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Computing Surface Reaction Rates by Adaptive Multilevel Splitting Combined with Machine Learning and Ab Initio Molecular Dynamics

Thomas Pigeon*
Thomas Pigeon
MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France
CERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France
IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
*E-mail: [email protected]
More by Thomas Pigeon
https://orcid.org/0000-0002-7828-5128
,
Gabriel Stoltz
Gabriel Stoltz
CERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France
MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France
More by Gabriel Stoltz
,
Manuel Corral-Valero
Manuel Corral-Valero
IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
More by Manuel Corral-Valero
https://orcid.org/0000-0002-4457-3914
,
Ani Anciaux-Sedrakian
Ani Anciaux-Sedrakian
IFP Energies Nouvelles, 1 et 4 avenue de Bois-Préau, F-92852 Rueil-Malmaison Cedex, France
More by Ani Anciaux-Sedrakian
,
Maxime Moreaud
Maxime Moreaud
IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
More by Maxime Moreaud
https://orcid.org/0000-0002-4908-401X
,
Tony Lelièvre*
Tony Lelièvre
CERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France
MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France
*E-mail: [email protected]
More by Tony Lelièvre
, and
Pascal Raybaud*
Pascal Raybaud
IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
*E-mail: [email protected]
More by Pascal Raybaud
https://orcid.org/0000-0003-4506-5062

Cite this: J. Chem. Theory Comput. 2023, 19, 12, 3538–3550

Publication Date (Web):June 5, 2023

https://doi.org/10.1021/acs.jctc.3c00280

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Abstract

Computing accurate rate constants for catalytic events occurring at the surface of a given material represents a challenging task with multiple potential applications in chemistry. To address this question, we propose an approach based on a combination of the rare event sampling method called adaptive multilevel splitting (AMS) and ab initio molecular dynamics. The AMS method requires a one-dimensional reaction coordinate to index the progress of the transition. Identifying a good reaction coordinate is difficult, especially for high dimensional problems such as those encountered in catalysis. We probe various approaches to build reaction coordinates such as support vector machine and path collective variables. The AMS is implemented so as to communicate with a density functional theory-plane wave code. A relevant case study in catalysis, the change of conformation and the dissociation of a water molecule chemisorbed on the (100) γ-alumina surface, is used to evaluate our approach. The calculated rate constants and transition mechanisms are discussed and compared to those obtained by a conventional static approach based on the Eyring–Polanyi equation with harmonic approximation. It is revealed that the AMS method may provide rate constants that are smaller than those provided by the static approach by up to 2 orders of magnitude due to entropic effects involved in the chemisorbed water molecule.

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1. Introduction

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The determination of chemical reaction rate constants is of tremendous importance to better understand and quantify the kinetics of molecular transformations. This can be a challenging task, especially in catalysis where multiple elementary steps are involved for one targeted reaction. Evaluating each of them by experimental methods being often out of reach, an alternative lies in the theoretical modeling of each of them. Thanks to the significant increase of computational resources, quantum simulation approaches are widely used nowadays to address numerous catalytic systems involved in petrochemistry, fine chemistry, and biomass conversion. (1−4)

However, at the simulation time scale, such chemical transformations are rare events. The typical time step for the integration of stochastic dynamics modeling the evolution of the system is of the order of 10^–15 s, while the frequency of most of the chemical reactions of interest is, at least, several orders of magnitude higher. Moreover, to accurately simulate catalytic activation of chemical bond breaking and formation, the simulation must include the explicit treatment of valence electrons and the quantum chemical calculation of the Hellmann–Feynman forces for each step of the dynamics. (5) Such an ab initio molecular dynamics (AIMD) approach becomes so computationally demanding that it is generally impossible to simulate a trajectory that is long enough to observe multiple reaction events, allowing the accurate quantification of rate constants.

Theoretical approaches most commonly used to explore chemical transformations are based on transition state theory (TST). (6) Within this formalism, the reactant and product are considered to be separated in phase space by a dynamical bottleneck, (7) which can be characterized as a surface in the configuration space. For a reaction with only one reactive path and only one energy barrier to cross, assuming momenta are not relevant for the transition process, this surface should contain the first order saddle points. The term transition state (TS) is versatile as it sometimes refers to a first order saddle point and sometimes to an isocommittor surface, as defined by IUPAC. (8) Considering TS to be surfaces, the reaction rate can be approximated as the frequency at which this surface is crossed. The most common approach to compute the reaction rate constant is called harmonic TST (hTST) as it allows to reduce the general TST expression into the “generalized” Eyring–Polanyi equation thanks to harmonic approximation of the potential energy surface. (6,9−11)

k_{h T S T} = κ (T) \frac{k_{B} T}{h} e^{- Δ G^{‡} / k_{B} T}

(1)

where ΔG^‡ is the free energy of activation computed as the difference of the free energy of the metastable basin and of the TS, k_B the Boltzmann constant, h the Planck constant, T the temperature, and κ(T) the transmission coefficient. This last quantity has to be between 0 and 1 and accounts for the recrossing of the surface. The free energy of activation is approximated via an harmonic approximation around the saddle point and the minima. Although hTST is one of the most popular methods to determine activation free energies and the rate constants of chemical events, particularly catalytic ones, it suffers from some weaknesses. Among them, the harmonic approximation of the potential energy surfaces as well as the determination of the prefactor κ in eq 1 might be questionable. In general, when the entropy of the metastable state and the transition state differ by a non negligible amount, the harmonic approximation can lead to significant errors. This can occur in various systems of interest in catalysis such as solid–liquid interfaces, zeolites, porous solids, and supported nanoparticles. (12) More general expressions for the TST rate, using a one-dimensional reaction coordinate (7,13) and relying on sampling methods to estimate free energies, (14−16) were proposed to overcome some limitations of eq 1. However, TST reaction rates contain a transmission coefficient κ ∈ (0, 1], accounting for the recrossing of the transition state surface, which is rather difficult to evaluate and which explains why bare TST overestimates the transition rate. (7,11,13,17−19)

There are of course alternative approaches to TST. The first one is based on the evolution of a time correlation function (13,20) that found applications in transition path sampling (TPS) (21) or other approaches such as the recent work relying on the Onsager–Machlup path probability distribution. (22) Another alternative, used in the present work, is provided by approaches based on the Hill relation: (23)

k_{H i l l} = Φ_{R} p_{R \to P} (\partial R)

(2)

where Φ_R is the flux of trajectories leaving the reactant state R and the committor probability at the boundary p_R→P(∂R), the probability of reaching the product state P before returning to R starting from the boundary ∂R. In other words, this relation states that the rate constant is the average rate at which the system attempts to leave the initial state times the probability of success. Equation 2 has been proven correct assuming that the reactant state R is metastable for systems evolving according to the overdamped Langevin dynamics (24) or Langevin dynamics. (25) The Hill relation is used in various approaches corresponding to so-called path sampling methods such as transition interface sampling (TIS), (26) forward flux sampling (FFS), (27) weighted ensembles (WE), (28) and adaptive multilevel splitting (AMS). (29) All these methods are designed to compute the probability p_R→P(∂R), which is the most difficult object to evaluate in eq 2. As a side product, these methods sample some reactive trajectories.

All the methods described previously exhibit different precisions and computational efficiencies. On the one hand, the hTST approach is, by far, the most inexpensive methodology in terms of computational resources, but, as mentioned above, it may lead to significant errors. On the other hand, the computational cost of enhanced sampling methods to estimate free energies is not negligible. The Hill relation has the advantage of being exact compared to TST approach, but the required computational cost can be high depending on the method to compute the probability p_R→P(∂R). Moreover, numerical methods sampling reactive trajectories offer the possibility of performing a more detailed analysis of reaction mechanisms.

Most methods to compute reaction rate constants require the definition of a collective variable (CV), either to define the states of the system, its free energy, or to use it as a one-dimensional reaction coordinate (RC) indexing the progress of the transition. In many situations, reactions go through one or a few channels in phase space. CVs should describe these channels with a minimal number of dimensions. Usually, CVs are defined thanks to chemical intuition or through the expert knowledge of the chemical system. They are typically based on key distances or angles associated with atoms central to the reaction mechanism. Nonetheless, this kind of heuristic approach can have some limitations especially when the studied mechanism is a priori unknown. Automatic and data based approaches using various machine learning (ML) methods offer very appealing perspectives in this context. Recent reviews (30−33) provide an overview of current options to propose CVs and discuss their advantages and drawbacks. These methods bear the promise of more systematic and efficient ways to define CVs, albeit at the expense of interpretability compared to intuitive CVs such as angles or distances. Nonetheless, machine-learned CVs are becoming common practice in the field. For example, support vector machine (SVM) models trained on a set of data generated by molecular dynamics were used for exploring the configurational transitions of model protein molecules. (34) In material sciences, the combination of SVM and AIMD was used for the mechanistic study of the diffusion of Al atoms on the Al(100) surface. (35) To the best of our knowledge, SVM has not been used to explore more complex reactive events, such as chemical bond breaking/formation catalyzed by an oxide material’s surface such as proposed in the present work.

To benchmark an innovative methodology based on the Hill relation for exploring reaction mechanisms occurring on catalytic materials, we chose in this work a relevant case study: the reactivity of water on the (100) orientation of γ-alumina, a widely used support in heterogeneous catalysis applied to biomass conversion. (36,37) Comprehensive density functional theory (DFT) based studies have revealed the versatile nature of active sites (Lewis Al and Bronsted Al–OH), their thermodynamic properties, (38−42) and their kinetic ones (TS and activation barriers) by using predominantly hTST calculations. (36,37,43,44) As for the study of many chemical reactions, especially in catalysis, most of the reaction rate constants have been computed so far within the TST framework. (4) Unbiased AIMD simulations have also been applied to decipher the γ-alumina’s reactivity, its local structure, and spectroscopic features, in the presence of liquid water in order to obtain a better understanding of phenomena occurring during the catalyst preparation or catalytic reaction. (45,46) TPS was used in particular for studying the catalytic reactivity of other oxide materials, (47,48) also in combination with the blue-moon ensemble formalism. (49)

Methods based on the Hill relation and rare event simulation methods are rarely used for studying chemical reactions (50) and, to the best of our knowledge, they have never been used to describe reactions in heterogeneous catalysis. Moreover, the AMS method has only been used for molecular dynamics applications to study the isomerization of small biomolecules (51) or a protein–ligand dissociation, (52) up to now.

Hence, the aim of the present work is to highlight how AMS applied to AIMD rare event sampling, combined with the ML approach, is able to compute reaction rate constants via the Hill relation in a relevant case study for heterogeneous catalysis. The CVs and RCs are built using SVM or path collective variables and well-chosen chemical descriptors. (53)

Considering the challenge of the chemical reactivity of the alumina catalysts highlighted before, we will aim at determining rate constants for a reaction network involving various water rotation, dissociation, and association events on the (100) γ-alumina surface.

This article is organized as follows. In the Methods section, the computational approach following a flowchart leading to the determination of rate constants is described. First, we present how the implementation of AMS coupled to a reference plane wave-DFT software enables the determination of rate constants. In a second part, numerical tools such as SVM and path collective variables (PCV) used to define CVs and RCs are presented. The Results and Discussion section first describes the catalytic model system of water activated on the γ-alumina surface, which was used to probe the theoretical approach. The constructions of CVs and RCs corresponding to the water molecule transformation path are then explained. Finally, the numerical values of reaction rates and the reactive trajectories are analyzed and compared with the standard hTST approach.

2. Methods

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The general flowchart of our approach is given in Figure 1. The first step is the definition of states defined as the ensemble of structures in the vicinity of a local potential energy minimum characterizing either a reactant or a product. In practice, these configurations are sampled by running a short AIMD simulation starting from minima identified on the potential energy surface (PES). Then, using this trajectory, the function numerically defining states is obtained by SVM and well-chosen chemical descriptors. Depending on the reaction rate constant to compute, each state has to be labeled as reactant or product. A reaction coordinate (RC) is then built, for instance by using the decision functions of the classifiers previously used to define states. Once states and a RC are defined, AMS is run to obtain an estimate of the reaction rate constant of the Langevin dynamics which is assumed to model accurately the system dynamics.

Figure 1. Global workflow to compute reaction rate constants with the Hill relation using adaptive multilevel splitting and support vector machine.

As a side remark, the definition of states and of the RC can be done using any type of collective variables describing the configurations of the system. In the present approach, the chosen collective variables are the 2100-dimensional descriptor of a selected atomic environment (see Section 2.2.1 for definition of the descriptor). The role of the SVM is to build a linear combination of these collective variables.

2.1. Reaction Rate Constant Estimation Using AMS

2.1.1. Motivation

To compute rate constants of rare events by using the Hill relation (eq 2), the flux of trajectories leaving the initial reactant state R (or the frequency at which trajectories leave R) must be evaluated. If the reactant state is properly defined, this quantity can be computed in a reasonably short time by unbiased MD. The difficulty lies in the estimation of the probability that a trajectory leaving R is reactive (i.e., goes to a product state P), since the probability p_R→P(∂R) is in most cases exceedingly small. The AMS algorithm is specifically designed to evaluate low probability events. (29) The key point of AMS is to propose a method that has a good behavior in terms of variance and computational efficiency to compute the probability p_R→P(∂R). This is achieved by first decomposing the rare event of interest into a succession of less unlikely events, the target probability to estimate being the product of the conditional probabilities associated with the subevents (see SI Section 1). Moreover, the subevents are built such that the associated conditional probabilities are all the same. This is indeed a desirable feature in order to reduce the overall variance of the estimator. (51) The mathematical analysis of the variance of the AMS estimator is provided in refs (29) and (54). We focus here on the presentation of the algorithm adapted to MD rare events and only mention that this algorithm is unbiased. (55) This means that, whatever the choice of the reaction coordinate ξ and the number of replicas of the system (see below), repeating the algorithm sufficiently many times will always provide the same result in average, and this average value coincides with the target probability. On the other hand, the variance of the probability estimator depends on the quality of ξ. This opens a way to define an iterative procedure to improve the definition of reaction coordinates, using the sampled reactive trajectories to define better reaction coordinates.

2.1.2. Computing the Flux and Sampling Initial Conditions

A separating surface Σ_R close to R is introduced for the estimation of the flux Φ_R, to determine actual exits out of R. (24) This surface has to enclose the reactant state, so that any trajectory going from R to P has to cross Σ_R (see Figure 2). Indeed, the location of this surface allows to select the trajectories that make actual excursions off the state R, in contrast to trajectories that would only wander out of R for a few steps and go back inside R right away. The approach to define the position of this surface is discussed precisely below. The flux Φ_R is then evaluated by starting a dynamics in the state R, counting the number of times

n_{l o o p - R Σ_{R} R}

it goes from R to Σ_R, crosses Σ_R and goes back to R, dividing this number by the overall time t_tot:

Φ_{R} = \frac{n_{l o o p - R Σ_{R} R}}{t_{t o t}} = \frac{1}{t_{l o o p - R Σ_{R} R}}

(3)

where

t_{l o o p - R Σ_{R} R}

is the average time that a trajectory takes to go out of R, cross Σ_R, and go back to R. Let us now focus on the second one: p_R→P(∂R). The computation of the flux Φ_R simultaneously allows one to generate some positions on the surface Σ_R, which will serve to estimate the probability p_R→P(Σ_R) instead of p_R→P(∂R). This approximation does not bias the result as far as R and Σ_R are within the same metastable basin. (24) These initial conditions must correspond to the first time a trajectory leaving R reaches the level Σ_R. As the efficient calculation of the flux and the sampling of initial conditions relies on parallelization strategies, a Fleming–Viot particle process is used in our implementation of this initialization procedure. (56) The particles undergo independent molecular dynamics, which means they can be run in parallel without requiring frequent communications.

Figure 2. First iteration of the AMS algorithm with k_min = 1 and N_rep = 3. Purple points represent the initial conditions on Σ_R. a) Identify the kill level $z_{k i l l}^{1} = z_{\max}^{k_{\min}, 0}$ and kill the replicas such that z_max^i,0 ≤ z_kill¹, i.e. the orange replica. b) Replace the killed replicas by the trajectory of one of the remaining replicas (the green one in this example) until the level z_kill¹ and continue the trajectory of the replica until it reaches either the state R or the state P.

2.1.3. AMS Requirements

To run an AMS estimation, the reactant state R and the product state P have to be defined. The surface Σ_R has to be placed such that each trajectory linking the reactant and the product state goes through Σ_R. Its distance to the boundary of R should be sufficiently small so that the sampling of initial conditions and the determination of the flux Φ_R is not exceedingly expensive in terms of computational cost. A number of replicas N_rep (or walkers) has to be defined, as well as a minimum number k_min of replicas to kill at each iteration of AMS. N_rep different initial conditions on the surface Σ_R are selected uniformly among the initial conditions sampled following the procedure described in the previous paragraph (purple points in Figure 2). Finally a reaction coordinate ξ should be defined to index the progression along the R → P transition. It has to be consistent with the states R and P, which can be generally enforced by setting ξ(q) = –∞ for q ∈ R and ξ(q) = +∞ for q ∈ P.

2.1.4. AMS Initialization

First, all the replicas are run from their initial conditions on Σ_R until either the R or P state is reached (see Figure 2a), which depicts an initialized set of three replicas). They are then iteratively updated until they all finish in the product state P. An illustration of an iteration is provided in Figure 2, the process being detailed in the next paragraph. In what follows, q_t^i,n denotes the position of the i-th replica at time t and iteration n. In particular,

{q_{0}^{i, n}}_{1 \leq i \leq N_{r e p}}

are initial conditions on Σ_R. The method to estimate the probability is also summarized in the pseudocode presented in SI Section 1.

2.1.5. AMS Iteration

Each iteration of the main AMS loop starts by defining the largest value of the RC for each replica at the n-th iteration as

z_{\max}^{i, n} = \sup_{t} (ξ (q_{t}^{i, n}))

. The replicas are then reordered by increasing values z_max^i,n (see Figure 2). According to the value of k_min, the level at which positions are killed is identified as an empirical quantile:

z_{k i l l}^{n + 1} = z_{\max}^{k_{\min}, n}

. This means that all the trajectories for which z_max^i,n ≤ z_killⁿ⁺¹ are killed. The number of killed trajectories at this iteration is denoted by η_killedⁿ⁺¹. Note that η_killedⁿ⁺¹ ≥ k_min by construction, but it could happen that η_killedⁿ⁺¹ ≥ k_min+1 when several trajectories reach exactly the same

z_{\max}^{k_{\min}, n}

. To keep the number of replica constant, η_killedⁿ⁺¹ trajectories have to be created by randomly branching η_killedⁿ⁺¹ trajectories among the remaining ones. More precisely, trajectories are duplicated until the first time they reach the level z_killⁿ⁺¹ and then the dynamics is ran from these points until it reaches R or P. In fact, at each iteration, the estimated probability

1 - \frac{η_{k i l l e d}^{n + 1}}{N_{r e p}}

is the probability for a trajectory to reach the surface

Σ_{z_{k i l l}^{n + 1}}

starting on the surface

Σ_{z_{k i l l}^{n}}

Any AMS iteration can be summarized by the succession of the steps illustrated in Figure 3.

Figure 3. Flowchart of one iteration of AMS. n is an iteration index and i a replica index.

2.1.6. AMS Termination and Probability Estimator

The AMS algorithm can terminate in two different manners. First, after a certain number of iterations, all the replicas reach the state P. In such a case, N_rep different reactive trajectories are obtained and the estimated transition probability is computed via:

{\hat{p}}_{R \to P} (Σ_{R}) = \prod_{n = 1}^{n_{\max}} (1 - \frac{η_{k i l l e d}^{n}}{N_{r e p}})

(4)

where n_max is the final number of iterations of the algorithm. The second option (not explicitly presented on Figure 3 since the RC is typically chosen so that this does not happen) is that at a certain iteration n, η_killedⁿ, is equal to the total number of replicas of the algorithm. This can happen if at some point all the copied replicas have the same value of z_max^i,n. This termination event is called “failure” as the algorithm is not able to provide reactive trajectories and the estimated probability is

{\hat{p}}_{R \to P} (Σ_{R}) = 0

, consistently with eq 4. Such a situation can be encountered if the system is stuck and all the replicas are progressively replaced by the copy of a single replica. It is also possible that the replicas reach their maximal ξ value in a zone of the phase space on which the reaction coordinate ξ remains constant while the trajectories are different.

It is possible to estimate the statistical error on the estimated probability

{\hat{p}}_{R \to P} (Σ_{R})

in eq 4 by repeating the estimation of the probability M_real times. These realizations should be independent and can take advantage of parallel architecture of current supercomputers. The confidence intervals presented in the results section all correspond to a 90% confidence. More details can be found in SI Section 2.

2.1.6.1. Multiple States Case

Defining the states R, P, and the surface Σ_R when there are multiple metastable states requires a specific treatment to compute state to state reaction rates. Two main approaches are proposed and made precise in Section 3 of the SI. The first one samples all possible trajectories starting from a given state. The second approach more specifically focuses on the targeted transition. An illustration and comparison of the two approaches is provided in the Results and Discussion section.

2.1.6.2. Implementation with a Plane Wave DFT Code

The AMS algorithm and the sampling of initial conditions was implemented in Python scripts calling the VASP software for AIMD simulations. (57,58) All simulations parameters are listed in SI Section 4. Some slight modifications have been implemented in the VASP code to allow for different stopping conditions of the VASP MD runs. More details concerning the implementation can be found in SI Section 5. The various repetitions M_real of the AMS estimation can be run independently in parallel. The Fleming–Viot particle scheme also allows for in dependent runs, communications are required only infrequently allowing arbitrary number of particles ran independently in parallel. The development of the scripts and the testing was mostly done on the ENER440 computer of IFPEN. Results presented in the following section come from simulations run on Joliot-Curie (Genci) and Topaze (CCRT).

2.2. Tools to Define States and Reaction Coordinates

This section describes the tools used to define the states and the reaction coordinates in the next section.

2.2.1. Representation of Chemical Structures

Reaction coordinates and states definitions must be invariant under rotation, translation, and symmetries of the system as well as by permutation of identical atoms. Since descriptions relying on Cartesian coordinates do not exhibit these properties, substantial work was conducted to find representations of atomic systems invariant by Galilean transformations and other symmetries, in particular in the field of ML empirical potentials. (59−62) We chose the smooth overlap of atomic positions (SOAP) (60) descriptor allowing us to capture enough information on atomic environments to reach errors of the order of 1 meV for potential energy surface fitting. (63,64) This descriptor turned out to be sufficient for our needs as illustrated in the Results and Discussion section. The detailed parameters used to compute SOAP descriptors using the DScribe Python package (65) can be found in SI Section 4.3.

2.2.2. Support Vector Machine

A linear SVM model is designed to find the highest margin separation plane between two sets of labeled points. The margin denotes the minimal distance between the plane and the labeled points. The details concerning this optimization problem can be found in ML textbooks (66) or the scikit-learn documentation. (67) The important result for this work is that, once the optimization problem is solved, only a certain subset of the total training set is used in the definition of the plane. These are the so-called support vectors which are the closest to the separation plane. The vector normal to this plane and the scalar defining its position is thus a linear combination of the support vectors. The classifier decision function is the algebraic distance to the plane multiplied by a scaling factor chosen so that the decision function value on support vectors which are not outliers is either 1 or −1. To define multiple states using SVM, the one versus all approach was chosen, as made precise later on in the result section dedicated to the definition of states. Linear SVM models were trained using the SVC routine of the scikit-learn package with a linear kernel. (67) The data were normalized using the standard scaler implemented in the same package. The regularization parameter was kept to the default value 1 as, after cross validation, the classification scores on the test sets were always 100%.

2.2.3. Path Collective Variables (PCVs)

The principle of PCVs is to first define a reference path for the transition as a sequence of structures

{R_{i}}_{0 \leq i \leq L - 1}

. These structures are represented here with the numerical SOAP descriptor. A reaction coordinate is then constructed as (53)

s (R) = \frac{\sum_{i = 0}^{L - 1} i e^{- λ d (R_{i}, R)}}{\sum_{i = 0}^{L - 1} e^{- λ d (R_{i}, R)}}

(5)

where d is a distance here the Euclidean norm. The parameter λ has to be of the order of variation of the inverse distances between two consecutive structures along the path. If the structures along the path are not evenly spaced along the path according to the distance d, a sequence of values λ_i can be used instead. In the present case, we chose λ_i as

{\begin{array}{l} λ_{i}^{- 1} = \frac{1}{2} (d (R_{i - 1}, R_{i}) + d (R_{i}, R_{i + 1})) \\ λ_{0}^{- 1} = d (R_{0}, R_{1}); λ_{L - 1}^{- 1} = d (R_{L - 2}, R_{L - 1}) \end{array}

(6)

PCVs were directly implemented in the Python scripts used for the reaction coordinate evaluation during the dynamics.

3. Results and Discussion

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3.1. γ-Al₂O₃ Models and Definition of States

3.1.1. Model of the Catalytic System

The catalytic case study chosen to benchmark the previously presented method is the transformation of a water molecule adsorbed on the (100) γ-alumina surface. (39,40,42) A representation of the γ-Al₂O₃ surface on which one water molecule is chemically adsorbed without dissociation on an aluminum Lewis site is given in Figure 4. More information about the alumina slab used is provided in SI Section 4.1.

Figure 4. Representation of one water molecule adsorbed on an aluminum site of the (100)-γ-alumina surface model (A₁ state). Surface atoms are represented as ball and sticks while subsurface ones are represented as lines. Color legend: red: oxygen, gray: aluminum, white: hydrogen, black lines: boundaries of the periodic cell. a) Top view; b) side view.

The first step is to identify various potential energy minima corresponding to the metastable states of the water molecule adsorbed on the surface either in a dissociative mode or a nondissociative mode. As described in what follows, the dissociative modes lead to the formation of two hydroxyl (OH) groups: the first one is formed upon the transfer of a H atom of the water molecule to a neighboring O site of the surface; the second one results from the native water molecule. This systematic exploration confirms previous DFT studies where the minima were identified by running multiple geometry optimizations starting from various initial conditions. (39,40)

3.1.2. Data Set Generation to Learn States

Once local minima are identified, the metastability of the basins surrounding them should be assessed because these local minima should be sufficiently separated from other local minima. To quantify this, two AIMD trajectories of 1 ps each were run starting from each minimum. The first AIMD was run with a friction parameter γ of 5 ps^–1 to thermalize the system faster, while the second one was run with γ = 0.5 ps^–1. If the system ends up in another potential energy well during this second part of the trajectory, then the initial well is not considered relevant to be qualified as a metastable state. At 300 K, multiple transitions between all basins were observed, thus all the potential wells cannot be considered metastable and relevant so as to mimic realistic chemical reactions. At 200 K, 8 genuine metastable states could be identified, indicating that at this temperature, the system better mimics chemical reaction conditions. The various identified states are named A_i or D_i depending on whether the state corresponds to a nondissociated adsorbed water molecule or to two surface hydroxyls after water dissociation, respectively (see Figure 5). Some of these states are in fact identical as there exists a plane symmetry in this structure and thus these metastable potential energy wells should be gathered in the same state. For example, the wells D₁ and D₃ are symmetrically identical.

Figure 5. Representation of the main different minimum energy structures corresponding to metastable states for the water molecule adsorbed on the (100)-γ-Al₂O₃ surface. Arrows represent transitions that might occur. Color legend: gray: aluminum, red; oxygen, white: hydrogen.

The numerical definitions of states A₁, A₂A₃, A₄, D₁D₃, and D₂D₄ were built using the one versus all (1-vs-all) linear SVM classifiers decision function f_X–vs–all. For instance, the state A₁ is defined as

{q | f_{A_{1} - v s - a l l} (S O A P (q)) \leq - 1}

. To train these models, the data used was a 1 ps MD trajectory at 50 K starting from each local minimum. The point of running a MD at a lower temperature was to obtain points close to the minimum of the potential energy well. The dynamics was run with a friction parameter γ of 5 ps^–1 during 1 ps for equilibration, then run during 1 ps with γ = 0.5 ps^–1.

The production runs of these trajectories were used to train the SVM classifiers. Only one SOAP descriptor centered on the oxygen atom of the adsorbed water molecule was used as features in the training set, thus SOAP is the function that maps the positions q to the SOAP descriptor of this oxygen atom environment SOAP(q). With the parameters mentioned in SI Section 4.4, this leads to an array of size 2100 to describe each structure. Before training the model, the variation of each dimension of the SOAP descriptors were scaled to have zero mean and unit variance. The test score of the SVM model was 100% in every case, which indicates that the set of structures represented with SOAP descriptors are linearly separable. On the other hand, trying to separate the SOAP descriptor of the trajectories starting from two symmetric minima such as D₁ and D₃ systematically led to smaller test scores. This indicates that the well surrounding these two minima are indeed similar in the sense of the SOAP descriptor.

As a side remark, in a situation where symmetries of states are unknown, using this kind of approach can help to identify some similarities. In Figure 6, an histogram of the decision function of a A₁-vs-D₁ SOAP-SVM classifier is plotted. The various colors represent the different labeled states. It is clear that this CV allows one to differentiate the A and D states. Moreover, according to this criterion, the A₂ and A₃ as well as well as the D₁/D₃ and D₂/D₄ groups of points bear some similarities due to the symmetry of the alumina surface.

Figure 6. Histogram of A₁-vs-D₁ SOAP-SVM CV on the whole labeled data set.

3.1.3. Definitions of the Boundary Surface Σ_R

For every metastable state X, a boundary Σ_X is defined from the 1 ps trajectory generated to assess the metastability of the basin X. This surface Σ_X is taken as a levelset of the decision function f_X–vs–all of the SVM classifier trained to distinguish this state from the others. This level is chosen so that during the 1 ps MD the trajectory goes 10 times above this level. In practice, for all the states A₁, A₂A₃, A₄, D₁D₃, and D₂D₄, the surface Σ_X was chosen as

{q | f_{A_{1} - v s - a l l} (S O A P (q)) = - 0.95}

. This choice implies that Σ_R for AMS is directly related to the choice of R since the two sets are determined from the level sets of the same function.

3.1.4. Definition of Reaction Coordinates (RCs)

The first RCs used to perform AMS simulations are the various 1-vs-all SVM decision functions. These RCs are therefore named “1-vs-all SOAP-SVM RC” in the following sections. Some more specific RCs are built using the same approach targeting a specific transition from a state to another. In this case, the decision function is obtained by separating only the two targeted states. The corresponding RCs are termed “1-vs-1 SOAP-SVM RC”. Finally, a PCV, termed “SOAP-PCV”, is also used as reaction coordinate to index the progression of AMS replicas. The SOAP-PCV RCs differ depending on the reference path. We consider here paths built by an interpolation of the Z-matrix representations of the minima of two metastable basins. (68) The associated RCs are termed “interpolated SOAP-PCV”.

3.2. Analysis of AMS Rate Constants

In this section, we analyze first the sensitivity of the reaction rates to two key parameters, the number of replicas (N_rep) and the number of repetitions of the probability estimation (M_real). These parameters also govern the computational cost and how this cost can be distributed on multiple CPUs, taking advantage of the parallel architecture of current supercomputers. Then, the other choices impacting the precision of the reaction rate constant are the RC and the states, also investigated in what follows. The reaction rate constants obtained for each observed transition are finally compared to values computed from hTST.

3.2.1. Parallel Calculations against Precision

The effect of the number of replicas (N_rep) and the number of AMS repetitions (M_real) is evaluated for a fixed number of initial conditions N_repM_real, which roughly corresponds to a fixed computational cost. Indeed, assuming that every branching during one AMS realization has the same cost in average and that η_killedⁿ is constant and equals k_min at all steps of the AMS realization, the cost of one AMS realization is given by the product of the number of AMS iterations (n_max) and the number of killed replicas (k_min). Under these assumptions, the AMS estimator (eq 4) writes:

\hat{p} = {(1 - \frac{k_{\min}}{N_{r e p}})}^{n_{\max}}

(7)

Assuming that

\frac{k_{\min}}{N_{r e p}}

is small, the computational cost of a single AMS simulation is

k_{\min} n_{\max} \approx - N_{r e p} \ln (\hat{p})

(8)

Taking into account the number of repetitions of the algorithm M_real, the final cost of a reaction rate constant estimation is

- M_{r e a l} N_{r e p} \ln (\hat{p})

. Considering the current implementation of AMS, M_real realizations of AMS can be run in parallel. The objective is to find the minimal value of N_rep to better distribute the computational cost on multiple parallel realizations. With too few replicas, the intrinsic variance of the AMS estimator can be so large that the confidence interval of the estimated probability contains 0, leading to not interpretable results. Table 1 reports the evolution of water rotation rate constants,

k_{A_{1} \to A_{2} A_{3}}

, calculated with AMS for various values of N_rep and M_real by using the “A₁-vs-all-SOAP-SVM” reaction coordinate and states defined as R = A₁ and P = A₂A₃ ∪ A₄ ∪ D₁D₃ ∪ D₂D₄.

Table 1. Estimation of Probability, Rate and the Corresponding 90% Confidence Intervals for Water Rotationa

M_real	N_rep	(fs)	p_{A₁→A₂A₃} (Σ_A₁)	k_{A₁→A₂A₃} (s^–1)
5	400	108 ± 5	(3.73 ± 3.03) 10^–3	(3.67 ± 2.99) 10¹⁰
10	200	110 ± 5	(3.38 ± 1.56) 10^–3	(3.08 ± 1.43) 10¹⁰
20	100	101 ± 5	(3.47 ± 1.96) 10^–3	(3.21 ± 1.82) 10¹⁰

^{^a}

The number of initial conditions M_realN_rep was varying M_real and N_rep. R = A₁, P = A₂A₃ ∪ A₄ ∪ D₁D₃ ∪ D₂D₄, ξ = A₁-vs-all SOAP SVM RC.

By definition t_loop is not impacted by N_rep or M_real. The average values of probability and rate are not much impacted in the present case, which is not the case for the variance. The choice of N_rep = 200 and M_real = 10 is sufficient to obtain a A₁ to A₂A₃ water rotation rate of 3.1 × 10¹⁰ s^–1 with the 90% confidence interval of [1.65 × 10¹⁰ s^–1, 4.51 × 10¹⁰ s^–1]. Similar precision can be obtained with N_rep = 100 and M_real = 20. Therefore, it is important to perform the AMS simulations a certain number of times (M_real) in order have a proper variance estimation. Hence, for a similar computational cost in CPU time, satisfactory accuracy can be obtained using M_real ≥ 10.

3.2.2. Impact of the Definition of Reaction Coordinates and States

The definitions of the states R and P determine the type of trajectories that can be sampled by the algorithm. The choice of the reaction coordinate impacts the quality of this sampling. For instance, exploring all types of trajectories from A₁ to any other states requires one to sample initial conditions on

Σ_{A_{1}}

, set R = A₁ and P = A₂A₃ ∪ A₄ ∪ D₁D₃ ∪ D₂D₄. Using the A₁-vs-All SOAP-SVM RC to sample trajectories ending in P leads to the results presented in Table 2. This approach allows the sampling of the transition A₁ → A₂A₃ with a reasonable accuracy according to the estimate of the rate constant’s variance. However, the less probable transitions (A₁ → D₁D₃ and A₁ → A₄) are under-sampled and the rate estimations are not precise enough as the 90% confidence interval contains 0. Moreover, the direct transition from A₁ to D₂D₄ is so rare that it has not even been sampled. To more accurately quantify the transition A₁ → D₁D₃, more specific RCs must be used. The results obtained with two other RCs are compared in Table 3. Changing the reaction coordinate A₁-vs-all SOAP-SVM into A₁-vs-D₁ SOAP-SVM for AMS does not significantly improve the rate constant precision as the estimated variance is still so large that 0 is contained in the confidence interval. This is due to the fact that in view of the definition of R and P, AMS still samples trajectories that are of no interest such as the rotation A₁ → A₂A₃. To observe only A₁ → D₁D₃ reactive trajectories, one possibility would be to set R = A₁ and P = D₁D₃. However, as the AMS iteration stops once the trajectories finish either in R or P, a trajectory including the A₁ → A₂A₃ rotation would consume too much computational time before going to R or P as the state A₂A₃ is metastable. Hence R and P must be defined differently. Considering a transition starting from A₁, with the choice R = A₁ ∪ A₂A₃ ∪ A₄ ∪ D₂D₄, P = D₁D₃, and initial conditions sampled on

Σ_{A_{1}}

, AMS is compelled to sample A₁ → D₁D₃ trajectories. The difference here with the previous case is that, if a rotation A₁ → A₂A₃ is observed in the course of the algorithm, then it will be stopped once it enters the A₂A₃ state and be considered as a non reactive trajectory. Such trajectories will ultimately be discarded and replaced by trajectories having higher z_max values of the chosen reaction coordinate defined in a way so as to enhance the sampling of trajectories between the desired metastable states. Both the quality of the reaction coordinate and the choice of the R and P states are important to obtain precise results for the A₁ → D₁D₃ transition (see Table 3). In our case study, the necessity to change the definition of R and P might be due to the difference of the transition probability between the rotation A₁ → A₂A₃ and the water dissociation A₁ → D₁D₃. Indeed, the half size of confidence intervals is larger than the average rate in the case where any type of rotations can be sampled, while constraining the AMS to sample only A₁ → D₁D₃ trajectories leads to smaller confidence intervals. In the present case the interpolated SOAP-PCV RC is not significantly better than the A₁-vs-D₁-SOAP-SVM RC in terms of variance as the 90% confidence error represents 97% of the average value while for the A₁-vs-D₁-SOAP-SVM RC it is 89%.

Table 2. Transition Rates Leaving A₁ Estimated Using A₁-vs-all SOAP-SVM RC, N_rep = 200, M_real = 10, R = A₁ and P = A₂A₃ ∪ A₄ ∪ D₁D₃ ∪ D₂D₄a

Transition		k_Transition (s^–1)
A₁ → A₂A₃	(3.38 ± 1.56) 10^–3	(3.17 ± 1.43) 10¹⁰
A₁ → D₁D₃	(1.79 ± 1.86) 10^–3	(1.63 ± 1.70) 10¹⁰
A₁ → A₄	(3.66 ± 6.02) 10^–7	(3.44 ± 5.50) 10⁶

^{^a}

As the results come from the same AMS is constant and equal to 110 ± 5 fs.

Table 3. Variation of the RC and Reactant/Product States R and P to Sample the A₁ → D₁D₃ Transition with N_rep = 200, M_real = 10 and Initial Conditions Sampled on

RC	(fs)	p_{A₁→D₁D₃} (Σ_A₁)	k_{A₁→D₁D₃} (s^–1)
R = A₁ ; P = A₂A₃ ∪ A₄ ∪ D₁D₃ ∪ D₂D₄
A₁-vs-all-SOAP-SVM	110 ± 5	(1.79 ± 1.86) 10^–3	(1.63 ± 1.70) 10¹⁰
A₁-vs-D₁-SOAP-SVM	105 ± 3	(1.81 ± 1.98) 10^–5	(1.72 ± 1.88) 10⁸
interpolated SOAP-PCV	104 ± 4	(1.95 ± 2.26) 10^–4	(1.87 ± 2.17) 10⁹
R = A₁ ∪ A₂A₃ ∪ A₄ ∪ D₂D₄ ; P = D₁D₃
A₁-vs-D₁-SOAP-SVM	105 ± 2	(3.31 ± 2.97) 10^–4	(3.15 ± 2.83) 10⁹
interpolated SOAP-PCV	108 ± 2	(1.78 ± 1.73) 10^–4	(1.64 ± 1.59) 10⁹

3.2.3. Comparison of the Rate Constants Calculated with AMS and with hTST

Various rate constants involved in the reaction networks of Figure 5 were computed using AMS. Various reaction coordinates and various definitions of the states R and P were used to obtain the results presented in Table 4. For the sake of clarity, the choice of R, P, RC, and AMS parameters for each transition are listed in SI Table 1. These rates obtained by AMS are directly compared to the reaction rate constants computed from the static hTST approach. Activation free energies calculated with hTST are reported in SI Table 3, and they qualitatively compare with previously published DFT data. (44)

Table 4. Transition Rate Constants for All the Transitions Observed in This Study with 90% Confidence Interval for AMS Results

Transition	k_{Transition–AMS} (s^–1)	k_{Transition–hTST} (s^–1)
Water rotations
A₁ → A₂A₃	(3.08 ± 1.43) 10¹⁰	7.55 × 10¹⁰
A₂A₃ → A₁	(1.49 ± 0.46) 10¹¹	2.06 × 10¹²
A₂A₃ → A₄	(4.33 ± 2.20) 10¹⁰	3.64 × 10¹⁰
A₄ → A₂A₃	(2.35 ± 0.87) 10¹¹	5.66 × 10¹¹
A₁ → A₄	(3.34 ± 6.56) 10⁶	2.04 × 10⁸
A₄ → A₁	(1.34 ± 0.68) 10¹⁰	8.65 × 10¹⁰
Hydroxyl rotation
D₁D₃ → D₂D₄	Ø	2.38 × 10⁹
D₂D₄ → D₁D₃	(2.86 ± 4.71) 10⁸	4.15 × 10⁹
Formation and dissociation of water
A₁ → D₁D₃	(1.64 ± 1.59) 10⁹	3.37 × 10¹¹
D₁D₃ → A₁	(2.32 ± 1.59) 10¹⁰	1.13 × 10¹²
A₂A₃ → D₂D₄	(7.86 ± 7.53) 10⁹	5.45 × 10¹³
D₂D₄ → A₂A₃	(1.28 ± 0.54) 10¹¹	1.17 × 10¹³
A₂A₃ → D₁D₃	Ø	Ø
D₁D₃ → A₂A₃	(2.33 ± 3.14) 10⁸	Ø

Reaction rate constants obtained by a harmonic approximation are consistently higher than those obtained via the Hill relation and AMS for the Langevin dynamics, with one single exception for the A₂A₃ → A₄ rotation. Assuming that the friction parameter is set so that Langevin dynamics reproduces accurately the system’s dynamics, the AMS rate constants should be more precise than the TST ones due to the intrinsic overestimation of rates of TST as mentioned in the Introduction. The harmonic approximation of the potential energy surface for fast approximations of free energies can lead to large errors. In particular, entropic effects are usually mistreated by hTST approaches as underlined by previous theoretical studies based on TPS and blue moon ensemble simulations (49) or other approaches. (12) In the present case, this might be the reason for the important overestimation of the rates of formation and dissociation events. Especially in the case of the A₂A₃ → D₂D₄ transition, the approximation of the TS free energy is so bad that the activation free energy is negative (as reported in SI Table 3), which leads to the large overestimation of the rates.

Under the assumption of a correctly parametrized dynamics, the values presented in Table 4 show that most water rotations are at least 1 order of magnitude faster than dissociation events. Only the direct A₁ → A₄ rotation seems to occur less frequently. The formation of water happens on the same time scale as the fast water rotations depending on the hydroxyls conformation. This ordering has to be compared to the one from hTST rate constants. The most frequent changes are the water formation and dissociation events. The slowest formation events occur as frequently as the fastest water rotation.

Using the presented approach to compute various reaction rate constants, especially those of forward and backward reactions, one can also deduce reaction free energies:

K_{R \to P} = \frac{k_{R \to P}}{k_{P \to R}}

(9)

and

Δ G_{R \to P} (T) = - N_{A} k_{B} T \ln (K_{R \to P})

(10)

where

N_{A}

is the Avogadro number and K_R→P is the reaction equilibrium constant.

The values of the reaction free energies of Table 5 allow one to identify that according to the harmonic approximation, the most stable state should be the D₁D₃, while the most stable one identified with the AMS method is A₁ for T = 200 K. Previous ab initio thermodynamic studies within harmonic approximations also identified that the dissociative state is more favored. (39,40) Here also, one may suspect that entropic contributions is at the origin of the change in the stability order. In particular, within the harmonic approximation, it is assumed that the adsorbed water molecule in A₁ state and in D₁D₃ has similar rotational and translational degree of freedoms. We cannot exclude that this assumption leads to errors as AIMD simulation reveals numerous rotational movements of the adsorbed water. This effect influences the entropy change and stabilizes the non dissociated A₁ state with respect to the dissociated one D₁D₃. This thermodynamic analysis may also be consistent with the previous kinetic observation. Indeed, the thermodynamic stabilization of the non dissociated reactant states with AMS induces that water dissociation rate constants are significantly smaller with AMS than with hTST.

Table 5. Reaction Heats at 200 K Computed from Table 4 and hTST

	AMS Value (kJ mol^–1)	hTST Value (kJ mol^–1)
Water rotations
	2.62 ± 2.66	5.50
	2.81 ± 2.83	4.56
	13.8 ± 4.43	10.1
Hydroxyl rotations
	Ø	0.93
Water dissociations
	4.41 ± 3.88	–2.56
	4.64 ± 3.54	2.01

3.3. Analysis of AMS Reactive Trajectories

In addition to computing reaction rates, we show in this section how the AMS method allows the sampling of reactive trajectories. The overall AMS trajectory lengths are on the order of 200 ps. Qualitatively speaking, some chemically relevant trends can be identified. We identify two pathways for the rotation A₄ → A₁. The first, and the less likely one, is similar to the path identified by the nudged elastic band (NEB) static approach (see SI Section 4.3). The second one seems to be more similar to a A₄ → A₂A₃ → A₁ rotation, where the trajectory does not actually enter the A₂A₃ state but approaches it for a few femtoseconds before continuing toward the A₁ state. The same type of paths are observed in the few trajectories where a transition D₁D₃ → A₂A₃ occurs. However, such a systematic analysis of each reactive trajectory might become rapidly tedious and not safe enough to capture the overall chemical trends, since more than 2000 A₁ → D₁D₃ trajectories are sampled by the AMS algorithm. An automated method is therefore necessary to analyze all of them and some dimensionality reduction is useful to this end.

3.3.1. Clustering Reactive Trajectories

In the case of the A₄ → A₁ rotation, two paths exist that can be identified by a visual inspection of many reactive trajectories. A more systematic way to proceed would be to rely on clustering methods, which are specially designed to identify groups within a data set. Among the various possible approaches, we used here an approach based on the K-means algorithm as implemented in scikit-learn. (67) To make the numerical representation of each trajectory independent on its length, each trajectory was represented as the intersections of the trajectory and five iso-levels of the A₄-vs-all SOAP-SVM RC. The details of the procedure to perform this clustering are presented in SI Section 7; the trajectories discussed in what folows are included as videos in the SI: Videos 2 and 3. It is important to mention that the K-means method requires to know a priori the number of clusters to find, thus various values should be tested. The two types of paths can be identified by visual inspection of the trajectory closest to each cluster’s centroid, even though all the trajectories are not perfectly assigned by this approach. Of course resorting to other clustering methods could be more efficient, but such a systematic study is beyond the scope of the present work. The “top” (green) path (see Figure 7; the trajectory is supplied in SI Video 2) qualitatively looks similar to the path found by the NEB. The fact that this path is less sampled than the “side” (blue) path (see Figure 7 and SI Video 3) indicates that this transition is rarer.

Figure 7. Schematic representation of the two types of paths for the A₄ → A₁ rotation. The first path (blue) is named “side” while the second one (green) is named “top”. The purple line represent the RC iso-levels used to represent the trajectories.

3.3.2. Stochastic Transition State estimation

One possibility is to consider only one structure per trajectory instead of the whole trajectory. The most important structure q along a trajectory can be defined as the one such that the committor probability is p_R→P(q) = 0.5, where p_R→P(q) is the probability that a molecular dynamics trajectory starting from q reaches first the P state rather than R. According to the IUPAC goldbook, (8) in the part of the TS definition referring to a surface, all the structures satisfying the p_R→P(q) = 0.5 conditions are part of the transition state. This definition of transition state as a “set of states (each characterized by its own geometry and energy)” is indeed not consistent with the following part of the definition “the transition state is characterized by one and only one imaginary frequency”, which presents it as a first order saddle point on the potential energy surface. The various structures q such that p_R→P(q) = 0.5 are not necessarily identical to the saddle points identified via the NEB method and harmonic frequencies calculations, although some resemblance is expected. We propose to investigate this point in what follows for one water dissociation on the alumina surface.

As mentioned in the Methods section, the estimated probability for a trajectory to reach the surface

Σ_{z_{k i l l}^{n + 1}}

starting on the surface

Σ_{z_{k i l l}^{n}}

1 - \frac{η_{k i l l e d}^{n + 1}}{N_{r e p}}

. By identifying the level n_0.5 such that

\prod_{n = n_{0.5}}^{n_{\max}} (1 - \frac{η_{k i l l e d}^{n}}{N_{r e p}}) = 0.5

(11)

one can define the iso-level Σ_0.5 of the reaction coordinate. The configurations corresponding to reactive trajectories crossing this surface are such that

{\hat{p}}_{R \to P} (q) = 0.5

. There should be at least one structure corresponding to this condition per reactive trajectory. Considering only the first structure crossing the iso-level Σ_0.5, the mean structure is computed, in the sense of the SOAP descriptor. This analysis was applied for the various realizations of AMS that were run.

3.3.3. Stochastic Transition State of Water Dissociation

For the dissociation event A₁ → D₁D₃, the interpolated SOAP-PCV reaction coordinate with the reactant and product states defined as R = A₁ ∪ A₂A₃ ∪ A₄ ∪ D₂D₄ and P = D₁D₃ leads to a mean structure of configurations such that p_R→P(q) = 0.5 qualitatively similar to the saddle point of the PES determined with the NEB method, as represented in Figure 8 (the corresponding trajectory is provided in SI: Video 1 ). From a quantitative viewpoint, some slight structural differences can be noted regarding the O–H distances involving the transferred H atom. For the saddle point, the broken O–H bond is 0.14 Å shorter than for AMS, whereas the newly formed O–H bond is 0.14 Å larger. This difference might come from the fact that momenta can bear a certain importance in the committor. Indeed, the committor values estimated bear dynamical information while the saddle point is defined only with the positions.

Figure 8. Ball and sticks representation of a) saddle point on the PES and b) mean structures such that p_R→P(q) = 0.5 on AMS using interpolated SOAP PCV RC. Color legend, red: oxygen, gray: aluminum, white: hydrogen.

The quality of this analysis depends on the quality of the sampling of the reaction path. Indeed, considering the reactive trajectories sampled from the AMS performed with R = A₁ and P = ∪ A₂A₃ ∪ A₄ ∪ D₁D₃ ∪ D₂D₄, the definition of the stochastic TS is of poor quality. This comes from the fact that this AMS mostly samples A₁ → A₂A₃ trajectories (and only rarely A₁ → D₁D₃ trajectories). The best approximation of a stochastic TS is on the most sampled path (Region 1 in Figure 9). This is in line with the obtained results for the confidence interval of the reaction rate constants (see Table 3). In Figure 9, the green curve represents the Σ_0.5 iso-level of the reaction coordinate, while the red one is the Σ_0.5 iso-level of the committor function. As these levels do not perfectly match on the whole space, the best approximation of the stochastic TS is in the region of space where most of the reactive trajectories concentrate (Region 1). This issue in the analysis of the reactive trajectories of less probable transitions is recurrent when multiple paths are sampled. An alternative approach to automatically identify whether multiple paths leading to a single product are present within a set of sampled trajectories would be desirable.

Figure 9. Schematic representation of poor match of the Σ_0.5 iso-level of the committor function (red) and the reaction coordinate (green). The green iso-level is placed after an AMS sampling of some reactive trajectories from R to P where a majority of the trajectories has gone via Region 1.

4. Conclusion

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Jump To

We proposed and implemented a theoretical approach based on the Hill relation to compute the exact reaction rate constants using rare event sampling and support vector machines. This approach is illustrated on various chemical events occurring at an oxide material surface. A key algorithm to this end is the AMS, which estimates reaction probabilities and samples reactive trajectories by using AIMD. For that purpose, SVM was used to define the chemically relevant states and reaction coordinates to index the transition from reactant to products. This allows the computation of the exact reaction rates for the dynamics at hand and makes possible a detailed analysis of reaction mechanisms via the inspection of reactive trajectories. The implementation tailored to communicate with a plane-wave DFT software allowed us to illustrate the approach by studying the reactivity of a water molecule adsorbed on the γ-alumina (100) surface. The computed reaction rate constants were discussed and compared to those of a static hTST approach.

The precision of the method is impacted by the choice of reaction coordinate, the choice of reactants and products in a multiple state situation, the number of repetitions of the probability estimation, and the number of replicas intrinsic to AMS. The hTST approach does not make assumptions on the system’s dynamics but relies on strong assumptions concerning the shape of the potential energy surface, implying an uncontrolled approximation of entropic contributions. The proposed methodology allows one to alleviate these limitations at the expense of an increased computational cost. Assuming that the Langevin dynamics accurately models the system’s dynamics (which involves in particular having a relevant value of the friction coefficient), the presented approach should be more precise than TST approaches. In the case considered here, hTST reaction rate constants are always higher than the ones estimated via AMS and the Hill relation. The relative stability of states is also different. In particular, we show that hTST underestimates the thermodynamic stability of adsorbed water molecules and simultaneously overestimates rate constants of water dissociation and formation. On top of that, the analysis of reactive trajectories allows one to identify possible paths that are not clearly identified via the NEB approach.

Depending on the system under investigation, this method used in combination with AIMD may become computationally expensive. This issue might be alleviated by the use of machine learning force fields (MLFFs), which can approach the accuracy of DFT force calculations at a much smaller computational cost. It also provides the opportunity to accurately describe nuclear quantum effects using path integral molecular dynamics such as in ref (69). Some active learning schemes to train MLFFs have been proposed recently, and they could articulate well with the present method. (70,71) In contrast to standard MD, the presented approach favors a sampling of transition regions that are crucial for the description of a chemical event. The study of a specific system could be done by first using jointly AMS and active learning to generate an accurate MLFF. Then, it could be used to evaluate accurately reaction rate constants and sample reactive trajectories.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00280.

Multilevel splitting estimator and AMS pseudo code, rate constant error estimation, state to state probability estimation in a multistate case, calculation parameters, implementation with VASP software, detailed numerical results, and clustering reactive trajectories (PDF)
Video 1 showing an A₁ → D₁ dissociation trajectory (AVI)
Video 2 showing an A₄ → A₁ “top” rotation trajectory (AVI)
Video 3 showing an A₄ → A₁ “side” rotation trajectory (AVI)

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Corresponding Authors
- Thomas Pigeon - MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France; CERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France; IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France; https://orcid.org/0000-0002-7828-5128; Email: [email protected]
- Tony Lelièvre - CERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France; MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France; Email: [email protected]
- Pascal Raybaud - IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France; https://orcid.org/0000-0003-4506-5062; Email: [email protected]
Authors
- Gabriel Stoltz - CERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France; MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, France
- Manuel Corral-Valero - IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France; https://orcid.org/0000-0002-4457-3914
- Ani Anciaux-Sedrakian - IFP Energies Nouvelles, 1 et 4 avenue de Bois-Préau, F-92852 Rueil-Malmaison Cedex, France
- Maxime Moreaud - IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France; https://orcid.org/0000-0002-4908-401X
Notes
The authors declare no competing financial interest.

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This project was realized in the framework of the joint laboratory IFPEN-Inria Convergence HPC/AI/HPDA for the energetic transition. Calculations were performed using the following HPC resources: Jean Zay and Occigen from GENCI-CINES, Joliot-Curie (Irene) from TGCC/CEA (Grant A0120806134), ENER440 from IFP Energies nouvelles and Topaze from CCRT-CEA. The work of T.L. and G.S. was funded in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (project EMC2, grant agreement no. 810367).

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A review. The unprecedented ability of computations to probe at.-level details of catalytic systems holds immense promise for the fundamentals-based bottom-up design of novel heterogeneous catalysts, which are at the heart of the chem. and energy sectors of industry. Here, we critically analyze recent advances in computational heterogeneous catalysis. First, we will survey the progress in electronic structure methods and atomistic catalyst models employed, which have enabled the catalysis community to build increasingly intricate, realistic, and accurate models of the active sites of supported transition-metal catalysts. We then review developments in microkinetic modeling, specifically mean-field microkinetic models and kinetic Monte Carlo simulations, which bridge the gap between nanoscale computational insights and macroscale exptl. kinetics data with increasing fidelity. We finally review the advancements in theor. methods for accelerating catalyst design and discovery. Throughout the review, we provide ample examples of applications, discuss remaining challenges, and provide our outlook for the near future.
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Allen, Rosalind J.; Warren, Patrick B.; Ten Wolde, Pieter Rein
Physical Review Letters (2005), 94 (1), 018104/1-018104/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
Bistable biochem. switches are widely found in gene regulatory networks and signal transduction pathways. Their switching dynamics are difficult to study, however, because switching events are rare, and the systems are out of equil. We present a simulation method for predicting the rate and mechanism of the flipping of these switches. We apply it to a genetic switch and find that it is highly efficient. The path ensembles for the forward and reverse processes do not coincide. The method is widely applicable to rare events and nonequil. processes.
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Huber, G.; Kim, S. Weighted-ensemble Brownian dynamics simulations for protein association reactions. Biophys. J. 1996, 70, 97– 110, DOI: 10.1016/S0006-3495(96)79552-8
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Weighted-ensemble Brownian dynamics simulations for protein association reactions
Huber, Gary A.; Kim, Sangtae
Biophysical Journal (1996), 70 (1), 97-110CODEN: BIOJAU; ISSN:0006-3495. (Biophysical Society)
A new method, weighted-ensemble Brownian dynamics, is proposed for the simulation of protein-assocn. reactions and other events whose frequencies of outcomes are constricted by free energy barriers. The method features a weighted ensemble of trajectories in configuration space with energy levels dictating the proper correspondence between "particles" and probability. Instead of waiting a very long time for an unlikely event to occur, the probability packets are split, and small packets of probability are allowed to diffuse almost immediately into regions of configuration space that are less likely to be sampled. The method was applied to the Northrup and Erickson (1992) model of docking-type diffusion-limited reactions and yields reaction rate consts. in agreement with those obtained by direct Brownian simulation, but at a fraction of the CPU time (10-4 to 10-3, depending on the model). Because the method is essentially a variant of std. Brownian dynamics algorithms, it is anticipated that weighted-ensemble Brownian dynamics, in conjunction with biophys. force models, can be applied to a large class of assocn. reactions of interest to the biophysics community.
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Cérou, F.; Guyader, A. Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 2007, 25, 417– 443, DOI: 10.1080/07362990601139628
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Glielmo, A.; Husic, B. E.; Rodriguez, A.; Clementi, C.; Noé, F.; Laio, A. Unsupervised learning methods for molecular simulation data. Chem. Rev. 2021, 121, 9722– 9758, DOI: 10.1021/acs.chemrev.0c01195
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Unsupervised Learning Methods for Molecular Simulation Data
Glielmo, Aldo; Husic, Brooke E.; Rodriguez, Alex; Clementi, Cecilia; Noe, Frank; Laio, Alessandro
Chemical Reviews (Washington, DC, United States) (2021), 121 (16), 9722-9758CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
A review. Unsupervised learning is becoming an essential tool to analyze the increasingly large amts. of data produced by atomistic and mol. simulations, in material science, solid state physics, biophysics, and biochem. In this Review, we provide a comprehensive overview of the methods of unsupervised learning that have been most commonly used to investigate simulation data and indicate likely directions for further developments in the field. In particular, we discuss feature representation of mol. systems and present state-of-the-art algorithms of dimensionality redn., d. estn., and clustering, and kinetic models. We divide our discussion into self-contained sections, each discussing a specific method. In each section, we briefly touch upon the math. and algorithmic foundations of the method, highlight its strengths and limitations, and describe the specific ways in which it has been used-or can be used-to analyze mol. simulation data.
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Chen, M. Collective variable-based enhanced sampling and machine learning. Eur. Phys. J. B 2021, 94, 211, DOI: 10.1140/epjb/s10051-021-00220-w
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Collective variable-based enhanced sampling and machine learning
Chen, Ming
European Physical Journal B: Condensed Matter and Complex Systems (2021), 94 (10), 211CODEN: EPJBFY; ISSN:1434-6028. (Springer)
Abstr.: Collective variable-based enhanced sampling methods have been widely used to study thermodn. properties of complex systems. Efficiency and accuracy of these enhanced sampling methods are affected by two factors: constructing appropriate collective variables for enhanced sampling and generating accurate free energy surfaces. Recently, many machine learning techniques have been developed to improve the quality of collective variables and the accuracy of free energy surfaces. Although machine learning has achieved great successes in improving enhanced sampling methods, there are still many challenges and open questions. In this perspective, we shall review recent developments on integrating machine learning techniques and collective variable-based enhanced sampling approaches. We also discuss challenges and future research directions including generating kinetic information, exploring high-dimensional free energy surfaces, and efficiently sampling all-atom configurations. Graphic abstr.: [graphic not available: see fulltext].
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Gkeka, P.; Stoltz, G.; Farimani, A. B.; Belkacemi, Z.; Ceriotti, M.; Chodera, J. D.; Dinner, A. R.; Ferguson, A. L.; Maillet, J.-B.; Minoux, H.; Peter, C.; Pietrucci, F.; Silveira, A.; Tkatchenko, A.; Trstanova, Z.; Wiewiora, R.; Lelièvre, T. Machine learning force fields and coarse-grained variables in molecular dynamics: Application to materials and biological systems. J. Chem. Theory Comput. 2020, 16, 4757– 4775, DOI: 10.1021/acs.jctc.0c00355
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Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems
Gkeka, Paraskevi; Stoltz, Gabriel; Barati Farimani, Amir; Belkacemi, Zineb; Ceriotti, Michele; Chodera, John D.; Dinner, Aaron R.; Ferguson, Andrew L.; Maillet, Jean-Bernard; Minoux, Herve; Peter, Christine; Pietrucci, Fabio; Silveira, Ana; Tkatchenko, Alexandre; Trstanova, Zofia; Wiewiora, Rafal; Lelievre, Tony
Journal of Chemical Theory and Computation (2020), 16 (8), 4757-4775CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
A review. Machine learning encompasses tools and algorithms that are now becoming popular in almost all scientific and technol. fields. This is true for mol. dynamics as well, where machine learning offers promises of extg. valuable information from the enormous amts. of data generated by simulation of complex systems. The authors provide here a review of the authors' current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab initio databases and the detn. of reaction coordinates for free energy computation and enhanced sampling.
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Ferguson, A. L. Machine learning and data science in soft materials engineering. J. Condens. Matter Phys. 2018, 30, 043002, DOI: 10.1088/1361-648X/aa98bd
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Machine learning and data science in soft materials engineering
Ferguson, Andrew L.
Journal of Physics: Condensed Matter (2018), 30 (4), 043002/1-043002/27CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
A review. In many branches of materials science it is now routine to generate data sets of such large size and dimensionality that conventional methods of anal. fail. Paradigms and tools from data science and machine learning can provide scalable approaches to identify and ext. trends and patterns within voluminous data sets, perform guided traversals of high-dimensional phase spaces, and furnish data-driven strategies for inverse materials design. This topical review provides an accessible introduction to machine learning tools in the context of soft and biol. materials by 'de-jargonizing' data science terminol., presenting a taxonomy of machine learning techniques, and surveying the math. underpinnings and software implementations of popular tools, including principal component anal., independent component anal., diffusion maps, support vector machines, and relative entropy. The authors present illustrative examples of machine learning applications in soft matter, including inverse design of self-assembling materials, nonlinear learning of protein folding landscapes, high- throughput antimicrobial peptide design, and data-driven materials design engines. The authors close with an outlook on the challenges and opportunities for the field.
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Sultan, M. M.; Pande, V. S. Automated design of collective variables using supervised machine learning. J. Chem. Phys. 2018, 149, 094106, DOI: 10.1063/1.5029972
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Automated design of collective variables using supervised machine learning
Sultan, Mohammad M.; Pande, Vijay S.
Journal of Chemical Physics (2018), 149 (9), 094106/1-094106/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
Selection of appropriate collective variables (CVs) for enhancing sampling of mol. simulations remains an unsolved problem in computational modeling. In particular, picking initial CVs is particularly challenging in higher dimensions. Which at. coordinates or transforms there of from a list of thousands should one pick for enhanced sampling runs. How does a modeler even begin to pick starting coordinates for investigation. This remains true even in the case of simple two state systems and only increases in difficulty for multi-state systems. In this work, we solve the "initial" CV problem using a data-driven approach inspired by the field of supervised machine learning (SML). In particular, we show how the decision functions in SML algorithms can be used as initial CVs (SMLcv) for accelerated sampling. Using solvated alanine dipeptide and Chignolin mini-protein as our test cases, we illustrate how the distance to the support vector machines' decision hyperplane, the output probability ests. from logistic regression, the outputs from shallow or deep neural network classifiers, and other classifiers may be used to reversibly sample slow structural transitions. We discuss the utility of other SML algorithms that might be useful for identifying CVs for accelerating mol. simulations. (c) 2018 American Institute of Physics.
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Pozun, Z. D.; Hansen, K.; Sheppard, D.; Rupp, M.; Müller, K.-R.; Henkelman, G. Optimizing transition states via kernel-based machine learning. J. Chem. Phys. 2012, 136, 174101, DOI: 10.1063/1.4707167
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Optimizing transition states via kernel-based machine learning
Pozun, Zachary D.; Hansen, Katja; Sheppard, Daniel; Rupp, Matthias; Mueller, Klaus-Robert; Henkelman, Graeme
Journal of Chemical Physics (2012), 136 (17), 174101/1-174101/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The authors present a method for optimizing transition state theory dividing surfaces with support vector machines. The resulting dividing surfaces require no a priori information or intuition about reaction mechanisms. To generate optimal dividing surfaces, the authors apply a cycle of machine-learning and refinement of the surface by mol. dynamics sampling. The machine-learned surfaces contain the relevant low-energy saddle points. The mechanisms of reactions may be extd. from the machine-learned surfaces to identify unexpected chem. relevant processes. Also, the machine-learned surfaces significantly increase the transmission coeff. for an adatom exchange involving many coupled degrees of freedom on a (100) surface when compared to a distance-based dividing surface. (c) 2012 American Institute of Physics.
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Christiansen, M. A.; Mpourmpakis, G.; Vlachos, D. G. Density functional theory - Computed mechanisms of ethylene and diethyl ether formation from ethanol on γ-Al₂O₃(100). ACS Catal. 2013, 3 (9), 1965– 1975, DOI: 10.1021/cs4002833
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Density Functional Theory-Computed Mechanisms of Ethylene and Diethyl Ether Formation from Ethanol on γ-Al2O3(100)
Christiansen, Matthew A.; Mpourmpakis, Giannis; Vlachos, Dionisios G.
ACS Catalysis (2013), 3 (9), 1965-1975CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
Multiple potential active sites on the surface of γ-Al2O3 have led to debate about the role of Lewis and/or Bronsted acidity in reactions of ethanol, while mechanistic insights into competitive prodn. of ethylene and di-Et ether are scarce. In this study, elementary adsorption and reaction mechanisms for ethanol dehydration and etherification are studied on the γ-Al2O3(100) surface using d. functional theory calcns. The O atom of adsorbed ethanol interacts strongly with surface Al (Lewis acid) sites, while adsorption is weak on Bronsted (surface H) and surface O sites. Water, a byproduct of both ethylene and di-Et ether formation, competes with ethanol for adsorption sites. Multiple pathways for ethylene formation from ethanol are explored, and a concerted Lewis-catalyzed elimination (E2) mechanism is found to be the energetically preferred pathway, with a barrier of Ea = 37 kcal/mol at the most stable site. Di-Et ether formation mechanisms presented for the first time on γ-Al2O3 indicate that the most favorable pathways involve Lewis-catalyzed SN2 reactions (Ea = 35 kcal/mol). Addnl. novel mechanisms for di-Et ether decompn. to ethylene are reported. Bronsted-catalyzed mechanisms for ethylene and ether formation are not favorable on the (100) facet because of weak adsorption on Bronsted sites. These results explain multiple exptl. observations, including the competition between ethylene and di-Et ether formation on alumina surfaces.
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Larmier, K.; Nicolle, A.; Chizallet, C.; Cadran, N.; Maury, S.; Lamic-Humblot, A.-F.; Marceau, E.; Lauron-Pernot, H. Influence of coadsorbed water and alcohol molecules on isopropyl alcohol dehydration on γ-alumina: Multiscale modeling of experimental kinetic profiles. ACS Catal. 2016, 6, 1905– 1920, DOI: 10.1021/acscatal.6b00080
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Influence of Coadsorbed Water and Alcohol Molecules on Isopropyl Alcohol Dehydration on γ-Alumina: Multiscale Modeling of Experimental Kinetic Profiles
Larmier, Kim; Nicolle, Andre; Chizallet, Celine; Cadran, Nicolas; Maury, Sylvie; Lamic-Humblot, Anne-Felicie; Marceau, Eric; Lauron-Pernot, Helene
ACS Catalysis (2016), 6 (3), 1905-1920CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
Successfully modeling the behavior of catalytic systems at different scales is a matter of importance not only for a fundamental understanding but also for a more rational design of catalysts and a more precise definition of the kinetic laws used as inputs in chem. engineering. We have developed here a multiscale modeling of the dehydration of iso-Pr alc. to propene and diisopropyl ether on γ-alumina catalysts, which clearly evidences and explains the central character of cooperative effects between coadsorbates in the kinetic network. The evolution of partial pressures with contact time was simulated using an original DFT-based microkinetic model based on a "macro site" centered on the main active site located on the (100) planes of alumina and comprising several neighboring adsorption sites. The formation of iso-Pr alc.-iso-Pr alc. or water-iso-Pr alc. dimers on the surface was required to correctly simulate the prodn. of the minor product, diisopropyl ether, and the evolution of the product partial pressures at high conversion. DFT calcns. were used to identify the structure of these dimers. In addn. to entropic effects, the selectivity to ether is ruled by (i) stabilizing interactions between coadsorbed iso-Pr alc. or water mols. and the nucleophilic alc. mol. reacting with the alcoholate intermediate, (ii) the formation of alcoholate-water dimers that selectively inhibit the formation of propene and increase the selectivity to ether at low conversion, and (iii) the reverse transformation of diisopropyl ether into propene and iso-Pr alc. that consumes ether at high conversion. The anal. expression of the reaction rate derived from this model and based on the existence of ensembles of interacting iso-Pr alc. and water mols. leads to a satisfactory modeling of the exptl. kinetic measurements at all conversions.
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Hass, K. C.; Schneider, W. F.; Curioni, A.; Andreoni, W. The chemistry of water on alumina surfaces: Reaction dynamics from first principles. Science 1998, 282, 265– 268, DOI: 10.1126/science.282.5387.265
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The chemistry of water on alumina surfaces: reaction dynamics from first principles
Hass, Kenneth C.; Schneider, William F.; Curioni, Alessandro; Andreoni, Wanda
Science (Washington, D. C.) (1998), 282 (5387), 265-268CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
Aluminas and their surface chem. play a vital role in many areas of modern technol. The behavior of adsorbed water is particularly important and poorly understood. Simulations of hydrated α-alumina (0001) surfaces with ab initio mol. dynamics elucidate many aspects of this problem, esp. the complex dynamics of water dissocn. and related surface reactions. At low water coverage, free energy profiles established that molecularly adsorbed water is metastable and dissocs. readily, even in the absence of defects, by a kinetically preferred pathway. Observations at higher water coverage revealed rapid dissocn. and unanticipated collective effects, including water-catalyzed dissocn. and proton transfer reactions between adsorbed water and hydroxide. The results provide a consistent interpretation of the measured coverage dependence of water heats of adsorption, hydroxyl vibrational spectra, and other expts.
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Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H. Hydroxyl groups on γ-alumina surfaces: A DFT study. J. Catal. 2002, 211, 1– 5, DOI: 10.1006/jcat.2002.3741
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Hydroxyl Groups on γ-Alumina Surfaces: A DFT Study
Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H.
Journal of Catalysis (2002), 211 (1), 1-5CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Science)
Despite numerous exptl. studies devoted to the acid-base properties of γ-alumina, the precise nature of surface acid sites remains unsolved. Using d. functional (DFT) calcns., we propose realistic models of γ-alumina (110) and (100) surfaces accounting for hydroxylation/dehydroxylation processes induced by temp. effects. The vibrational anal., based on DFT calcns., leads to an accurate assignment of the OH stretching frequencies obsd. by IR spectroscopy. The extension to chlorinated surfaces, which brings new insights into the understanding of the role of dopes, is also addressed.
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Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P. Use of DFT to achieve a rational understanding of acido-basic properties of γ-alumina surfaces. J. Catal. 2004, 226, 54– 68, DOI: 10.1016/j.jcat.2004.04.020
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Use of DFT to achieve a rational understanding of acid-basic properties of γ-alumina surfaces
Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H.
Journal of Catalysis (2004), 226 (1), 54-68CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Science)
In a recent priority communication [M. Digne et al., J. Catal. 211 (2002) 1], we proposed the first ab initio constructed models of γ-alumina surfaces. Using the same d.-functional approach, we investigate in further detail the acid-basic properties of the three relevant γ-alumina (100), (110), and (111) surfaces, taking into account the temp.-dependent hydroxyl surface coverages. The simulations, compared fruitfully with many available exptl. data, enable us to solve the challenging assignment of the OH-stretching frequencies, as obtained from IR spectroscopy. The precise nature of the acid surface sites (concns. and strengths) is also detd. The acid strengths are quantified by simulating the adsorption of relevant probe mols. such as CO and pyridine in correlation with surface electronic properties. These results seriously challenge the historical model of a defective spinel for γ-alumina and establish the basis for a more rigorous description of the acid-basic properties of γ-alumina.
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Wischert, R.; Laurent, P.; Copéret, C.; Delbecq, F.; Sautet, P. γ-Alumina: The essential and unexpected role of water for the structure, stability, and reactivity of ”defect” sites. J. Am. Chem. Soc. 2012, 134, 14430– 14449, DOI: 10.1021/ja3042383
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γ-Alumina: The Essential and Unexpected Role of Water for the Structure, Stability, and Reactivity of Defect Sites
Wischert, Raphael; Laurent, Pierre; Coperet, Christophe; Delbecq, Francoise; Sautet, Philippe
Journal of the American Chemical Society (2012), 134 (35), 14430-14449CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
Combining expts. and DFT calcns., we show that tricoordinate AlIII Lewis acid sites, which are present as metastable species exclusively on the major (110) termination of γ- and δ-Al2O3 particles, correspond to the defect sites, which are held responsible for the unique properties of activated (thermally pretreated) alumina. These defects are, in fact, largely responsible for the adsorption of N2 and the splitting of CH4 and H2. In contrast, five-coordinate Al surface sites of the minor (100) termination cannot account for the obsd. reactivity. The AlIII sites, which are formed upon partial dehydroxylation of the surface (the optimal pretreatment temp. being 700 °C for all probes), can coordinate N2 selectively. In combination with specific O atoms, they form extremely reactive Al,O Lewis acid-base pairs that trigger the low-temp. heterolytic splitting of CH4 and H2 to yield Al-CH3 and Al-H species, resp. H2 is found overall more reactive than CH4 because of its higher acidity, hence it also reacts on four-coordinate sites of the (110) termination. Water has the dual role of stabilizing the (110) termination and modifying (often increasing) both the Lewis acidity of the aluminum and the basicity of nearby oxygens, hence the high reactivity of partially dehydroxylated alumina surfaces. In addn., we demonstrate that the presence of water enhances the acidity of certain four-coordinate Al atoms, which leads to strong coordination of the CO mol. with a spectroscopic signature similar to that on AlIII sites, thus showing the limits of this widely used probe for the acidity of oxides. Overall, the dual role of water translates into optimal water coverage, and this probably explains why in many catalyst prepns., optimal pretreatment temps. are typically obsd. in the activation step of alumina.
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Pigeon, T.; Chizallet, C.; Raybaud, P. Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces. J. Catal. 2022, 405, 140– 151, DOI: 10.1016/j.jcat.2021.11.011
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Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces
Pigeon, Thomas; Chizallet, Celine; Raybaud, Pascal
Journal of Catalysis (2022), 405 (), 140-151CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Inc.)
The rational understanding of γ-alumina (γ-Al2O3) supported catalysts requires an ever more improved at. scale detn. of the support's surface properties. By using d. functional theory (DFT) calcns., we show how the structural and energetic surface properties of alumina crystallites intrinsically depend on its synthesis pathway. Considering the case study of the topotactic transformation of boehmite (γ-AlOOH) into γ-Al2O3 taking place during calcination, we propose a methodol. to mimic this pathway by reconstructing relevant slabs of boehmite into γ-alumina slabs following 3 steps: dehydration, contraction/translation and Al migration into spinel or non-spinel sites. On the one hand, we confirm the reliability of some earlier 100, 110 and 111 surface structures detd. by std. bulk cleavage approach. Moreover, we find new γ-alumina surfaces harboring Bronsted acid sites (BAS) and Lewis acid sites (LAS) with specific local structures. More strikingly, we find that the basal (110)b surface of alumina inherited from the (0 1 0) basal surface of boehmite, exhibits a larger no. of isolated μ2-OH groups than the lateral( 110)l surface. For the lateral (110)l (resp. 111) orientation, four (resp. three) thermodynamically competing surfaces are identified, including models earlier proposed. These results are induced by finite size and morphol. effects during the topotactic transformation of boehmite crystallites. Thanks to a thorough comparative anal. of morphol. and nature of BAS and LAS as a function of thermal treatment and water pressure for each surface, we identify coherent chem. families of surfaces across the main crystallog. orientations. These features open the door to a better differentiation of the reactivity of the basal alumina surfaces from the lateral ones.
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Lu, Y.-H.; Wu, S.-Y.; Chen, H.-T. H₂O Adsorption/Dissociation and H₂ generation by the reaction of H₂O with Al₂O₃ materials: A first-principles investigation. J. Phys. Chem. C 2016, 120, 21561– 21570, DOI: 10.1021/acs.jpcc.6b07191
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H2O Adsorption/Dissociation and H2 Generation by the Reaction of H2O with Al2O3 Materials: A First-Principles Investigation
Lu, Yu-Huan; Wu, Shiuan-Yau; Chen, Hsin-Tsung
Journal of Physical Chemistry C (2016), 120 (38), 21561-21570CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
The microscopic reaction mechanisms for the water adsorption/dissocn. and hydrogen generation processes on the α-Al2O3(0001) surface are clarified by using spin-polarized d. functional theory with the projected augmented wave approach. The adsorptions of OH, O, and H species are also examd. Calcns. show that the H2O, OH, O, and H species prefer to adsorb at the Al(II)-top, Al(I, II)-top, Al(I, II)-bridge, and Al(II)-top sites with adsorption energies of -1.34, -5.91, -8.22, and -3.14 eV on the Al-terminated surface, whereas those are Al-top, Al-top, Al-top, and O-top sites with adsorption energies of -1.11, -2.79, -2.00, and -2.23 eV for the Al, O-terminated surface. Geometries of the mol. adsorbed intermediates, transition states, and the hydroxylated products as well as the energetic reaction routes are fully elucidated. Hydrogen generation and full dissocn. of water are found to occur on the Al-terminated surface with overall exothermicities of 2.37 and 4.22 eV, whereas only the prodn. of coadsorbed H(ads) + OH(ads) is obsd. on the Al, O-terminated surface with an overall exothermicity of 1.06-1.64 eV. In addn., the local d. of states and Bader charge calcns. are carried out to study the interaction between the adsorbate and surface along the reaction.
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Pan, Y.; Liu, C.-J.; Ge, Q. Adsorption and protonation of CO₂ on partially hydroxylated γ-Al₂O₃ surfaces: A density functional theory study. Langmuir 2008, 24, 12410– 12419, DOI: 10.1021/la802295x
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Adsorption and Protonation of CO2 on Partially Hydroxylated γ-Al2O3 Surfaces: A Density Functional Theory Study
Pan, Yunxiang; Liu, Chang-jun; Ge, Qingfeng
Langmuir (2008), 24 (21), 12410-12419CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
Adsorption and protonation of CO2 on the (110) and (100) surfaces of γ-Al2O3 were studied using d. functional theory slab calcns. On the dry (110) and (100) surfaces, the O-Al bridge sites are energetically favorable for CO2 adsorption. The adsorbed CO2 was bound in a bidentate configuration across the O-Al bridge sites, forming a carbonate species. The strongest binding with an adsorption energy of 0.80 eV occurs at the O3c-Al5c bridge site of the (100) surface. Dissocn. of water across the O-Al bridge sites resulted in partially hydroxylated surfaces, and the dissocn. is energetically favorable on both surfaces. Water dissocn. on the (110) surface has a barrier of 0.42 eV, but the same process on the (100) surface has no barrier with respect to the isolated water mol. On the partially hydroxylated γ-Al2O3 surfaces, a bicarbonate species was formed by protonating the carbonate species with the protons from neighboring hydroxyl groups. The energy difference between the bicarbonate species and the coadsorbed bidentate carbonate species and hydroxyls is only 0.04 eV on the (110) surface, but the difference reaches 0.97 eV on the (100) surface. The activation barrier for forming the bicarbonate species on the (100) surface, 0.42 eV, is also lower than that on the (110) surface (0.53 eV).
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Ngouana-Wakou, B. F.; Cornette, P.; Valero, M. C.; Costa, D.; Raybaud, P. An atomistic description of the γ-alumina/water interface revealed by ab initio molecular dynamics. J. Phys. Chem. C 2017, 121, 10351– 10363, DOI: 10.1021/acs.jpcc.7b00101
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An Atomistic Description of the γ-Alumina/Water Interface Revealed by Ab Initio Molecular Dynamics
Ngouana-Wakou, B. F.; Cornette, P.; Corral Valero, M.; Costa, D.; Raybaud, P.
Journal of Physical Chemistry C (2017), 121 (19), 10351-10363CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
The authors report ab initio mol. dynamics (AIMD) simulations of the (100) and (110) γ-Al2O3/water interfaces at 300 K, using two sets of supercell models for each surface and two time lengths of simulation (10 and 40 ps). The authors first show that the effect of liq. water on the vibrational frequencies of hydroxyl groups at the interface varies according to the type of surface. This trend is explained by two key parameters affecting the interaction of both surfaces with water: the nature of the OH groups (i.e., μ1-OH, μ1-H2O, μ2-OH, and μ3-OH) and H-bond network among surface OH groups. The hydroxylated (110) surface favors the local structuration of water at the interface and the solvation of its μ1-OH and μ1-H2O groups by water similarly as in bulk liq. water. By contrast, on the (100) surface, a stronger H-bond network among μ1-OH and μ1-H2O groups reduces the water/surface interaction. The authors illustrate also how the interfacial interacting sites are spatially organized on the surfaces by two-dimensional maps of O-H distances. On both surfaces, the interfacial water layer orientation is predominantly Hup-Hdown. For long AIMD simulation time, Grotthuss-like mechanisms are identified on the (110) surface.
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Réocreux, R.; Jiang, T.; Iannuzzi, M.; Michel, C.; Sautet, P. Structuration and dynamics of interfacial liquid water at hydrated γ-alumina determined by ab initio molecular simulations: Implications for nanoparticle stability. ACS Appl. Nano Mater. 2018, 1, 191– 199, DOI: 10.1021/acsanm.7b00100
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Structuration and Dynamics of Interfacial Liquid Water at Hydrated γ-Alumina Determined by ab Initio Molecular Simulations: Implications for Nanoparticle Stability
Reocreux, Romain; Jiang, Tao; Iannuzzi, Marcella; Michel, Carine; Sautet, Philippe
ACS Applied Nano Materials (2018), 1 (1), 191-199CODEN: AANMF6; ISSN:2574-0970. (American Chemical Society)
Liq. water/solid interfaces are central in catalytic nanomaterials, from their prepn. to their chem. stability under harsh catalytic conditions such as the hot aq. medium used in biomass valorization. Here we report an ab initio mol. dynamics (AIMD) study of the γ-Al2O3 (110)/water interface using the most recent surface model available in the literature. The size of the simulation box and the duration of the AIMD simulation enables us to characterize the whole interface at the at. scale. The simulation evidences a redistribution of protons within the chemisorbed water layer. The influence of γ-Al2O3 (110) is also important on the water mols. that are not bound to the surface: it is only above 10 Å that water recovers its bulk liq. behavior. The influence of alumina is structural, with preferred angular orientations for water mols., and also dynamical. The translational self-diffusivity of water is diminished by up to 2 orders of magnitude, and the angular relaxation time increased up to a factor of 6. The influence of the interface on chemisorbed water mols. is also characterized with an IR spectrum (fully simulated at the d. functional theory level) that shows two distinct regions (3500 and 3200 cm-1) assigned to two different interfacial environments. This full characterization of the nanoscale interfacial zone highlights the specific physicochem. features of water that arise in contact with γ-Al2O3 and opens the door to an improved prepn. of supported catalysts (from templating agents to protective coatings).
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Lo, C. S.; Radhakrishnan, R.; Trout, B. L. Application of transition path sampling methods in catalysis: A new mechanism for CC bond formation in the methanol coupling reaction in Chabazite. Catal. Today 2005, 105, 93– 105, DOI: 10.1016/j.cattod.2005.04.005
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Application of transition path sampling methods in catalysis: A new mechanism for C-C bond formation in the methanol coupling reaction in chabazite
Lo, Cynthia S.; Radhakrishnan, Ravi; Trout, Bernhardt L.
Catalysis Today (2005), 105 (1), 93-105CODEN: CATTEA; ISSN:0920-5861. (Elsevier B.V.)
We describe the application of transition path sampling methods to the methanol coupling reaction in the zeolite chabazite; these methods have only been recently applied to complex chem. systems. Using these methods, we have found a new mechanism for the formation of the first C-C bond. In our mechanism, the reaction, at 400 °C, proceeds via a two-step process: (1) the breaking of the C-O bond of the chemisorbed methoxonium cation, followed by the transfer of a hydride ion from the remaining methanol mol. to the Me cation, resulting in the formation of H2O, CH4 , and CH2OH+ and (2) a simultaneous proton transfer from methane to water, and direct C-C bond formation between the Me anion and CH2OH+, resulting in the formation of ethanol. The C - C bond forming process has the higher barrier, with an activation energy of about 100.49 kJ/mol.
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Bucko, T.; Benco, L.; Dubay, O.; Dellago, C.; Hafner, J. Mechanism of alkane dehydrogenation catalyzed by acidic zeolites: Ab initio transition path sampling. J. Chem. Phys. 2009, 131, 214508, DOI: 10.1063/1.3265715
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Mechanism of alkane dehydrogenation catalyzed by acidic zeolites: ab initio transition path sampling
Bucko, Tomas; Benco, Lubomir; Dubay, Orest; Dellago, Christoph; Hafner, Juergen
Journal of Chemical Physics (2009), 131 (21), 214508/1-214508/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The dehydrogenation of propane over acidic chabazite has been studied using ab initio d.-functional simulations in combination with static transition-state searches and dynamic transition path sampling (TPS) methods at elevated temps. The acidic zeolite has been modeled both using a small cluster and a large periodic model consisting of two unit cells, the TPS simulations allow to account for the effect of temp. and entropy. In agreement with exptl. observations we find propene as the dominant reaction product and that the barrier for the dehydrogenation of a Me group is higher than that for a methylene group. However, whereas all studies based on small cluster models (including the present one) conclude that the reaction proceeds via the formation of an alkoxy intermediate, our TPS studies based on a large periodic model lead to the conclusion that propene formation occurs via the formation of various forms of Pr cations stabilized by entropy, while the formation of an alkoxy species is a relatively rare event. It was obsd. only in 15% of the reactive trajectories for Me dehydrogenation and even in only 8% of the methylene dehydrogenation reactions. Our studies demonstrate the importance of entropic effects and the need to account for the structure and flexibility of the zeolitic framework by using large periodic models. (c) 2009 American Institute of Physics.
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Rey, J.; Bignaud, C.; Raybaud, P.; Bucko, T.; Chizallet, C. Dynamic features of transition states for beta-scission reactions of alkenes over acid zeolites revealed by AIMD simulations. Angew. Chem., Int. Ed. Engl. 2020, 59, 18938– 18942, DOI: 10.1002/anie.202006065
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Dynamic Features of Transition States for β-Scission Reactions of Alkenes over Acid Zeolites Revealed by AIMD Simulations
Rey Jerome; Bignaud Charles; Raybaud Pascal; Chizallet Celine; Bignaud Charles; Bucko Tomas; Bucko Tomas
Angewandte Chemie (International ed. in English) (2020), 59 (43), 18938-18942 ISSN:.
Zeolite-catalyzed alkene cracking is key to optimize the size of hydrocarbons. The nature and stability of intermediates and transition states (TS) are, however, still debated. We combine transition path sampling and blue moon ensemble density functional theory simulations to unravel the behavior of C7 alkenes in CHA zeolite. Free energy profiles are determined, linking π-complexes, alkoxides and carbenium ions, for B1 (secondary to tertiary) and B2 (tertiary to secondary) β-scissions. B1 is found to be easier than B2 . The TS for B1 occurs at the breaking of the C-C bond, while for B2 it is the proton transfer from propenium to the zeolite. We highlight the dynamic behaviors of the various intermediates along both pathways, which reduce activation energies with respect to those previously evaluated by static approaches. We finally revisit the ranking of isomerization and cracking rate constants, which are crucial for future kinetic studies.
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Roet, S.; Daub, C. D.; Riccardi, E. Chemistrees: Data-driven identification of reaction pathways via machine learning. J. Chem. Theory Comput. 2021, 17, 6193– 6202, DOI: 10.1021/acs.jctc.1c00458
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Chemistrees: Data-Driven Identification of Reaction Pathways via Machine Learning
Roet, Sander; Daub, Christopher D.; Riccardi, Enrico
Journal of Chemical Theory and Computation (2021), 17 (10), 6193-6202CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
We propose to analyze mol. dynamics (MD) output via a supervised machine learning (ML) algorithm, the decision tree. The approach aims to identify the predominant geometric features which correlate with trajectories that transition between two arbitrarily defined states. The data-driven algorithm aims to identify these features without the bias of human "chem. intuition". We demonstrate the method by analyzing the proton exchange reactions in formic acid solvated in small water clusters. The simulations were performed with ab initio MD combined with a method to efficiently sample the rare event, path sampling. Our ML anal. identified relevant geometric variables involved in the proton transfer reaction and how they may change as the no. of solvating water mols. changes.
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Lopes, L. J. S.; Lelièvre, T. Analysis of the adaptive multilevel splitting method on the isomerization of alanine dipeptide. J. Comput. Chem. 2019, 40, 1198– 1208, DOI: 10.1002/jcc.25778
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Analysis of the adaptive multilevel splitting method on the isomerization of alanine dipeptide
Lopes, Laura J. S.; Lelievre, Tony
Journal of Computational Chemistry (2019), 40 (11), 1198-1208CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
We apply the adaptive multilevel splitting (AMS) method to the Ceq → Cax transition of alanine dipeptide in vacuum. Some properties of the algorithm are numerically illustrated, such as the unbiasedness of the probability estimator and the robustness of the method with respect to the reaction coordinate. We also calc. the transition time obtained via the probability estimator, using an appropriate ensemble of initial conditions. Finally, we show how the AMS method can be used to compute an approxn. of the committor function.
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Teo, I.; Mayne, C. G.; Schulten, K.; Lelièvre, T. Adaptive multilevel mplitting method for molecular dynamics calculation of benzamidine-trypsin dissociation time. J. Chem. Theory Comput. 2016, 12, 2983– 2989, DOI: 10.1021/acs.jctc.6b00277
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Adaptive Multilevel Splitting Method for Molecular Dynamics Calculation of Benzamidine-Trypsin Dissociation Time
Teo, Ivan; Mayne, Christopher G.; Schulten, Klaus; Lelievre, Tony
Journal of Chemical Theory and Computation (2016), 12 (6), 2983-2989CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
Adaptive multilevel splitting (AMS) is a rare event sampling method that requires minimal parameter tuning and allows unbiased sampling of transition pathways of a given rare event. Previous simulation studies have verified the efficiency and accuracy of AMS in the calcn. of transition times for simple systems in both Monte Carlo and mol. dynamics (MD) simulations. Now, AMS is applied for the first time to an MD simulation of protein-ligand dissocn., representing a leap in complexity from the previous test cases. Of interest is the dissocn. rate, which is typically too low to be accessible to conventional MD. The present study joins other recent efforts to develop advanced sampling techniques in MD to calc. dissocn. rates, which are gaining importance in the pharmaceutical field as indicators of drug efficacy. The system investigated here, benzamidine bound to trypsin, is an example common to many of these efforts. The AMS est. of the dissocn. rate was found to be (2.6 ± 2.4) × 102 s-1, which compares well with the exptl. value.
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Branduardi, D.; Gervasio, F. L.; Parrinello, M. From A to B in free energy space. J. Chem. Phys. 2007, 126, 054103, DOI: 10.1063/1.2432340
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From A to B in free energy space
Branduardi, Davide; Gervasio, Francesco Luigi; Parrinello, Michele
Journal of Chemical Physics (2007), 126 (5), 054103/1-054103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
The authors present a new method for searching low free energy paths in complex mol. systems at finite temp. They introduce two variables that are able to describe the position of a point in configurational space relative to a preassigned path. With the help of these two variables the authors combine features of approaches such as metadynamics or umbrella sampling with those of path based methods. This allows global searches in the space of paths to be performed and a new variational principle for the detn. of low free energy paths to be established. Contrary to metadynamics or umbrella sampling the path can be described by an arbitrary large no. of variables, still the energy profile along the path can be calcd. The authors exemplify the method numerically by studying the conformational changes of alanine dipeptide.
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Cérou, F.; Delyon, B.; Guyader, A.; Rousset, M. On the Asymptotic Normality of Adaptive Multilevel Splitting. SIAM-ASA J. Uncertain. Quantif. 2019, 7, 1– 30, DOI: 10.1137/18M1187477
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Bréhier, C.-E.; Gazeau, M.; Goudenège, L.; Lelièvre, T.; Rousset, M. Unbiasedness of some generalized adaptive multilevel splitting algorithms. J. Appl. Probab. 2016, 26, 3559– 3601, DOI: 10.1214/16-AAP1185
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Binder, A.; Lelièvre, T.; Simpson, G. A generalized parallel replica dynamics. J. Comput. Phys. 2015, 284, 595– 616, DOI: 10.1016/j.jcp.2015.01.002
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Kresse, G.; Hafner, J. Ab-initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558– 561, DOI: 10.1103/PhysRevB.47.558
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Ab initio molecular dynamics of liquid metals
Kresse, G.; Hafner, J.
Physical Review B: Condensed Matter and Materials Physics (1993), 47 (1), 558-61CODEN: PRBMDO; ISSN:0163-1829.
The authors present ab initio quantum-mech. mol.-dynamics calcns. based on the calcn. of the electronic ground state and of the Hellmann-Feynman forces in the local-d. approxn. at each mol.-dynamics step. This is possible using conjugate-gradient techniques for energy minimization, and predicting the wave functions for new ionic positions using sub-space alignment. This approach avoids the instabilities inherent in quantum-mech. mol.-dynamics calcns. for metals based on the use of a factitious Newtonian dynamics for the electronic degrees of freedom. This method gives perfect control of the adiabaticity and allows one to perform simulations over several picoseconds.
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Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999, 59, 1758– 1775, DOI: 10.1103/PhysRevB.59.1758
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From ultrasoft pseudopotentials to the projector augmented-wave method
Kresse, G.; Joubert, D.
Physical Review B: Condensed Matter and Materials Physics (1999), 59 (3), 1758-1775CODEN: PRBMDO; ISSN:0163-1829. (American Physical Society)
The formal relationship between ultrasoft (US) Vanderbilt-type pseudopotentials and Blochl's projector augmented wave (PAW) method is derived. The total energy functional for US pseudopotentials can be obtained by linearization of two terms in a slightly modified PAW total energy functional. The Hamilton operator, the forces, and the stress tensor are derived for this modified PAW functional. A simple way to implement the PAW method in existing plane-wave codes supporting US pseudopotentials is pointed out. In addn., crit. tests are presented to compare the accuracy and efficiency of the PAW and the US pseudopotential method with relaxed-core all-electron methods. These tests include small mols. (H2, H2O, Li2, N2, F2, BF3, SiF4) and several bulk systems (diamond, Si, V, Li, Ca, CaF2, Fe, Co, Ni). Particular attention is paid to the bulk properties and magnetic energies of Fe, Co, and Ni.
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Behler, J.; Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 2007, 98, 146401, DOI: 10.1103/PhysRevLett.98.146401
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Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces
Behler, Jorg; Parrinello, Michele
Physical Review Letters (2007), 98 (14), 146401/1-146401/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
The accurate description of chem. processes often requires the use of computationally demanding methods like d.-functional theory (DFT), making long simulations of large systems unfeasible. In this Letter we introduce a new kind of neural-network representation of DFT potential-energy surfaces, which provides the energy and forces as a function of all at. positions in systems of arbitrary size and is several orders of magnitude faster than DFT. The high accuracy of the method is demonstrated for bulk silicon and compared with empirical potentials and DFT. The method is general and can be applied to all types of periodic and nonperiodic systems.
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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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Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials. Phys. Rev. B 2019, 99, 014104, DOI: 10.1103/PhysRevB.99.014104
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61
Atomic cluster expansion for accurate and transferable interatomic potentials
Drautz, Ralf
Physical Review B (2019), 99 (1), 014104CODEN: PRBHB7; ISSN:2469-9969. (American Physical Society)
The at. cluster expansion is developed as a complete descriptor of the local at. environment, including multicomponent materials, and its relation to a no. of other descriptors and potentials is discussed. The effort for evaluating the at. cluster expansion is shown to scale linearly with the no. of neighbors, irresp. of the order of the expansion. Application to small Cu clusters demonstrates smooth convergence of the at. cluster expansion to meV accuracy. By introducing nonlinear functions of the at. cluster expansion an interat. potential is obtained that is comparable in accuracy to state-of-the-art machine learning potentials. Because of the efficient convergence of the at. cluster expansion relevant subspaces can be sampled uniformly and exhaustively. This is demonstrated by testing against a large database of d. functional theory calcns. for copper.
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Chen, C.; Zuo, Y.; Ye, W.; Li, X.; Deng, Z.; Ong, S. P. A critical review of machine learning of energy materials. Adv. Energy Mater. 2020, 10, 1903242, DOI: 10.1002/aenm.201903242
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A Critical Review of Machine Learning of Energy Materials
Chen, Chi; Zuo, Yunxing; Ye, Weike; Li, Xiangguo; Deng, Zhi; Ong, Shyue Ping
Advanced Energy Materials (2020), 10 (8), 1903242CODEN: ADEMBC; ISSN:1614-6840. (Wiley-Blackwell)
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Computer Physics Communications (2020), 247 (), 106949CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
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Bocus, M.; Goeminne, R.; Lamaire, A.; Cools-Ceuppens, M.; Verstraelen, T.; Van Speybroeck, V. Nuclear quantum effects on zeolite proton hopping kinetics explored with machine learning potentials and path integral molecular dynamics. Nat. Commun. 2023, 14, 1008, DOI: 10.1038/s41467-023-36666-y
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Nuclear quantum effects on zeolite proton hopping kinetics explored with machine learning potentials and path integral molecular dynamics
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Nature Communications (2023), 14 (1), 1008CODEN: NCAOBW; ISSN:2041-1723. (Nature Portfolio)
Proton hopping is a key reactive process within zeolite catalysis. However, the accurate detn. of its kinetics poses major challenges both for theoreticians and experimentalists. Nuclear quantum effects (NQEs) are known to influence the structure and dynamics of protons, but their rigorous inclusion through the path integral mol. dynamics (PIMD) formalism was so far beyond reach for zeolite catalyzed processes due to the excessive computational cost of evaluating all forces and energies at the D. Functional Theory (DFT) level. Herein, we overcome this limitation by training first a reactive machine learning potential (MLP) that can reproduce with high fidelity the DFT potential energy surface of proton hopping around the first Al coordination sphere in the H-CHA zeolite. The MLP offers an immense computational speedup, enabling us to derive accurate reaction kinetics beyond std. transition state theory for the proton hopping reaction. Overall, more than 0.6μs of simulation time was needed, which is far beyond reach of any std. DFT approach. NQEs are found to significantly impact the proton hopping kinetics up to ∼473 K. Moreover, PIMD simulations with deuterium can be performed without any addnl. training to compute kinetic isotope effects over a broad range of temps.
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Vandermause, J.; Torrisi, S. B.; Batzner, S.; Xie, Y.; Sun, L.; Kolpak, A. M.; Kozinsky, B. On-the-fly active learning of interpretable Bayesian force fields for atomistic rare events. Npj Comput. Mater. 2020, 6. DOI: 10.1038/s41524-020-0283-z
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Jinnouchi, R.; Miwa, K.; Karsai, F.; Kresse, G.; Asahi, R. On-the-fly active learning of interatomic potentials for large-scale atomistic simulations. J. Phys. Chem. Lett. 2020, 11, 6946– 6955, DOI: 10.1021/acs.jpclett.0c01061
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On-the-Fly Active Learning of Interatomic Potentials for Large-Scale Atomistic Simulations
Jinnouchi, Ryosuke; Miwa, Kazutoshi; Karsai, Ferenc; Kresse, Georg; Asahi, Ryoji
Journal of Physical Chemistry Letters (2020), 11 (17), 6946-6955CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
The on-the-fly generation of machine-learning force fields by active-learning schemes attracts a great deal of attention in the community of atomistic simulations. The algorithms allow the machine to self-learn an interat. potential and construct machine-learned models on the fly during simulations. State-of-the-art query strategies allow the machine to judge whether new structures are out of the training data set or not. Only when the machine judges the necessity of updating the data set with the new structures are first-principles calcns. carried out. Otherwise, the yet available machine-learned model was used to update the at. positions. In this manner, most of the first-principles calcns. are bypassed during training, and overall, simulations are accelerated by several orders of magnitude while retaining almost first-principles accuracy. In this Perspective, after describing essential components of the active-learning algorithms, the authors demonstrate the power of the schemes by presenting recent applications.
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Abstract
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Figure 1
Figure 1. Global workflow to compute reaction rate constants with the Hill relation using adaptive multilevel splitting and support vector machine.
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Figure 2
Figure 2. First iteration of the AMS algorithm with k_min = 1 and N_rep = 3. Purple points represent the initial conditions on Σ_R. a) Identify the kill level $z_{k i l l}^{1} = z_{\max}^{k_{\min}, 0}$ and kill the replicas such that z_max^i,0 ≤ z_kill¹, i.e. the orange replica. b) Replace the killed replicas by the trajectory of one of the remaining replicas (the green one in this example) until the level z_kill¹ and continue the trajectory of the replica until it reaches either the state R or the state P.
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Figure 3
Figure 3. Flowchart of one iteration of AMS. n is an iteration index and i a replica index.
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Figure 4
Figure 4. Representation of one water molecule adsorbed on an aluminum site of the (100)-γ-alumina surface model (A₁ state). Surface atoms are represented as ball and sticks while subsurface ones are represented as lines. Color legend: red: oxygen, gray: aluminum, white: hydrogen, black lines: boundaries of the periodic cell. a) Top view; b) side view.
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Figure 5
Figure 5. Representation of the main different minimum energy structures corresponding to metastable states for the water molecule adsorbed on the (100)-γ-Al₂O₃ surface. Arrows represent transitions that might occur. Color legend: gray: aluminum, red; oxygen, white: hydrogen.
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Figure 6
Figure 6. Histogram of A₁-vs-D₁ SOAP-SVM CV on the whole labeled data set.
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Figure 7
Figure 7. Schematic representation of the two types of paths for the A₄ → A₁ rotation. The first path (blue) is named “side” while the second one (green) is named “top”. The purple line represent the RC iso-levels used to represent the trajectories.
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Figure 8
Figure 8. Ball and sticks representation of a) saddle point on the PES and b) mean structures such that p_R→P(q) = 0.5 on AMS using interpolated SOAP PCV RC. Color legend, red: oxygen, gray: aluminum, white: hydrogen.
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Figure 9
Figure 9. Schematic representation of poor match of the Σ_0.5 iso-level of the committor function (red) and the reaction coordinate (green). The green iso-level is placed after an AMS sampling of some reactive trajectories from R to P where a majority of the trajectories has gone via Region 1.
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  Time correlation function methods are used to discuss classical isomerization reactions of small nonrigid mols. in liq. solvents. Mol. expressions are derived for a macroscopic phenomenol. rate const. The form of several of these equations depends upon what ensemble is used when performing avs. over initial conditions. All of these formulas, however, reduce to 1 final phys. expression whose value is manifestly independent of ensemble. The validity of the phys. expression hinges on a sepn. of time scales and the plateau value problem. The approxns. needed to obtain transition state theory are described and the errors involved are estd. The coupling of the reaction coordinate to the liq. medium provides the dissipation necessary for the existence of a plateau value for the rate const., but it also leads to failure of Wigner's fundamental assumption for transition state theory. For many isomerization reactions, the transmission coeff. will differ significantly from unity and the difference will be a strong function of the thermodn. state of the liq. solvent.
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  Yang, Y. I.; Shao, Q.; Zhang, J.; Yang, L.; Gao, Y. Q. Enhanced sampling in molecular dynamics. J. Chem. Phys. 2019, 151, 070902, DOI: 10.1063/1.5109531
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  Enhanced sampling in molecular dynamics
  Yang, Yi Isaac; Shao, Qiang; Zhang, Jun; Yang, Lijiang; Gao, Yi Qin
  Journal of Chemical Physics (2019), 151 (7), 070902/1-070902/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  A review. Although mol. dynamics simulations have become a useful tool in essentially all fields of chem., condensed matter physics, materials science, and biol., there is still a large gap between the time scale which can be reached in mol. dynamics simulations and that obsd. in expts. To address the problem, many enhanced sampling methods were introduced, which effectively extend the time scale being approached in simulations. In this perspective, we review a variety of enhanced sampling methods. We first discuss collective-variables-based methods including metadynamics and variationally enhanced sampling. Then, collective variable free methods such as parallel tempering and integrated tempering methods are presented. At last, we conclude with a brief introduction of some newly developed combinatory methods. We summarize in this perspective not only the theor. background and numerical implementation of these methods but also the new challenges and prospects in the field of the enhanced sampling. (c) 2019 American Institute of Physics.
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  Transition state theory: Variational formulation, dynamical corrections, and error estimates
  Vanden-Eijnden, Eric; Tal, Fabio A.
  Journal of Chemical Physics (2005), 123 (18), 184103/1-184103/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Transition state theory (TST) is revisited, as well as evolutions upon TST such as variational TST in which the TST dividing surface is optimized so as to minimize the rate of recrossing through this surface and methods which aim at computing dynamical corrections to the TST transition rate const. The theory is discussed from an original viewpoint. It is shown how to compute exactly the mean frequency of transition between two predefined sets which either partition phase space (as in TST) or are taken to be well-sepd. metastable sets corresponding to long-lived conformation states (as necessary to obtain the actual transition rate consts. between these states). Exact and approx. criterions for the optimal TST dividing surface with min. recrossing rate are derived. Some issues about the definition and meaning of the free energy in the context of TST are also discussed. Finally precise error ests. for the numerical procedure to evaluate the transmission coeff. κS of the TST dividing surface are given, and it is shown that the relative error on κS scales as 1/√κS when κS is small. This implies that dynamical corrections to the TST rate const. can be computed efficiently if and only if the TST dividing surface has a transmission coeff. κS which is not too small. In particular, the TST dividing surface must be optimized upon (for otherwise κS is generally very small), but this may not be sufficient to make the procedure numerically efficient (because the optimal dividing surface has max. κS, but this coeff. may still be very small).
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  Miller, William H.; Schwartz, Steven D.; Tromp, John W.
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  Several formally exact expressions for quantum-mech. rate consts. (i.e., bimol. reactive cross sections suitably averaged and summed over initial and final states) are derived, and their relation to one another analyzed. They may provide a useful means for calcg. quantum-mech. rate consts. accurately without having to solve the complete state-to-state quantum-mech. reactive-scattering problem. Several ways are discussed for evaluating the quantum-mech. traces involved in these expressions, including a path-integral evaluation of the Boltzmann operator/time propagator and a discrete basis-set approxn. Both these methods are applied to a 1-dimensional test problem (the Eckart barrier).
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  Mandelli, D.; Hirshberg, B.; Parrinello, M. Metadynamics of paths. Phys. Rev. Lett. 2020, 125, 026001, DOI: 10.1103/PhysRevLett.125.026001
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  Mandelli, Davide; Hirshberg, Barak; Parrinello, Michele
  Physical Review Letters (2020), 125 (2), 026001CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)
  We present a method to sample reactive pathways via biased mol. dynamics simulations in trajectory space. We show that the use of enhanced sampling techniques enables unconstrained exploration of multiple reaction routes. Time correlation functions are conveniently computed via reweighted avs. along a single trajectory and kinetic rates are accessed at no addnl. cost. These abilities are illustrated analyzing a model potential and the umbrella inversion of NH3 in water. The algorithm allows a parallel implementation and promises to be a powerful tool for the study of rare events.
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  Baudel, M.; Guyader, A.; Lelièvre, T. On the Hill relation and the mean reaction time for metastable processes. Stoch Process Their Appl. 2023, 155, 393– 436, DOI: 10.1016/j.spa.2022.10.014
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  Lelièvre, T.; Ramil, M.; Reygner, J. Estimation of statistics of transitions and Hill relation for Langevin dynamics. arXiv:2206.13264 [math.PR] . 2022, to appear in Annales de l’Institut Henri Poincaré. DOI: 10.48550/arXiv.2206.13264
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  van Erp, T. S.; Moroni, D.; Bolhuis, P. G. A novel path sampling method for the calculation of rate constants. J. Chem. Phys. 2003, 118, 7762– 7774, DOI: 10.1063/1.1562614
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  A novel path sampling method for the calculation of rate constants
  van Erp, Titus S.; Moroni, Daniele; Bolhuis, Peter G.
  Journal of Chemical Physics (2003), 118 (17), 7762-7774CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  We derive a novel efficient scheme to measure the rate const. of transitions between stable states sepd. by high free energy barriers in a complex environment within the framework of transition path sampling. The method is based on directly and simultaneously measuring the fluxes through many phase space interfaces and increases the efficiency with at least a factor of 2 with respect to existing transition path sampling rate const. algorithms. The new algorithm is illustrated on the isomerization of a diat. mol. immersed in a simple fluid.
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  Allen, R. J.; Warren, P. B.; ten Wolde, P. R. Sampling rare switching events in biochemical networks. Phys. Rev. Lett. 2005, 94, 018104, DOI: 10.1103/PhysRevLett.94.018104
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  Sampling Rare Switching Events in Biochemical Networks
  Allen, Rosalind J.; Warren, Patrick B.; Ten Wolde, Pieter Rein
  Physical Review Letters (2005), 94 (1), 018104/1-018104/4CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
  Bistable biochem. switches are widely found in gene regulatory networks and signal transduction pathways. Their switching dynamics are difficult to study, however, because switching events are rare, and the systems are out of equil. We present a simulation method for predicting the rate and mechanism of the flipping of these switches. We apply it to a genetic switch and find that it is highly efficient. The path ensembles for the forward and reverse processes do not coincide. The method is widely applicable to rare events and nonequil. processes.
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  Huber, G.; Kim, S. Weighted-ensemble Brownian dynamics simulations for protein association reactions. Biophys. J. 1996, 70, 97– 110, DOI: 10.1016/S0006-3495(96)79552-8
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  Weighted-ensemble Brownian dynamics simulations for protein association reactions
  Huber, Gary A.; Kim, Sangtae
  Biophysical Journal (1996), 70 (1), 97-110CODEN: BIOJAU; ISSN:0006-3495. (Biophysical Society)
  A new method, weighted-ensemble Brownian dynamics, is proposed for the simulation of protein-assocn. reactions and other events whose frequencies of outcomes are constricted by free energy barriers. The method features a weighted ensemble of trajectories in configuration space with energy levels dictating the proper correspondence between "particles" and probability. Instead of waiting a very long time for an unlikely event to occur, the probability packets are split, and small packets of probability are allowed to diffuse almost immediately into regions of configuration space that are less likely to be sampled. The method was applied to the Northrup and Erickson (1992) model of docking-type diffusion-limited reactions and yields reaction rate consts. in agreement with those obtained by direct Brownian simulation, but at a fraction of the CPU time (10-4 to 10-3, depending on the model). Because the method is essentially a variant of std. Brownian dynamics algorithms, it is anticipated that weighted-ensemble Brownian dynamics, in conjunction with biophys. force models, can be applied to a large class of assocn. reactions of interest to the biophysics community.
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  Cérou, F.; Guyader, A. Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 2007, 25, 417– 443, DOI: 10.1080/07362990601139628
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  Glielmo, A.; Husic, B. E.; Rodriguez, A.; Clementi, C.; Noé, F.; Laio, A. Unsupervised learning methods for molecular simulation data. Chem. Rev. 2021, 121, 9722– 9758, DOI: 10.1021/acs.chemrev.0c01195
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  Unsupervised Learning Methods for Molecular Simulation Data
  Glielmo, Aldo; Husic, Brooke E.; Rodriguez, Alex; Clementi, Cecilia; Noe, Frank; Laio, Alessandro
  Chemical Reviews (Washington, DC, United States) (2021), 121 (16), 9722-9758CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
  A review. Unsupervised learning is becoming an essential tool to analyze the increasingly large amts. of data produced by atomistic and mol. simulations, in material science, solid state physics, biophysics, and biochem. In this Review, we provide a comprehensive overview of the methods of unsupervised learning that have been most commonly used to investigate simulation data and indicate likely directions for further developments in the field. In particular, we discuss feature representation of mol. systems and present state-of-the-art algorithms of dimensionality redn., d. estn., and clustering, and kinetic models. We divide our discussion into self-contained sections, each discussing a specific method. In each section, we briefly touch upon the math. and algorithmic foundations of the method, highlight its strengths and limitations, and describe the specific ways in which it has been used-or can be used-to analyze mol. simulation data.
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  Chen, M. Collective variable-based enhanced sampling and machine learning. Eur. Phys. J. B 2021, 94, 211, DOI: 10.1140/epjb/s10051-021-00220-w
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  Collective variable-based enhanced sampling and machine learning
  Chen, Ming
  European Physical Journal B: Condensed Matter and Complex Systems (2021), 94 (10), 211CODEN: EPJBFY; ISSN:1434-6028. (Springer)
  Abstr.: Collective variable-based enhanced sampling methods have been widely used to study thermodn. properties of complex systems. Efficiency and accuracy of these enhanced sampling methods are affected by two factors: constructing appropriate collective variables for enhanced sampling and generating accurate free energy surfaces. Recently, many machine learning techniques have been developed to improve the quality of collective variables and the accuracy of free energy surfaces. Although machine learning has achieved great successes in improving enhanced sampling methods, there are still many challenges and open questions. In this perspective, we shall review recent developments on integrating machine learning techniques and collective variable-based enhanced sampling approaches. We also discuss challenges and future research directions including generating kinetic information, exploring high-dimensional free energy surfaces, and efficiently sampling all-atom configurations. Graphic abstr.: [graphic not available: see fulltext].
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32. 32
  Gkeka, P.; Stoltz, G.; Farimani, A. B.; Belkacemi, Z.; Ceriotti, M.; Chodera, J. D.; Dinner, A. R.; Ferguson, A. L.; Maillet, J.-B.; Minoux, H.; Peter, C.; Pietrucci, F.; Silveira, A.; Tkatchenko, A.; Trstanova, Z.; Wiewiora, R.; Lelièvre, T. Machine learning force fields and coarse-grained variables in molecular dynamics: Application to materials and biological systems. J. Chem. Theory Comput. 2020, 16, 4757– 4775, DOI: 10.1021/acs.jctc.0c00355
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  Machine Learning Force Fields and Coarse-Grained Variables in Molecular Dynamics: Application to Materials and Biological Systems
  Gkeka, Paraskevi; Stoltz, Gabriel; Barati Farimani, Amir; Belkacemi, Zineb; Ceriotti, Michele; Chodera, John D.; Dinner, Aaron R.; Ferguson, Andrew L.; Maillet, Jean-Bernard; Minoux, Herve; Peter, Christine; Pietrucci, Fabio; Silveira, Ana; Tkatchenko, Alexandre; Trstanova, Zofia; Wiewiora, Rafal; Lelievre, Tony
  Journal of Chemical Theory and Computation (2020), 16 (8), 4757-4775CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  A review. Machine learning encompasses tools and algorithms that are now becoming popular in almost all scientific and technol. fields. This is true for mol. dynamics as well, where machine learning offers promises of extg. valuable information from the enormous amts. of data generated by simulation of complex systems. The authors provide here a review of the authors' current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab initio databases and the detn. of reaction coordinates for free energy computation and enhanced sampling.
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33. 33
  Ferguson, A. L. Machine learning and data science in soft materials engineering. J. Condens. Matter Phys. 2018, 30, 043002, DOI: 10.1088/1361-648X/aa98bd
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  Machine learning and data science in soft materials engineering
  Ferguson, Andrew L.
  Journal of Physics: Condensed Matter (2018), 30 (4), 043002/1-043002/27CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)
  A review. In many branches of materials science it is now routine to generate data sets of such large size and dimensionality that conventional methods of anal. fail. Paradigms and tools from data science and machine learning can provide scalable approaches to identify and ext. trends and patterns within voluminous data sets, perform guided traversals of high-dimensional phase spaces, and furnish data-driven strategies for inverse materials design. This topical review provides an accessible introduction to machine learning tools in the context of soft and biol. materials by 'de-jargonizing' data science terminol., presenting a taxonomy of machine learning techniques, and surveying the math. underpinnings and software implementations of popular tools, including principal component anal., independent component anal., diffusion maps, support vector machines, and relative entropy. The authors present illustrative examples of machine learning applications in soft matter, including inverse design of self-assembling materials, nonlinear learning of protein folding landscapes, high- throughput antimicrobial peptide design, and data-driven materials design engines. The authors close with an outlook on the challenges and opportunities for the field.
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34. 34
  Sultan, M. M.; Pande, V. S. Automated design of collective variables using supervised machine learning. J. Chem. Phys. 2018, 149, 094106, DOI: 10.1063/1.5029972
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  Automated design of collective variables using supervised machine learning
  Sultan, Mohammad M.; Pande, Vijay S.
  Journal of Chemical Physics (2018), 149 (9), 094106/1-094106/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  Selection of appropriate collective variables (CVs) for enhancing sampling of mol. simulations remains an unsolved problem in computational modeling. In particular, picking initial CVs is particularly challenging in higher dimensions. Which at. coordinates or transforms there of from a list of thousands should one pick for enhanced sampling runs. How does a modeler even begin to pick starting coordinates for investigation. This remains true even in the case of simple two state systems and only increases in difficulty for multi-state systems. In this work, we solve the "initial" CV problem using a data-driven approach inspired by the field of supervised machine learning (SML). In particular, we show how the decision functions in SML algorithms can be used as initial CVs (SMLcv) for accelerated sampling. Using solvated alanine dipeptide and Chignolin mini-protein as our test cases, we illustrate how the distance to the support vector machines' decision hyperplane, the output probability ests. from logistic regression, the outputs from shallow or deep neural network classifiers, and other classifiers may be used to reversibly sample slow structural transitions. We discuss the utility of other SML algorithms that might be useful for identifying CVs for accelerating mol. simulations. (c) 2018 American Institute of Physics.
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35. 35
  Pozun, Z. D.; Hansen, K.; Sheppard, D.; Rupp, M.; Müller, K.-R.; Henkelman, G. Optimizing transition states via kernel-based machine learning. J. Chem. Phys. 2012, 136, 174101, DOI: 10.1063/1.4707167
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  Optimizing transition states via kernel-based machine learning
  Pozun, Zachary D.; Hansen, Katja; Sheppard, Daniel; Rupp, Matthias; Mueller, Klaus-Robert; Henkelman, Graeme
  Journal of Chemical Physics (2012), 136 (17), 174101/1-174101/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The authors present a method for optimizing transition state theory dividing surfaces with support vector machines. The resulting dividing surfaces require no a priori information or intuition about reaction mechanisms. To generate optimal dividing surfaces, the authors apply a cycle of machine-learning and refinement of the surface by mol. dynamics sampling. The machine-learned surfaces contain the relevant low-energy saddle points. The mechanisms of reactions may be extd. from the machine-learned surfaces to identify unexpected chem. relevant processes. Also, the machine-learned surfaces significantly increase the transmission coeff. for an adatom exchange involving many coupled degrees of freedom on a (100) surface when compared to a distance-based dividing surface. (c) 2012 American Institute of Physics.
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36. 36
  Christiansen, M. A.; Mpourmpakis, G.; Vlachos, D. G. Density functional theory - Computed mechanisms of ethylene and diethyl ether formation from ethanol on γ-Al₂O₃(100). ACS Catal. 2013, 3 (9), 1965– 1975, DOI: 10.1021/cs4002833
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  Density Functional Theory-Computed Mechanisms of Ethylene and Diethyl Ether Formation from Ethanol on γ-Al2O3(100)
  Christiansen, Matthew A.; Mpourmpakis, Giannis; Vlachos, Dionisios G.
  ACS Catalysis (2013), 3 (9), 1965-1975CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
  Multiple potential active sites on the surface of γ-Al2O3 have led to debate about the role of Lewis and/or Bronsted acidity in reactions of ethanol, while mechanistic insights into competitive prodn. of ethylene and di-Et ether are scarce. In this study, elementary adsorption and reaction mechanisms for ethanol dehydration and etherification are studied on the γ-Al2O3(100) surface using d. functional theory calcns. The O atom of adsorbed ethanol interacts strongly with surface Al (Lewis acid) sites, while adsorption is weak on Bronsted (surface H) and surface O sites. Water, a byproduct of both ethylene and di-Et ether formation, competes with ethanol for adsorption sites. Multiple pathways for ethylene formation from ethanol are explored, and a concerted Lewis-catalyzed elimination (E2) mechanism is found to be the energetically preferred pathway, with a barrier of Ea = 37 kcal/mol at the most stable site. Di-Et ether formation mechanisms presented for the first time on γ-Al2O3 indicate that the most favorable pathways involve Lewis-catalyzed SN2 reactions (Ea = 35 kcal/mol). Addnl. novel mechanisms for di-Et ether decompn. to ethylene are reported. Bronsted-catalyzed mechanisms for ethylene and ether formation are not favorable on the (100) facet because of weak adsorption on Bronsted sites. These results explain multiple exptl. observations, including the competition between ethylene and di-Et ether formation on alumina surfaces.
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37. 37
  Larmier, K.; Nicolle, A.; Chizallet, C.; Cadran, N.; Maury, S.; Lamic-Humblot, A.-F.; Marceau, E.; Lauron-Pernot, H. Influence of coadsorbed water and alcohol molecules on isopropyl alcohol dehydration on γ-alumina: Multiscale modeling of experimental kinetic profiles. ACS Catal. 2016, 6, 1905– 1920, DOI: 10.1021/acscatal.6b00080
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  Influence of Coadsorbed Water and Alcohol Molecules on Isopropyl Alcohol Dehydration on γ-Alumina: Multiscale Modeling of Experimental Kinetic Profiles
  Larmier, Kim; Nicolle, Andre; Chizallet, Celine; Cadran, Nicolas; Maury, Sylvie; Lamic-Humblot, Anne-Felicie; Marceau, Eric; Lauron-Pernot, Helene
  ACS Catalysis (2016), 6 (3), 1905-1920CODEN: ACCACS; ISSN:2155-5435. (American Chemical Society)
  Successfully modeling the behavior of catalytic systems at different scales is a matter of importance not only for a fundamental understanding but also for a more rational design of catalysts and a more precise definition of the kinetic laws used as inputs in chem. engineering. We have developed here a multiscale modeling of the dehydration of iso-Pr alc. to propene and diisopropyl ether on γ-alumina catalysts, which clearly evidences and explains the central character of cooperative effects between coadsorbates in the kinetic network. The evolution of partial pressures with contact time was simulated using an original DFT-based microkinetic model based on a "macro site" centered on the main active site located on the (100) planes of alumina and comprising several neighboring adsorption sites. The formation of iso-Pr alc.-iso-Pr alc. or water-iso-Pr alc. dimers on the surface was required to correctly simulate the prodn. of the minor product, diisopropyl ether, and the evolution of the product partial pressures at high conversion. DFT calcns. were used to identify the structure of these dimers. In addn. to entropic effects, the selectivity to ether is ruled by (i) stabilizing interactions between coadsorbed iso-Pr alc. or water mols. and the nucleophilic alc. mol. reacting with the alcoholate intermediate, (ii) the formation of alcoholate-water dimers that selectively inhibit the formation of propene and increase the selectivity to ether at low conversion, and (iii) the reverse transformation of diisopropyl ether into propene and iso-Pr alc. that consumes ether at high conversion. The anal. expression of the reaction rate derived from this model and based on the existence of ensembles of interacting iso-Pr alc. and water mols. leads to a satisfactory modeling of the exptl. kinetic measurements at all conversions.
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  open URL
38. 38
  Hass, K. C.; Schneider, W. F.; Curioni, A.; Andreoni, W. The chemistry of water on alumina surfaces: Reaction dynamics from first principles. Science 1998, 282, 265– 268, DOI: 10.1126/science.282.5387.265
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  The chemistry of water on alumina surfaces: reaction dynamics from first principles
  Hass, Kenneth C.; Schneider, William F.; Curioni, Alessandro; Andreoni, Wanda
  Science (Washington, D. C.) (1998), 282 (5387), 265-268CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)
  Aluminas and their surface chem. play a vital role in many areas of modern technol. The behavior of adsorbed water is particularly important and poorly understood. Simulations of hydrated α-alumina (0001) surfaces with ab initio mol. dynamics elucidate many aspects of this problem, esp. the complex dynamics of water dissocn. and related surface reactions. At low water coverage, free energy profiles established that molecularly adsorbed water is metastable and dissocs. readily, even in the absence of defects, by a kinetically preferred pathway. Observations at higher water coverage revealed rapid dissocn. and unanticipated collective effects, including water-catalyzed dissocn. and proton transfer reactions between adsorbed water and hydroxide. The results provide a consistent interpretation of the measured coverage dependence of water heats of adsorption, hydroxyl vibrational spectra, and other expts.
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39. 39
  Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H. Hydroxyl groups on γ-alumina surfaces: A DFT study. J. Catal. 2002, 211, 1– 5, DOI: 10.1006/jcat.2002.3741
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  Hydroxyl Groups on γ-Alumina Surfaces: A DFT Study
  Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H.
  Journal of Catalysis (2002), 211 (1), 1-5CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Science)
  Despite numerous exptl. studies devoted to the acid-base properties of γ-alumina, the precise nature of surface acid sites remains unsolved. Using d. functional (DFT) calcns., we propose realistic models of γ-alumina (110) and (100) surfaces accounting for hydroxylation/dehydroxylation processes induced by temp. effects. The vibrational anal., based on DFT calcns., leads to an accurate assignment of the OH stretching frequencies obsd. by IR spectroscopy. The extension to chlorinated surfaces, which brings new insights into the understanding of the role of dopes, is also addressed.
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40. 40
  Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P. Use of DFT to achieve a rational understanding of acido-basic properties of γ-alumina surfaces. J. Catal. 2004, 226, 54– 68, DOI: 10.1016/j.jcat.2004.04.020
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  Use of DFT to achieve a rational understanding of acid-basic properties of γ-alumina surfaces
  Digne, M.; Sautet, P.; Raybaud, P.; Euzen, P.; Toulhoat, H.
  Journal of Catalysis (2004), 226 (1), 54-68CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Science)
  In a recent priority communication [M. Digne et al., J. Catal. 211 (2002) 1], we proposed the first ab initio constructed models of γ-alumina surfaces. Using the same d.-functional approach, we investigate in further detail the acid-basic properties of the three relevant γ-alumina (100), (110), and (111) surfaces, taking into account the temp.-dependent hydroxyl surface coverages. The simulations, compared fruitfully with many available exptl. data, enable us to solve the challenging assignment of the OH-stretching frequencies, as obtained from IR spectroscopy. The precise nature of the acid surface sites (concns. and strengths) is also detd. The acid strengths are quantified by simulating the adsorption of relevant probe mols. such as CO and pyridine in correlation with surface electronic properties. These results seriously challenge the historical model of a defective spinel for γ-alumina and establish the basis for a more rigorous description of the acid-basic properties of γ-alumina.
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41. 41
  Wischert, R.; Laurent, P.; Copéret, C.; Delbecq, F.; Sautet, P. γ-Alumina: The essential and unexpected role of water for the structure, stability, and reactivity of ”defect” sites. J. Am. Chem. Soc. 2012, 134, 14430– 14449, DOI: 10.1021/ja3042383
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  41
  γ-Alumina: The Essential and Unexpected Role of Water for the Structure, Stability, and Reactivity of Defect Sites
  Wischert, Raphael; Laurent, Pierre; Coperet, Christophe; Delbecq, Francoise; Sautet, Philippe
  Journal of the American Chemical Society (2012), 134 (35), 14430-14449CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
  Combining expts. and DFT calcns., we show that tricoordinate AlIII Lewis acid sites, which are present as metastable species exclusively on the major (110) termination of γ- and δ-Al2O3 particles, correspond to the defect sites, which are held responsible for the unique properties of activated (thermally pretreated) alumina. These defects are, in fact, largely responsible for the adsorption of N2 and the splitting of CH4 and H2. In contrast, five-coordinate Al surface sites of the minor (100) termination cannot account for the obsd. reactivity. The AlIII sites, which are formed upon partial dehydroxylation of the surface (the optimal pretreatment temp. being 700 °C for all probes), can coordinate N2 selectively. In combination with specific O atoms, they form extremely reactive Al,O Lewis acid-base pairs that trigger the low-temp. heterolytic splitting of CH4 and H2 to yield Al-CH3 and Al-H species, resp. H2 is found overall more reactive than CH4 because of its higher acidity, hence it also reacts on four-coordinate sites of the (110) termination. Water has the dual role of stabilizing the (110) termination and modifying (often increasing) both the Lewis acidity of the aluminum and the basicity of nearby oxygens, hence the high reactivity of partially dehydroxylated alumina surfaces. In addn., we demonstrate that the presence of water enhances the acidity of certain four-coordinate Al atoms, which leads to strong coordination of the CO mol. with a spectroscopic signature similar to that on AlIII sites, thus showing the limits of this widely used probe for the acidity of oxides. Overall, the dual role of water translates into optimal water coverage, and this probably explains why in many catalyst prepns., optimal pretreatment temps. are typically obsd. in the activation step of alumina.
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42. 42
  Pigeon, T.; Chizallet, C.; Raybaud, P. Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces. J. Catal. 2022, 405, 140– 151, DOI: 10.1016/j.jcat.2021.11.011
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  42
  Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces
  Pigeon, Thomas; Chizallet, Celine; Raybaud, Pascal
  Journal of Catalysis (2022), 405 (), 140-151CODEN: JCTLA5; ISSN:0021-9517. (Elsevier Inc.)
  The rational understanding of γ-alumina (γ-Al2O3) supported catalysts requires an ever more improved at. scale detn. of the support's surface properties. By using d. functional theory (DFT) calcns., we show how the structural and energetic surface properties of alumina crystallites intrinsically depend on its synthesis pathway. Considering the case study of the topotactic transformation of boehmite (γ-AlOOH) into γ-Al2O3 taking place during calcination, we propose a methodol. to mimic this pathway by reconstructing relevant slabs of boehmite into γ-alumina slabs following 3 steps: dehydration, contraction/translation and Al migration into spinel or non-spinel sites. On the one hand, we confirm the reliability of some earlier 100, 110 and 111 surface structures detd. by std. bulk cleavage approach. Moreover, we find new γ-alumina surfaces harboring Bronsted acid sites (BAS) and Lewis acid sites (LAS) with specific local structures. More strikingly, we find that the basal (110)b surface of alumina inherited from the (0 1 0) basal surface of boehmite, exhibits a larger no. of isolated μ2-OH groups than the lateral( 110)l surface. For the lateral (110)l (resp. 111) orientation, four (resp. three) thermodynamically competing surfaces are identified, including models earlier proposed. These results are induced by finite size and morphol. effects during the topotactic transformation of boehmite crystallites. Thanks to a thorough comparative anal. of morphol. and nature of BAS and LAS as a function of thermal treatment and water pressure for each surface, we identify coherent chem. families of surfaces across the main crystallog. orientations. These features open the door to a better differentiation of the reactivity of the basal alumina surfaces from the lateral ones.
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43. 43
  Lu, Y.-H.; Wu, S.-Y.; Chen, H.-T. H₂O Adsorption/Dissociation and H₂ generation by the reaction of H₂O with Al₂O₃ materials: A first-principles investigation. J. Phys. Chem. C 2016, 120, 21561– 21570, DOI: 10.1021/acs.jpcc.6b07191
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  43
  H2O Adsorption/Dissociation and H2 Generation by the Reaction of H2O with Al2O3 Materials: A First-Principles Investigation
  Lu, Yu-Huan; Wu, Shiuan-Yau; Chen, Hsin-Tsung
  Journal of Physical Chemistry C (2016), 120 (38), 21561-21570CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
  The microscopic reaction mechanisms for the water adsorption/dissocn. and hydrogen generation processes on the α-Al2O3(0001) surface are clarified by using spin-polarized d. functional theory with the projected augmented wave approach. The adsorptions of OH, O, and H species are also examd. Calcns. show that the H2O, OH, O, and H species prefer to adsorb at the Al(II)-top, Al(I, II)-top, Al(I, II)-bridge, and Al(II)-top sites with adsorption energies of -1.34, -5.91, -8.22, and -3.14 eV on the Al-terminated surface, whereas those are Al-top, Al-top, Al-top, and O-top sites with adsorption energies of -1.11, -2.79, -2.00, and -2.23 eV for the Al, O-terminated surface. Geometries of the mol. adsorbed intermediates, transition states, and the hydroxylated products as well as the energetic reaction routes are fully elucidated. Hydrogen generation and full dissocn. of water are found to occur on the Al-terminated surface with overall exothermicities of 2.37 and 4.22 eV, whereas only the prodn. of coadsorbed H(ads) + OH(ads) is obsd. on the Al, O-terminated surface with an overall exothermicity of 1.06-1.64 eV. In addn., the local d. of states and Bader charge calcns. are carried out to study the interaction between the adsorbate and surface along the reaction.
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44. 44
  Pan, Y.; Liu, C.-J.; Ge, Q. Adsorption and protonation of CO₂ on partially hydroxylated γ-Al₂O₃ surfaces: A density functional theory study. Langmuir 2008, 24, 12410– 12419, DOI: 10.1021/la802295x
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  44
  Adsorption and Protonation of CO2 on Partially Hydroxylated γ-Al2O3 Surfaces: A Density Functional Theory Study
  Pan, Yunxiang; Liu, Chang-jun; Ge, Qingfeng
  Langmuir (2008), 24 (21), 12410-12419CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)
  Adsorption and protonation of CO2 on the (110) and (100) surfaces of γ-Al2O3 were studied using d. functional theory slab calcns. On the dry (110) and (100) surfaces, the O-Al bridge sites are energetically favorable for CO2 adsorption. The adsorbed CO2 was bound in a bidentate configuration across the O-Al bridge sites, forming a carbonate species. The strongest binding with an adsorption energy of 0.80 eV occurs at the O3c-Al5c bridge site of the (100) surface. Dissocn. of water across the O-Al bridge sites resulted in partially hydroxylated surfaces, and the dissocn. is energetically favorable on both surfaces. Water dissocn. on the (110) surface has a barrier of 0.42 eV, but the same process on the (100) surface has no barrier with respect to the isolated water mol. On the partially hydroxylated γ-Al2O3 surfaces, a bicarbonate species was formed by protonating the carbonate species with the protons from neighboring hydroxyl groups. The energy difference between the bicarbonate species and the coadsorbed bidentate carbonate species and hydroxyls is only 0.04 eV on the (110) surface, but the difference reaches 0.97 eV on the (100) surface. The activation barrier for forming the bicarbonate species on the (100) surface, 0.42 eV, is also lower than that on the (110) surface (0.53 eV).
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45. 45
  Ngouana-Wakou, B. F.; Cornette, P.; Valero, M. C.; Costa, D.; Raybaud, P. An atomistic description of the γ-alumina/water interface revealed by ab initio molecular dynamics. J. Phys. Chem. C 2017, 121, 10351– 10363, DOI: 10.1021/acs.jpcc.7b00101
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  45
  An Atomistic Description of the γ-Alumina/Water Interface Revealed by Ab Initio Molecular Dynamics
  Ngouana-Wakou, B. F.; Cornette, P.; Corral Valero, M.; Costa, D.; Raybaud, P.
  Journal of Physical Chemistry C (2017), 121 (19), 10351-10363CODEN: JPCCCK; ISSN:1932-7447. (American Chemical Society)
  The authors report ab initio mol. dynamics (AIMD) simulations of the (100) and (110) γ-Al2O3/water interfaces at 300 K, using two sets of supercell models for each surface and two time lengths of simulation (10 and 40 ps). The authors first show that the effect of liq. water on the vibrational frequencies of hydroxyl groups at the interface varies according to the type of surface. This trend is explained by two key parameters affecting the interaction of both surfaces with water: the nature of the OH groups (i.e., μ1-OH, μ1-H2O, μ2-OH, and μ3-OH) and H-bond network among surface OH groups. The hydroxylated (110) surface favors the local structuration of water at the interface and the solvation of its μ1-OH and μ1-H2O groups by water similarly as in bulk liq. water. By contrast, on the (100) surface, a stronger H-bond network among μ1-OH and μ1-H2O groups reduces the water/surface interaction. The authors illustrate also how the interfacial interacting sites are spatially organized on the surfaces by two-dimensional maps of O-H distances. On both surfaces, the interfacial water layer orientation is predominantly Hup-Hdown. For long AIMD simulation time, Grotthuss-like mechanisms are identified on the (110) surface.
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46. 46
  Réocreux, R.; Jiang, T.; Iannuzzi, M.; Michel, C.; Sautet, P. Structuration and dynamics of interfacial liquid water at hydrated γ-alumina determined by ab initio molecular simulations: Implications for nanoparticle stability. ACS Appl. Nano Mater. 2018, 1, 191– 199, DOI: 10.1021/acsanm.7b00100
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  46
  Structuration and Dynamics of Interfacial Liquid Water at Hydrated γ-Alumina Determined by ab Initio Molecular Simulations: Implications for Nanoparticle Stability
  Reocreux, Romain; Jiang, Tao; Iannuzzi, Marcella; Michel, Carine; Sautet, Philippe
  ACS Applied Nano Materials (2018), 1 (1), 191-199CODEN: AANMF6; ISSN:2574-0970. (American Chemical Society)
  Liq. water/solid interfaces are central in catalytic nanomaterials, from their prepn. to their chem. stability under harsh catalytic conditions such as the hot aq. medium used in biomass valorization. Here we report an ab initio mol. dynamics (AIMD) study of the γ-Al2O3 (110)/water interface using the most recent surface model available in the literature. The size of the simulation box and the duration of the AIMD simulation enables us to characterize the whole interface at the at. scale. The simulation evidences a redistribution of protons within the chemisorbed water layer. The influence of γ-Al2O3 (110) is also important on the water mols. that are not bound to the surface: it is only above 10 Å that water recovers its bulk liq. behavior. The influence of alumina is structural, with preferred angular orientations for water mols., and also dynamical. The translational self-diffusivity of water is diminished by up to 2 orders of magnitude, and the angular relaxation time increased up to a factor of 6. The influence of the interface on chemisorbed water mols. is also characterized with an IR spectrum (fully simulated at the d. functional theory level) that shows two distinct regions (3500 and 3200 cm-1) assigned to two different interfacial environments. This full characterization of the nanoscale interfacial zone highlights the specific physicochem. features of water that arise in contact with γ-Al2O3 and opens the door to an improved prepn. of supported catalysts (from templating agents to protective coatings).
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47. 47
  Lo, C. S.; Radhakrishnan, R.; Trout, B. L. Application of transition path sampling methods in catalysis: A new mechanism for CC bond formation in the methanol coupling reaction in Chabazite. Catal. Today 2005, 105, 93– 105, DOI: 10.1016/j.cattod.2005.04.005
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  Application of transition path sampling methods in catalysis: A new mechanism for C-C bond formation in the methanol coupling reaction in chabazite
  Lo, Cynthia S.; Radhakrishnan, Ravi; Trout, Bernhardt L.
  Catalysis Today (2005), 105 (1), 93-105CODEN: CATTEA; ISSN:0920-5861. (Elsevier B.V.)
  We describe the application of transition path sampling methods to the methanol coupling reaction in the zeolite chabazite; these methods have only been recently applied to complex chem. systems. Using these methods, we have found a new mechanism for the formation of the first C-C bond. In our mechanism, the reaction, at 400 °C, proceeds via a two-step process: (1) the breaking of the C-O bond of the chemisorbed methoxonium cation, followed by the transfer of a hydride ion from the remaining methanol mol. to the Me cation, resulting in the formation of H2O, CH4 , and CH2OH+ and (2) a simultaneous proton transfer from methane to water, and direct C-C bond formation between the Me anion and CH2OH+, resulting in the formation of ethanol. The C - C bond forming process has the higher barrier, with an activation energy of about 100.49 kJ/mol.
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48. 48
  Bucko, T.; Benco, L.; Dubay, O.; Dellago, C.; Hafner, J. Mechanism of alkane dehydrogenation catalyzed by acidic zeolites: Ab initio transition path sampling. J. Chem. Phys. 2009, 131, 214508, DOI: 10.1063/1.3265715
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  Mechanism of alkane dehydrogenation catalyzed by acidic zeolites: ab initio transition path sampling
  Bucko, Tomas; Benco, Lubomir; Dubay, Orest; Dellago, Christoph; Hafner, Juergen
  Journal of Chemical Physics (2009), 131 (21), 214508/1-214508/11CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
  The dehydrogenation of propane over acidic chabazite has been studied using ab initio d.-functional simulations in combination with static transition-state searches and dynamic transition path sampling (TPS) methods at elevated temps. The acidic zeolite has been modeled both using a small cluster and a large periodic model consisting of two unit cells, the TPS simulations allow to account for the effect of temp. and entropy. In agreement with exptl. observations we find propene as the dominant reaction product and that the barrier for the dehydrogenation of a Me group is higher than that for a methylene group. However, whereas all studies based on small cluster models (including the present one) conclude that the reaction proceeds via the formation of an alkoxy intermediate, our TPS studies based on a large periodic model lead to the conclusion that propene formation occurs via the formation of various forms of Pr cations stabilized by entropy, while the formation of an alkoxy species is a relatively rare event. It was obsd. only in 15% of the reactive trajectories for Me dehydrogenation and even in only 8% of the methylene dehydrogenation reactions. Our studies demonstrate the importance of entropic effects and the need to account for the structure and flexibility of the zeolitic framework by using large periodic models. (c) 2009 American Institute of Physics.
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49. 49
  Rey, J.; Bignaud, C.; Raybaud, P.; Bucko, T.; Chizallet, C. Dynamic features of transition states for beta-scission reactions of alkenes over acid zeolites revealed by AIMD simulations. Angew. Chem., Int. Ed. Engl. 2020, 59, 18938– 18942, DOI: 10.1002/anie.202006065
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  Dynamic Features of Transition States for β-Scission Reactions of Alkenes over Acid Zeolites Revealed by AIMD Simulations
  Rey Jerome; Bignaud Charles; Raybaud Pascal; Chizallet Celine; Bignaud Charles; Bucko Tomas; Bucko Tomas
  Angewandte Chemie (International ed. in English) (2020), 59 (43), 18938-18942 ISSN:.
  Zeolite-catalyzed alkene cracking is key to optimize the size of hydrocarbons. The nature and stability of intermediates and transition states (TS) are, however, still debated. We combine transition path sampling and blue moon ensemble density functional theory simulations to unravel the behavior of C7 alkenes in CHA zeolite. Free energy profiles are determined, linking π-complexes, alkoxides and carbenium ions, for B1 (secondary to tertiary) and B2 (tertiary to secondary) β-scissions. B1 is found to be easier than B2 . The TS for B1 occurs at the breaking of the C-C bond, while for B2 it is the proton transfer from propenium to the zeolite. We highlight the dynamic behaviors of the various intermediates along both pathways, which reduce activation energies with respect to those previously evaluated by static approaches. We finally revisit the ranking of isomerization and cracking rate constants, which are crucial for future kinetic studies.
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50. 50
  Roet, S.; Daub, C. D.; Riccardi, E. Chemistrees: Data-driven identification of reaction pathways via machine learning. J. Chem. Theory Comput. 2021, 17, 6193– 6202, DOI: 10.1021/acs.jctc.1c00458
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  Chemistrees: Data-Driven Identification of Reaction Pathways via Machine Learning
  Roet, Sander; Daub, Christopher D.; Riccardi, Enrico
  Journal of Chemical Theory and Computation (2021), 17 (10), 6193-6202CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
  We propose to analyze mol. dynamics (MD) output via a supervised machine learning (ML) algorithm, the decision tree. The approach aims to identify the predominant geometric features which correlate with trajectories that transition between two arbitrarily defined states. The data-driven algorithm aims to identify these features without the bias of human "chem. intuition". We demonstrate the method by analyzing the proton exchange reactions in formic acid solvated in small water clusters. The simulations were performed with ab initio MD combined with a method to efficiently sample the rare event, path sampling. Our ML anal. identified relevant geometric variables involved in the proton transfer reaction and how they may change as the no. of solvating water mols. changes.
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51. 51
  Lopes, L. J. S.; Lelièvre, T. Analysis of the adaptive multilevel splitting method on the isomerization of alanine dipeptide. J. Comput. Chem. 2019, 40, 1198– 1208, DOI: 10.1002/jcc.25778
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  Analysis of the adaptive multilevel splitting method on the isomerization of alanine dipeptide
  Lopes, Laura J. S.; Lelievre, Tony
  Journal of Computational Chemistry (2019), 40 (11), 1198-1208CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
  We apply the adaptive multilevel splitting (AMS) method to the Ceq → Cax transition of alanine dipeptide in vacuum. Some properties of the algorithm are numerically illustrated, such as the unbiasedness of the probability estimator and the robustness of the method with respect to the reaction coordinate. We also calc. the transition time obtained via the probability estimator, using an appropriate ensemble of initial conditions. Finally, we show how the AMS method can be used to compute an approxn. of the committor function.
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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00280.
- Multilevel splitting estimator and AMS pseudo code, rate constant error estimation, state to state probability estimation in a multistate case, calculation parameters, implementation with VASP software, detailed numerical results, and clustering reactive trajectories (PDF)
- Video 1 showing an A₁ → D₁ dissociation trajectory (AVI)
- Video 2 showing an A₄ → A₁ “top” rotation trajectory (AVI)
- Video 3 showing an A₄ → A₁ “side” rotation trajectory (AVI)
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Computing Surface Reaction Rates by Adaptive Multilevel Splitting Combined with Machine Learning and Ab Initio Molecular Dynamics

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Abstract

1. Introduction

2. Methods

Figure 1

2.1. Reaction Rate Constant Estimation Using AMS

2.1.1. Motivation

2.1.2. Computing the Flux and Sampling Initial Conditions

Figure 2

2.1.3. AMS Requirements

2.1.4. AMS Initialization

2.1.5. AMS Iteration

Figure 3

2.1.6. AMS Termination and Probability Estimator

2.1.6.1. Multiple States Case

2.1.6.2. Implementation with a Plane Wave DFT Code

2.2. Tools to Define States and Reaction Coordinates

2.2.1. Representation of Chemical Structures

2.2.2. Support Vector Machine

2.2.3. Path Collective Variables (PCVs)

3. Results and Discussion

3.1. γ-Al2O3 Models and Definition of States

3.1.1. Model of the Catalytic System

Figure 4

3.1.2. Data Set Generation to Learn States

Figure 5

Figure 6

3.1.3. Definitions of the Boundary Surface ΣR

3.1.4. Definition of Reaction Coordinates (RCs)

3.2. Analysis of AMS Rate Constants

3.2.1. Parallel Calculations against Precision

3.2.2. Impact of the Definition of Reaction Coordinates and States

3.2.3. Comparison of the Rate Constants Calculated with AMS and with hTST

3.3. Analysis of AMS Reactive Trajectories

3.3.1. Clustering Reactive Trajectories

Figure 7

3.3.2. Stochastic Transition State estimation

3.3.3. Stochastic Transition State of Water Dissociation

Figure 8

Figure 9

4. Conclusion

Supporting Information

Terms & Conditions

Author Information

Acknowledgments

References

Cited By

Abstract

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

References

Supporting Information

Supporting Information

Terms & Conditions

STEP 1:

STEP 2:

3.1. γ-Al₂O₃ Models and Definition of States

3.1.3. Definitions of the Boundary Surface Σ_R