Static surface temperature effects on the dissociation of H2 and D2 on Cu(111)

On the quantum dynamical treatment of surface vibrational modes for reactive scattering of H2 from Cu(111) at 925 K

We construct the effective Hartree potential for H2 on Cu(111) as introduced in our earlier work [Dutta et al., J. Chem. Phys. 154, 104103 (2021), and Dutta et al., J. Chem. Phys. 157, 194112 (2022)] starting from the same gas-metal interaction potential obtained for 0 K. Unlike in that work, we now explicitly account for surface expansion at 925 K and investigate different models to describe the surface vibrational modes: (i) a cluster model yielding harmonic normal modes at 0 K and (ii) slab models resulting in phonons at 0 and 925 K according to the quasi-harmonic approximation-all consistently calculated at the density functional theory level with the same exchange-correlation potential. While performing dynamical calculations for the H2(v = 0, j = 0)-Cu(111) system employing Hartree potential constructed with 925 K phonons and surface temperature, (i) the calculated chemisorption probabilities are the highest compared to the other approaches over the energy domain and (ii) the threshold for the reaction probability is the lowest, in close agreement with the experiment. Although the survival probabilities (v' = 0) depict the expected trend (lower in magnitude), the excitation probabilities (v' = 1) display a higher magnitude since the 925 K phonons and surface temperature are more effective for the excitation process compared to the phonons/normal modes obtained from the other approaches investigated to describe the surface.

Dissociative chemisorption of O2 on Al(111): dynamics on a potential energy surface computed with a non-self-consistent screened hybrid density functional approach

Density functional theory (DFT) at the generalized gradient approximation (GGA) level is often considered the best compromise between feasibility and accuracy for reactions of molecules on metal surfaces. Recent work, however, strongly suggests that density functionals (DFs) based on GGA exchange are not able to describe molecule-metal surface reactions for which the work function of the metal surface minus the electron affinity of the molecule is less than 7 eV. Systems for which this is true exhibit an increased charge transfer from the metal to the molecule at the transition state, increasing the delocalisation of the electron density. This enlarged delocalisation can cause GGA-DFT to underestimate energy values relative to the gas-phase and thus underestimate the barrier height, similar to what has been observed for several gas-phase reactions. An example of such a molecule-metal surface system is O2 + Al(111). Following a similar strategy as for gas-phase reactions, previous work showed results of increased accuracy when using a screened hybrid DF for O2 + Al(111). However, even screened hybrid DFs are computationally expensive to use for metal surfaces. To resolve this, we test a non-self-consistent field (NSCF) screened hybrid DF approach. This approach computes screened hybrid DFT energies based on self-consistent-field (SCF) GGA electronic densities. Here, we explore the accuracy of the NSCF screened hybrid DF approach by implementing the NSCF HSE03-1/3x@RPBE DF for O2 + Al(111). We compute and analyse molecular beam sticking probabilities as well as a set of sticking probabilities for rotationally aligned O2. Our results show that the NSCF approach results in reaction probability curves that reproduce SCF results with near-chemical accuracy, suggesting that the NSCF approach can be used advantageously for exploratory purposes. An analysis of the potential energy surface and the barriers gives insight into the cause of the disagreement between the SCF and NSCF reaction probabilities and into the changes needed in theoretical modelling to further improve the description of the O2 + Al(111) system. Finally, the hole model yields fair agreement with dynamics results for the reaction probability curve, but results in an increased slope of the reaction probability curve compared to the molecular dynamics, with a shift to lower or higher energies depending on whether the vibrational energy of the molecule is included in the initial energy of the molecule or not.

Simulating Highly Activated Sticking of H2 on Al(110): Quantum versus Quasi-Classical Dynamics

We evaluate the importance of quantum effects on the sticking of H₂ on Al(110) for conditions that are close to those of molecular beam experiments that have been done on this system. Calculations with the quasi-classical trajectory (QCT) method and with quantum dynamics (QD) are performed using a model in which only motion in the six molecular degrees of freedom is allowed. The potential energy surface used has a minimum barrier height close to the value recently obtained with the quantum Monte Carlo method. Monte Carlo averaging over the initial rovibrational states allowed the QD calculations to be done with an order of magnitude smaller computational expense. The sticking probability curve computed with QD is shifted to lower energies relative to the QCT curve by 0.21 to 0.05 kcal/mol, with the highest shift obtained for the lowest incidence energy. Quantum effects are therefore expected to play a small role in calculations that would evaluate the accuracy of electronic structure methods for determining the minimum barrier height to dissociative chemisorption for H₂ + Al(110) on the basis of the standard procedure for comparing results of theory with molecular beam experiments.

The quantum dynamics of H2 on Cu(111) at a surface temperature of 925 K: Comparing state-of-the-art theory to state-of-the-art experiments 2

State-of-the-art 6D quantum dynamics simulations for the dissociative chemisorption of H₂ on a thermally distorted Cu(111) surface, using the static corrugation model, were analyzed to produce several (experimentally available) observables. The expected error, especially important for lower reaction probabilities, was quantified using wavepackets on several different grids as well as two different analysis approaches to obtain more accurate results in the region where a slow reaction channel was experimentally shown to be dominant. The lowest reaction barrier sites for different thermally distorted surface slabs are shown to not just be energetically, but also geometrically, different between surface configurations, which can be used to explain several dynamical effects found when including surface temperature effects. Direct comparison of simulated time-of-flight spectra to those obtained from state-of-the-art desorption experiments showed much improved agreement compared to the perfect lattice BOSS approach. Agreement with experimental rotational and vibrational efficacies also somewhat improved when thermally excited surfaces were included in the theoretical model. Finally, we present clear quantum effects in the rotational quadrupole alignment parameters found for the lower rotationally excited states, which underlines the importance of careful quantum dynamical analyses of this system.

Accurate Description of the Quantum Dynamical Surface Temperature Effects on the Dissociative Chemisorption of H2 from Cu(111)

Accurately describing surface temperature effects for the dissociative scattering of H2 on a metal surface on a quantum dynamical level is currently one of the open challenges for theoretical surface scientists. We present the first quantum dynamical (QD) simulations of hydrogen dissociating on a Cu(111) surface which accurately describe all relevant surface temperature effects, using the static corrugation model (SCM). The reaction probabilities we obtain show very good agreement with those found using quasi-classical dynamics (QCD), both for individual surface slabs and for an averaged, thus Monte-Carlo sampled, set of thermally distorted surface configurations. Rovibrationally elastic scattering probabilities show a much clearer difference between the QCD and QD results, which appears to be traceable back towards thermally distorted surface configurations with very low dissociation probabilities and underlines the importance of investigating more observables than just dissociation. By reducing the number of distorted surface atoms included in the dynamical model, we also show that only including one, or even three, surface atoms is generally not enough to accurately describe the effects of surface temperature on dissociation and elastic scattering. These results are a major step forward in accurately describing hydrogen scattering from a thermally excited Cu(111) surface, and open up a pathway to better describe reaction and scattering from other relevant crystal facets, such as stepped surfaces, at moderately elevated surface temperatures where quantum effects are expected to play a more important role in the dissociation of H2 on Cu.

Stopping molecular rotation using coherent ultra-low-energy magnetic manipulations

Rotational motion lies at the heart of intermolecular, molecule-surface chemistry and cold molecule science, motivating the development of methods to excite and de-excite rotations. Existing schemes involve perturbing the molecules with photons or electrons which supply or remove energy comparable to the rotational level spacing. Here, we study the possibility of de-exciting the molecular rotation of a D₂ molecule, from J = 2 to the non-rotating J = 0 state, without using an energy-matched perturbation. We show that passing the beam through a 1 m long magnetic field, which splits the rotational projection states by only 10⁻¹²eV, can change the probability that a molecule-surface collision will stop a molecule from rotating and lose rotational energy which is 9 orders larger than that of the magnetic manipulation. Calculations confirm that different rotational orientations have different de-excitation probabilities but underestimate rotational flips (∆m_J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ne$$\end{document}≠0), highlighting the importance of the results as a sensitive benchmark for further developing theoretical models of molecule-surface interactions.

Manipulating the rotational motions of molecules may provide a tool for controlling chemical processes. Here the authors demonstrate that the rotation of a D₂ molecule can be stopped, upon collision with a metal surface, by a magnetic field that affects the rotational levels to a much smaller extent than the energy difference upon de-excitation.

Performance of Made Simple Meta-GGA Functionals with rVV10 Nonlocal Correlation for H2 + Cu(111), D2 + Ag(111), H2 + Au(111), and D2 + Pt(111)

Accurately modeling heterogeneous catalysis requires accurate descriptions of rate-controlling elementary reactions of molecules on metal surfaces, but standard density functionals (DFs) are not accurate enough for this. The problem can be solved with the specific reaction parameter approach to density functional theory (SRP-DFT), but the transferability of SRP DFs among chemically related systems is limited. We combine the MS-PBEl, MS-B86bl, and MS-RPBEl semilocal made simple (MS) meta-generalized gradient approximation (GGA) (mGGA) DFs with rVV10 nonlocal correlation, and we evaluate their performance for the hydrogen (H2) + Cu(111), deuterium (D2) + Ag(111), H2 + Au(111), and D2 + Pt(111) gas-surface systems. The three MS mGGA DFs that have been combined with rVV10 nonlocal correlation were not fitted to reproduce particular experiments, nor has the b parameter present in rVV10 been reoptimized. Of the three DFs obtained the MS-PBEl-rVV10 DF yields an excellent description of van der Waals well geometries. The three original MS mGGA DFs gave a highly accurate description of the metals, which was comparable in quality to that obtained with the PBEsol DF. Here, we find that combining the three original MS mGGA DFs with rVV10 nonlocal correlation comes at the cost of a slightly less accurate description of the metal. However, the description of the metal obtained in this way is still better than the descriptions obtained with SRP DFs specifically optimized for individual systems. Using the Born-Oppenheimer static surface (BOSS) model, simulations of molecular beam dissociative chemisorption experiments yield chemical accuracy for the D2 + Ag(111) and D2 + Pt(111) systems. A comparison between calculated and measured E 1/2(ν, J) parameters describing associative desorption suggests chemical accuracy for the associative desorption of H2 from Au(111) as well. Our results suggest that ascending Jacob's ladder to the mGGA rung yields increasingly more accurate results for gas-surface reactions of H2 (D2) interacting with late transition metals.

Searching Configurations in Uncertainty Space: Active Learning of High-Dimensional Neural Network Reactive Potentials

Neural network (NN) potential energy surfaces (PESs) have been widely used in atomistic simulations with ab initio accuracy. While constructing NN PESs, their training data points are often sampled by molecular dynamics trajectories. This strategy can be however inefficient for reactive systems involving rare events. Here, we develop an uncertainty-driven active learning strategy to automatically and efficiently generate high-dimensional NN-based reactive potentials, taking a gas-surface reaction as an example. The difference between two independent NN models is used as a simple and differentiable uncertainty metric, allowing us to quickly search in the uncertainty space and place new samples at which the PES is less reliable. By interfacing this algorithm with the first-principles simulation package, we demonstrate that a globally accurate NN potential of the H₂ + Ag(111) system can be constructed with merely ∼150 data points. This PES can be further refined to describe H₂ dissociation on Ag(100) by adding ∼130 more configurations on this facet. The entire process is completely automatic and self-terminated once the relative error criterion is fulfilled. Impressively, data points sampled by this uncertainty-driven strategy are substantially fewer than by the traditional trajectory-based sampling. The final NN PES not only converges well the quantum dissociation probability of the molecule but also well-reproduces the phonon properties of the substrate and is capable of describing surface temperature effects. These results show the potential of this active learning approach in developing high-dimensional NN reactive potentials in gas and condensed phases.

Computational approaches to dissociative chemisorption on metals: towards chemical accuracy

We review the state-of-the-art in the theory of dissociative chemisorption (DC) of small gas phase molecules on metal surfaces, which is important to modeling heterogeneous catalysis for practical reasons, and for achieving an understanding of the wealth of experimental information that exists for this topic, for fundamental reasons. We first give a quick overview of the experimental state of the field. Turning to the theory, we address the challenge that barrier heights (E_b, which are not observables) for DC on metals cannot yet be calculated with chemical accuracy, although embedded correlated wave function theory and diffusion Monte-Carlo are moving in this direction. For benchmarking, at present chemically accurate E_b can only be derived from dynamics calculations based on a semi-empirically derived density functional (DF), by computing a sticking curve and demonstrating that it is shifted from the curve measured in a supersonic beam experiment by no more than 1 kcal mol^-1. The approach capable of delivering this accuracy is called the specific reaction parameter (SRP) approach to density functional theory (DFT). SRP-DFT relies on DFT and on dynamics calculations, which are most efficiently performed if a potential energy surface (PES) is available. We therefore present a brief review of the DFs that now exist, also considering their performance on databases for E_b for gas phase reactions and DC on metals, and for adsorption to metals. We also consider expressions for SRP-DFs and briefly discuss other electronic structure methods that have addressed the interaction of molecules with metal surfaces. An overview is presented of dynamical models, which make a distinction as to whether or not, and which dissipative channels are modeled, the dissipative channels being surface phonons and electronically non-adiabatic channels such as electron-hole pair excitation. We also discuss the dynamical methods that have been used, such as the quasi-classical trajectory method and quantum dynamical methods like the time-dependent wave packet method and the reaction path Hamiltonian method. Limits on the accuracy of these methods are discussed for DC of diatomic and polyatomic molecules on metal surfaces, paying particular attention to reduced dimensionality approximations that still have to be invoked in wave packet calculations on polyatomic molecules like CH₄. We also address the accuracy of fitting methods, such as recent machine learning methods (like neural network methods) and the corrugation reducing procedure. In discussing the calculation of observables we emphasize the importance of modeling the properties of the supersonic beams in simulating the sticking probability curves measured in the associated experiments. We show that chemically accurate barrier heights have now been extracted for DC in 11 molecule-metal surface systems, some of which form the most accurate core of the only existing database of E_b for DC reactions on metal surfaces (SBH10). The SRP-DFs (or candidate SRP-DFs) that have been derived show transferability in many cases, i.e., they have been shown also to yield chemically accurate E_b for chemically related systems. This can in principle be exploited in simulating rates of catalyzed reactions on nano-particles containing facets and edges, as SRP-DFs may be transferable among systems in which a molecule dissociates on low index and stepped surfaces of the same metal. In many instances SRP-DFs have allowed important conclusions regarding the mechanisms underlying observed experimental trends. An important recent observation is that SRP-DFT based on semi-local exchange DFs has so far only been successful for systems for which the difference of the metal work function and the molecule's electron affinity exceeds 7 eV. A main challenge to SRP-DFT is to extend its applicability to the other systems, which involve a range of important DC reactions of e.g. O₂, H₂O, NH₃, CO₂, and CH₃OH. Recent calculations employing a PES based on a screened hybrid exchange functional suggest that the road to success may be based on using exchange functionals of this category.

Effect of surface temperature on quantum dynamics of H2 on Cu(111) using a chemically accurate potential energy surface

The effect of surface atom vibrations on H₂ scattering from a Cu(111) surface at different temperatures is being investigated for hydrogen molecules in their rovibrational ground state (v = 0, j = 0). We assume weakly correlated interactions between molecular degrees of freedom and surface modes through a Hartree product type wavefunction. While constructing the six-dimensional effective Hamiltonian, we employ (a) a chemically accurate potential energy surface according to the static corrugation model [M. Wijzenbroek and M. F. Somers, J. Chem. Phys. 137, 054703 (2012)]; (b) normal mode frequencies and displacement vectors calculated with different surface atom interaction potentials within a cluster approximation; and (c) initial state distributions for the vibrational modes according to Bose-Einstein probability factors. We carry out 6D quantum dynamics with the so-constructed effective Hamiltonian and analyze sticking and state-to-state scattering probabilities. The surface atom vibrations affect the chemisorption dynamics. The results show physically meaningful trends for both reaction and scattering probabilities compared to experimental and other theoretical results.

Beyond the static corrugation model: Dynamic surfaces with the embedded atom method

The D₂ on Cu(111) system has for many years been one of the major benchmark systems for surface scientists. Generating surface configurations using the embedded atom method (EAM), we investigate the quality of the chemically accurate static corrugation model (SCM) for including surface temperature effects, with a focus on the random displacement approach to its distorted surface generation. With this EAM potential, we also treat the Cu(111) surface of our system fully dynamically and shine a further light on not only the quality of the SCM sudden approach but also the limited effect of energy exchange with the surface. Reaction and (in)elastic scattering probability curves, as well as simulated time-of-flight spectra, show good agreement with both earlier works and experimental results, with surface reactions showing a preference for surface atoms displaced away from the incoming molecule. The good agreement with the non-static surface model also further establishes the limited effect of energy exchange on not only the reaction but also on the elastic and inelastic scattering probabilities, even though some molecular translational energy is deposited into the surface.

Designing new SRP density functionals including non-local vdW-DF2 correlation for H2 + Cu(111) and their transferability to H2 + Ag(111), Au(111) and Pt(111)

Specific reaction parameter density functionals (SRP-DFs) that can describe dissociative chemisorption molecular beam experiments of hydrogen (H₂) on cold transition metal surfaces with chemical accuracy have so far been shown to be only transferable among different facets of the same metal, but not among different metals. We design new SRP-DFs that include non-local vdW-DF2 correlation for the H₂ + Cu(111) system, and evaluate their transferability to the highly activated H₂ + Ag(111) and H₂ + Au(111) systems and the non-activated H₂ + Pt(111) system. We design our functionals for the H₂ + Cu(111) system since it is the best studied system both theoretically and experimentally. Here we demonstrate that a SRP-DF fitted to reproduce molecular beam sticking experiments for H₂ + Cu(111) with chemical accuracy can also describe such experiments for H₂ + Pt(111) with chemical accuracy, and vice versa. Chemically accurate functionals have been obtained that perform very well with respect to reported van der Waals well geometries, and which improve the description of the metal over current generalized gradient approximation (GGA) based SRP-DFs. From a systematic comparison of our new SRP-DFs that include non-local correlation to previously developed SRP-DFs, for both activated and non-activated systems, we identify non-local correlation as a key ingredient in the construction of transferable SRP-DFs for H₂ interacting with transition metals. Our results are in excellent agreement with experiment when accurately measured observables are available. It is however clear from our analysis that, except for the H₂ + Cu(111) system, there is a need for more, more varied, and more accurately described experiments in order to further improve the design of SRP-DFs. Additionally, we confirm that, when including non-local correlation, the sticking of H₂ on Cu(111) is still well described quasi-classically.

First principles rates for surface chemistry employing exact transition state theory: application to recombinative desorption of hydrogen from Cu(111)

We present first principles calculations of the reactive flux for thermal recombinative desorption of hydrogen from Cu(111).

Unified and transferable description of dynamics of H2 dissociative adsorption on multiple copper surfaces via machine learning

Schematic of the developed neural network potential energy surface enabling a unified and transferable description of dynamics of H₂ dissociative adsorption on multiple copper surfaces.

Quantum Dynamics of Dissociative Chemisorption of H2 on the Stepped Cu(211) Surface

Reactions on stepped surfaces are relevant to heterogeneous catalysis, in which a reaction often takes place at the edges of nanoparticles where the edges resemble steps on single-crystal stepped surfaces. Previous results on H₂ + Cu(211) showed that, in this system, steps do not enhance the reactivity and raised the question of whether this effect could be, in any way, related to the neglect of quantum dynamical effects in the theory. To investigate this, we present full quantum dynamical molecular beam simulations of sticking of H₂ on Cu(211), in which all important rovibrational states populated in a molecular beam experiment are taken into account. We find that the reaction of H₂ with Cu(211) is very well described with quasi-classical dynamics when simulating molecular beam sticking experiments, in which averaging takes place over a large number of rovibrational states and over translational energy distributions. Our results show that the stepped Cu(211) surface is distinct from its component Cu(111) terraces and Cu(100) steps and cannot be described as a combination of its component parts with respect to the reaction dynamics when considering the orientational dependence. Specifically, we present evidence that, at translational energies close to the reaction threshold, vibrationally excited molecules show a negative rotational quadrupole alignment parameter on Cu(211), which is not found on Cu(111) and Cu(100). The effect arises because these molecules react with a site-specific reaction mechanism at the step, that is, inelastic rotational enhancement, which is only effective for molecules with a small absolute value of the magnetic rotation quantum number. From a comparison to recent associative desorption experiments as well as Born-Oppenheimer molecular dynamics calculations, it follows that the effects of surface atom motion and electron-hole pair excitation on the reactivity fall within chemical accuracy, that is, modeling these effect shifts extracted reaction probability curves by less than 1 kcal/mol translational energy. We found no evidence in our fully state-resolved calculations for the "slow" reaction channel that was recently reported for associative desorption of H₂ from Cu(111) and Cu(211), but our results for the fast channel are in good agreement with the experiments on H₂ + Cu(211).

Quantum and classical dynamics of reactive scattering of H2 from metal surfaces

State-of-the-art theoretical models allow nowadays an accurate description of H₂/metal surface systems and phenomena relative to heterogeneous catalysis. Here we review the most relevant ones investigated during the last 10 years.

Toward a Database of Chemically Accurate Barrier Heights for Reactions of Molecules with Metal Surfaces

Being able to calculate reaction barrier heights to within chemical accuracy (errors < 1 kcal/mol) is crucial to the accurate modeling of chemical reactions. Although accurate databases exist that can help theorists with benchmarking new electronic structure theories on gas-phase chemical reactions, no such databases exist for reactions of molecules with metal surfaces. Nonetheless, most chemicals are made in heterogeneously catalyzed processes, of which many take place over metal particles. Presently, barrier heights for molecule-metal surface reactions have been determined with chemical accuracy for only two systems, that is, H2 + Cu(111) and H2 + Cu(100). This has been done with semiempirically determined density functionals, which were fitted through comparisons of dynamics results with molecular beam sticking probabilities. The prospects of extending the database with chemically accurate data for other molecule-metal reactions, either with the use of semiempirical density functional theory or with first-principles theory, are discussed.