Effect of Anharmonicity on Adsorption Thermodynamics

Importance sampling within configuration space integration for adsorbate thermophysical properties: a case study for CH3/Ni(111)

A new strategy is presented for computing anharmonic partition functions for the motion of adsorbates relative to a catalytic surface. Importance sampling is compared with conventional Monte Carlo. The importance sampling is significantly more efficient. This new approach is applied to CH3* on Ni(111) as a test case. The motion of methyl relative to the nickel surface is found to be anharmonic, with significantly higher entropy compared to the standard harmonic oscillator model. The new method is freely available as part of the Minima-Preserving Neural Network within the ADTHERM package.

Multilevel quantum mechanical calculations show the role of promoter molecules in the dehydration of methanol to dimethyl ether in H-ZSM-5

Methyl carboxylate esters promote the formation of dimethyl ether (DME) from the dehydration of methanol in H-ZSM-5 zeolite. We employ a multilevel quantum method to explore the possible associative and dissociative mechanisms in the presence, and absence, of six methyl ester promoters. This hybrid method combines density functional theory, with dispersion corrections (DFT-D3), for the full periodic system, with second-order Møller-Plesset perturbation theory (MP2) for small clusters representing the reaction site, and coupled cluster with single, double, and perturbative triple substitution (CCSD(T)) for the reacting molecules. The calculated adsorption enthalpy of methanol, and reaction enthalpies of the dehydration of methanol to DME within H-ZSM-5, agree with experiment to within chemical accuracy (∼4 kJ mol-1). For the promoters, a reaction pathway via an associative mechanism gives lower overall reaction enthalpies and barriers compared to the reaction with methanol only. Each stage of this mechanism is explored and related to experimental data. We provide evidence that suggests the promoter's adsorption to the Brønsted acid site is the most important factor dictating its efficiency.

A Critical Assessment on Calculating Vibrational Spectra in Nanostructured Materials

Vibrational spectroscopy is an omnipresent spectroscopic technique to characterize functional nanostructured materials such as zeolites, metal-organic frameworks (MOFs), and metal-halide perovskites (MHPs). The resulting experimental spectra are usually complex, with both low-frequency framework modes and high-frequency functional group vibrations. Therefore, theoretically calculated spectra are often an essential element to elucidate the vibrational fingerprint. In principle, there are two possible approaches to calculate vibrational spectra: (i) a static approach that approximates the potential energy surface (PES) as a set of independent harmonic oscillators and (ii) a dynamic approach that explicitly samples the PES around equilibrium by integrating Newton's equations of motions. The dynamic approach considers anharmonic and temperature effects and provides a more genuine representation of materials at true operating conditions; however, such simulations come at a substantially increased computational cost. This is certainly true when forces and energy evaluations are performed at the quantum mechanical level. Molecular dynamics (MD) techniques have become more established within the field of computational chemistry. Yet, for the prediction of infrared (IR) and Raman spectra of nanostructured materials, their usage has been less explored and remain restricted to some isolated successes. Therefore, it is currently not a priori clear which methodology should be used to accurately predict vibrational spectra for a given system. A comprehensive comparative study between various theoretical methods and experimental spectra for a broad set of nanostructured materials is so far lacking. To fill this gap, we herein present a concise overview on which methodology is suited to accurately predict vibrational spectra for a broad range of nanostructured materials and formulate a series of theoretical guidelines to this purpose. To this end, four different case studies are considered, each treating a particular material aspect, namely breathing in flexible MOFs, characterization of defects in the rigid MOF UiO-66, anharmonic vibrations in the metal-halide perovskite CsPbBr3, and guest adsorption on the pores of the zeolite H-SSZ-13. For all four materials, in their guest- and defect-free state and at sufficiently low temperatures, both the static and dynamic approach yield qualitatively similar spectra in agreement with experimental results. When the temperature is increased, the harmonic approximation starts to fail for CsPbBr3 due to the presence of anharmonic phonon modes. Also, the spectroscopic fingerprints of defects and guest species are insufficiently well predicted by a simple harmonic model. Both phenomena flatten the potential energy surface (PES), which facilitates the transitions between metastable states, necessitating dynamic sampling. On the basis of the four case studies treated in this Review, we can propose the following theoretical guidelines to simulate accurate vibrational spectra of functional solid-state materials: (i) For nanostructured crystalline framework materials at low temperature, insights into the lattice dynamics can be obtained using a static approach relying on a few points on the PES and an independent set of harmonic oscillators. (ii) When the material is evaluated at higher temperatures or when additional complexity enters the system, e.g., strong anharmonicity, defects, or guest species, the harmonic regime breaks down and dynamic sampling is required for a correct prediction of the phonon spectrum. These guidelines and their illustrations for prototype material classes can help experimental and theoretical researchers to enhance the knowledge obtained from a lattice dynamics study.

Free and internal energies for the adsorption of short alkanes into the zeolite SSZ-13 from ab initio molecular dynamics simulations

Electronic structure calculations have become a valuable tool in understanding chemical reactions of hydrocarbons in zeolite pores. However, commonly applied approaches to calculate free energies based on static electronic structure calculations significantly overestimate the entropic penalty for molecular adsorption into zeolite pores. Here, we use ab initio molecular dynamics (AIMD) simulations to model the adsorption of methane, ethane, and propane to purely siliceous and protonated SSZ-13. In our analyses we focus on the internal and Helmholtz free energies of adsorption of each molecule and compare our results to various approaches for the calculation of free energies based on static calculations. We find that only an approach that retains two thirds of the translational entropy of the adsorbate upon adsorption compares favorably with AIMD simulations. However, comparison to experimental measurements of Gibbs free energies of adsorption reported in the literature implies that we might not have captured the full complexity of alkane adsorption in our model. We expect that results in this work will help to develop a better understanding of alkane adsorption in zeolites, and that the provided data will serve as a benchmark for free energy calculations of alkane adsorption in zeolites in the future.

Operando Modeling of Zeolite-Catalyzed Reactions Using First-Principles Molecular Dynamics Simulations

Within this Perspective, we critically reflect on the role of first-principles molecular dynamics (MD) simulations in unraveling the catalytic function within zeolites under operating conditions. First-principles MD simulations refer to methods where the dynamics of the nuclei is followed in time by integrating the Newtonian equations of motion on a potential energy surface that is determined by solving the quantum-mechanical many-body problem for the electrons. Catalytic solids used in industrial applications show an intriguing high degree of complexity, with phenomena taking place at a broad range of length and time scales. Additionally, the state and function of a catalyst critically depend on the operating conditions, such as temperature, moisture, presence of water, etc. Herein we show by means of a series of exemplary cases how first-principles MD simulations are instrumental to unravel the catalyst complexity at the molecular scale. Examples show how the nature of reactive species at higher catalytic temperatures may drastically change compared to species at lower temperatures and how the nature of active sites may dynamically change upon exposure to water. To simulate rare events, first-principles MD simulations need to be used in combination with enhanced sampling techniques to efficiently sample low-probability regions of phase space. Using these techniques, it is shown how competitive pathways at operating conditions can be discovered and how broad transition state regions can be explored. Interestingly, such simulations can also be used to study hindered diffusion under operating conditions. The cases shown clearly illustrate how first-principles MD simulations reveal insights into the catalytic function at operating conditions, which could not be discovered using static or local approaches where only a few points are considered on the potential energy surface (PES). Despite these advantages, some major hurdles still exist to fully integrate first-principles MD methods in a standard computational catalytic workflow or to use the output of MD simulations as input for multiple length/time scale methods that aim to bridge to the reactor scale. First of all, methods are needed that allow us to evaluate the interatomic forces with quantum-mechanical accuracy, albeit at a much lower computational cost compared to currently used density functional theory (DFT) methods. The use of DFT limits the currently attainable length/time scales to hundreds of picoseconds and a few nanometers, which are much smaller than realistic catalyst particle dimensions and time scales encountered in the catalysis process. One solution could be to construct machine learning potentials (MLPs), where a numerical potential is derived from underlying quantum-mechanical data, which could be used in subsequent MD simulations. As such, much longer length and time scales could be reached; however, quite some research is still necessary to construct MLPs for the complex systems encountered in industrially used catalysts. Second, most currently used enhanced sampling techniques in catalysis make use of collective variables (CVs), which are mostly determined based on chemical intuition. To explore complex reactive networks with MD simulations, methods are needed that allow the automatic discovery of CVs or methods that do not rely on a priori definition of CVs. Recently, various data-driven methods have been proposed, which could be explored for complex catalytic systems. Lastly, first-principles MD methods are currently mostly used to investigate local reactive events. We hope that with the rise of data-driven methods and more efficient methods to describe the PES, first-principles MD methods will in the future also be able to describe longer length/time scale processes in catalysis. This might lead to a consistent dynamic description of all steps-diffusion, adsorption, and reaction-as they take place at the catalyst particle level.

Evaluating the Role of Anharmonic Vibrations in Zeolite β Materials

The characterization of zeolitic materials is often facilitated by spectroscopic analysis of vibrations, which informs about the bonding character of the substrate and any adsorbents. Computational simulations aid the interpretation of the spectra but often ignore anharmonic effects that can affect the spectral characteristics significantly. Here, the impact of anharmonicity is demonstrated with a combination of dynamical and static simulations applied to the structures formed during the synthesis of Sn-BEA via solid-state incorporation (SSI): the initial siliceous BEA (Si-β), aluminosilicate BEA (H-β), dealuminated BEA (deAl-β), and Sn-BEA (Sn-β). Heteroatom and defect-containing BEA are shown to have strong anharmonic vibrational contributions, with atomic and elemental resolution highlighting particularly the prevalence for H atoms (H-β, deAl-β) as well as localization to heteroatoms at defect sites. We simulate the vibrational spectra of BEA accounting for anharmonic contributions and observe an improved agreement with experimental data compared to harmonic methods, particularly at wavenumbers below 1500 cm-1. The results demonstrate the importance of incorporating anharmonic effects in simulations of vibrational spectra, with consequences toward future characterization and application of zeolitic materials.

Challenges in modelling dynamic processes in realistic nanostructured materials at operating conditions

The question is addressed in how far current modelling strategies are capable of modelling dynamic phenomena in realistic nanostructured materials at operating conditions. Nanostructured materials used in applications are far from perfect; they possess a broad range of heterogeneities in space and time extending over several orders of magnitude. Spatial heterogeneities from the subnanometre to the micrometre scale in crystal particles with a finite size and specific morphology, impact the material's dynamics. Furthermore, the material's functional behaviour is largely determined by the operating conditions. Currently, there exists a huge length-time scale gap between attainable theoretical length-time scales and experimentally relevant scales. Within this perspective, three key challenges are highlighted within the molecular modelling chain to bridge this length-time scale gap. Methods are needed that enable (i) building structural models for realistic crystal particles having mesoscale dimensions with isolated defects, correlated nanoregions, mesoporosity, internal and external surfaces; (ii) the evaluation of interatomic forces with quantum mechanical accuracy albeit at much lower computational cost than the currently used density functional theory methods and (iii) derivation of the kinetics of phenomena taking place in a multi-length-time scale window to obtain an overall view of the dynamics of the process. This article is part of a discussion meeting issue 'Supercomputing simulations of advanced materials'.

Reactivity of Single-Atom Alloy Nanoparticles: Modeling the Dehydrogenation of Propane

Physical catalysts often have multiple sites where reactions can take place. One prominent example is single-atom alloys, where the reactive dopant atoms can preferentially locate in the bulk or at different sites on the surface of the nanoparticle. However, ab initio modeling of catalysts usually only considers one site of the catalyst, neglecting the effects of multiple sites. Here, nanoparticles of copper doped with single-atom rhodium or palladium are modeled for the dehydrogenation of propane. Single-atom alloy nanoparticles are simulated at 400-600 K, using machine learning potentials trained on density functional theory calculations, and then the occupation of different single-atom active sites is identified using a similarity kernel. Further, the turnover frequency for all possible sites is calculated for propane dehydrogenation to propene through microkinetic modeling using density functional theory calculations. The total turnover frequencies of the whole nanoparticle are then described from both the population and the individual turnover frequency of each site. Under operating conditions, rhodium as a dopant is found to almost exclusively occupy (111) surface sites while palladium as a dopant occupies a greater variety of facets. Undercoordinated dopant surface sites are found to tend to be more reactive for propane dehydrogenation compared to the (111) surface. It is found that considering the dynamics of the single-atom alloy nanoparticle has a profound effect on the calculated catalytic activity of single-atom alloys by several orders of magnitude.

Anharmonic Correction to Free Energy Barriers from DFT-Based Molecular Dynamics Using Constrained Thermodynamic Integration

For the calculation of anharmonic contributions to free energy barriers, constrained thermodynamic λ-path integration (λ-TI) from a harmonic reference force field to density functional theory is presented as an alternative to the established Blue Moon ensemble method (ξ-TI), in which free energy gradients along the reaction coordinate ξ are integrated. With good agreement in all cases, the λ-TI method is benchmarked against the ξ-TI method for several reactions, including the internal CH₃ group rotation in ethane, a nucleophilic substitution of CH₃Cl, a retro-Diels-Alder reaction, and a proton transfer in zeolite H-SSZ-13. An advantage of λ-TI is that one can use virtually any reference state to compute anharmonic contributions to reaction free energies or free energy barriers. This is particularly relevant for catalysis, where it is now possible to compute anharmonic corrections to the free energy of a transition state relative to any reference, for example, the most stable state of the active site and the reactants in the gas phase. This is in contrast to ξ-TI, where free energy barriers can only be computed relative to an initial state with all reactants coadsorbed. Finally, the Bennett acceptance ratio method combined with λ-TI is demonstrated to reduce the number of required integration grid points with tolerable accuracy, favoring thus λ-TI over ξ-TI in terms of computational efficiency.

Transition-state correlations for predicting thermochemistry of adsorbates and surface reactions

Estimating thermochemical properties from linear correlations may provide a pathway to circumvent expensive density functional theory (DFT) calculations for quantities such as pre-exponentials and temperature corrections to DFT energies. Here, we construct thermochemical scaling relations between C₁-C₆n-alkanes in the gas phase and adsorbed alkyl chains extending from several transition metal surfaces, and examine changes in the slope and fit between metals and adsorption sites. We subsequently add -OH, -NH₂, CO, and CC functional groups to the C₁-C₆ molecules and demonstrate strong linear correlations for thermochemistry across all species. We broaden the correlations to incorporate transition states of C₁-C₆n-alkane dehydrogenation reactions, where thermochemistry for computationally prohibitive transition-state calculations can be quickly assessed. Additionally, we rationalize the linearity of thermochemical correlations based on the composition of the homologous series and theoretical assessments. As an application of the correlations, we estimate pre-exponentials for elementary surface reactions of ethane and propane hydrogenolysis on Ru(0001), which is of relevance to plastic hydrogenolysis. Depending on kinetically important steps, entropic contributions may be necessary to include in certain reaction mechanisms; in contrasting examples, entropies are found to be relatively insignificant for ethane hydrogenolysis but pertinent for propane hydrogenolysis.

Dual-Site Model for Ab Initio 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.

An understanding of coronavirus and exploring the molecular dynamics simulations to find promising candidates against the Mpro of nCoV to combat the COVID-19: A systematic review

The first infection case of new coronavirus was reported at the end of 2019 and after then, the cases are reported in all nations across the world in a very short period. Further, the regular news of mutations in the virus has made life restricted with appropriate behavior. To date, a new strain (Omicron and its new subvariant Omicron XE) has brought fear amongst us due to a higher trajectory of increase in the number of cases. The researchers thus started giving attention to this viral infection and discovering drug-like candidates to cure the infections. Finding a drug for any viral infection is not an easy task and takes plenty of time. Therefore, computational chemistry/bioinformatics is followed to get promising molecules against viral infection. Molecular dynamics (MD) simulations are being explored to get drug candidates in a short period. The molecules are screened via molecular docking, which provides preliminary information which can be further verified by molecular dynamics (MD) simulations. To understand the change in structure, MD simulations generated several trajectories such as root mean square deviation (RMSD), root mean square fluctuation (RMSF), hydrogen bonding, and radius of gyration for the main protease (Mpro) of the new coronavirus (nCoV) in the presence of small molecules. Additionally, change in free energy for the formation of complex of Mpro of nCoV with the small molecule can be determined by applying molecular mechanics with generalized born and surface area solvation (MM-GBSA). Thus, the promising molecules can be further explored for clinical trials to combat coronavirus disease-19 (COVID-19).

Comparative Analysis of Uncoupled Mode Approximations for Molecular Thermochemistry and Kinetics

The accurate prediction of thermochemistry and kinetic parameters is an important task for reaction modeling. Unfortunately, the commonly used harmonic oscillator model is often not accurate enough due to the absence of anharmonic effects. In this work, we improve the representation of an anharmonic potential energy surface (PES) using uncoupled mode (UM) approximations, which model the full-dimensional PES as a sum of one-dimensional potentials of each mode. We systematically analyze different PES sampling schemes and coordinate systems for constructing the one-dimensional potentials, and benchmark the performance of UM methods on data sets of molecular thermochemistry and kinetic properties. The results show that the accuracy of the UM approach strongly depends on how the one-dimensional potentials are defined. If one-dimensional potentials are constructed by sampling along normal mode directions (UM-N) or along the directions that minimize intermode coupling (E- and E'-optimized), the accuracies of the predicted properties are not significantly improved compared to the harmonic oscillator model. However, significant improvements can be achieved by sampling the torsional modes separately from the vibrational modes (UM-T and UM-VT). In this work, three types of coordinate systems are examined, including redundant internal coordinates (RIC), hybrid internal coordinates (HIC), and translation-rotation-internal coordinates (TRIC). The HIC and TRIC coordinate systems can outperform RIC since transition state species may contain large-amplitude interfragmentary motions that regular internal coordinates can not describe adequately. Among all the methods we examined, the activation energies and pre-exponential factors calculated using UM-VT with either TRIC or HIC best agree with the reference values. Since UM-VT requires only a number of additional single point energy calculations for each independent mode, the scaling of computational costs of UM-VT is the same as that of the standard harmonic oscillator model, making UM-VT an appealing way of calculating the thermochemistry and kinetic properties for large-size systems.

Rigid Body Approximation for the Anharmonic Description of Molecule-Surface Vibrations

We present an anharmonic approach for molecule-surface vibrations that employs rigid body coordinates, based on the Rodrigues rotation formula, to describe curvilinear displacements (rotations and translations) of the molecule along normal modes. These displacements are used to calculate energy data points from which one-dimensional polynomial potentials are fitted using cubic splines. In these potentials, for each of the six rigid body modes separately, one-dimensional Schrödinger equations are solved with harmonic oscillator or Fourier functions (for pure rotations) as basis sets. The anharmonic vibrational energies obtained are used to calculate partition functions and from them enthalpies, entropies, and Gibbs free energies of adsorption. Our numerical implementation has been successfully tested for Morse and cosine potentials with known analytical solutions. The methods have been applied to adsorption of CH₄ on the hydroxyl group of the proton form of the chabazite zeolite (H-CHA), as well as to adsorption of CH₄ and CO on the Mg²⁺ ions of the metal-organic framework (MOF) Mg₂(dobdc). To obtain the best estimates for thermodynamic functions, we include the coupling between molecule-surface vibrations and intrasystem vibrations at the harmonic level. The calculated Gibbs free energies show that the coupling is small for CH₄/H-CHA and CO/MOF (between -0.7 and +0.1 kJ/mol) but substantial for CH₄/MOF (-3.4 kJ/mol). The predicted anharmonic effect on the Gibbs free energy of adsorption for CH₄/H-CHA, CH₄/MOF, and CO/MOF is -4.7, 0.3 ± 0.7, and -2.4 ± 0.6 kJ/mol, respectively, which results in +4.2, +0.9 ± 0.7, and -0.4 ± 0.6 kJ/mol, respectively, for the deviation from experiment. This is well within chemical accuracy limits (±4.2 kJ/mol) for the adsorption of CH₄ and CO in the MOF. The larger deviation for CH₄/H-CHA, at the edge of the chemical accuracy range, is most likely due to contributions from soft zeolite modes which are neglected in our approach.

Portable Models for Entropy Effects on Kinetic Selectivity

Differences in entropies of competing transition states can direct kinetic selectivity. Understanding and modeling such entropy differences at the molecular level is complicated by the fact that entropy is statistical in nature; i.e., it depends on multiple vibrational states of transition structures, the existence of multiple dynamically accessible pathways past these transition structures, and contributions from multiple transition structures differing in conformation/configuration. The difficulties associated with modeling each of these contributors are discussed here, along with possible solutions, all with an eye toward the development of portable qualitative models of use to experimentalists aiming to design reactions that make use of entropy to control kinetic selectivity.

Kinetic Monte Carlo simulations for heterogeneous catalysis: Fundamentals, current status, and challenges

Kinetic Monte Carlo (KMC) simulations in combination with first-principles (1p)-based calculations are rapidly becoming the gold-standard computational framework for bridging the gap between the wide range of length scales and time scales over which heterogeneous catalysis unfolds. 1p-KMC simulations provide accurate insights into reactions over surfaces, a vital step toward the rational design of novel catalysts. In this Perspective, we briefly outline basic principles, computational challenges, successful applications, as well as future directions and opportunities of this promising and ever more popular kinetic modeling approach.

Ab initio molecular dynamics with enhanced sampling in heterogeneous catalysis

Enhanced sampling ab initio simulations enable to study chemical phenomena in catalytic systems including thermal effects & anharmonicity, & collective dynamics describing enthalpic & entropic contributions, which can significantly impact on reaction free energy landscapes.

Chemically Accurate Vibrational Free Energies of Adsorption from Density Functional Theory Molecular Dynamics: Alkanes in Zeolites

We present a methodology to compute, at reduced computational cost, Gibbs free energies, enthalpies, and entropies of adsorption from molecular dynamics. We calculate vibrational partition functions from vibrational energies, which we obtain from the vibrational density of states by projection on the normal modes. The use of a set of well-chosen reference structures along the trajectories accounts for the anharmonicities of the modes. For the adsorption of methane, ethane, and propane in the H-CHA zeolite, we limit our treatment to a set of vibrational modes localized at the adsorption site (zeolitic OH group) and the alkane molecule interacting with it. Only two short trajectories (1–20 ps) are required to reach convergence (<1 kJ/mol) for the thermodynamic functions. The mean absolute deviations from the experimentally measured values are 2.6, 2.8, and 4.7 kJ/mol for the Gibbs free energy, the enthalpy, and the entropy term (−TΔS), respectively. In particular, the entropy terms show a major improvement compared to the harmonic approximation and almost reach the accuracy of the previous use of anharmonic frequencies obtained with curvilinear distortions of individual modes. The thermodynamic functions so obtained follow the trend of the experimental values for methane, ethane, and propane, and the Gibbs free energy of adsorption at experimental conditions is correctly predicted to change from positive for methane (5.9 kJ/mol) to negative for ethane (−4.8 kJ/mol) and propane (−7.1 kJ/mol).

Calculation of absolute molecular entropies and heat capacities made simple

We propose a fully-automated composite scheme for the accurate and numerically stable calculation of molecular entropies by efficiently combining density-functional theory (DFT), semi-empirical methods (SQM), and force-field (FF) approximations. The scheme is systematically expandable and can be integrated seamlessly with continuum-solvation models. Anharmonic effects are included through the modified rigid-rotor-harmonic-oscillator (msRRHO) approximation and the Gibbs-Shannon formula for extensive conformer ensembles (CEs), which are generated by a metadynamics search algorithm and are extrapolated to completeness. For the first time, variations of the ro-vibrational entropy over the CE are consistently accounted-for through a Boltzmann-population average. Extensive tests of the protocol with the two standard DFT approaches B97-3c and B3LYP-D3 reveal an unprecedented accuracy with mean deviations <1 cal mol-1 K-1 (about <1-2%) for the total gas phase molecular entropy of medium-sized molecules. Even for the hardship case of extremely flexible linear alkanes (C14H30-C16H34), errors are only about 3 cal mol-1 K-1. Comprehensive tests indicate a relatively strong variation of the conformational entropy on the underlying level of theory for typical drug molecules, inferring the complex potential energy surfaces as the main source of error. Furthermore, we show some application examples for the calculation of free energy differences in typical chemical reactions.

Anharmonic Correction to Adsorption Free Energy from DFT-Based MD Using Thermodynamic Integration

Adsorption processes are often governed by weak interactions for which the estimation of entropy contributions by means of the harmonic approximation is prone to be inaccurate. Thermodynamic integration (TI) from the harmonic to the fully interacting system (λ-path integration) can be used to compute anharmonic corrections. Here, we combine TI with (curvilinear) internal coordinates in periodic systems to make the formalism available in computational studies. Our implementation of ab initio molecular dynamics in VASP is independent of the reaction path and can be thus applied to study adsorption processes relative to the gas phase and does hence provide a useful tool for computational catalysis. We discuss the application of the approach on three model systems for which exact semianalytical solutions exist and illustrate and quantify the importance of anharmonic vibrations, hindered rotations, and hindered translations (dissociation). Eventually, we apply the method to study the adsorption of small adsorbates in a zeolite (H-SSZ-13).

Thermal behaviour of hindered rotor in static electric and laser field

We solve exactly the time independent Schrödinger equation (TISE) for hindered rotor confined in well shaped potential. Afterwards, the confined system is subjected to interact with laser and static electric field. TISE for the dressed confined rotor is solved using standard numerical method to get energy spectrum and eigenfunctions. Dependence of orientation and alignment parameters on static electric field and confining potential is studied. We also compute thermal properties like entropy and heat capacity of the system under consideration using canonical partition within statistical thermodynamic formalism. We have also studied the effect of external field parameters, confinement strength and temperature on these thermal properties.

Computational Investigation of Correlations in Adsorbate Entropy for Pure-Silica Zeolite Adsorbents

Vast numbers of unstudied hypothetical porous frameworks continue to spark interest in optimizing adsorption and catalytic processes. Evaluating the use of such materials depends on the accessibility of thermodynamic metrics such as the free energy, which, in turn, depend on the satisfactory estimation or calculation of the adsorption entropy, which often remains elusive. Previous works using simulations and experimental data have demonstrated relationships between the entropy and system descriptors, allowing for sensible predictions based on more-easily obtained physical parameters. However, the resultant conclusions were either based on experimental data for industrially relevant alkanes or lacked a significant sample size. In this paper, we evaluate correlations between gas-phase and adsorbed-phase entropies for a larger and more chemically diverse set of adsorbate molecules by using force fields and statistical mechanical expressions to calculate those entropies. In total, we perform calculations for 37 molecules across 10 chemical categories available in the TraPPE force field set, as adsorbed in five siliceous zeolites. Our results show that linear correlations between the gas- and adsorbed-phase entropies persist for the larger and diverse set of adsorbate molecules studied here, proving a broader applicability and justifying the use of simple correlations for many adsorbates and, presumably, adsorbent materials.

Predicting adsorption selectivities from pure gas isotherms for gas mixtures in metal-organic frameworks

We perform Grand Canonical Monte Carlo simulations on a lattice of Mg²⁺ sites (GCMC) for adsorption of four binary A/B mixtures, CH₄/N₂, CO/N₂, CO₂/N₂, and CO₂/CH₄, in the metal–organic framework Mg₂(2,5-dioxidobenzedicarboxylate), also known as CPO-27–Mg or Mg–MOF-74. We present a mean field co-adsorption isotherm model and show that its predictions agree with the GCMC results if the same quantum chemical ab initio data are used for Gibbs free energies of adsorption at the individual sites and for lateral interaction energies between the same, A⋯A and B⋯B, and unlike, A⋯B, adsorbed molecules. We use both approaches to test the assumption underlying Ideal Adsorbed Solution Theory (IAST), namely approximating A⋯B interaction energies as the arithmetic mean of A⋯A and B⋯B interaction energies. While IAST works well for mixtures with weak lateral interactions, CH₄/N₂ and CO/N₂, the deviations are large for mixtures with stronger lateral interactions, CO₂/N₂ and CO₂/CH₄. Motivated by the theory of London dispersion forces, we propose use of the geometric mean instead of the arithmetic mean and achieve substantial improvements. For CO₂/CH₄, the lateral interactions become anisotropic. To include this in the geometric mean co-adsorption model, we introduce an anisotropy factor. We propose a protocol, named co-adsorption mean field theory (CAMT), for co-adsorption selectivity prediction from known (experiment or simulation) pure component isotherms which is similar to the IAST protocol but uses the geometric mean to approximate mixed pair interaction energies and yields improved results for non-ideal mixtures.

A new mixing rule (geometric mean) is proposed with substantial improvements compared to the widely used ideal adsorbed solution theory for adsorbates with strong lateral interactions.

Towards operando computational modeling in heterogeneous catalysis

An increased synergy between experimental and theoretical investigations in heterogeneous catalysis has become apparent during the last decade. Experimental work has extended from ultra-high vacuum and low temperature towards operando conditions. These developments have motivated the computational community to move from standard descriptive computational models, based on inspection of the potential energy surface at 0 K and low reactant concentrations (0 K/UHV model), to more realistic conditions. The transition from 0 K/UHV to operando models has been backed by significant developments in computer hardware and software over the past few decades. New methodological developments, designed to overcome part of the gap between 0 K/UHV and operando conditions, include (i) global optimization techniques, (ii) ab initio constrained thermodynamics, (iii) biased molecular dynamics, (iv) microkinetic models of reaction networks and (v) machine learning approaches. The importance of the transition is highlighted by discussing how the molecular level picture of catalytic sites and the associated reaction mechanisms changes when the chemical environment, pressure and temperature effects are correctly accounted for in molecular simulations. It is the purpose of this review to discuss each method on an equal footing, and to draw connections between methods, particularly where they may be applied in combination.

Uncertainty quantification for quantum chemical models of complex reaction networks

For the quantitative understanding of complex chemical reaction mechanisms, it is, in general, necessary to accurately determine the corresponding free energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a general (complex) reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to still determine quantitative fluxes in a reaction network if one is interested in specific time scales. In this paper, we present discrete-time kinetic simulations in discrete state space taking free energy uncertainties into account. The method builds upon thermo-chemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated for the example of the formose reaction. An important application of our approach is the detection of regions in a reaction network which require further investigation, given the uncertainties introduced by both approximate electronic structure methods and kinetic models. Such cases can then be studied in greater detail with more sophisticated first-principles calculations and kinetic simulations.

Entrapped Single Tungstate Site in Zeolite for Cooperative Catalysis of Olefin Metathesis with Brønsted Acid Site

Industrial olefin metathesis catalysts generally suffer from low reaction rates and require harsh reaction conditions for moderate activities. This is due to their inability to prevent metathesis active sites (MASs) from aggregation and their intrinsic poor adsorption and activation of olefin molecules. Here, isolated tungstate species as single molecular MASs are immobilized inside zeolite pores by Brønsted acid sites (BASs) on the inner surface. It is demonstrated that unoccupied BASs in atomic proximity to MASs enhance olefin adsorption and facilitate the formation of metallocycle intermediates in a stereospecific manner. Thus, effective cooperative catalysis takes place over the BAS-MAS pair inside the zeolite cavity. In consequence, for the cross-metathesis of ethene and trans-2-butene to propene, under mild reaction conditions, the propene production rate over WO _x/USY is ca. 7300 times that over the industrial WO₃/SiO₂-based catalyst. A propene yield up to 79% (80% selectivity) without observable deactivation was obtained over WO _x/USY for a wide range of reaction conditions.

Understanding Brønsted-Acid Catalyzed Monomolecular Reactions of Alkanes in Zeolite Pores by Combining Insights from Experiment and Theory

Acidic zeolites are effective catalysts for the cracking of large hydrocarbon molecules into lower molecular weight products required for transportation fuels. However, the ways in which the zeolite structure affects the catalytic activity at Brønsted protons are not fully understood. One way to characterize the influence of the zeolite structure on the catalysis is to study alkane cracking and dehydrogenation at very low conversion, conditions for which the kinetics are well defined. To understand the effects of zeolite structure on the measured rate coefficient (k_app ), it is necessary to identify the equilibrium constant for adsorption into the reactant state (K_ads-H+ ) and the intrinsic rate coefficient of the reaction (k_int ) at reaction temperatures, since k_app is proportional to the product of K_ads-H+ and k_int . We show that K_ads-H+ cannot be calculated from experimental adsorption data collected near ambient temperature, but can, however, be estimated accurately from configurational-bias Monte Carlo (CBMC) simulations. Using monomolecular cracking and dehydrogenation of C₃ -C₆ alkanes as an example, we review recent efforts aimed at elucidating the influence of the acid site location and the zeolite framework structure on the observed values of k_app and its components, K_ads-H+ and k_int .

Zeolite Adsorption Free Energies from ab Initio Potentials of Mean Force

The ability of metal-exchanged zeolites to chemisorb small gas molecules is key to their performance as heterogeneous catalysts and gas-separating agents. Here, we propose and evaluate an ab initio potential of mean force (PMF) method for computing adsorption free energies of representative small molecules to Cu-exchanged sites in SSZ-13 zeolite. We show that Cu ions are mobilized by adsorbates and, as a result, computed free energies are significantly more negative than those obtained from a conventional harmonic oscillator model. PMF-derived free energies are consistent with available experiment and, in many cases, with a dynamics-based quasi-harmonic analysis (QHA). The PMF approach avoids the artificial partitioning of degrees of freedom intrinsic to the QHA. On the basis of the PMF results, we propose a simple correlation to estimate free energies from computed adsorption energies and gas-phase entropies.

Ab initio study of methanol and ethanol adsorption on Brønsted sites in zeolite H-MFI

Ethanol in zeolite H-MFI: passing from DFT to MP2 and taking anharmonicity into account changes enthalpies of adsorption significantly.