LYP+D* slightly overestimates them by about 2 kJ/mol. With MP2:B3LYP+D*, also the mean absolute error is somewhat smaller, 3.8 kJ/mol compared to 5.6 kJ/mol for B3LYP+D*. Both the B3LYP+D* and the hybrid MP2/CBS:B3LYP+D* method predict the same sequence of binding energies for carbon monoxide (Ni > Mg > Zn) and carbon dioxide (Mg > Ni > Zn) adsorption on open metal cation sites in the CPO-27 metal–organic frameworks

Ab Initio Prediction of Adsorption Isotherms for Small Molecules in Metal–Organic Frameworks

For CO and N₂ on Mg²⁺ sites of the metal–organic framework CPO-27-Mg (Mg-MOF-74), ab initio calculations of Gibbs free energies of adsorption have been performed. Combined with the Bragg-Williams/Langmuir model and taking into account the experimental site availability (76.5%), we obtained adsorption isotherms in close agreement with those in experiment. The remaining deviations in the Gibbs free energy (about 1 kJ/mol) are significantly smaller than the “chemical accuracy” limit of about 4 kJ/mol. The presented approach uses (i) a DFT dispersion method (PBE+D2) to optimize the structure and to calculate <i>anharmonic frequencies</i> for vibrational partition functions and (ii) a “hybrid MP2:(PBE+D2)+ΔCCSD­(T)” method to determine electronic energies. With the achieved accuracy (estimated uncertainty ±1.4 kJ/mol), the ab initio energies become useful benchmarks for assessing different DFT + dispersion methods (PBE+D2, B3LYP+D*, and vdW-D2), whereas the ab initio heats, entropies, and Gibbs free energies of adsorption are used to assess the reliability of experimental values derived from fitting isotherms or from variable-temperature IR studies