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Condensed Matter, Interfaces, and MaterialsSeptember 25, 2023

Estimating Free Energy Barriers for Heterogeneous Catalytic Reactions with Machine Learning Potentials and Umbrella Integration
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Sina Stocker
Sina Stocker
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
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Hyunwook Jung
Hyunwook Jung
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
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Gábor Csányi
Gábor Csányi
Engineering Laboratory, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
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https://orcid.org/0000-0002-8180-2034
C. Franklin Goldsmith
C. Franklin Goldsmith
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
School of Engineering, Brown University, Providence, Rhode Island 02912, United States
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https://orcid.org/0000-0002-2212-0172
Karsten Reuter
Karsten Reuter
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
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https://orcid.org/0000-0001-8473-8659
Johannes T. Margraf*
Johannes T. Margraf
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
*Email: [email protected]
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https://orcid.org/0000-0002-0862-5289

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Journal of Chemical Theory and Computation

Cite this: J. Chem. Theory Comput. 2023, 19, 19, 6796–6804

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https://doi.org/10.1021/acs.jctc.3c00541

Published September 25, 2023

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Abstract

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Predicting the rate constants of elementary reaction steps is key for the computational modeling of catalytic processes. Within transition state theory (TST), this requires an accurate estimation of the corresponding free energy barriers. While sophisticated methods for estimating free energy differences exist, these typically require extensive (biased) molecular dynamics simulations that are computationally prohibitive with the first-principles electronic structure methods that are typically used in catalysis research. In this contribution, we show that machine-learning (ML) interatomic potentials can be trained in an automated iterative workflow to perform such free energy calculations at a much reduced computational cost as compared to a direct density functional theory (DFT) based evaluation. For the decomposition of CHO on Rh(111), we find that thermal effects are substantial and lead to a decrease in the free energy barrier, which can be vanishingly small, depending on the DFT functional used. This is in stark contrast to previously reported estimates based on a harmonic TST approximation, which predicted an increase in the barrier at elevated temperatures. Since CHO is the reactant of the putative rate limiting reaction step in syngas conversion on Rh(111) and essential for the selectivity toward oxygenates containing multiple carbon atoms (C₂₊ oxygenates), our results call into question the reported mechanism established by microkinetic models.

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Subjects

I. Introduction

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It is well-known that the rate, selectivity, and yield of a chemical process can be controlled by the use of catalysts. (1) In simple terms, this works via a modulation of the relative energetics of the reaction intermediates and transition states, which partake in the overall process leading from the reactants to the products. Here, the energetic barriers for elementary reaction steps are of particular interest, since (according to transition state theory, TST) the rate constant of each step is proportional to the exponential of the free energy barrier. (2,3) The accurate computational prediction of free energy barriers for elementary reactions is thus essential for understanding the detailed mechanisms of catalytic processes and for designing new catalysts. (4−6)

Predicting free energy barriers accurately is notoriously difficult, however. This is because the simple picture of the barrier as an energy difference between a single minimum configuration and a transition state does not apply at elevated temperatures. Instead, a rigorous free energy calculation requires extensive sampling of the configuration space along a suitable reaction coordinate, e.g., via Transition Path Sampling, (7) Metadynamics, (8) or Umbrella Sampling. (9,10)

These methods are commonly applied in biomolecular simulations, e.g., to study the binding affinities of drug candidates to certain enzymes. Here, computationally efficient empirical force fields are available so that extensive sampling is not an insurmountable issue. Unfortunately, this is not the case in heterogeneous catalysis, where the surface of a solid catalyst must be accurately modeled. This requires the use of computationally expensive first-principles methods like density functional theory (DFT) with the consequence that rigorous free energy calculations are rarely performed in this context.

To avoid this computational bottleneck, DFT based catalysis research instead typically relies on a more approximate treatment of free energy barriers. In the simplest case, temperature effects on the barrier are simply neglected so that the Helmholtz free energy surface (FES) is approximated by the 0 K potential energy surface (PES). This simplifies the problem to finding the transition state between the initial and final minimum configuration of the reaction on the PES, which can be achieved, e.g., with the Climbing Image Nudged Elastic Band (CI-NEB) approach and related methods. (11)

Additionally, free energy corrections to the PES barrier can be computed. These rely on calculating the vibrational modes of the reactants in the initial, final, and transition state configurations and are thus somewhat more involved. (12) Most commonly, the harmonic approximation (HA) is used in this context, meaning that a second-order Taylor expansion of the PES around the configurations of interest is performed, from which the finite-temperature free energy corrections can be calculated analytically. While applying the HA is relatively straightforward in principle, it is known to be inadequate at higher temperatures and for low-frequency modes, such as hindered translations or rotations. (12,13) These factors lead to some ambiguity on how to treat (or whether to neglect) slow modes and overall add to the uncertainty of predicted free energy barriers. (14)

Fortunately, the success of machine-learning (ML) interatomic potentials in chemistry and materials science in recent years has opened a new perspective for free energy calculations in heterogeneous catalysis. Machine-learning potentials can provide fast and accurate surrogate models of the DFT PES, (15−17) so that the kind of extensive sampling that is required for rigorous free energy calculations becomes feasible.

In this contribution, we demonstrate this for the initial hydrogenation step of CO on Rh(111), focusing on the kinetic stability of the CHO reaction intermediate (Figure 1). This system is of particular interest since Rh-based catalysts are unique in their ability to convert syngas (CO and H₂) to higher oxygenates like ethanol and acetaldehyde with appreciable selectivity. (18−21) We focus on CHO because it is the reactant in the putative rate limiting step for ethanol and acetaldehyde synthesis, according to some microkinetic models. (19) Furthermore, the relative formation rates of CHO and COH ultimately determine the selectivity for the C₂₊ oxygenates.

Figure 1. Top and side views of local minimum configurations for CHO and CO+H on the Rh(111) surface. The geometric measures d and θ, which define the collective variable used below, are indicated for CHO.

We present a workflow that combines the iterative training of Gaussian Approximation Potentials (GAP) with the Umbrella Integration (UI) approach, yielding a fast and accurate method for free energy calculations in surface catalysis. Subsequently, the results are compared to previously reported harmonic estimates of the free energy barrier, and the influence of different density functional approximations is discussed.

II. Methods

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II.1. Umbrella Integration

The UI approach is a hybrid between two popular free energy methods, namely, Umbrella Sampling (US) and Thermodynamic Integration. (10,22,23) As in the US, (9) a set of harmonic biasing potentials that span the scope of a collective variable (CV) ξ are defined:

w_{i} (R) = \frac{1}{2} k {(ξ^{'} (R) - ξ_{i})}^{2}

(1)

Here, w_i is the ith biasing potential with a spring constant k, which restrains the sampling around the window center ξ_i. ξ′(R) is a function that maps the coordinates of system R to the collective variable ξ, which can be an interatomic distance, angle, coordination number, or more complex function of the atomic coordinates. Note that ξ can, in principle, be higher dimensional, although we only consider the 1D case herein. The biasing potential restrains simulations to a region in phase space close to ξ_i, which is referred to as the ith window.

With these potentials, a series of biased molecular dynamics (MD) simulations are performed, where the biases ensure that energetically unfavorable parts of the PES are sampled sufficiently. In conventional US, the FES along ξ is then typically obtained from the corresponding MD ensembles using the weighted histogram analysis method (WHAM). (24−26) In UI, the FES is instead obtained by estimating the gradients of the free energy F with respect to ξ. (10,22,23) To this end, one merely needs the mean value of ξ (ξ̅_i) for each biased ensemble:

{\frac{\partial F}{\partial ξ} |}_{{\bar{ξ}}_{i}} \approx - k ({\bar{ξ}}_{i} - ξ_{i})

(2)

This approximation holds under the assumption that the distribution in ξ is unimodal and symmetric, which is generally the case if k is sufficiently large, so the main remaining source of uncertainty is the statistical error on ξ̅_i. The FES can then be obtained by integrating

\frac{\partial F}{\partial ξ}

. Stecher et al. showed that this can be done in an uncertainty aware fashion using Gaussian Process Regression (GPR) and we follow this approach herein. (27) All UI simulations below are performed with the atomic simulation environment (ASE) and a Langevin thermostat. (28)

II.2. Gaussian Approximation Potentials

GAP models are a well established class of ML interatomic potentials that are also based on GPR. (29) Since the GAP approach was extensively reviewed recently, we only briefly summarize the details of the potential used herein. (15) All hyperparameters as well as the potential itself are provided as Supporting Information in this article.

We express the total formation energy of the surface–adsorbate system in terms of a two-body and a many-body contribution, with the latter being described by the Superposition of Atomic Densities (SOAP) representation. (30) The potential is trained in an iterative fashion by running UI simulations and NEB calculations to generate new configurations with the GAP model and augmenting the training set accordingly (see below for details). This provides a data-efficient and automated workflow for generating a training set that covers the relevant parts of the phase space. To avoid redundancies in the configurations that are added to the training set, these are selected according to the kernel distance between new configurations and the current training set (so-called farthest point sampling, FPS). (31)

II.3. Density Functional Theory

In computational catalysis at metal surfaces, semilocal functionals based on the Generalized Gradient Approximation (GGA) are most commonly used. This is due to their relatively high computational efficiency compared to those of hybrid DFT and explicit many-body methods. Additionally, these functionals are fairly accurate in describing the energetics of molecular adsorbates on transition metal surfaces, particularly when combined with appropriate dispersion corrections. (32,33)

To explore the influence of different computational setups, two separate GAP models were trained with different reference data. On one hand, the revPBE (34) functional and vdW^surf dispersion correction (35) were used, as implemented in the full-potential numerical atomic orbital code FHI-aims. (36) Here, light integration settings and a tier-1 basis set were used, as is usually done for ab initio MD simulations. On the other hand, the BEEF-vdW (37) functional was used, as implemented in the plane-wave code QuantumEspresso, using ultrasoft pseudopotentials and a kinetic energy cutoff of 500 eV for orbitals and 5000 eV for the density. (38) Note that the revPBE setup was used in the iterative training scheme, while the BEEF-vdW potential was subsequently trained in the same configurations. In the following, all figures show revPBE-based results unless otherwise noted. All simulations were performed in a 3 × 3 × 4 Rh(111) surface slab (where the lower two slab layers were constrained during all simulations). A 4 × 4 × 1 k-grid was used to sample the Brillouin zone.

For harmonic free energy corrections, vibrational frequencies were obtained at the DFT level via finite-difference Hessians of the adsorbates in the initial and transition state geometries. Since vibrational frequencies are more sensitive to the convergence of geometry optimizations than total energies, ground state geometries were tightly converged to maximum force norms of 0.01 eV/Å. The CI-NEB transition state geometries were refined using the iterative Hessian diagonalization algorithm implemented in the Sella package. (39)

As is common practice in computational catalysis, the vibrational frequencies were computed by generating the finite differences of the Hessian for the adsorbate atoms only. This was done to ensure consistency with the literature, in particular ref (19). For the UI calculations, only the lower two layers of the metal slab were constrained. This introduces a slight inconsistency between the UI and the HA calculations. To evaluate the effect of this inconsistency, vibrational calculations were repeated under inclusion of the top two metal layers for the BEEF-vdW functional. From this, we find that including these atoms in the Hessian only has a marginal effect on the harmonic free energy barrier, which increases by 0.01 eV.

III. Results

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III.1. Minimum Energy Path and Collective Variable

The focus of this work will be on the first hydrogenation step of CO on Rh(111) to form CHO (Figure 1). Here, the small barrier for the reverse reaction is of particular interest, since it determines the stability of the CHO intermediate on the surface. The geometric changes between the reactant and product can be described in terms of the angle between the CO bond and the surface normal (θ, in Radians) and the C–H distance (d, in Å). While these parameters would in principle form adequate CVs for this reaction, the computational effort for free energy calculations rises substantially with each dimension that is considered. We therefore first define an effective one-dimensional CV.

To this end, we take advantage of the fact that the CI-NEB method allows calculating minimum energy paths for chemical reactions on the PES. This yields a series of configurations (termed images) that connect reactant and product and include the transition state. In Figure 2, the NEB images of CHO formation are plotted with respect to d and θ. This reveals that the reaction first proceeds by a gradual decrease of d and increase of θ, until the transition state is reached. Subsequently, θ further increases, until the product geometry is obtained.

Figure 2. Nudged elastic band images plotted in terms of the angle between the CO and the surface normal (θ) and the C–H distance (d). The color bar shows the relative energies of the images in eV, with the yellow points being close to the transition state. The collective variable used herein is a linear combination of θ and d, indicated by the gray line.

Based on this path, we define the CV ξ used in the following as a linear combination of d and θ, connecting the reactant minimum configuration and the transition state. Here, the units of the parameters are chosen to render the overall CV unit-less. This CV is plotted as a gray line in Figure 2, with the projection of each NEB image indicated by the dotted lines and diamonds. Note that any parallel line in this plot effectively corresponds to the same CV. This shows that all NEB images are well separated on this scale, indicating that ξ is a suitable reaction coordinate. For the following free energy calculations, 50 evenly spaced biasing potentials are defined in the range ξ = −0.2–0.85, with a spring constant k = 50 eV.

III.2. Potential Training and Validation

The GAP potential used to run the UI simulations is trained in an iterative fashion. Specifically, we define an initial training set of 50 configurations, consisting of the images from the DFT based NEB calculation shown in Figure 2, dimer curves of the light element pairs (CO, CC, OO, HH, OH, and CH), as well as optimized and rattled configurations of the pristine surface. This set is used to train an initial, coarse GAP model.

While far from chemically accurate, this potential can directly be used to run biased MD simulations at 573 K in order to explore the phase space relevant to the UI simulations. As there is some redundancy between neighboring windows, we randomly select ten windows from which to sample from. Subsequently, a diverse set of ten structures is extracted from these configurations via FPS and evaluated with single-point DFT calculations. This data is used to validate the accuracy of the current potential and added to the training set for the next iteration.

We find that energies and forces are well converged in 17 iterations, with mean energy errors well below 10 meV per atom and force errors below 75 meV/Å. Note that after initial convergence was observed in iteration 12, the potentials are additionally used to perform NEB calculations to ensure that the minimum energy paths of the GAP and the underlying DFT functional coincide. The evolution of the corresponding training and validation errors is shown in Figure 3. Importantly, the validation errors are for unseen configurations from the exact kind of simulation that we intend to run with this model. This gives us high confidence that the potential will be sufficiently accurate and stable for this purpose. (40)

Figure 3. Evolution of mean absolute errors (MAE) on energies (top) and forces (bottom) during iterative training. At each iteration, new configurations are generated via Umbrella Sampling and are used as a validation set. These structures are added to the training set for the next iteration.

As indicated above, the energies and forces on the final training set were additionally recalculated at the BEEF-vdW level. This data was used to train a second potential without performing additional training iterations. This allows us to explore the impact of using different computational setups for the free energy calculations.

III.3. Free Energy Calculations

The converged potentials were used to run extensive US calculations with 100 ps trajectories per window, at 523 K, which is the experimental operating temperature in ref (19). This amounts to 16 ns of dynamics overall, underscoring why this type of simulation is computationally prohibitive at the first-principles level. The resulting FESs at the revPBE+vdW^surf and BEEF+vdW levels are shown in Figure 4, along with the PESs obtained with NEB calculations.

Figure 4. Potential energy surface from Nudged Elastic Band calculations (left) and free energy surface from Umbrella Integration (center). Umbrella Integration estimates the relative free energies from the derivative of the free energy with respect to the collective variable (right). The estimated barriers using harmonic free energy corrections are indicated by the stars in the central plot. Calculations were performed with GAP models trained on the revPBE+vdW^surf and BEEF+vdW data, respectively. The gray region in the rightmost plot indicates the standard deviation of the Gaussian Process model used for Umbrella Integration.

A comparison of the PESs and FESs reveals that thermal effects decrease the barrier for decomposition of CHO significantly. Indeed, the barrier nearly vanishes at the revPBE+vdW^surf level, indicating that CHO is not a stable intermediate at 523 K at all. At the BEEF-vdW level, a small barrier of 0.13 eV remains. Figure 4 also shows the free energy derivatives, as defined in eq 2. This shows that the GAP-based MD simulations provide good coverage of the CV range. More importantly, the derivatives display no discontinuities, which would point to lack of convergence or broken ergodicity.

III.4. Rate Constants

For catalytic applications, the free energy barrier is mainly of interest as an intermediate quantity for computing rate constant k. The most common framework for computing k from first-principles is Transition State Theory (TST). In this context, the rate constant is defined via the Eyring equation as

k = \frac{k_{B} T}{h} \exp (\frac{- Δ G^{‡}}{k_{B} T})

(3)

where ΔG is the free energy barrier. One can illustrate the importance of thermal effects on rates by plugging in the potential energy barrier ΔE instead of the free energy barrier. At the BEEF-vdW level the corresponding rates differ by a factor of 85, i.e., by almost 2 orders of magnitude.

Equivalently, the Eyring equation can also be formulated in terms of the partition functions of the initial and transition states, Q_IS and Q_TS:

k = \frac{k_{B} T}{h} \frac{Q_{TS}^{‡}}{Q_{IS}} \exp (\frac{- Δ E}{k_{B} T})

(4)

This equation is the basis of the commonly used Harmonic Approximation (HA) for computing rate constants. This assumes that the adsorbate is tightly enough bound to the surface so that all degrees of freedom can be described as vibrations. In this case, the corresponding partition functions can be computed from the vibrational modes of the adsorbate in the initial and transition state configurations. Depending on whether the vibrations are described as classical or quantum harmonic oscillators, the corresponding expressions are

Q_{HA, class} = \prod_{i} \frac{k_{B} T}{h ν_{i}}

(5)

and

Q_{HA, quant} = \prod_{i} \frac{\exp (\frac{- h ν_{i}}{2 k_{B} T})}{1 - \exp (\frac{- h ν_{i}}{k_{B} T})}

(6)

where the product runs over all 3N vibrational modes and ν_i is the frequency of mode i.

We can now compare different approximate rate constants for the decomposition of CHO on Rh(111) (Figure 6). Within the HA, they can be computed by using the classical and quantum partition functions, yielding k_H,class and k_H,quant. According to the correspondence principle, the classical and quantum models should yield equivalent results in the high-temperature limit. At which temperature this limit is reached in practice depends on the specific vibrational frequencies of the system, however. This is illustrated in Figure 5, where the ratios of the quantum and classical rate constants are plotted as a function of temperature. This reveals that k_H,quant exceeds k_H,class by a factor of 1.5 to 2 at 523 K, depending on the DFT functional. While this is a relatively small difference given that rate constants tend to vary by several orders of magnitude in catalytic reaction networks, quantum nuclear effects are clearly not negligible for this reaction.

Figure 5. Ratio between harmonic rate constants computed with quantum and classical partition functions. The dotted line marks 523 K, which is the temperature used in the Umbrella Integration simulations.

Next, we can compare the rate constants obtained from the HA with those obtained via the UI free energy barriers (k_UI). These are shown in Figure 6, revealing that the classical, fully anharmonic description of the free energy barrier obtained with UI yields significantly larger rate constants than both the classical and quantum HA. Indeed, the enhancement from k_H,class to k_UI is much larger than the enhancement from k_H,class to k_H,quant (30 and 5-fold, for BEEF-vdW and revPBE+vdW^surf, respectively).

Figure 6. Transition state theory rate constants obtained with the harmonic approximation (HA) using classical and quantum partition functions and using umbrella integration (UI) free energy barriers. In addition to the classical UI rate constants, the influence of quantum nuclear effects can be estimated from the HA using the Pitzer–Gwinn (PG) correction, as detailed in the main text. As a baseline, the dotted lines indicate the TST rate constant when ignoring quantum and thermal effects completely.

Both thermal and quantum effects thus lead to a significant increase in the rate constants relative to the naive baseline of plugging ΔE into eq 3 (the dotted lines in Figure 6). To estimate the combined effect of (anharmonic) thermal and quantum effects, the Pitzer–Gwinn (PG) approximation for the partition functions can be used. (41) To this end, we assume that

Q_{UI, quant} \approx Q_{UI, class} \frac{Q_{H, quant}}{Q_{H, class}}

(7)

from which it follows that

k_{UI, PG} = k_{UI} \frac{k_{H, quant}}{k_{H, class}}

(8)

This leads to estimated rate constants of 1 × 10¹² s^–1 (BEEF-vdW) and 9 × 10¹² s^–1 (revPBE+vdW^surf). These rate constants can be understood more intuitively in terms of the half-lives

τ_{1 / 2}

of CHO they imply. These are 550 and 80 fs for BEEF-vdW and revPBE+vdW^surf, respectively. Under these conditions, CHO decomposition is thus hardly a rare event.

More generally, it is notable that the inclusion of thermal and nuclear quantum effects changes the rate constants by 1.5 to 2 orders of magnitude, depending on the functional. These effects are thus on the same order of magnitude as the difference between the functionals. Since the potential energy barriers ΔE enter eq 4 exponentially while the partition functions are part of the prefactor, the accuracy of DFT transition state energetics has been a prime focus in computational catalysis methods development. (42) Our results indicate that thermal effects can be equally important for certain reactions.

III.5. Harmonic Free Energy Corrections

Analogously to the HA for rate constants, it is also possible to define a harmonic free energy approximation by exploiting the equivalence of eq 3 and eq 4. In Figure 4, the HA free energy barriers (based on quantum partition functions) for BEEF-vdW and revPBE+vdW^surf are indicated as stars in the central plot. An appealing feature of these harmonic free energy corrections is that they can be decomposed into physically interpretable contributions, namely, the integrated heat capacity C_vib^0→T, the entropic contribution −TS_vib, and the zero point vibrational energy (ZPVE). These contributions to the barrier are shown in Table 1.

Table 1. Individual Contributions to Harmonic Free Energy Corrections of the DFT Reaction Barriers at 523 K^a

Contribution	revPBE+vdW^surf	BEEF-vdW
C_vib^0→T	–16	–18
–ΔTS_vib	25	16
ΔZPVE	–60	–74
Sum	–51	–76

^{^a}

The final line is the overall correction. All values are given in meV.

This reveals that the lowering of the HA barriers is mainly due to the ZPVE, which is by far the most negative contribution for both functionals. Meanwhile, the thermal contributions (C_vib^0→T and −TS_vib) have small and compensating effects on the barrier, with the entropic contributions to the barrier being positive and of similar magnitude to the negative contributions of the integrated heat capacity. This is in stark contrast to the UI predictions, which exclusively cover thermal effects and lead to a much stronger decrease in the barrier.

Indeed, the HA is known to be inadequate for low frequency modes, for which empirical corrections or separate treatments must be used. (12,14) For this reaction, all real vibrational modes in the initial and transition states display frequencies above 90 cm^–1, however, so that they would not be considered to be particularly pathological (see SI). Nevertheless, the PES obviously displays considerable anharmonicity. This is likely related to the small reaction barrier and the small geometric changes between the initial and transition states. As a consequence, application of the HA cannot be recommended in such situations.

It should also be noted that HA is remarkably sensitive to small changes in the geometry for this reaction. Specifically, in ref (19), the corresponding free energy barrier is estimated to be 0.49 eV at the BEEF-vdW level (compared to 0.25 eV found herein). In other words, the free energy correction is found to substantially increase the barrier therein, in contrast to our findings at both the UI and HA level. This discrepancy is likely caused by spurious low frequency modes, which disappear when tightly converging the geometry optimization. While such problems can be identified by manual inspection, they are challenging to diagnose in high-throughput settings such as ref (19).

III.6. Density Functional Comparison

Figure 6 implies a remarkably large influence of using different dispersion-corrected GGA functionals on the predicted rates. Here, further analysis indicates that this is to some extent an artifact of using the combination of revPBE and the vdW^surf dispersion correction. Specifically, the DFT reaction barrier was recomputed using single point calculations with different functionals and dispersion corrections at the BEEF-vdW and revPBE+vdW^surf initial and transition state geometries (see Figure 7). Here, we considered the pure PBE and revPBE functionals as well as PBE with both the vdW^surf and conventional Tkatchenko–Scheffler dispersion corrections (see the SI for details). This reveals revPBE+vdW^surf to be something of an outlier (see the SI for the individual barriers of each functional).

Figure 7. Density functional theory energy barriers obtained with single point calculations using different functionals and dispersion corrections at the BEEF-vdW and revPBE+vdW^surf initial and transition state geometries. In both cases, BEEF-vdW and revPBE+vdW^surf represent the upper and lower bounds of the estimated barrier, respectively, as indicated by the labels. Details for the other functionals are given in the main text and the SI. The shaded bar indicates the standard deviation of the BEEF ensemble uncertainty estimate.

This appears to be related to the fact that revPBE is more repulsive than other GGA functionals like BEEF and PBE, meaning that the onset of the damping function connecting the exchange-correlation functional and the dispersion correction must be set to smaller interatomic distances. (43) While this works reasonably well for noncovalent interactions in molecular dimers and liquid water, (43) the performance for adsorbates on surfaces deteriorates. (44)

On the other hand, it is also notable that BEEF-vdW predicts the highest of all barriers, although the other predictions lie within the standard deviation of the BEEF ensemble uncertainty. (37) Consequently, the BEEF-vdW and revPBE+vdW^surf results discussed above represent the upper and lower bounds of the expected barrier within the scope of GGA DFT. Overall, the qualitative behavior of both functionals is analogous, in that the inclusion of thermal effects, anharmonicity, and nuclear quantum effects all lead to a significant increase in the predicted rate constants. In terms of the absolute rates we can conclude that the BEEF-vdW numbers are likely more reliable, though they may be slightly overestimating the barrier.

III.7. Limitations

The focus of the current article is on obtaining ML potentials for the computation of free-energy profiles (FEPs) in surface catalysis. It should be emphasized, however, that obtaining the FEP itself is not sufficient to obtain the exact rate constants for a given level of theory. There are several further approximations involved in both the computation of the free energy barrier and in TST.

Regarding the former, calculating the free energy barrier from the minimum and maximum of the FEP is an approximation, because the entropy along the CV is neglected. As a consequence, the barrier computed this way is also somewhat dependent on the choice of CV. Dietschreit et al. recently reported formulas for computing exact free energy barriers. (45) This requires the reconstruction of the reweighted ensemble of all configurations, which is beyond the scope of the current study.

Regarding the use of TST, in particular, the presence of recrossing events can be problematic. A recrossing occurs when a trajectory reaches the transition state from the initial state and subsequently returns to the initial state before it visits the final state. TST assumes that this does not happen, so that the presence of such unsuccessful transitions means that the TST rate constants are too large. This can be addressed by introducing an additional transmission coefficient κ < 1 in eq 3, to account for recrossing.

Given the UI/TST estimated half-lives of 1138 and 124 fs for the two functionals (without the PG correction for nuclear quantum effects), the validity of TST can be tested by running unbiased MD simulations seeded in the initial state. Specifically, when 120 configurations are drawn from MD simulations constrained around the initial state with a biasing potential and subsequently running 2.5 ps long unbiased MD trajectories, reactive events are frequently observed. From these simulations, the half-lives of CHO can empirically be estimated as 725 and 140 fs, respectively. These times are in good agreement with the ones estimated from UI/TST. Indeed, this agreement may be somewhat fortuitous, and it should be noted that the unbiased MD estimates are an imperfect benchmark. In addition to significant statistical uncertainty (see SI), the initial structures are drawn from a biased ensemble and equilibration to the correct initial state distribution is hindered by frequent reactive events.

An alternative approach is to directly estimate the transmission coefficient κ from simulations. To this end, unbiased MD simulations were seeded both at the refined NEB TS and at the maximum of the free energy profile, for the BEEF-vdW based potential. In the latter case, configurations were drawn from restrained MD simulations close to the free energy maximum. In both cases, 240 MD trajectories were started by using 120 random velocities drawn from the Maxwell–Boltzmann distribution and integrating forward and backward in time until either the initial or final state basin is reached (defined as ξ < – 0.2 and ξ > 2.0, respectively).

This yields transmission coefficients of κ = 0.87 and κ = 0.81, respectively. In other words, a moderate amount of recrossing is indeed observed, so that the reported rate constants in Figure 6 are too large by 10–20%. Notably, transmission coefficients on this order to account for recrossing are not uncommon for barrierless reactions in gas-phase kinetics (see, for example, ref (46) and references therein, in which a dynamical correction of 15% was applied).

IV. Conclusions

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The current work demonstrates that state-of-the-art free energy calculations can be performed for heterogeneous catalytic reactions with DFT-quality potential energy surfaces. This is achieved by using ML interatomic potentials and a data-efficient iterative training scheme. Overall, the presented approach allows for a more rigorous treatment of thermal effects in computational catalysis and provides a valuable benchmark for approximate free energy corrections.

The present results indicate that CHO is an unstable (or at least extremely short-lived) species on Rh(111). While the formation of this species does not appear to be rate limiting in recent microkinetic simulations of syngas conversion, this finding does have some consequences for the mechanistic understanding of this process. On one hand, the rate limiting step in ref (19) is the subsequent hydrogenation step from CHO to CHOH, which assumes the presence of CHO on the surface. On the other hand, the relative stabilities of CHO and COH determine the selectivity of syngas conversion to different products.

More generally, our results fundamentally call into question the accuracy of the free energy barriers currently used in computational heterogeneous catalysis. We therefore aim to re-examine the reaction network of syngas chemistry on Rh more broadly in future work. In this context, it should be noted that the ML potentials used herein are single-purpose models trained and used for one specific elementary reaction. When treating a full reaction network, there are potentially synergy effects, in the sense that new data for a specific elementary step will likely improve the model for all steps in the network. Similarly, a potential could be trained for different CO and H coverages simultaneously. Indeed, lateral interactions are known to be important for this process. (21)

From a methodological perspective, our analysis furthermore indicates that nuclear quantum effects lead to a further decrease in this reaction barrier. In this context, a more rigorous treatment beyond the PG approximation would be of interest, for example, using path-integral MD (47) or multicomponent DFT. (48)

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00541.

Detailed description of the GAP model hyperparameters, training procedure, vibrational frequencies and density functional differences (PDF)

ct3c00541_si_001.pdf (1.88 MB)

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Author Information

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Corresponding Author
- Johannes T. Margraf - Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany; https://orcid.org/0000-0002-0862-5289; Email: [email protected]
Authors
- Sina Stocker - Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
- Hyunwook Jung - Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany
- Gábor Csányi - Engineering Laboratory, University of Cambridge, Cambridge CB2 1PZ, United Kingdom; https://orcid.org/0000-0002-8180-2034
- C. Franklin Goldsmith - Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany; School of Engineering, Brown University, Providence, Rhode Island 02912, United States; https://orcid.org/0000-0002-2212-0172
- Karsten Reuter - Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany; https://orcid.org/0000-0001-8473-8659
Funding
Open access funded by Max Planck Society. H.J. and C.F.G. gratefully acknowledge support from the Alexander-von-Humboldt (AvH) Foundation.
Notes
The authors declare no competing financial interest.

Acknowledgments

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S.S. gratefully acknowledges Wenbin Xu for the support with the Quantum Espresso calculations.

References

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Nature communications (2017), 8 (), 14621 ISSN:.
Surface reaction networks involving hydrocarbons exhibit enormous complexity with thousands of species and reactions for all but the very simplest of chemistries. We present a framework for optimization under uncertainty for heterogeneous catalysis reaction networks using surrogate models that are trained on the fly. The surrogate model is constructed by teaching a Gaussian process adsorption energies based on group additivity fingerprints, combined with transition-state scaling relations and a simple classifier for determining the rate-limiting step. The surrogate model is iteratively used to predict the most important reaction step to be calculated explicitly with computationally demanding electronic structure theory. Applying these methods to the reaction of syngas on rhodium(111), we identify the most likely reaction mechanism. Propagating uncertainty throughout this process yields the likelihood that the final mechanism is complete given measurements on only a subset of the entire network and uncertainty in the underlying density functional theory calculations.
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Deimel, M.; Prats, H.; Seibt, M.; Reuter, K.; Andersen, M. Selectivity Trends and Role of Adsorbate–Adsorbate Interactions in CO Hydrogenation on Rhodium Catalysts. ACS Catal. 2022, 12, 7907– 7917, DOI: 10.1021/acscatal.2c02353
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Selectivity Trends and Role of Adsorbate-Adsorbate Interactions in CO Hydrogenation on Rhodium Catalysts
Deimel, Martin; Prats, Hector; Seibt, Michael; Reuter, Karsten; Andersen, Mie
ACS Catalysis (2022), 12 (13), 7907-7917CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
Predictive-quality computational modeling of heterogeneously catalyzed reactions has emerged as an important tool for the anal. and assessment of activity and activity trends. In contrast, more subtle selectivities and selectivity trends still pose a significant challenge to prevalent microkinetic modeling approaches that typically employ a mean-field approxn. (MFA). Here, we focus on CO hydrogenation on Rh catalysts with the possible products methane, acetaldehyde, ethanol, and water. This reaction has already been subjected to a no. of exptl. and theor. studies with conflicting views on the factors controlling activity and selectivity toward the more valuable higher oxygenates. Using accelerated first-principles kinetic Monte Carlo simulations and explicitly and systematically accounting for adsorbate-adsorbate interactions through a cluster expansion approach, we model the reaction on the low-index Rh(111) and stepped Rh(211) surfaces. We find that the Rh(111) facet is selective toward methane, while the Rh(211) facet exhibits a similar selectivity toward methane and acetaldehyde. This is consistent with the exptl. selectivity obsd. for larger, predominantly (111)-exposing Rh nanoparticles and resolves the discrepancy with earlier first-principles MFA microkinetic work that found the Rh(111) facet to be selective toward acetaldehyde. While the latter work tried to approx. account for lateral interactions through coverage-dependent rate expressions, our anal. demonstrates that this fails to sufficiently capture concomitant correlations among the adsorbed reaction intermediates that crucially det. the overall selectivity.
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Kästner, J.; Thiel, W. Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: “Umbrella integration. J. Chem. Phys. 2005, 123, 144104, DOI: 10.1063/1.2052648
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Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: "Umbrella integration"
Kastner, Johannes; Thiel, Walter
Journal of Chemical Physics (2005), 123 (14), 144104/1-144104/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
We present a method to analyze biased mol.-dynamics and Monte Carlo simulations, also known as umbrella sampling. In the limiting case of a strong bias, this method is equiv. to thermodn. integration. It employs only quantities with easily controllable equilibration and greatly reduces the statistical errors compared to the std. weighted histogram anal. method. We show the success of our approach for two examples, one analytic function, and one biol. system.
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Kästner, J.; Thiel, W. Analysis of the statistical error in umbrella sampling simulations by umbrella integration. J. Chem. Phys. 2006, 124, 234106, DOI: 10.1063/1.2206775
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Analysis of the statistical error in umbrella sampling simulations by umbrella integration
Kastner, Johannes; Thiel, Walter
Journal of Chemical Physics (2006), 124 (23), 234106/1-234106/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Umbrella sampling simulations, or biased mol. dynamics, can be used to calc. the free-energy change of a chem. reaction. We investigate the sources of different sampling errors and derive approx. expressions for the statistical errors when using harmonic restraints and umbrella integration anal. This leads to generally applicable rules for the choice of the bias potential and the sampling parameters. Numerical results for simulations on an anal. model potential are presented for validation. While the derivations are based on umbrella integration anal., the final error est. is evaluated from the raw simulation data, and it may therefore be generally applicable as indicated by tests using the weighted histogram anal. method.
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Roux, B. The calculation of the potential of mean force using computer simulations. Comput. Phys. Commun. 1995, 91, 275– 282, DOI: 10.1016/0010-4655(95)00053-I
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The calculation of the potential of mean force using computer simulations
Roux, Benoit
Computer Physics Communications (1995), 91 (1-3), 275-82CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier)
The problem of unbiasing and combining the results of umbrella sampling calcns. is reviewed. The weighted histogram anal. method (WHAM) of S. Kumar et al. (J. Comp. Chem. 13 (1992) 1011) is described and compared with other approaches. The method is illustrated with mol. dynamics simulations of the alanine dipeptide for one- and two-dimensional free energy surfaces. The results show that the WHAM approach simplifies considerably the task of recombining the various windows in complex systems.
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Kumar, S.; Rosenberg, J. M.; Bouzida, D.; Swendsen, R. H.; Kollman, P. A. The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J. Comput. Chem. 1992, 13, 1011– 1021, DOI: 10.1002/jcc.540130812
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The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method
Kumar, Shankar; Bouzida, Djamal; Swendsen, Robert H.; Kollman, Peter A.; Rosenberg, John M.
Journal of Computational Chemistry (1992), 13 (8), 1011-21CODEN: JCCHDD; ISSN:0192-8651.
The Weighted Histogram Anal. Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomol. Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calcns. This algorithm possesses the following advantages over methods that are currently employed: (1) it provides a built-in est. of sampling errors thereby yielding objective ests. of the optimal location and length of addnl. simulations needed to achieve a desired level of precision; (2) it yields the "best" value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addn. to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better ests. of the free-energy differences. By recasting the Ferrenberg-Swendsen Multiple Histogram equations in a form suitable for mol. mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine.
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Hub, J. S.; De Groot, B. L.; Van der Spoel, D. g_wham─A Free Weighted Histogram Analysis Implementation Including Robust Error and Autocorrelation Estimates. J. Chem. Theory Comput. 2010, 6, 3713– 3720, DOI: 10.1021/ct100494z
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g_wham: A Free Weighted Histogram Analysis Implementation Including Robust Error and Autocorrelation Estimates
Hub, Jochen S.; de Groot, Bert L.; van der Spoel, David
Journal of Chemical Theory and Computation (2010), 6 (12), 3713-3720CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
The Weighted Histogram Anal. Method (WHAM) is a std. technique used to compute potentials of mean force (PMFs) from a set of umbrella sampling simulations. Here, the authors present a new WHAM implementation, termed g_wham, which is distributed freely with the GROMACS mol. simulation suite. G_wham ests. statistical errors using the technique of bootstrap anal. Three bootstrap methods are supported: (i) bootstrapping new trajectories based on the umbrella histograms, (ii) bootstrapping of complete histograms, and (iii) Bayesian bootstrapping of complete histograms, i.e., bootstrapping via the assignment of random wts. to the histograms. Because methods ii and iii consider only complete histograms as independent data points, these methods do not require the accurate calcn. of autocorrelation times. The authors demonstrate that, given sufficient sampling, bootstrapping new trajectories allows for an accurate error est. In the presence of long autocorrelations, however, (Bayesian) bootstrapping of complete histograms yields a more reliable error est., whereas bootstrapping of new trajectories may underestimate the error. In addn., the authors emphasize that the incorporation of autocorrelations into WHAM reduces the bias from limited sampling, in particular, when computing periodic PMFs in inhomogeneous systems such as solvated lipid membranes or protein channels.
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Stecher, T.; Bernstein, N.; Csányi, G. Free Energy Surface Reconstruction from Umbrella Samples Using Gaussian Process Regression. J. Chem. Theory Comput. 2014, 10, 4079– 4097, DOI: 10.1021/ct500438v
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Free Energy Surface Reconstruction from Umbrella Samples Using Gaussian Process Regression
Stecher, Thomas; Bernstein, Noam; Csanyi, Gabor
Journal of Chemical Theory and Computation (2014), 10 (9), 4079-4097CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We demonstrate how the Gaussian process regression approach can be used to efficiently reconstruct free energy surfaces from umbrella sampling simulations. By making a prior assumption of smoothness and taking account of the sampling noise in a consistent fashion, we achieve a significant improvement in accuracy over the state of the art in two or more dimensions or, equivalently, a significant cost redn. to obtain the free energy surface within a prescribed tolerance in both regimes of spatially sparse data and short sampling trajectories. Stemming from its Bayesian interpretation the method provides meaningful error bars without significant addnl. computation. A software implementation is made available on www.libatoms.org.
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Hjorth Larsen, A.; Jørgen Mortensen, J.; Blomqvist, J.; Castelli, I. E.; Christensen, R.; Dułak, M.; Friis, J.; Groves, M. N.; Hammer, B.; Hargus, C.; Hermes, E. D.; Jennings, P. C.; Bjerre Jensen, P.; Kermode, J.; Kitchin, J. R.; Leonhard Kolsbjerg, E.; Kubal, J.; Kaasbjerg, K.; Lysgaard, S.; Bergmann Maronsson, J.; Maxson, T.; Olsen, T.; Pastewka, L.; Peterson, A.; Rostgaard, C.; Schiøtz, J.; Schütt, O.; Strange, M.; Thygesen, K. S.; Vegge, T.; Vilhelmsen, L.; Walter, M.; Zeng, Z.; Jacobsen, K. W. The atomic simulation environment─a Python library for working with atoms. J. Phys.: Condens. Matter 2017, 29, 273002, DOI: 10.1088/1361-648X/aa680e
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The atomic simulation environment-a Python library for working with atoms
Hjorth Larsen Ask; Jorgen Mortensen Jens; Blomqvist Jakob; Castelli Ivano E; Christensen Rune; Dulak Marcin; Friis Jesper; Groves Michael N; Hammer Bjork; Hargus Cory; Hermes Eric D; Jennings Paul C; Bjerre Jensen Peter; Kermode James; Kitchin John R; Leonhard Kolsbjerg Esben; Kubal Joseph; Kaasbjerg Kristen; Lysgaard Steen; Bergmann Maronsson Jon; Maxson Tristan; Olsen Thomas; Pastewka Lars; Peterson Andrew; Rostgaard Carsten; Schiotz Jakob; Schutt Ole; Strange Mikkel; Thygesen Kristian S; Vegge Tejs; Vilhelmsen Lasse; Walter Michael; Zeng Zhenhua; Jacobsen Karsten W
Journal of physics. Condensed matter : an Institute of Physics journal (2017), 29 (27), 273002 ISSN:.
The atomic simulation environment (ASE) is a software package written in the Python programming language with the aim of setting up, steering, and analyzing atomistic simulations. In ASE, tasks are fully scripted in Python. The powerful syntax of Python combined with the NumPy array library make it possible to perform very complex simulation tasks. For example, a sequence of calculations may be performed with the use of a simple 'for-loop' construction. Calculations of energy, forces, stresses and other quantities are performed through interfaces to many external electronic structure codes or force fields using a uniform interface. On top of this calculator interface, ASE provides modules for performing many standard simulation tasks such as structure optimization, molecular dynamics, handling of constraints and performing nudged elastic band calculations.
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Bartók, A. P.; Payne, M. C.; Kondor, R.; Csányi, G. Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons. Phys. Rev. Lett. 2010, 104, 136403, DOI: 10.1103/PhysRevLett.104.136403
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Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons
Bartok, Albert P.; Payne, Mike C.; Kondor, Risi; Csanyi, Gabor
Physical Review Letters (2010), 104 (13), 136403/1-136403/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
We introduce a class of interat. potential models that can be automatically generated from data consisting of the energies and forces experienced by atoms, as derived from quantum mech. calcns. The models do not have a fixed functional form and hence are capable of modeling complex potential energy landscapes. They are systematically improvable with more data. We apply the method to bulk crystals, and test it by calcg. properties at high temps. Using the interat. potential to generate the long mol. dynamics trajectories required for such calcns. saves orders of magnitude in computational cost.
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Bartók, A. P.; Kondor, R.; Csányi, G. On representing chemical environments. Phys. Rev. B 2013, 87, 184115, DOI: 10.1103/PhysRevB.87.184115
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On representing chemical environments
Bartok, Albert P.; Kondor, Risi; Csanyi, Gabor
Physical Review B: Condensed Matter and Materials Physics (2013), 87 (18), 184115/1-184115/16CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
We review some recently published methods to represent at. neighborhood environments, and analyze their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave nos. are used to expand the at. neighborhood d. function. Using the example system of small clusters, we quant. show that this expansion needs to be carried to higher and higher wave nos. as the no. of neighbors increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether different approach, called Smooth Overlap of Atomic Positions, that sidesteps these difficulties by directly defining the similarity between any two neighborhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.
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Bartók, A. P.; De, S.; Poelking, C.; Bernstein, N.; Kermode, J. R.; Csányi, G.; Ceriotti, M. Machine learning unifies the modeling of materials and molecules. Sci. Adv. 2017, 3, e1701816, DOI: 10.1126/sciadv.1701816
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Machine learning unifies the modeling of materials and molecules
Bartok, Albert P.; De, Sandip; Poelking, Carl; Bernstein, Noam; Kermode, James R.; Csanyi, Gabor; Ceriotti, Michele
Science Advances (2017), 3 (12), e1701816/1-e1701816/8CODEN: SACDAF; ISSN:2375-2548. (American Association for the Advancement of Science)
Detg. the stability of mols. and condensed phases is the cornerstone of atomistic modeling, underpinning our understanding of chem. and materials properties and transformations. We show that a machine-learning model, based on a local description of chem. environments and Bayesian statistical learning, provides a unified framework to predict at.-scale properties. It captures the quantum mech. effects governing the complex surface reconstructions of silicon, predicts the stability of different classes of mols. with chem. accuracy, and distinguishes active and inactive protein ligands with more than 99% reliability. The universality and the systematic nature of our framework provide new insight into the potential energy surface of materials and mols.
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Hammer, B.; Hansen, L. B.; Nørskov, J. K. Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals. Phys. Rev. B 1999, 59, 7413– 7421, DOI: 10.1103/PhysRevB.59.7413
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Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals
Hammer, B.; Hansen, L. B.; Norskov, J. K.
Physical Review B: Condensed Matter and Materials Physics (1999), 59 (11), 7413-7421CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
A simple formulation of a generalized gradient approxn. for the exchange and correlation energy of electrons has been proposed by J. Perdew et al. (1996). Subsequently, Y. Zhang and W. Wang (1998) have shown that a slight revision of the Perdew-Burke-Ernzerhof (PBE) functional systematically improves the atomization energies for a large database of small mols. In the present work, we show that the Zhang and Yang functional (revPBE) also improves the chemisorption energetics of atoms and mols. on transition-metal surfaces. Our test systems comprise at. and mol. adsorption of oxygen, CO, and NO on Ni(100), Ni(111), Rh(100), Pd(100), and Pd(111) surfaces. As the revPBE functional may locally violate the Lieb-Oxford criterion, we further develop an alternative revision of the PBE functional, RPBE, which gives the same improvement of the chemisorption energies as the revPBE functional at the same time as it fulfills the Lieb-Oxford criterion locally.
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Wellendorff, J.; Silbaugh, T. L.; Garcia-Pintos, D.; Nørskov, J. K.; Bligaard, T.; Studt, F.; Campbell, C. T. A benchmark database for adsorption bond energies to transition metal surfaces and comparison to selected DFT functionals. Surf. Sci. 2015, 640, 36– 44, DOI: 10.1016/j.susc.2015.03.023
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A benchmark database for adsorption bond energies to transition metal surfaces and comparison to selected DFT functionals
Wellendorff, Jess; Silbaugh, Trent L.; Garcia-Pintos, Delfina; Noerskov, Jens K.; Bligaard, Thomas; Studt, Felix; Campbell, Charles T.
Surface Science (2015), 640 (), 36-44CODEN: SUSCAS; ISSN:0039-6028. (Elsevier B.V.)
A literature collection of exptl. adsorption energies over late transition metal surfaces is presented for systems where the authors believe that the energy measurements are particularly accurate, and the at.-scale adsorption geometries are particularly well established. This could become useful for benchmarking theor. methods for calcg. adsorption processes. The authors compare the exptl. results to six commonly used electron d. functionals, including some (RPBE, BEEF-vdW) which were specifically developed to treat adsorption processes. The comparison shows that there is ample room for improvements in the theor. descriptions.
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Zhang, Y.; Yang, W. Comment on “Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1998, 80, 890– 890, DOI: 10.1103/PhysRevLett.80.890
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Comment on "Generalized Gradient Approximation Made Simple"
Zhang, Yingkai; Yang, Weitao
Physical Review Letters (1998), 80 (4), 890CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
A Comment on the Letter by John P. Perdew, Kieron Burke, and Matthias Ernzerhof, Phys. 77, 3865 (1996). The authors of the Letter offer a Reply.
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Ruiz, V. G.; Liu, W.; Tkatchenko, A. Density-functional theory with screened van der Waals interactions applied to atomic and molecular adsorbates on close-packed and non-close-packed surfaces. Phys. Rev. B 2016, 93, 035118, DOI: 10.1103/PhysRevB.93.035118
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Density-functional theory with screened van der Waals interactions applied to atomic and molecular adsorbates on close-packed and non-close-packed surfaces
Ruiz, Victor G.; Liu, Wei; Tkatchenko, Alexandre
Physical Review B (2016), 93 (3), 035118/1-035118/17CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
Modeling the adsorption of atoms and mols. on surfaces requires efficient electronic-structure methods that are able to capture both covalent and noncovalent interactions in a reliable manner. In order to tackle this problem, we have developed a method within d.-functional theory (DFT) to model screened van der Waals interactions (vdW) for atoms and mols. on surfaces (the so-called DFT+vdWsurf method). The relatively high accuracy of the DFT+vdWsurf method in the calcn. of both adsorption distances and energies, as well as the high degree of its reliability across a wide range of adsorbates, indicates the importance of the collective electronic effects within the extended substrate for the calcn. of the vdW energy tail. We examine in detail the theor. background of the method and assess its performance for adsorption phenomena including the physisorption of Xe on selected close-packed transition metal surfaces and 3,4,9,10-perylene-tetracarboxylic acid dianhydride (PTCDA) on Au(111). We also address the performance of DFT+vdWsurf in the case of non-close-packed surfaces by studying the adsorption of Xe on Cu(110) and the interfaces formed by the adsorption of a PTCDA monolayer on the Ag(111), Ag(100), and Ag(110) surfaces. We conclude by discussing outstanding challenges in the modeling of vdW interactions for studying at. and mol. adsorbates on inorg. substrates.
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Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab initio molecular simulations with numeric atom-centered orbitals. Comput. Phys. Commun. 2009, 180, 2175– 2196, DOI: 10.1016/j.cpc.2009.06.022
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Ab initio molecular simulations with numeric atom-centered orbitals
Blum, Volker; Gehrke, Ralf; Hanke, Felix; Havu, Paula; Havu, Ville; Ren, Xinguo; Reuter, Karsten; Scheffler, Matthias
Computer Physics Communications (2009), 180 (11), 2175-2196CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
We describe a complete set of algorithms for ab initio mol. simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of mol. and materials properties from quantum-mech. first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute "ab initio mol. simulations" (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on d.-functional theory (DFT) in the local and semilocal (generalized gradient) approxns., but an extension to hybrid functionals, Hartree-Fock theory, and MP2/GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron/full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio mol. dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calcn. to range from qual. tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs.
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Wellendorff, J.; Lundgaard, K. T.; Møgelhøj, A.; Petzold, V.; Landis, D. D.; Nørskov, J. K.; Bligaard, T.; Jacobsen, K. W. Density functionals for surface science: Exchange-correlation model development with Bayesian error estimation. Phys. Rev. B 2012, 85, 235149, DOI: 10.1103/PhysRevB.85.235149
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Density functionals for surface science: exchange-correlation model development with Bayesian error estimation
Wellendorff, Jess; Lundgaard, Keld T.; Moegelhoej, Andreas; Petzold, Vivien; Landis, David D.; Noerskov, Jens K.; Bligaard, Thomas; Jacobsen, Karsten W.
Physical Review B: Condensed Matter and Materials Physics (2012), 85 (23), 235149/1-235149/23CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
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Advanced capabilities for materials modelling with QUANTUM ESPRESSO
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Journal of Physics: Condensed Matter (2017), 29 (46), 465901/1-465901/30CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
QUANTUM ESPRESSO is an integrated suite of open-source computer codes for quantum simulations of materials using state-of-the-art electronic-structure techniques, based on d.-functional theory, d.-functional perturbation theory, and many-body perturbation theory, within the plane-wave pseudopotential and projector-augmented-wave approaches. QUANTUM ESPRESSO owes its popularity to the wide variety of properties and processes it allows to simulate, to its performance on an increasingly broad array of hardware architectures, and to a community of researchers that rely on its capabilities as a core open-source development platform to implement their ideas. In this paper we describe recent extensions and improvements, covering new methodologies and property calculators, improved parallelization, code modularization, and extended interoperability both within the distribution and with external software.
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The Journal of chemical physics (2020), 153 (10), 104701 ISSN:.
Molecular adsorption at organic/metal interfaces depends on a range of mechanisms: covalent bonds, charge transfer, Pauli repulsion, and van der Waals (vdW) interactions shape the potential energy surface (PES), making it key to understanding organic/metal interfaces. Describing such interfaces with density functional theory requires carefully selecting the exchange correlation (XC) functional and vdW correction scheme. To explore the reproducibility of the PES with respect to the choice of method, we present a benchmark of common local, semi-local, and non-local XC functionals in combination with various vdW corrections. We benchmark these methods using perylene-tetracarboxylic dianhydride on Ag(111), one of the most frequently studied organic/metal interfaces. For each method, we determine the PES using a Gaussian process regression algorithm, which requires only about 50 density functional theory calculations as input. This allows a detailed analysis of the PESs' features, such as the positions and energies of minima and saddle points. Comparing the results from different combinations of XC functionals and vdW corrections enables us to identify trends and differences between the approaches. PESs for different computation methods are in qualitative agreement but also display significant quantitative differences. In particular, the lateral positions of adsorption geometries agree well with experiment, while adsorption heights, energies, and barriers show larger discrepancies.
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Journal of Chemical Physics (2022), 157 (8), 084113CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Given a chem. reaction going from reactant (R) to the product (P) on a potential energy surface (PES) and a collective variable (CV) discriminating between R and P, we define the free-energy profile (FEP) as the logarithm of the marginal Boltzmann distribution of the CV. This FEP is not a true free energy. Nevertheless, it is common to treat the FEP as the "free-energy" analog of the min. potential energy path and to take the activation free energy, ΔF‡RP, as the difference between the max. at the transition state and the min. at R. We show that this approxn. can result in large errors. The FEP depends on the CV and is, therefore, not unique. For the same reaction, different discriminating CVs can yield different ΔF‡RP. We derive an exact expression for the activation free energy that avoids this ambiguity. We find ΔF‡RP to be a combination of the probability of the system being in the reactant state, the probability d. on the dividing surface, and the thermal de Broglie wavelength assocd. with the transition. We apply our formalism to simple analytic models and realistic chem. systems and show that the FEP-based approxn. applies only at low temps. for CVs with a small effective mass. Most chem. reactions occur on complex, high-dimensional PES that cannot be treated anal. and pose the added challenge of choosing a good CV. We study the influence of that choice and find that, while the reaction free energy is largely unaffected, ΔF‡RP is quite sensitive. (c) 2022 American Institute of Physics.
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Temperature and Pressure-Dependent Rate Coefficients for the Reaction of Vinyl Radical with Molecular Oxygen
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Journal of Physical Chemistry A (2015), 119 (28), 7766-7779CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
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Nature Reviews Chemistry (2018), 2 (3), 0109CODEN: NRCAF7; ISSN:2397-3358. (Nature Research)
A review. Atomistic simulations of chem., biol. and materials systems have become increasingly precise and predictive owing to the development of accurate and efficient techniques that describe the quantum mech. behavior of electrons. Nevertheless, the overwhelming majority of such simulations still assumes that the nuclei behave as classical particles. Historically, this approxn. could sometimes be justified owing to the complexity and computational overhead. However, neglecting nuclear quantum effects has become one of the largest sources of error, esp. when systems contg. light atoms are treated using current state-of-the-art descriptions of chem. interactions. Over the past decade, this realization has spurred a series of methodol. advances that have dramatically reduced the cost of including these important phys. effects in the structure and dynamics of chem. systems. Here, we discuss how these developments are now allowing nuclear quantum effects to become a mainstream feature of mol. simulations. These advances have led to new insights into phenomena that are relevant to different areas of science - from biochem. to condensed matter - and open the door to many exciting future opportunities.
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Pavošević, F.; Culpitt, T.; Hammes-Schiffer, S. Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects via the Nuclear–Electronic Orbital Method. Chem. Rev. 2020, 120, 4222– 4253, DOI: 10.1021/acs.chemrev.9b00798
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Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects via the Nuclear-Electronic Orbital Method
Pavosevic, Fabijan; Culpitt, Tanner; Hammes-Schiffer, Sharon
Chemical Reviews (Washington, DC, United States) (2020), 120 (9), 4222-4253CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
A review. In multicomponent quantum chem., more than one type of particle is treated quantum mech. with either d. functional theory or wave function based methods. In particular, the nuclear-electronic orbital (NEO) approach treats specified nuclei, typically hydrogen nuclei, on the same level as the electrons. This approach enables the incorporation of nuclear quantum effects, such as nuclear delocalization, anharmonicity, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects directly into quantum chem. calcns. Such effects impact optimized geometries, mol. vibrational frequencies, reaction paths, isotope effects, and dynamical simulations. Multicomponent d. functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) achieve an optimal balance between computational efficiency and accuracy for computing ground and excited state properties, resp. Multicomponent wave function based methods, such as the coupled cluster singles and doubles (NEO-CCSD) method for ground states and the equation-of-motion counterpart (NEO-EOM-CCSD) for excited states, attain similar accuracy without requiring any parametrization and can be systematically improved but are more computationally expensive. Variants of the orbital-optimized perturbation theory (NEO-OOMP2) method achieve nearly the accuracy of NEO-CCSD for ground states with significantly lower computational cost. Addnl. approaches for computing excited electronic, vibrational, and vibronic states include the delta SCF (NEO-ΔSCF), complete active space SCF (NEO-CASSCF), and nonorthogonal CI methods. Multireference methods are particularly important for describing hydrogen tunneling processes. Other types of multicomponent systems, such as those contg. electrons and positrons, have also been studied within the NEO framework. The NEO approach allows the incorporation of nuclear quantum effects and non-Born-Oppenheimer effects for specified nuclei into quantum chem. calcns. in an accessible and computationally efficient manner.
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Abstract
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Figure 1
Figure 1. Top and side views of local minimum configurations for CHO and CO+H on the Rh(111) surface. The geometric measures d and θ, which define the collective variable used below, are indicated for CHO.
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Figure 2
Figure 2. Nudged elastic band images plotted in terms of the angle between the CO and the surface normal (θ) and the C–H distance (d). The color bar shows the relative energies of the images in eV, with the yellow points being close to the transition state. The collective variable used herein is a linear combination of θ and d, indicated by the gray line.
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Figure 3
Figure 3. Evolution of mean absolute errors (MAE) on energies (top) and forces (bottom) during iterative training. At each iteration, new configurations are generated via Umbrella Sampling and are used as a validation set. These structures are added to the training set for the next iteration.
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Figure 4
Figure 4. Potential energy surface from Nudged Elastic Band calculations (left) and free energy surface from Umbrella Integration (center). Umbrella Integration estimates the relative free energies from the derivative of the free energy with respect to the collective variable (right). The estimated barriers using harmonic free energy corrections are indicated by the stars in the central plot. Calculations were performed with GAP models trained on the revPBE+vdW^surf and BEEF+vdW data, respectively. The gray region in the rightmost plot indicates the standard deviation of the Gaussian Process model used for Umbrella Integration.
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Figure 5
Figure 5. Ratio between harmonic rate constants computed with quantum and classical partition functions. The dotted line marks 523 K, which is the temperature used in the Umbrella Integration simulations.
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Figure 6
Figure 6. Transition state theory rate constants obtained with the harmonic approximation (HA) using classical and quantum partition functions and using umbrella integration (UI) free energy barriers. In addition to the classical UI rate constants, the influence of quantum nuclear effects can be estimated from the HA using the Pitzer–Gwinn (PG) correction, as detailed in the main text. As a baseline, the dotted lines indicate the TST rate constant when ignoring quantum and thermal effects completely.
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Figure 7
Figure 7. Density functional theory energy barriers obtained with single point calculations using different functionals and dispersion corrections at the BEEF-vdW and revPBE+vdW^surf initial and transition state geometries. In both cases, BEEF-vdW and revPBE+vdW^surf represent the upper and lower bounds of the estimated barrier, respectively, as indicated by the labels. Details for the other functionals are given in the main text and the SI. The shaded bar indicates the standard deviation of the BEEF ensemble uncertainty estimate.
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  Norskov, J. K.; Bligaard, T.; Rossmeisl, J.; Christensen, C. H.
  Nature Chemistry (2009), 1 (1), 37-46CODEN: NCAHBB; ISSN:1755-4330. (Nature Publishing Group)
  A review; over the past decade the theor. description of surface reactions has undergone a radical development. Advances in d. functional theory mean it is now possible to describe catalytic reactions at surfaces with the detail and accuracy required for computational results to compare favorably with expts. Theor. methods can be used to describe surface chem. reactions in detail and to understand variations in catalytic activity from one catalyst to another. Here, we review the first steps towards using computational methods to design new catalysts. Examples include screening for catalysts with increased activity and catalysts with improved selectivity. We discuss how, in the future, such methods may be used to engineer the electronic structure of the active surface by changing its compn. and structure.
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6. 6
  Margraf, J. T.; Jung, H.; Scheurer, C.; Reuter, K. Exploring Catalytic Reaction Networks with Machine Learning. Nat. Catal. 2023, 6, 112– 121, DOI: 10.1038/s41929-022-00896-y
  There is no corresponding record for this reference.
7. 7
  Dellago, C.; Bolhuis, P. G.; Csajka, F. S.; Chandler, D. Transition path sampling and the calculation of rate constants. J. Chem. Phys. 1998, 108, 1964– 1977, DOI: 10.1063/1.475562
  7
  Transition path sampling and the calculation of rate constants
  Dellago, Christoph; Bolhuis, Peter G.; Csajka, Felix S.; Chandler, David
  Journal of Chemical Physics (1998), 108 (5), 1964-1977CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We have developed a method to study transition pathways for rare events in complex systems. The method can be used to det. rate consts. for transitions between stable states by turning the calcn. of reactive flux correlation functions into the computation of an isomorphic reversible work. In contrast to previous dynamical approaches, the method relies neither on prior knowledge nor on explicit specification of transition states. Rather, it provides an importance sampling from which transition states can be characterized statistically. A simple model is analyzed to illustrate the methodol.
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8. 8
  Bussi, G.; Laio, A. Using metadynamics to explore complex free-energy landscapes. Nat. Rev. Phys. 2020, 2, 200– 212, DOI: 10.1038/s42254-020-0153-0
  There is no corresponding record for this reference.
9. 9
  Torrie, G. M.; Valleau, J. P. Monte Carlo free energy estimates using non-Boltzmann sampling: Application to the sub-critical Lennard-Jones fluid. Chem. Phys. Lett. 1974, 28, 578– 581, DOI: 10.1016/0009-2614(74)80109-0
  9
  Monte-Carlo free energy estimates using non-Boltzmann sampling. Application to the subcritical Lennard-Jones fluid
  Torrie, Glenn M.; Valleau, John P.
  Chemical Physics Letters (1974), 28 (4), 578-81CODEN: CHPLBC; ISSN:0009-2614.
  A Monte Carlo technique is presented for estn. of the free energies of fluids by sampling on distributions designed for this purpose, rather than on the usual Boltzmann distribution. As an illustration of its use, the free energy of a Lennard-Jones fluid in the liq.-vapour coexistence region was estd. by relating it to that of the inverse-12 (soft sphere) fluid, which itself shows no condensation.
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10. 10
  Kästner, J. Umbrella sampling. Wiley Interdiscip. Rev. Comput. Mol. Sci. 2011, 1, 932– 942, DOI: 10.1002/wcms.66
  There is no corresponding record for this reference.
11. 11
  Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys. 2000, 113, 9901– 9904, DOI: 10.1063/1.1329672
  11
  A climbing image nudged elastic band method for finding saddle points and minimum energy paths
  Henkelman, Graeme; Uberuaga, Blas P.; Jonsson, Hannes
  Journal of Chemical Physics (2000), 113 (22), 9901-9904CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A modification of the nudged elastic band method for finding min. energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring consts. are used to increase the d. of images near the top of the energy barrier to get an improved est. of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir(111) and H2 on Si(100) using plane wave based d. functional theory are presented.
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12. 12
  Andersen, M.; Cingolani, J. S.; Reuter, K. Ab Initio Thermodynamics of Hydrocarbons Relevant to Graphene Growth at Solid and Liquid Cu Surfaces. J. Phys. Chem. C 2019, 123, 22299– 22310, DOI: 10.1021/acs.jpcc.9b05642
  12
  Ab Initio Thermodynamics of Hydrocarbons Relevant to Graphene Growth at Solid and Liquid Cu Surfaces
  Andersen, Mie; Cingolani, Juan Santiago; Reuter, Karsten
  Journal of Physical Chemistry C (2019), 123 (36), 22299-22310CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
  Using ab initio thermodn., the stability of a wide range of hydrocarbon adsorbates under various chem. vapor deposition (CVD) conditions (temp., methane and hydrogen pressures) used in exptl. graphene growth protocols at solid and liq. Cu surfaces has been explored. At the employed high growth temps. around the m.p. of Cu, we find that commonly used thermodn. models such as the harmonic oscillator model may no longer be accurate. Instead, we account for the translational and rotational mobility of adsorbates using a recently developed hindered translator and rotator model or a two-dimensional ideal gas model. The thermodn. considerations turn out to be crucial for explaining exptl. results and allow us to improve and extend the findings of earlier theor. studies regarding the role of hydrogen and hydrocarbon species in CVD. In particular, we find that smaller hydrocarbons will completely dehydrogenate under most CVD conditions. For larger clusters, our results show that metal-terminated and hydrogen-terminated edges have very similar stabilities. While both cluster types might thus form during the expt., we show that the low binding strength of clusters with hydrogen-terminated edges could result in instability toward desorption.
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13. 13
  Bajpai, A.; Mehta, P.; Frey, K.; Lehmer, A. M.; Schneider, W. F. Benchmark First-Principles Calculations of Adsorbate Free Energies. ACS Catal. 2018, 8, 1945– 1954, DOI: 10.1021/acscatal.7b03438
  13
  Benchmark First-Principles Calculations of Adsorbate Free Energies
  Bajpai, Anshumaan; Mehta, Prateek; Frey, Kurt; Lehmer, Andrew M.; Schneider, William F.
  ACS Catalysis (2018), 8 (3), 1945-1954CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
  Adsorbate free energies are fundamental quantities in the microkinetic modeling of catalytic reactions. In first-principles modeling, finite-temp. free energies are generally obtained by combining d. functional theory energies with std. approx. models, such as the harmonic oscillator, the hindered translator, or the two-dimensional ideal gas. In this work, we calc. accurate free energies directly from first-principles potential energy surfaces combined with exact quantum mech. solns. for the translational energy states to benchmark the reliability of common approxns. Through a series of case studies of monat. adsorbates on metal surfaces, we show that no one free energy model performs satisfactorily in all cases. Moreover, even combinations of different approxns. sometimes deviate significantly from the free energies calcd. by our first-principles approach. Using observations from these case studies, we discuss how a full quantum mech. approach can be extended to calc. accurate free energies for arbitrary adsorbate potential energy surfaces at computational cost similar to std. models.
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14. 14
  Brogaard, R. Y.; Henry, R.; Schuurman, Y.; Medford, A. J.; Moses, P. G.; Beato, P.; Svelle, S.; Nørskov, J. K.; Olsbye, U. Methanol-to-hydrocarbons conversion: The alkene methylation pathway. J. Catal. 2014, 314, 159– 169, DOI: 10.1016/j.jcat.2014.04.006
  14
  Methanol-to-hydrocarbons conversion: The alkene methylation pathway
  Brogaard, Rasmus Y.; Henry, Reynald; Schuurman, Yves; Medford, Andrew J.; Moses, Poul Georg; Beato, Pablo; Svelle, Stian; Noerskov, Jens K.; Olsbye, Unni
  Journal of Catalysis (2014), 314 (), 159-169CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Inc.)
  Research on zeolite-catalyzed methanol-to-hydrocarbons (MTH) conversion has long been concerned with the mechanism of the reaction between methanol and alkenes. Two pathways were debated: (1) the stepwise, proceeding through a surface-methoxy intermediate and (2) the concerted, in which the alkenes react directly with methanol. This work addresses the debate through micro-kinetic modeling based on d. functional theory calcns. of both pathways, as well as expts. employing temporal anal. of products to study the kinetics of the stepwise pathway for alkenes in H-ZSM-22 zeolite. The model predicts the stepwise pathway to prevail at typical MTH reaction temps., due to a higher entropy loss in the concerted as compared to the stepwise pathway. The entropy difference results from intermediate release of water in the stepwise pathway. These findings lead one to suggest that the stepwise pathway should also be considered when modeling MTH conversion in other zeolites.
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15. 15
  Deringer, V. L.; Bartók, A. P.; Bernstein, N.; Wilkins, D. M.; Ceriotti, M.; Csányi, G. Gaussian Process Regression for Materials and Molecules. Chem. Rev. 2021, 121, 10073– 10141, DOI: 10.1021/acs.chemrev.1c00022
  15
  Gaussian Process Regression for Materials and Molecules
  Deringer, Volker L.; Bartok, Albert P.; Bernstein, Noam; Wilkins, David M.; Ceriotti, Michele; Csanyi, Gabor
  Chemical Reviews (Washington, DC, United States) (2021), 121 (16), 10073-10141CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
  A review. We provide an introduction to Gaussian process regression (GPR) machine-learning methods in computational materials science and chem. The focus of the present review is on the regression of atomistic properties: in particular, on the construction of interat. potentials, or force fields, in the Gaussian Approxn. Potential (GAP) framework; beyond this, we also discuss the fitting of arbitrary scalar, vectorial, and tensorial quantities. Methodol. aspects of ref. data generation, representation and regression, as well as the question how a data-driven model may be validated, are reviewed and critically discussed. A survey of applications to a variety of research questions in chem. and materials science illustrates the rapid growth in the field. A vision is outlined for the development of the methodol. in the years to come.
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16. 16
  Behler, J. Four Generations of High-Dimensional Neural Network Potentials. Chem. Rev. 2021, 121, 10037– 10072, DOI: 10.1021/acs.chemrev.0c00868
  16
  Four Generations of High-Dimensional Neural Network Potentials
  Behler, Joerg
  Chemical Reviews (Washington, DC, United States) (2021), 121 (16), 10037-10072CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
  A review. Since their introduction about 25 years ago, machine learning (ML) potentials have become an important tool in the field of atomistic simulations. After the initial decade, in which neural networks were successfully used to construct potentials for rather small mol. systems, the development of high-dimensional neural network potentials (HDNNPs) in 2007 opened the way for the application of ML potentials in simulations of large systems contg. thousands of atoms. To date, many other types of ML potentials have been proposed continuously increasing the range of problems that can be studied. In this review, the methodol. of the family of HDNNPs including new recent developments will be discussed using a classification scheme into four generations of potentials, which is also applicable to many other types of ML potentials. The first generation is formed by early neural network potentials designed for low-dimensional systems. High-dimensional neural network potentials established the second generation and are based on three key steps: first, the expression of the total energy as a sum of environment-dependent at. energy contributions; second, the description of the at. environments by atom-centered symmetry functions as descriptors fulfilling the requirements of rotational, translational, and permutation invariance; and third, the iterative construction of the ref. electronic structure data sets by active learning. In third-generation HDNNPs, in addn., long-range interactions are included employing environment-dependent partial charges expressed by at. neural networks. In fourth-generation HDNNPs, which are just emerging, in addn., nonlocal phenomena such as long-range charge transfer can be included. The applicability and remaining limitations of HDNNPs are discussed along with an outlook at possible future developments.
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17. 17
  Behler, J.; Csányi, G. Machine learning potentials for extended systems: a perspective. Eur. Phys. J. B 2021, 94, 142, DOI: 10.1140/epjb/s10051-021-00156-1
  17
  Machine learning potentials for extended systems: a perspective
  Behler, Joerg; Csanyi, Gabor
  European Physical Journal B: Condensed Matter and Complex Systems (2021), 94 (7), 142CODEN: EPJBFY; ISSN:1434-6028. (Springer)
  Abstr.: In the past two and a half decades machine learning potentials have evolved from a special purpose soln. to a broadly applicable tool for large-scale atomistic simulations. By combining the efficiency of empirical potentials and force fields with an accuracy close to first-principles calcns. they now enable computer simulations of a wide range of mols. and materials. In this perspective, we summarize the present status of these new types of models for extended systems, which are increasingly used for materials modeling. There are several approaches, but they all have in common that they exploit the locality of at. properties in some form. Long-range interactions, most prominently electrostatic interactions, can also be included even for systems in which non-local charge transfer leads to an electronic structure that depends globally on all at. positions. Remaining challenges and limitations of current approaches are discussed. Graphic Abstr.: [graphic not available: see fulltext].
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18. 18
  Bwoker, M. On the mechanism of ethanol synthesis on rhodium. Catal. Today 1992, 15, 77– 100, DOI: 10.1016/0920-5861(92)80123-5
  There is no corresponding record for this reference.
19. 19
  Yang, N.; Medford, A. J.; Liu, X.; Studt, F.; Bligaard, T.; Bent, S. F.; Nørskov, J. K. Intrinsic Selectivity and Structure Sensitivity of Rhodium Catalysts for C2+Oxygenate Production. J. Am. Chem. Soc. 2016, 138, 3705– 3714, DOI: 10.1021/jacs.5b12087
  19
  Intrinsic Selectivity and Structure Sensitivity of Rhodium Catalysts for C2+ Oxygenate Production
  Yang, Nuoya; Medford, Andrew J.; Liu, Xinyan; Studt, Felix; Bligaard, Thomas; Bent, Stacey F.; Noerskov, Jens K.
  Journal of the American Chemical Society (2016), 138 (11), 3705-3714CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
  Synthesis gas (CO + H2) conversion is a promising route to converting coal, natural gas, or biomass into synthetic liq. fuels. Rhodium has long been studied as it is the only elemental catalyst that has demonstrated selectivity to ethanol and other C2+ oxygenates. However, the fundamentals of syngas conversion over rhodium are still debated. In this work a microkinetic model is developed for conversion of CO and H2 into methane, ethanol, and acetaldehyde on the Rh (211) and (111) surfaces, chosen to describe steps and close-packed facets on catalyst particles. The model is based on DFT calcns. using the BEEF-vdW functional. The mean-field kinetic model includes lateral adsorbate-adsorbate interactions, and the BEEF-vdW error estn. ensemble is used to propagate error from the DFT calcns. to the predicted rates. The model shows the Rh(211) surface to be ∼6 orders of magnitude more active than the Rh(111) surface, but highly selective toward methane, while the Rh(111) surface is intrinsically selective toward acetaldehyde. A variety of Rh/SiO2 catalysts are synthesized, tested for catalytic oxygenate prodn., and characterized using TEM. The exptl. results indicate that the Rh(111) surface is intrinsically selective toward acetaldehyde, and a strong inverse correlation between catalytic activity and oxygenate selectivity is obsd. Furthermore, iron impurities are shown to play a key role in modulating the selectivity of Rh/SiO2 catalysts toward ethanol. The exptl. observations are consistent with the structure-sensitivity predicted from theory. This work provides an improved at.-scale understanding and new insight into the mechanism, active site, and intrinsic selectivity of syngas conversion over rhodium catalysts and may also guide rational design of alloy catalysts made from more abundant elements.
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20. 20
  Ulissi, Z. W.; Medford, A. J.; Bligaard, T.; Nørskov, J. K. To address surface reaction network complexity using scaling relations machine learning and DFT calculations. Nat. Commun. 2017, 8, 14621, DOI: 10.1038/ncomms14621
  20
  To address surface reaction network complexity using scaling relations machine learning and DFT calculations
  Ulissi Zachary W; Norskov Jens K; Medford Andrew J; Bligaard Thomas
  Nature communications (2017), 8 (), 14621 ISSN:.
  Surface reaction networks involving hydrocarbons exhibit enormous complexity with thousands of species and reactions for all but the very simplest of chemistries. We present a framework for optimization under uncertainty for heterogeneous catalysis reaction networks using surrogate models that are trained on the fly. The surrogate model is constructed by teaching a Gaussian process adsorption energies based on group additivity fingerprints, combined with transition-state scaling relations and a simple classifier for determining the rate-limiting step. The surrogate model is iteratively used to predict the most important reaction step to be calculated explicitly with computationally demanding electronic structure theory. Applying these methods to the reaction of syngas on rhodium(111), we identify the most likely reaction mechanism. Propagating uncertainty throughout this process yields the likelihood that the final mechanism is complete given measurements on only a subset of the entire network and uncertainty in the underlying density functional theory calculations.
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21. 21
  Deimel, M.; Prats, H.; Seibt, M.; Reuter, K.; Andersen, M. Selectivity Trends and Role of Adsorbate–Adsorbate Interactions in CO Hydrogenation on Rhodium Catalysts. ACS Catal. 2022, 12, 7907– 7917, DOI: 10.1021/acscatal.2c02353
  21
  Selectivity Trends and Role of Adsorbate-Adsorbate Interactions in CO Hydrogenation on Rhodium Catalysts
  Deimel, Martin; Prats, Hector; Seibt, Michael; Reuter, Karsten; Andersen, Mie
  ACS Catalysis (2022), 12 (13), 7907-7917CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
  Predictive-quality computational modeling of heterogeneously catalyzed reactions has emerged as an important tool for the anal. and assessment of activity and activity trends. In contrast, more subtle selectivities and selectivity trends still pose a significant challenge to prevalent microkinetic modeling approaches that typically employ a mean-field approxn. (MFA). Here, we focus on CO hydrogenation on Rh catalysts with the possible products methane, acetaldehyde, ethanol, and water. This reaction has already been subjected to a no. of exptl. and theor. studies with conflicting views on the factors controlling activity and selectivity toward the more valuable higher oxygenates. Using accelerated first-principles kinetic Monte Carlo simulations and explicitly and systematically accounting for adsorbate-adsorbate interactions through a cluster expansion approach, we model the reaction on the low-index Rh(111) and stepped Rh(211) surfaces. We find that the Rh(111) facet is selective toward methane, while the Rh(211) facet exhibits a similar selectivity toward methane and acetaldehyde. This is consistent with the exptl. selectivity obsd. for larger, predominantly (111)-exposing Rh nanoparticles and resolves the discrepancy with earlier first-principles MFA microkinetic work that found the Rh(111) facet to be selective toward acetaldehyde. While the latter work tried to approx. account for lateral interactions through coverage-dependent rate expressions, our anal. demonstrates that this fails to sufficiently capture concomitant correlations among the adsorbed reaction intermediates that crucially det. the overall selectivity.
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22. 22
  Kästner, J.; Thiel, W. Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: “Umbrella integration. J. Chem. Phys. 2005, 123, 144104, DOI: 10.1063/1.2052648
  22
  Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: "Umbrella integration"
  Kastner, Johannes; Thiel, Walter
  Journal of Chemical Physics (2005), 123 (14), 144104/1-144104/5CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We present a method to analyze biased mol.-dynamics and Monte Carlo simulations, also known as umbrella sampling. In the limiting case of a strong bias, this method is equiv. to thermodn. integration. It employs only quantities with easily controllable equilibration and greatly reduces the statistical errors compared to the std. weighted histogram anal. method. We show the success of our approach for two examples, one analytic function, and one biol. system.
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23. 23
  Kästner, J.; Thiel, W. Analysis of the statistical error in umbrella sampling simulations by umbrella integration. J. Chem. Phys. 2006, 124, 234106, DOI: 10.1063/1.2206775
  23
  Analysis of the statistical error in umbrella sampling simulations by umbrella integration
  Kastner, Johannes; Thiel, Walter
  Journal of Chemical Physics (2006), 124 (23), 234106/1-234106/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Umbrella sampling simulations, or biased mol. dynamics, can be used to calc. the free-energy change of a chem. reaction. We investigate the sources of different sampling errors and derive approx. expressions for the statistical errors when using harmonic restraints and umbrella integration anal. This leads to generally applicable rules for the choice of the bias potential and the sampling parameters. Numerical results for simulations on an anal. model potential are presented for validation. While the derivations are based on umbrella integration anal., the final error est. is evaluated from the raw simulation data, and it may therefore be generally applicable as indicated by tests using the weighted histogram anal. method.
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24. 24
  Roux, B. The calculation of the potential of mean force using computer simulations. Comput. Phys. Commun. 1995, 91, 275– 282, DOI: 10.1016/0010-4655(95)00053-I
  24
  The calculation of the potential of mean force using computer simulations
  Roux, Benoit
  Computer Physics Communications (1995), 91 (1-3), 275-82CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier)
  The problem of unbiasing and combining the results of umbrella sampling calcns. is reviewed. The weighted histogram anal. method (WHAM) of S. Kumar et al. (J. Comp. Chem. 13 (1992) 1011) is described and compared with other approaches. The method is illustrated with mol. dynamics simulations of the alanine dipeptide for one- and two-dimensional free energy surfaces. The results show that the WHAM approach simplifies considerably the task of recombining the various windows in complex systems.
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25. 25
  Kumar, S.; Rosenberg, J. M.; Bouzida, D.; Swendsen, R. H.; Kollman, P. A. The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J. Comput. Chem. 1992, 13, 1011– 1021, DOI: 10.1002/jcc.540130812
  25
  The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method
  Kumar, Shankar; Bouzida, Djamal; Swendsen, Robert H.; Kollman, Peter A.; Rosenberg, John M.
  Journal of Computational Chemistry (1992), 13 (8), 1011-21CODEN: JCCHDD; ISSN:0192-8651.
  The Weighted Histogram Anal. Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomol. Hamiltonians. The method is presented here as an extension of the Umbrella Sampling method for free-energy and Potential of Mean Force calcns. This algorithm possesses the following advantages over methods that are currently employed: (1) it provides a built-in est. of sampling errors thereby yielding objective ests. of the optimal location and length of addnl. simulations needed to achieve a desired level of precision; (2) it yields the "best" value of free energies by taking into account all the simulations so as to minimize the statistical errors; (3) in addn. to optimizing the links between simulations, it also allows multiple overlaps of probability distributions for obtaining better ests. of the free-energy differences. By recasting the Ferrenberg-Swendsen Multiple Histogram equations in a form suitable for mol. mechanics type Hamiltonians, we have demonstrated the feasibility and robustness of this method by applying it to a test problem of the generation of the Potential of Mean Force profile of the pseudorotation phase angle of the sugar ring in deoxyadenosine.
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26. 26
  Hub, J. S.; De Groot, B. L.; Van der Spoel, D. g_wham─A Free Weighted Histogram Analysis Implementation Including Robust Error and Autocorrelation Estimates. J. Chem. Theory Comput. 2010, 6, 3713– 3720, DOI: 10.1021/ct100494z
  26
  g_wham: A Free Weighted Histogram Analysis Implementation Including Robust Error and Autocorrelation Estimates
  Hub, Jochen S.; de Groot, Bert L.; van der Spoel, David
  Journal of Chemical Theory and Computation (2010), 6 (12), 3713-3720CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  The Weighted Histogram Anal. Method (WHAM) is a std. technique used to compute potentials of mean force (PMFs) from a set of umbrella sampling simulations. Here, the authors present a new WHAM implementation, termed g_wham, which is distributed freely with the GROMACS mol. simulation suite. G_wham ests. statistical errors using the technique of bootstrap anal. Three bootstrap methods are supported: (i) bootstrapping new trajectories based on the umbrella histograms, (ii) bootstrapping of complete histograms, and (iii) Bayesian bootstrapping of complete histograms, i.e., bootstrapping via the assignment of random wts. to the histograms. Because methods ii and iii consider only complete histograms as independent data points, these methods do not require the accurate calcn. of autocorrelation times. The authors demonstrate that, given sufficient sampling, bootstrapping new trajectories allows for an accurate error est. In the presence of long autocorrelations, however, (Bayesian) bootstrapping of complete histograms yields a more reliable error est., whereas bootstrapping of new trajectories may underestimate the error. In addn., the authors emphasize that the incorporation of autocorrelations into WHAM reduces the bias from limited sampling, in particular, when computing periodic PMFs in inhomogeneous systems such as solvated lipid membranes or protein channels.
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27. 27
  Stecher, T.; Bernstein, N.; Csányi, G. Free Energy Surface Reconstruction from Umbrella Samples Using Gaussian Process Regression. J. Chem. Theory Comput. 2014, 10, 4079– 4097, DOI: 10.1021/ct500438v
  27
  Free Energy Surface Reconstruction from Umbrella Samples Using Gaussian Process Regression
  Stecher, Thomas; Bernstein, Noam; Csanyi, Gabor
  Journal of Chemical Theory and Computation (2014), 10 (9), 4079-4097CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We demonstrate how the Gaussian process regression approach can be used to efficiently reconstruct free energy surfaces from umbrella sampling simulations. By making a prior assumption of smoothness and taking account of the sampling noise in a consistent fashion, we achieve a significant improvement in accuracy over the state of the art in two or more dimensions or, equivalently, a significant cost redn. to obtain the free energy surface within a prescribed tolerance in both regimes of spatially sparse data and short sampling trajectories. Stemming from its Bayesian interpretation the method provides meaningful error bars without significant addnl. computation. A software implementation is made available on www.libatoms.org.
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28. 28
  Hjorth Larsen, A.; Jørgen Mortensen, J.; Blomqvist, J.; Castelli, I. E.; Christensen, R.; Dułak, M.; Friis, J.; Groves, M. N.; Hammer, B.; Hargus, C.; Hermes, E. D.; Jennings, P. C.; Bjerre Jensen, P.; Kermode, J.; Kitchin, J. R.; Leonhard Kolsbjerg, E.; Kubal, J.; Kaasbjerg, K.; Lysgaard, S.; Bergmann Maronsson, J.; Maxson, T.; Olsen, T.; Pastewka, L.; Peterson, A.; Rostgaard, C.; Schiøtz, J.; Schütt, O.; Strange, M.; Thygesen, K. S.; Vegge, T.; Vilhelmsen, L.; Walter, M.; Zeng, Z.; Jacobsen, K. W. The atomic simulation environment─a Python library for working with atoms. J. Phys.: Condens. Matter 2017, 29, 273002, DOI: 10.1088/1361-648X/aa680e
  28
  The atomic simulation environment-a Python library for working with atoms
  Hjorth Larsen Ask; Jorgen Mortensen Jens; Blomqvist Jakob; Castelli Ivano E; Christensen Rune; Dulak Marcin; Friis Jesper; Groves Michael N; Hammer Bjork; Hargus Cory; Hermes Eric D; Jennings Paul C; Bjerre Jensen Peter; Kermode James; Kitchin John R; Leonhard Kolsbjerg Esben; Kubal Joseph; Kaasbjerg Kristen; Lysgaard Steen; Bergmann Maronsson Jon; Maxson Tristan; Olsen Thomas; Pastewka Lars; Peterson Andrew; Rostgaard Carsten; Schiotz Jakob; Schutt Ole; Strange Mikkel; Thygesen Kristian S; Vegge Tejs; Vilhelmsen Lasse; Walter Michael; Zeng Zhenhua; Jacobsen Karsten W
  Journal of physics. Condensed matter : an Institute of Physics journal (2017), 29 (27), 273002 ISSN:.
  The atomic simulation environment (ASE) is a software package written in the Python programming language with the aim of setting up, steering, and analyzing atomistic simulations. In ASE, tasks are fully scripted in Python. The powerful syntax of Python combined with the NumPy array library make it possible to perform very complex simulation tasks. For example, a sequence of calculations may be performed with the use of a simple 'for-loop' construction. Calculations of energy, forces, stresses and other quantities are performed through interfaces to many external electronic structure codes or force fields using a uniform interface. On top of this calculator interface, ASE provides modules for performing many standard simulation tasks such as structure optimization, molecular dynamics, handling of constraints and performing nudged elastic band calculations.
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29. 29
  Bartók, A. P.; Payne, M. C.; Kondor, R.; Csányi, G. Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons. Phys. Rev. Lett. 2010, 104, 136403, DOI: 10.1103/PhysRevLett.104.136403
  29
  Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons
  Bartok, Albert P.; Payne, Mike C.; Kondor, Risi; Csanyi, Gabor
  Physical Review Letters (2010), 104 (13), 136403/1-136403/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  We introduce a class of interat. potential models that can be automatically generated from data consisting of the energies and forces experienced by atoms, as derived from quantum mech. calcns. The models do not have a fixed functional form and hence are capable of modeling complex potential energy landscapes. They are systematically improvable with more data. We apply the method to bulk crystals, and test it by calcg. properties at high temps. Using the interat. potential to generate the long mol. dynamics trajectories required for such calcns. saves orders of magnitude in computational cost.
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30. 30
  Bartók, A. P.; Kondor, R.; Csányi, G. On representing chemical environments. Phys. Rev. B 2013, 87, 184115, DOI: 10.1103/PhysRevB.87.184115
  30
  On representing chemical environments
  Bartok, Albert P.; Kondor, Risi; Csanyi, Gabor
  Physical Review B: Condensed Matter and Materials Physics (2013), 87 (18), 184115/1-184115/16CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
  We review some recently published methods to represent at. neighborhood environments, and analyze their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave nos. are used to expand the at. neighborhood d. function. Using the example system of small clusters, we quant. show that this expansion needs to be carried to higher and higher wave nos. as the no. of neighbors increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether different approach, called Smooth Overlap of Atomic Positions, that sidesteps these difficulties by directly defining the similarity between any two neighborhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal.
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31. 31
  Bartók, A. P.; De, S.; Poelking, C.; Bernstein, N.; Kermode, J. R.; Csányi, G.; Ceriotti, M. Machine learning unifies the modeling of materials and molecules. Sci. Adv. 2017, 3, e1701816, DOI: 10.1126/sciadv.1701816
  31
  Machine learning unifies the modeling of materials and molecules
  Bartok, Albert P.; De, Sandip; Poelking, Carl; Bernstein, Noam; Kermode, James R.; Csanyi, Gabor; Ceriotti, Michele
  Science Advances (2017), 3 (12), e1701816/1-e1701816/8CODEN: SACDAF; ISSN:2375-2548. (American Association for the Advancement of Science)
  Detg. the stability of mols. and condensed phases is the cornerstone of atomistic modeling, underpinning our understanding of chem. and materials properties and transformations. We show that a machine-learning model, based on a local description of chem. environments and Bayesian statistical learning, provides a unified framework to predict at.-scale properties. It captures the quantum mech. effects governing the complex surface reconstructions of silicon, predicts the stability of different classes of mols. with chem. accuracy, and distinguishes active and inactive protein ligands with more than 99% reliability. The universality and the systematic nature of our framework provide new insight into the potential energy surface of materials and mols.
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32. 32
  Hammer, B.; Hansen, L. B.; Nørskov, J. K. Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals. Phys. Rev. B 1999, 59, 7413– 7421, DOI: 10.1103/PhysRevB.59.7413
  32
  Improved adsorption energetics within density-functional theory using revised Perdew-Burke-Ernzerhof functionals
  Hammer, B.; Hansen, L. B.; Norskov, J. K.
  Physical Review B: Condensed Matter and Materials Physics (1999), 59 (11), 7413-7421CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
  A simple formulation of a generalized gradient approxn. for the exchange and correlation energy of electrons has been proposed by J. Perdew et al. (1996). Subsequently, Y. Zhang and W. Wang (1998) have shown that a slight revision of the Perdew-Burke-Ernzerhof (PBE) functional systematically improves the atomization energies for a large database of small mols. In the present work, we show that the Zhang and Yang functional (revPBE) also improves the chemisorption energetics of atoms and mols. on transition-metal surfaces. Our test systems comprise at. and mol. adsorption of oxygen, CO, and NO on Ni(100), Ni(111), Rh(100), Pd(100), and Pd(111) surfaces. As the revPBE functional may locally violate the Lieb-Oxford criterion, we further develop an alternative revision of the PBE functional, RPBE, which gives the same improvement of the chemisorption energies as the revPBE functional at the same time as it fulfills the Lieb-Oxford criterion locally.
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33. 33
  Wellendorff, J.; Silbaugh, T. L.; Garcia-Pintos, D.; Nørskov, J. K.; Bligaard, T.; Studt, F.; Campbell, C. T. A benchmark database for adsorption bond energies to transition metal surfaces and comparison to selected DFT functionals. Surf. Sci. 2015, 640, 36– 44, DOI: 10.1016/j.susc.2015.03.023
  33
  A benchmark database for adsorption bond energies to transition metal surfaces and comparison to selected DFT functionals
  Wellendorff, Jess; Silbaugh, Trent L.; Garcia-Pintos, Delfina; Noerskov, Jens K.; Bligaard, Thomas; Studt, Felix; Campbell, Charles T.
  Surface Science (2015), 640 (), 36-44CODEN: SUSCAS; ISSN:0039-6028. (Elsevier B.V.)
  A literature collection of exptl. adsorption energies over late transition metal surfaces is presented for systems where the authors believe that the energy measurements are particularly accurate, and the at.-scale adsorption geometries are particularly well established. This could become useful for benchmarking theor. methods for calcg. adsorption processes. The authors compare the exptl. results to six commonly used electron d. functionals, including some (RPBE, BEEF-vdW) which were specifically developed to treat adsorption processes. The comparison shows that there is ample room for improvements in the theor. descriptions.
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34. 34
  Zhang, Y.; Yang, W. Comment on “Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1998, 80, 890– 890, DOI: 10.1103/PhysRevLett.80.890
  34
  Comment on "Generalized Gradient Approximation Made Simple"
  Zhang, Yingkai; Yang, Weitao
  Physical Review Letters (1998), 80 (4), 890CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  A Comment on the Letter by John P. Perdew, Kieron Burke, and Matthias Ernzerhof, Phys. 77, 3865 (1996). The authors of the Letter offer a Reply.
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35. 35
  Ruiz, V. G.; Liu, W.; Tkatchenko, A. Density-functional theory with screened van der Waals interactions applied to atomic and molecular adsorbates on close-packed and non-close-packed surfaces. Phys. Rev. B 2016, 93, 035118, DOI: 10.1103/PhysRevB.93.035118
  35
  Density-functional theory with screened van der Waals interactions applied to atomic and molecular adsorbates on close-packed and non-close-packed surfaces
  Ruiz, Victor G.; Liu, Wei; Tkatchenko, Alexandre
  Physical Review B (2016), 93 (3), 035118/1-035118/17CODEN: PRBHB7; ISSN:2469-9950. (American Physical Society)
  Modeling the adsorption of atoms and mols. on surfaces requires efficient electronic-structure methods that are able to capture both covalent and noncovalent interactions in a reliable manner. In order to tackle this problem, we have developed a method within d.-functional theory (DFT) to model screened van der Waals interactions (vdW) for atoms and mols. on surfaces (the so-called DFT+vdWsurf method). The relatively high accuracy of the DFT+vdWsurf method in the calcn. of both adsorption distances and energies, as well as the high degree of its reliability across a wide range of adsorbates, indicates the importance of the collective electronic effects within the extended substrate for the calcn. of the vdW energy tail. We examine in detail the theor. background of the method and assess its performance for adsorption phenomena including the physisorption of Xe on selected close-packed transition metal surfaces and 3,4,9,10-perylene-tetracarboxylic acid dianhydride (PTCDA) on Au(111). We also address the performance of DFT+vdWsurf in the case of non-close-packed surfaces by studying the adsorption of Xe on Cu(110) and the interfaces formed by the adsorption of a PTCDA monolayer on the Ag(111), Ag(100), and Ag(110) surfaces. We conclude by discussing outstanding challenges in the modeling of vdW interactions for studying at. and mol. adsorbates on inorg. substrates.
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36. 36
  Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab initio molecular simulations with numeric atom-centered orbitals. Comput. Phys. Commun. 2009, 180, 2175– 2196, DOI: 10.1016/j.cpc.2009.06.022
  36
  Ab initio molecular simulations with numeric atom-centered orbitals
  Blum, Volker; Gehrke, Ralf; Hanke, Felix; Havu, Paula; Havu, Ville; Ren, Xinguo; Reuter, Karsten; Scheffler, Matthias
  Computer Physics Communications (2009), 180 (11), 2175-2196CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
  We describe a complete set of algorithms for ab initio mol. simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of mol. and materials properties from quantum-mech. first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute "ab initio mol. simulations" (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on d.-functional theory (DFT) in the local and semilocal (generalized gradient) approxns., but an extension to hybrid functionals, Hartree-Fock theory, and MP2/GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron/full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio mol. dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calcn. to range from qual. tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs.
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37. 37
  Wellendorff, J.; Lundgaard, K. T.; Møgelhøj, A.; Petzold, V.; Landis, D. D.; Nørskov, J. K.; Bligaard, T.; Jacobsen, K. W. Density functionals for surface science: Exchange-correlation model development with Bayesian error estimation. Phys. Rev. B 2012, 85, 235149, DOI: 10.1103/PhysRevB.85.235149
  37
  Density functionals for surface science: exchange-correlation model development with Bayesian error estimation
  Wellendorff, Jess; Lundgaard, Keld T.; Moegelhoej, Andreas; Petzold, Vivien; Landis, David D.; Noerskov, Jens K.; Bligaard, Thomas; Jacobsen, Karsten W.
  Physical Review B: Condensed Matter and Materials Physics (2012), 85 (23), 235149/1-235149/23CODEN: PRBMDO; ISSN:1098-0121. (American Physical Society)
  A methodol. for semiempirical d. functional optimization, using regularization and cross-validation methods from machine learning, is developed. We demonstrate that such methods enable well-behaved exchange-correlation approxns. in very flexible model spaces, thus avoiding the overfitting found when std. least-squares methods are applied to high-order polynomial expansions. A general-purpose d. functional for surface science and catalysis studies should accurately describe bond breaking and formation in chem., solid state physics, and surface chem., and should preferably also include van der Waals dispersion interactions. Such a functional necessarily compromises between describing fundamentally different types of interactions, making transferability of the d. functional approxn. a key issue. We investigate this trade-off between describing the energetics of intramol. and intermol., bulk solid, and surface chem. bonding, and the developed optimization method explicitly handles making the compromise based on the directions in model space favored by different materials properties. The approach is applied to designing the Bayesian error estn. functional with van der Waals correlation (BEEF-vdW), a semilocal approxn. with an addnl. nonlocal correlation term. Furthermore, an ensemble of functionals around BEEF-vdW comes out naturally, offering an est. of the computational error. An extensive assessment on a range of data sets validates the applicability of BEEF-vdW to studies in chem. and condensed matter physics. Applications of the approxn. and its Bayesian ensemble error est. to two intricate surface science problems support this.
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38. 38
  Giannozzi, P.; Andreussi, O.; Brumme, T.; Bunau, O.; Buongiorno Nardelli, M.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Cococcioni, M.; Colonna, N.; Carnimeo, I.; Dal Corso, A.; De Gironcoli, S.; Delugas, P.; Distasio, R. A.; Ferretti, A.; Floris, A.; Fratesi, G.; Fugallo, G.; Gebauer, R.; Gerstmann, U.; Giustino, F.; Gorni, T.; Jia, J.; Kawamura, M.; Ko, H.-Y.; Kokalj, A.; Küçükbenli, E.; Lazzeri, M.; Marsili, M.; Marzari, N.; Mauri, F.; Nguyen, N. L.; Nguyen, H.-V.; Otero-de-la Roza, A.; Paulatto, L.; Poncé, S.; Rocca, D.; Sabatini, R.; Santra, B.; Schlipf, M.; Seitsonen, A. P.; Smogunov, A.; Timrov, I.; Thonhauser, T.; Umari, P.; Vast, N.; Wu, X.; Baroni, S. Advanced capabilities for materials modelling with Quantum Espresso. J. Phys.: Condens. Matter 2017, 29, 465901, DOI: 10.1088/1361-648X/aa8f79
  38
  Advanced capabilities for materials modelling with QUANTUM ESPRESSO
  Giannozzi, P.; Andreussi, O.; Brumme, T.; Bunau, O.; Buongiorno Nardelli, M.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Cococcioni, M.; Colonna, N.; Carnimeo, I.; Dal Corso, A.; de Gironcoli, S.; Delugas, P.; Di Stasio, R. A., Jr.; Ferretti, A.; Floris, A.; Fratesi, G.; Fugallo, G.; Gebauer, R.; Gerstmann, U.; Giustino, F.; Gorni, T.; Jia, J.; Kawamura, M.; Ko, H.-Y.; Kokalj, A.; Kucukbenli, E.; Lazzeri, M.; Marsili, M.; Marzari, N.; Mauri, F.; Nguyen, N. L.; Nguyen, H.-V.; Otero-de-la-Roza, A.; Paulatto, L.; Ponce, S.; Rocca, D.; Sabatini, R.; Santra, B.; Schlipf, M.; Seitsonen, A. P.; Smogunov, A.; Timrov, I.; Thonhauser, T.; Umari, P.; Vast, N.; Wu, X.; Baroni, S.
  Journal of Physics: Condensed Matter (2017), 29 (46), 465901/1-465901/30CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
  QUANTUM ESPRESSO is an integrated suite of open-source computer codes for quantum simulations of materials using state-of-the-art electronic-structure techniques, based on d.-functional theory, d.-functional perturbation theory, and many-body perturbation theory, within the plane-wave pseudopotential and projector-augmented-wave approaches. QUANTUM ESPRESSO owes its popularity to the wide variety of properties and processes it allows to simulate, to its performance on an increasingly broad array of hardware architectures, and to a community of researchers that rely on its capabilities as a core open-source development platform to implement their ideas. In this paper we describe recent extensions and improvements, covering new methodologies and property calculators, improved parallelization, code modularization, and extended interoperability both within the distribution and with external software.
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39. 39
  Hermes, E.; Sargsyan, K.; Najm, H.; Zádor, J. Sella, an open-source automation-friendly molecular saddle point optimizer. ChemRxiv, DOI: 10.26434/chemrxiv-2022-44r17 , 2022.
  There is no corresponding record for this reference.
40. 40
  Stocker, S.; Gasteiger, J.; Becker, F.; Günnemann, S.; Margraf, J. T. How robust are modern graph neural network potentials in long and hot molecular dynamics simulations?. Mach. Learn.: Sci. Technol. 2022, 3, 045010, DOI: 10.1088/2632-2153/ac9955
  There is no corresponding record for this reference.
41. 41
  Pitzer, K. S.; Gwinn, W. D. Energy Levels and Thermodynamic Functions for Molecules with Internal Rotation I. Rigid Frame with Attached Tops. J. Chem. Phys. 1942, 10, 428– 440, DOI: 10.1063/1.1723744
  41
  Energy levels and thermodynamic functions for molecules with internal rotation. I. Rigid frame with attached tops
  Pitzer, Kenneth S.; Gwinn, Wm. D.
  Journal of Chemical Physics (1942), 10 (), 428-40CODEN: JCPSA6; ISSN:0021-9606.
  Math.
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42. 42
  Mallikarjun Sharada, S.; Bligaard, T.; Luntz, A. C.; Kroes, G.-J.; Nørskov, J. K. SBH10: A Benchmark Database of Barrier Heights on Transition Metal Surfaces. J. Phys. Chem. C 2017, 121, 19807– 19815, DOI: 10.1021/acs.jpcc.7b05677
  42
  SBH10: A Benchmark Database of Barrier Heights on Transition Metal Surfaces
  Mallikarjun Sharada, Shaama; Bligaard, Thomas; Luntz, Alan C.; Kroes, Geert-Jan; Noerskov, Jens K.
  Journal of Physical Chemistry C (2017), 121 (36), 19807-19815CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
  While the performance of d. functional approxns. (DFAs) for gas phase reaction energetics has been extensively benchmarked, their reliability for activation barriers on surfaces is not fully understood. The primary reason for this is the absence of well-defined, chem. accurate benchmark databases for chem. on surfaces. We present a database of 10 surface barrier heights for dissocn. of small mols., SBH10, based on carefully chosen refs. from mol. beam scattering, laser assisted associative desorption, and thermal expts. Our benchmarking study compares the performance of a dispersion-cor. generalized gradient approxn. (GGA-vdW), BEEF-vdW, a meta-GGA, MS2, and a screened hybrid functional, HSE06. In stark contrast to gas phase reactions for which GGAs systematically underestimate barrier heights and hybrids tend to be most accurate, the BEEF-vdW functional dets. barriers accurately to within 0.14 eV of expts., while MS2 and HSE06 underestimate barrier heights on surfaces. Higher accuracy of BEEF-vdW stems from the fact that the functional is trained on chemisorption systems, and transition states for dissocn. on surfaces closely resemble the final, chemisorbed states. Therefore, a functional that can describe chemisorption accurately can also reliably predict barrier heights on surfaces.
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43. 43
  Caro, M. A. Parametrization of the Tkatchenko-Scheffler dispersion correction scheme for popular exchange-correlation density functionals: effect on the description of liquid water. arXiv , 1704.00761v2, 2017.
  There is no corresponding record for this reference.
44. 44
  Hörmann, L.; Jeindl, A.; Hofmann, O. T. Reproducibility of potential energy surfaces of organic/metal interfaces on the example of PTCDA on Ag(111). J. Chem. Phys. 2020, 153, 104701, DOI: 10.1063/5.0020736
  44
  Reproducibility of potential energy surfaces of organic/metal interfaces on the example of PTCDA on Ag(111)
  Hormann Lukas; Jeindl Andreas; Hofmann Oliver T
  The Journal of chemical physics (2020), 153 (10), 104701 ISSN:.
  Molecular adsorption at organic/metal interfaces depends on a range of mechanisms: covalent bonds, charge transfer, Pauli repulsion, and van der Waals (vdW) interactions shape the potential energy surface (PES), making it key to understanding organic/metal interfaces. Describing such interfaces with density functional theory requires carefully selecting the exchange correlation (XC) functional and vdW correction scheme. To explore the reproducibility of the PES with respect to the choice of method, we present a benchmark of common local, semi-local, and non-local XC functionals in combination with various vdW corrections. We benchmark these methods using perylene-tetracarboxylic dianhydride on Ag(111), one of the most frequently studied organic/metal interfaces. For each method, we determine the PES using a Gaussian process regression algorithm, which requires only about 50 density functional theory calculations as input. This allows a detailed analysis of the PESs' features, such as the positions and energies of minima and saddle points. Comparing the results from different combinations of XC functionals and vdW corrections enables us to identify trends and differences between the approaches. PESs for different computation methods are in qualitative agreement but also display significant quantitative differences. In particular, the lateral positions of adsorption geometries agree well with experiment, while adsorption heights, energies, and barriers show larger discrepancies.
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45. 45
  Dietschreit, J. C. B.; Diestler, D. J.; Hulm, A.; Ochsenfeld, C.; Gómez-Bombarelli, R. From free-energy profiles to activation free energies. J. Chem. Phys. 2022, 157, 084113, DOI: 10.1063/5.0102075
  45
  From free-energy profiles to activation free energies
  Dietschreit, Johannes C. B.; Diestler, Dennis J.; Hulm, Andreas; Ochsenfeld, Christian; Gomez-Bombarelli, Rafael
  Journal of Chemical Physics (2022), 157 (8), 084113CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Given a chem. reaction going from reactant (R) to the product (P) on a potential energy surface (PES) and a collective variable (CV) discriminating between R and P, we define the free-energy profile (FEP) as the logarithm of the marginal Boltzmann distribution of the CV. This FEP is not a true free energy. Nevertheless, it is common to treat the FEP as the "free-energy" analog of the min. potential energy path and to take the activation free energy, ΔF‡RP, as the difference between the max. at the transition state and the min. at R. We show that this approxn. can result in large errors. The FEP depends on the CV and is, therefore, not unique. For the same reaction, different discriminating CVs can yield different ΔF‡RP. We derive an exact expression for the activation free energy that avoids this ambiguity. We find ΔF‡RP to be a combination of the probability of the system being in the reactant state, the probability d. on the dividing surface, and the thermal de Broglie wavelength assocd. with the transition. We apply our formalism to simple analytic models and realistic chem. systems and show that the FEP-based approxn. applies only at low temps. for CVs with a small effective mass. Most chem. reactions occur on complex, high-dimensional PES that cannot be treated anal. and pose the added challenge of choosing a good CV. We study the influence of that choice and find that, while the reaction free energy is largely unaffected, ΔF‡RP is quite sensitive. (c) 2022 American Institute of Physics.
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46. 46
  Goldsmith, C. F.; Harding, L. B.; Georgievskii, Y.; Miller, J. A.; Klippenstein, S. J. Temperature and Pressure-Dependent Rate Coefficients for the Reaction of Vinyl Radical with Molecular Oxygen. J. Phys. Chem. A 2015, 119, 7766– 7779, DOI: 10.1021/acs.jpca.5b01088
  46
  Temperature and Pressure-Dependent Rate Coefficients for the Reaction of Vinyl Radical with Molecular Oxygen
  Goldsmith, C. Franklin; Harding, Lawrence B.; Georgievskii, Yuri; Miller, James A.; Klippenstein, Stephen J.
  Journal of Physical Chemistry A (2015), 119 (28), 7766-7779CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
  State-of-the-art calcns. of the C2H3O2 potential energy surface are presented. A new method is described for computing the interaction potential for R + O2 reactions. The method, which combines accurate detn. of the quartet potential along the doublet min. energy path with multireference calcns. of the doublet/quartet splitting, decreases the uncertainty in the doublet potential and thence the rate consts. by more than a factor of 2. The temp.- and pressure-dependent rate coeffs. are computed using variable reaction coordinate transition-state theory, variational transition-state theory, and conventional transition-state theory, as implemented in a new RRKM/ME code. The main bimol. product channels are CH2O + HCO at lower temps. and CH2CHO + O at higher temps. Above 10 atm, the collisional stabilization of CH2CHOO directly competes with these two product channels. CH2CHOO decomps. primarily to CH2O + HCO. The next two most significant bimol. products are OCHCHO + H and 3CHCHO + OH, and not C2H2 + HO2. C2H3 + O2 will be predominantly chain branching above 1700 K. Uncertainty anal. is presented for the two most important transition states. The uncertainties in these two barrier heights result in a significant uncertainty in the temp. at which CH2CHO + O overtakes all other product channels.
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47. 47
  Markland, T. E.; Ceriotti, M. Nuclear quantum effects enter the mainstream. Nat. Rev. Chem. 2018, 2, 0109, DOI: 10.1038/s41570-017-0109
  47
  Nuclear quantum effects enter the mainstream
  Markland, Thomas E.; Ceriotti, Michele
  Nature Reviews Chemistry (2018), 2 (3), 0109CODEN: NRCAF7; ISSN:2397-3358. (Nature Research)
  A review. Atomistic simulations of chem., biol. and materials systems have become increasingly precise and predictive owing to the development of accurate and efficient techniques that describe the quantum mech. behavior of electrons. Nevertheless, the overwhelming majority of such simulations still assumes that the nuclei behave as classical particles. Historically, this approxn. could sometimes be justified owing to the complexity and computational overhead. However, neglecting nuclear quantum effects has become one of the largest sources of error, esp. when systems contg. light atoms are treated using current state-of-the-art descriptions of chem. interactions. Over the past decade, this realization has spurred a series of methodol. advances that have dramatically reduced the cost of including these important phys. effects in the structure and dynamics of chem. systems. Here, we discuss how these developments are now allowing nuclear quantum effects to become a mainstream feature of mol. simulations. These advances have led to new insights into phenomena that are relevant to different areas of science - from biochem. to condensed matter - and open the door to many exciting future opportunities.
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48. 48
  Pavošević, F.; Culpitt, T.; Hammes-Schiffer, S. Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects via the Nuclear–Electronic Orbital Method. Chem. Rev. 2020, 120, 4222– 4253, DOI: 10.1021/acs.chemrev.9b00798
  48
  Multicomponent Quantum Chemistry: Integrating Electronic and Nuclear Quantum Effects via the Nuclear-Electronic Orbital Method
  Pavosevic, Fabijan; Culpitt, Tanner; Hammes-Schiffer, Sharon
  Chemical Reviews (Washington, DC, United States) (2020), 120 (9), 4222-4253CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
  A review. In multicomponent quantum chem., more than one type of particle is treated quantum mech. with either d. functional theory or wave function based methods. In particular, the nuclear-electronic orbital (NEO) approach treats specified nuclei, typically hydrogen nuclei, on the same level as the electrons. This approach enables the incorporation of nuclear quantum effects, such as nuclear delocalization, anharmonicity, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects directly into quantum chem. calcns. Such effects impact optimized geometries, mol. vibrational frequencies, reaction paths, isotope effects, and dynamical simulations. Multicomponent d. functional theory (NEO-DFT) and time-dependent DFT (NEO-TDDFT) achieve an optimal balance between computational efficiency and accuracy for computing ground and excited state properties, resp. Multicomponent wave function based methods, such as the coupled cluster singles and doubles (NEO-CCSD) method for ground states and the equation-of-motion counterpart (NEO-EOM-CCSD) for excited states, attain similar accuracy without requiring any parametrization and can be systematically improved but are more computationally expensive. Variants of the orbital-optimized perturbation theory (NEO-OOMP2) method achieve nearly the accuracy of NEO-CCSD for ground states with significantly lower computational cost. Addnl. approaches for computing excited electronic, vibrational, and vibronic states include the delta SCF (NEO-ΔSCF), complete active space SCF (NEO-CASSCF), and nonorthogonal CI methods. Multireference methods are particularly important for describing hydrogen tunneling processes. Other types of multicomponent systems, such as those contg. electrons and positrons, have also been studied within the NEO framework. The NEO approach allows the incorporation of nuclear quantum effects and non-Born-Oppenheimer effects for specified nuclei into quantum chem. calcns. in an accessible and computationally efficient manner.
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Supporting Information
Supporting Information
The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00541.
- Detailed description of the GAP model hyperparameters, training procedure, vibrational frequencies and density functional differences (PDF)
- ct3c00541_si_001.pdf (1.88 MB)
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Estimating Free Energy Barriers for Heterogeneous Catalytic Reactions with Machine Learning Potentials and Umbrella IntegrationClick to copy article linkArticle link copied!

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I. Introduction

Figure 1

II. Methods

II.1. Umbrella Integration

II.2. Gaussian Approximation Potentials

II.3. Density Functional Theory

III. Results

III.1. Minimum Energy Path and Collective Variable

Figure 2

III.2. Potential Training and Validation

Figure 3

III.3. Free Energy Calculations

Figure 4

III.4. Rate Constants

Figure 5

Figure 6

III.5. Harmonic Free Energy Corrections

III.6. Density Functional Comparison

Figure 7

III.7. Limitations

IV. Conclusions

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Estimating Free Energy Barriers for Heterogeneous Catalytic Reactions with Machine Learning Potentials and Umbrella Integration
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