Elastic Polarizable Environment Cluster Embedding Approach for Covalent Oxides and Zeolites Based on a Density Functional Method

PyDFT-QMMM: A modular, extensible software framework for DFT-based QM/MM molecular dynamics

PyDFT-QMMM is a Python-based package for performing hybrid quantum mechanics/molecular mechanics (QM/MM) simulations at the density functional level of theory. The program is designed to treat short-range and long-range interactions through user-specified combinations of electrostatic and mechanical embedding procedures within periodic simulation domains, providing necessary interfaces to external quantum chemistry and molecular dynamics software. To enable direct embedding of long-range electrostatics in periodic systems, we have derived and implemented force terms for our previously described QM/MM/PME approach [Pederson and McDaniel, J. Chem. Phys. 156, 174105 (2022)]. Communication with external software packages Psi4 and OpenMM is facilitated through Python application programming interfaces (APIs). The core library contains basic utilities for running QM/MM molecular dynamics simulations, and plug-in entry-points are provided for users to implement custom energy/force calculation and integration routines, within an extensible architecture. The user interacts with PyDFT-QMMM primarily through its Python API, allowing for complex workflow development with Python scripting, for example, interfacing with PLUMED for free energy simulations. We provide benchmarks of forces and energy conservation for the QM/MM/PME and alternative QM/MM electrostatic embedding approaches. We further demonstrate a simple example use case for water solute in a water solvent system, for which radial distribution functions are computed from 100 ps QM/MM simulations; in this example, we highlight how the solvation structure is sensitive to different basis-set choices due to under- or over-polarization of the QM water molecule's electron density.

Molecules in Environments: Toward Systematic Quantum Embedding of Electrons and Drude Oscillators

We develop a quantum embedding method that enables accurate and efficient treatment of interactions between molecules and an environment, while explicitly including many-body correlations. The molecule is composed of classical nuclei and quantum electrons, whereas the environment is modeled via charged quantum harmonic oscillators. We construct a general Hamiltonian and introduce a variational Ansatz for the correlated ground state of the fully interacting molecule-environment system. This wave function is optimized via the variational Monte Carlo method and the ground state energy is subsequently estimated through the diffusion Monte Carlo method. The proposed scheme allows an explicit many-body treatment of electrostatic, polarization, and dispersion interactions between the molecule and the environment. We study solvation energies and excitation energies of benzene derivatives, obtaining excellent agreement with explicit ab initio calculations and experiments.

EE-ONIOM-CT Method to Efficiently Account for the Missing Interactions in ONIOM: Energies and Analytic Gradients

Hybrid methods such as ONIOM (QM:QM) are widely used for the study of local processes in large systems. However, the intrinsic need for system partitioning often leads to a less-than-desirable performance for some important chemical processes. This is due to the missing interactions in the chemically important model region (i.e., active site) of the high-level theory. The missing interactions can be categorized into two classes, viz. charge transfer (i.e., charge redistribution) and long-range electrostatic interactions. Our group presented two entirely different methods to treat these deficiencies individually. ONIOM-CT and ONIOM-EE methods have been demonstrated to improve the performance of ONIOM by incorporating charge transfer and missing electrostatic interactions, respectively. In general, the inclusion of the missing interactions separately in two different calculations may not be sufficient to reach a high accuracy. Thus, it is highly desirable to develop a method to correct both deficiencies simultaneously. In this work, we aim to connect the methods ONIOM-CT and ONIOM-EE for a more comprehensive treatment. A "stepwise" model was found to be necessary for a robust performance. This model employs a stepwise procedure by first satisfying the ONIOM-CT condition for charge balance before accounting for the electrostatic interactions from the rest of the system perturbatively. This has the advantage of easy interpretation due to the clear separation of the two effects. We demonstrate the performance of our method using embedding charges determined from a Mulliken population analysis. An efficient analytic gradient expression for this method is derived and implemented by requiring three sets of z-vector self-consistent equations. The performance of our method is assessed against full system calculations in high-level theory for a set of three proton transfer reactions representing different degrees of electrostatic embedding.

The application of QM/MM simulations in heterogeneous catalysis

The QM/MM simulation method is provenly efficient for the simulation of biological systems, where an interplay of extensive environment and delicate local interactions drives a process of interest through a funnel on a complex energy landscape. Recent advances in quantum chemistry and force-field methods present opportunities for the adoption of QM/MM to simulate heterogeneous catalytic processes, and their related systems, where similar intricacies exist on the energy landscape. Herein, the fundamental theoretical considerations for performing QM/MM simulations, and the practical considerations for setting up QM/MM simulations of catalytic systems, are introduced; then, areas of heterogeneous catalysis are explored where QM/MM methods have been most fruitfully applied. The discussion includes simulations performed for adsorption processes in solvent at metallic interfaces, reaction mechanisms within zeolitic systems, nanoparticles, and defect chemistry within ionic solids. We conclude with a perspective on the current state of the field and areas where future opportunities for development and application exist.

ONIOM Method with Charge Transfer Corrections (ONIOM-CT): Analytic Gradients and Benchmarking

Hybrid methods such as ONIOM that treat different regions of a large molecule using different methods are widely used to investigate chemical reactions in a variety of materials and biological systems. However, there are inherent sources of significant errors due to the standard treatment of the boundary between the regions using hydrogen link atoms. In particular, an unbalanced charge distribution in the chemically important model region is a potential source of such problems. We have previously suggested ONIOM-CT (ONIOM with charge transfer corrections) which addresses this issue by applying a potential in the form of point charges to obtain a desired charge redistribution. The metric for charge redistribution relies on the type of population analysis used to obtain the charges. ONIOM-CT has been implemented using Mulliken and Löwdin population analyses and has been shown to improve computed reaction energies for illustrative chemical reactions. In this work, we derive and implement the analytic gradients for ONIOM-CT that requires solving two sets of coupled-perturbed self-consistent equations, one each for the model system and the full system. However, both are needed only at the low level of theory, allowing for an efficient formulation and implementation for both Mulliken and Löwdin population analyses. Benchmarking and illustrative geometry optimizations have been carried out for a previously studied set of reactions involving a single link atom between regions. Additionally, we have generalized our method for the treatment of model systems involving multiple link atoms to enable applications for a broader set of problems. The generalized methods are illustrated for both charge models. Furthermore, we have studied a set of three proton transfer reactions and demonstrate that significant improvement is achieved by ONIOM-CT over ONIOM using both Mulliken and Löwdin population analyses.

General embedded cluster protocol for accurate modeling of oxygen vacancies in metal-oxides

The O vacancy (Ov) formation energy, EOv, is an important property of a metal-oxide, governing its performance in applications such as fuel cells or heterogeneous catalysis. These defects are routinely studied with density functional theory (DFT). However, it is well-recognized that standard DFT formulations (e.g., the generalized gradient approximation) are insufficient for modeling the Ov, requiring higher levels of theory. The embedded cluster method offers a promising approach to compute EOv accurately, giving access to all electronic structure methods. Central to this approach is the construction of quantum(-mechanically treated) clusters placed within suitable embedding environments. Unfortunately, current approaches to constructing the quantum clusters either require large system sizes, preventing application of high-level methods, or require significant manual input, preventing investigations of multiple systems simultaneously. In this work, we present a systematic and general quantum cluster design protocol that can determine small converged quantum clusters for studying the Ov in metal-oxides with accurate methods, such as local coupled cluster with single, double, and perturbative triple excitations. We apply this protocol to study the Ov in the bulk and surface planes of rutile TiO2 and rock salt MgO, producing the first accurate and well-converged determinations of EOv with this method. These reference values are used to benchmark exchange–correlation functionals in DFT, and we find that all the studied functionals underestimate EOv, with the average error decreasing along the rungs of Jacob’s ladder. This protocol is automatable for high-throughput calculations and can be generalized to study other point defects or adsorbates.

Embedding Methods for Quantum Chemistry: Applications from Materials to Life Sciences

Quantum mechanical embedding methods hold the promise to transform not just the way calculations are performed, but to significantly reduce computational costs and improve scaling for macro-molecular systems containing hundreds if not thousands of atoms. The field of embedding has grown increasingly broad with many approaches of different intersecting flavors. In this perspective, we lay out the methods into two streams: QM:MM and QM:QM, showcasing the advantages and disadvantages of both. We provide a review of the literature, the underpinning theories including our contributions, and we highlight current applications with select examples spanning both materials and life sciences. We conclude with prospects and future outlook on embedding, and our view on the use of universal test case scenarios for cross-comparisons of the many available (and future) embedding theories.

C–C coupling at a zeolite-supported Rh(i) complex. DFT search for the mechanism

DFT modelling suggests a metallacycle mechanism for the dimerization of ethene over a faujasite-supported Rh(i) complex, rationalizing the experimental selectivity.

Interfacing the Core-Shell or the Drude Polarizable Force Field With Car-Parrinello Molecular Dynamics for QM/MM Simulations

We report a quantum mechanics/polarizable–molecular mechanics (QM/p–MM) potential based molecular dynamics (MD) technique where the core–shell (or the Drude) type polarizable MM force field is interfaced with the plane-wave density functional theory based QM force field which allows Car–Parrinello MD for the QM subsystem. In the QM/p-MM Lagrangian proposed here, the shell (or the Drude) MM variables are treated as extended degrees of freedom along with the Kohn–Sham (KS) orbitals describing the QM wavefunction. The shell and the KS orbital degrees of freedom are then adiabatically decoupled from the nuclear degrees of freedom. In this respect, we also present here the Nosé–Hoover Chain thermostat implementation for the dynamical subsystems. Our approach is then used to investigate the effect of MM polarization on the QM/MM results. Especially, the consequence of MM polarization on reaction free energy barriers, defect formation energy, and structural and dynamical properties are investigated. A low point charge polarizable potential (p–MZHB) for pure siliceous systems is also reported here.

Modelling metal centres, acid sites and reaction mechanisms in microporous catalysts

We discuss the role of QM/MM (embedded cluster) computational techniques in catalytic science, in particular their application to microporous catalysis. We describe the methodologies employed and illustrate their utility by briefly summarising work on metal centres in zeolites. We then report a detailed investigation into the behaviour of methanol at acidic sites in zeolites H-ZSM-5 and H-Y in the context of the methanol-to-hydrocarbons/olefins process. Studying key initial steps of the reaction (the adsorption and subsequent methoxylation), we probe the effect of framework topology and Brønsted acid site location on the energetics of these initial processes. We find that although methoxylation is endothermic with respect to the adsorbed system (by 17-56 kJ mol(-1) depending on the location), there are intriguing correlations between the adsorption/reaction energies and the geometries of the adsorbed species, of particular significance being the coordination of methyl hydrogens. These observations emphasise the importance of adsorbate coordination with the framework in zeolite catalysed conversions, and how this may vary with framework topology and site location, particularly suited to investigation by QM/MM techniques.

CPMD/GULP QM/MM interface for modeling periodic solids: Implementation and its application in the study of Y-zeolite supported Rhn clusters

We report here the development of hybrid quantum mechanics/molecular mechanics (QM/MM) interface between the plane-wave density functional theory based CPMD code and the empirical force-field based GULP code for modeling periodic solids and surfaces. The hybrid QM/MM interface is based on the electrostatic coupling between QM and MM regions. The interface is designed for carrying out full relaxation of all the QM and MM atoms during geometry optimizations and molecular dynamics simulations, including the boundary atoms. Both Born-Oppenheimer and Car-Parrinello molecular dynamics schemes are enabled for the QM part during the QM/MM calculations. This interface has the advantage of parallelization of both the programs such that the QM and MM force evaluations can be carried out in parallel to model large systems. The interface program is first validated for total energy conservation and parallel scaling performance is benchmarked. Oxygen vacancy in α-cristobalite is then studied in detail and the results are compared with a fully QM calculation and experimental data. Subsequently, we use our implementation to investigate the structure of rhodium cluster (Rhn ; n = 2 to 6) formed from Rh(C2 H4 )2 complex adsorbed within a cavity of Y-zeolite in a reducible atmosphere of H2 gas. © 2016 Wiley Periodicals, Inc.

A potential with low point charges for pure siliceous zeolites

A modified CHARMM force-field (ZHB potential) with low point charges for silica was previously proposed by Zimmerman et al. (J. Chem. Theory Comput. 2011, 7, 1695). The ZHB potential is advantageous for quantum mechanics/molecular mechanics simulations as it minimizes the electron spill-out problems. In the same spirit, here we propose a modified ZHB potential (MZHB) by reformulating its bonding potential, while retaining the nonbonding potential as in the ZHB force-field. We show that several structural and dynamic properties of silica, like the IR spectrum, distribution functions, mechanical properties, and negative thermal expansion computed using the MZHB potential agree well with experimental data. Further, transferability of MZHB is also tested for reproducing the crystallographic structures of several polymorphs of silica. © 2015 Wiley Periodicals, Inc.

Predicting adsorption enthalpies on silicalite and HZSM-5: A benchmark study on DFT strategies addressing dispersion interactions

We evaluated the accuracy of periodic density functional calculations for adsorption enthalpies of water, alkanes, and alcohols in silicalite and HZSM-5 zeolites using a gradient-corrected density functional with empirical dispersion corrections (PBE-D) as well as a nonlocal correlation functional (vdW-DF2). Results of both approaches agree in acceptable fashion with experimental adsorption energies of alcohols in silicalite, but the adsorption energies for n-alkanes in both zeolite models are overestimated, by 21-46 kJ mol(-1). For PBE-D calculations, the adsorption of alkanes is exclusively determined by the empirical dispersion term, while the generalized gradient approximation-DFT part is purely repulsive, preventing the molecule to come too close to the zeolite walls. The vdW-DF2 results are comparable to those of PBE-D calculations, but the latter values are slightly closer to the experiment in most cases. Thus, both computational approaches are unable to reproduce available experimental adsorption energies of alkanes in silicalite and HZSM-5 zeolite with chemical accuracy.

Electron and hole stability in GaN and ZnO

We assess the thermodynamic doping limits of GaN and ZnO on the basis of point defect calculations performed using the embedded cluster approach and employing a hybrid non-local density functional for the quantum mechanical region. Within this approach we have calculated a staggered (type-II) valence band alignment between the two materials, with the N 2p states contributing to the lower ionization potential of GaN. With respect to the stability of free electron and hole carriers, redox reactions resulting in charge compensation by ionic defects are found to be largely endothermic (unfavourable) for electrons and exothermic (favourable) for holes, which is consistent with the efficacy of electron conduction in these materials. Approaches for overcoming these fundamental thermodynamic limits are discussed.

Quantum chemical modeling of uranyl adsorption on mineral surfaces

We overview quantum mechanical simulations that model the adsorption of actinide ions at solvated mineral surfaces. Pertinent examples illustrate the status of this emerging field of computational chemistry. In particular, we describe our own studies on uranyl adsorption on kaolinite. Already the few available results, from applications of density functional methods to cluster models or periodic slab models, show that such calculations are a useful complement to experimental investigations. Detailed information at the atomic level from accurate electronic structure calculations on well defined model systems helps to refine current interpretations of the chemical nature of uranyl adsorption species and to discover new features of these interface systems. Results from quantum mechanical simulations also provide a valuable reference for future experimental investigations.

Advances in correlated electronic structure methods for solids, surfaces, and nanostructures

Calculations of the electronic structure of solids began decades ago, but only recently have solid-state quantum techniques become sufficiently reliable that their application is nearly as routine as quantum chemistry is for molecules. We aim to introduce chemists to the pros and cons of first-principles methods that can provide atomic-scale insight into the properties and chemistry of bulk materials, interfaces, and nanostructures. The techniques we review include the ubiquitous density functional theory (DFT), which is often sufficient, especially for metals; extensions such as DFT + U and hybrid DFT, which incorporate exact exchange to rid DFT of its spurious self-interactions (critical for some semiconductors and strongly correlated materials); many-body Green's function (GW and Bethe-Salpeter) methods for excited states; quantum Monte Carlo, in principle an exact theory but for which forces (hence structure optimization and dynamics) are problematic; and embedding theories that locally refine the quantum treatment to improve accuracy.

Thermodynamics of reversible gas adsorption on alkali-metal exchanged zeolites—the interplay of infrared spectroscopy and theoretical calculations

Detailed understanding of weak solid-gas interactions giving rise to reversible gas adsorption on zeolites and related materials is relevant to both, fundamental studies on gas adsorption and potential improvement on a number of (adsorption based) technological processes. Combination of variable-temperature infrared spectroscopy with theoretical calculations constitutes a fruitful approach towards both of these aims. Such an approach is demonstrated here (mainly) by reviewing recent studies on hydrogen and carbon monoxide adsorption (at a low temperature) on alkali-metal exchanged ferrierite. However, the methodology discussed, which involves the interplay of experimental measurements and theoretical calculations at the periodic DFT level, should be equally valid for many other gas-solid systems. Specific aspects considered are the identification of gas adsorption complexes and thermodynamic studies related to standard adsorption enthalpy and entropy.

Density Functional Embedded Cluster Study of Cu4, Ag4and Au4Species Interacting with Oxygen Vacancies on the MgO(001) Surface

Cu(4), Ag(4), and Au(4) species adsorbed on an MgO(001) surface that exhibits neutral (F(s)) and charged (F(s) (+)) oxygen vacancies have been studied using a density functional approach and advanced embedding models. The gas-phase rhombic-planar structure of the coinage metal tetramers is only moderately affected by adsorption. In the most stable surface configuration, the plane of the tetramers is oriented perpendicular to the MgO(001) surface; one metal atom is attached to an oxygen vacancy and another one is bound to a nearby surface oxygen anion. A very similar structural motif was recently found on defect-free MgO(001), where two O(2-) ions serve as adsorption sites. Following the trend of the interactions with the regular MgO(001) surface, Au(4) and Cu(4) bind substantially stronger to F(s) and F(s) (+) sites than Ag(4). This stronger adsorption interaction at oxygen vacancies, in particular at F(s), is partly due to a notable accumulation of electron density on the adsorbates. We also examined the propensity of small supported metal species to aggregate to adsorbed di-, tri- and tetramers. Furthermore, we demonstrated that core-level ionization potentials offer the possibility for detecting experimentally supported metal tetramers and characterizing them structurally with the help of calculated data.

Rings and Strain in Pure Silica Zeolites

The stability of rings in pure silica zeolite frameworks is investigated through a new method. The energetic terms influencing the stability of rings are evaluated through the short-range plus electrostatics corresponding to SiO bonds and OSiO and SiOSi angles. Several atomistic force fields have been evaluated, and a reparametrization of the Vessal-Leslie-Catlow force field has been chosen because of its physical meaning and good structural accuracy. The ring energetics are analyzed and discussed in terms of local strain in the framework and in terms of geometric variables. This methodology allows to differentiate between the strain of rings of a given size in different zeolite structures. In particular, it is found that double four rings (D4Rs) are not necessarily, as previously stated, a strained secondary building unit. The analyses of D4Rs in the topologies AST, BEC, and LTA allow the calculation of its stability in the different structures showing high energy in BEC and LTA and low energy in AST. Implications of these results on nucleation of zeolites are drawn regarding the facility with which D4Rs are inserted in different frameworks.

Comparison of all sites for Ti substitution in zeolite TS-1 by an accurate embedded-cluster method

We studied the preferential location of Ti centers in the framework of the Ti-containing MFI zeolite TS-1 using a hybrid DFT/MM embedding method developed recently. This "covalent elastic polarizable environment" (covEPE) cluster embedding allows a complete and self-consistent treatment of solid covalent systems such as zeolites. For the present study, we used a gradient-corrected density functional approach. The resulting structural features of both Si- and Ti-substituted forms of the zeolite framework fit well with available experimental information. The calculated substitution energy of Ti at the 12 crystallographically different tetrahedral sites of the MFI structure vary within 19 kJ/mol with T12 and T2 as most and least preferred sites, respectively. On the basis of these computational results and the preferential sites for Ti substitution reported from different experimental investigations, we concluded that the Ti distribution in the TS-1 framework is not governed by the thermodynamic stability of the pure material.

A QUICKSTEP-based quantum mechanics/molecular mechanics approach for silica

Quantum mechanics/molecular mechanics (QM/MM) approaches are currently used to describe several properties of silica-based systems, which are local in nature and require a quantum description of only a small number of atoms around the site of interest, e.g., local chemical reactivity or spectroscopic properties of point defects. We present a QM/MM scheme for silica suitable to be implemented in the general QM/MM framework recently developed for large scale molecular dynamics simulations, within the QUICKSTEP approach to the description of the quantum region. Our scheme has been validated by computing the structural and dynamical properties of an oxygen vacancy in alpha-quartz, a prototypical defect in silica. We have found that good convergence in the Si-Si bond length and formation energy is achieved by using a quantum cluster of only eight atoms in size. We check the suitability of the method for molecular dynamics and evaluate the Si-Si bond frequency from the velocity-velocity correlation function.

Gold Atoms and Dimers on Amorphous SiO2: Calculation of Optical Properties and Cavity Ringdown Spectroscopy Measurements

We report on the optical absorption spectra of gold atoms and dimers deposited on amorphous silica in size-selected fashion. Experimental spectra were obtained by cavity ringdown spectroscopy. Issues on soft-landing, fragmentation, and thermal diffusion are discussed on the basis of the experimental results. In parallel, cluster and periodic supercell density functional theory (DFT) calculations were performed to model atoms and dimers trapped on various defect sites of amorphous silica. Optically allowed electronic transitions were calculated, and comparisons with the experimental spectra show that silicon dangling bonds [[triple bond]Si(.-)], nonbridging oxygen [[triple bond]Si-O(.-)], and the silanolate group [[triple bond]Si-O(-)] act as trapping centers for the gold particles. The results are not only important for understanding the chemical bonding of atoms and clusters on oxide surfaces, but they will also be of fundamental interest for photochemical studies of size-selected clusters on surfaces.

Single d-Metal Atoms on Fs and Fs+ Defects of MgO(001): A Theoretical Study across the Periodic Table

Single d-metal atoms on oxygen defects F(s) and F(s+) of the MgO(001) surface were studied theoretically. We employed an accurate density functional method combined with cluster models, embedded in an elastic polarizable environment, and we applied two gradient-corrected exchange-correlation functionals. In this way, we quantified how 17 metal atoms from groups 6-11 of the periodic table (Cu, Ag, Au; Ni, Pd, Pt; Co, Rh, Ir; Fe, Ru, Os; Mn, Re; and Cr, Mo, W) interact with terrace sites of MgO. We found bonding with F(s) and F(s+) defects to be in general stronger than that with O2- sites, except for Mn-, Re-, and Fe/F(s) complexes. In M/F(s) systems, electron density is accumulated on the metal center in a notable fashion. The binding energy on both kinds of O defects increases from 3d- to 4d- to 5d-atoms of a given group, at variance with the binding energy trend established earlier for the M/O2- complexes, 4d < 3d < 5d. Regarding the evolution of the binding energy along a period, group 7 atoms are slightly destabilized compared to their group 6 congeners in both the F(s) and F(s+) complexes; for later transition elements, the binding energy increases gradually up to group 10 and finally decreases again in group 11, most strongly on the F(s) site. This trend is governed by the negative charge on the adsorbed atoms. We discuss implications for an experimental detection of metal atoms on oxide supports based on computed core-level energies.