ACS Publications. Most Trusted. Most Cited. Most Read
CONTENT TYPES

Figure 1Loading Img
RETURN TO ISSUEPREVReaction MechanismsNEXT

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]
  • 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
  • Manuel Corral-Valero
    Manuel Corral-Valero
    IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
  • Ani Anciaux-Sedrakian
    Ani Anciaux-Sedrakian
    IFP Energies Nouvelles, 1 et 4 avenue de Bois-Préau, F-92852 Rueil-Malmaison Cedex, France
  • Maxime Moreaud
    Maxime Moreaud
    IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
  • 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]
  • , and 
  • Pascal Raybaud*
    Pascal Raybaud
    IFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, France
    *E-mail: [email protected]
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
Copyright © 2023 American Chemical Society
  • Subscribed

Article Views

332

Altmetric

-

Citations

-
LEARN ABOUT THESE METRICS
PDF (5 MB) open URL
Supporting Info (4)»

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.

This publication is licensed under the terms of your institutional subscription. Request reuse permissions.

1. Introduction

ARTICLE SECTIONS
Jump To

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)
khTST=κ(T)kBTheΔG/kBT
(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, kB 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)
kHill=ΦRpRP(R)
(2)
where ΦR is the flux of trajectories leaving the reactant state R and the committor probability at the boundary pRP(∂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 pRP(∂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 pRP(∂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

ARTICLE SECTIONS
Jump To

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

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 pRP(∂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 pRP(∂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 ΣR close to R is introduced for the estimation of the flux ΦR, 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 ΣR (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 ΦR is then evaluated by starting a dynamics in the state R, counting the number of times nloopRΣRR it goes from R to ΣR, crosses ΣR and goes back to R, dividing this number by the overall time ttot:
ΦR=nloopRΣRRttot=1tloopRΣRR
(3)
where tloopRΣRR is the average time that a trajectory takes to go out of R, cross ΣR, and go back to R. Let us now focus on the second one: pRP(∂R). The computation of the flux ΦR simultaneously allows one to generate some positions on the surface ΣR, which will serve to estimate the probability pRPR) instead of pRP(∂R). This approximation does not bias the result as far as R and ΣR are within the same metastable basin. (24) These initial conditions must correspond to the first time a trajectory leaving R reaches the level ΣR. 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

Figure 2. First iteration of the AMS algorithm with kmin = 1 and Nrep = 3. Purple points represent the initial conditions on ΣR. a) Identify the kill level zkill1=zmaxkmin,0 and kill the replicas such that zmaxi,0zkill1, 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 zkill1 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 ΣR has to be placed such that each trajectory linking the reactant and the product state goes through ΣR. Its distance to the boundary of R should be sufficiently small so that the sampling of initial conditions and the determination of the flux ΦR is not exceedingly expensive in terms of computational cost. A number of replicas Nrep (or walkers) has to be defined, as well as a minimum number kmin of replicas to kill at each iteration of AMS. Nrep different initial conditions on the surface ΣR 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 RP transition. It has to be consistent with the states R and P, which can be generally enforced by setting ξ(q) = – for qR and ξ(q) = + for qP.

2.1.4. AMS Initialization

First, all the replicas are run from their initial conditions on ΣR 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, qti,n denotes the position of the i-th replica at time t and iteration n. In particular, {q0i,n}1iNrep are initial conditions on ΣR. 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 zmaxi,n=supt(ξ(qti,n)). The replicas are then reordered by increasing values zmaxi,n (see Figure 2). According to the value of kmin, the level at which positions are killed is identified as an empirical quantile: zkilln+1=zmaxkmin,n. This means that all the trajectories for which zmaxi,nzkilln+1 are killed. The number of killed trajectories at this iteration is denoted by ηkilledn+1. Note that ηkilledn+1kmin by construction, but it could happen that ηkilledn+1kmin+1 when several trajectories reach exactly the same zmaxkmin,n. To keep the number of replica constant, ηkilledn+1 trajectories have to be created by randomly branching ηkilledn+1 trajectories among the remaining ones. More precisely, trajectories are duplicated until the first time they reach the level zkilln+1 and then the dynamics is ran from these points until it reaches R or P. In fact, at each iteration, the estimated probability 1ηkilledn+1Nrep is the probability for a trajectory to reach the surface Σzkilln+1 starting on the surface Σzkilln.
Any AMS iteration can be summarized by the succession of the steps illustrated in Figure 3.

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, Nrep different reactive trajectories are obtained and the estimated transition probability is computed via:
p^RP(ΣR)=n=1nmax(1ηkillednNrep)
(4)
where nmax 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, ηkilledn, 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 zmaxi,n. This termination event is called “failure” as the algorithm is not able to provide reactive trajectories and the estimated probability is p^RP(ΣR)=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 p^RP(ΣR) in eq 4 by repeating the estimation of the probability Mreal 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 ΣR 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 Mreal 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 {Ri}0iL1. These structures are represented here with the numerical SOAP descriptor. A reaction coordinate is then constructed as (53)
s(R)=i=0L1ieλd(Ri,R)i=0L1eλd(Ri,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 λi can be used instead. In the present case, we chose λi as
{λi1=12(d(Ri1,Ri)+d(Ri,Ri+1))λ01=d(R0,R1);λL11=d(RL2,RL1)
(6)
PCVs were directly implemented in the Python scripts used for the reaction coordinate evaluation during the dynamics.

3. Results and Discussion

ARTICLE SECTIONS
Jump To

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

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 Ai or Di 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

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 fX–vs–all. For instance, the state A1 is defined as {q|fA1vsall(SOAP(q))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

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

3.1.3. Definitions of the Boundary Surface ΣR

For every metastable state X, a boundary ΣX is defined from the 1 ps trajectory generated to assess the metastability of the basin X. This surface ΣX is taken as a levelset of the decision function fX–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 ΣX was chosen as {q|fA1vsall(SOAP(q))=0.95}. This choice implies that ΣR 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 (Nrep) and the number of repetitions of the probability estimation (Mreal). 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 (Nrep) and the number of AMS repetitions (Mreal) is evaluated for a fixed number of initial conditions NrepMreal, 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 ηkilledn is constant and equals kmin at all steps of the AMS realization, the cost of one AMS realization is given by the product of the number of AMS iterations (nmax) and the number of killed replicas (kmin). Under these assumptions, the AMS estimator (eq 4) writes:
p^=(1kminNrep)nmax
(7)
Assuming that kminNrep is small, the computational cost of a single AMS simulation is
kminnmaxNrepln(p^)
(8)
Taking into account the number of repetitions of the algorithm Mreal, the final cost of a reaction rate constant estimation is MrealNrepln(p^). Considering the current implementation of AMS, Mreal realizations of AMS can be run in parallel. The objective is to find the minimal value of Nrep 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,kA1A2A3, calculated with AMS for various values of Nrep and Mreal by using the “A1-vs-all-SOAP-SVM” reaction coordinate and states defined as R = A1 and P = A2A3A4D1D3D2D4.
Table 1. Estimation of Probability, Rate and the Corresponding 90% Confidence Intervals for Water Rotationa
MrealNrep
(fs)
pA1→A2A3A1)kA1→A2A3 (s–1)
5400108 ± 5(3.73 ± 3.03) 10–3(3.67 ± 2.99) 1010
10200110 ± 5(3.38 ± 1.56) 10–3(3.08 ± 1.43) 1010
20100101 ± 5(3.47 ± 1.96) 10–3(3.21 ± 1.82) 1010
a

The number of initial conditions MrealNrep was varying Mreal and Nrep. R = A1, P = A2A3A4D1D3D2D4, ξ = A1-vs-all SOAP SVM RC.

By definition tloop is not impacted by Nrep or Mreal. 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 Nrep = 200 and Mreal = 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 Nrep = 100 and Mreal = 20. Therefore, it is important to perform the AMS simulations a certain number of times (Mreal) in order have a proper variance estimation. Hence, for a similar computational cost in CPU time, satisfactory accuracy can be obtained using Mreal ≥ 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 ΣA1, set R = A1 and P = A2A3A4D1D3D2D4. 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 A1A2A3 with a reasonable accuracy according to the estimate of the rate constant’s variance. However, the less probable transitions (A1D1D3 and A1A4) 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 A1D1D3, 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 A1A2A3. To observe only A1D1D3 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 A1A2A3 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 = A1A2A3A4D2D4, P = D1D3, and initial conditions sampled on ΣA1, AMS is compelled to sample A1D1D3 trajectories. The difference here with the previous case is that, if a rotation A1A2A3 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 zmax 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 A1D1D3 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 A1A2A3 and the water dissociation A1D1D3. 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 A1D1D3 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, Nrep = 200, Mreal = 10, R = A1 and P = A2A3A4D1D3D2D4a
Transition
kTransition (s–1)
A1A2A3(3.38 ± 1.56) 10–3(3.17 ± 1.43) 1010
A1D1D3(1.79 ± 1.86) 10–3(1.63 ± 1.70) 1010
A1A4(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 A1D1D3 Transition with Nrep = 200, Mreal = 10 and Initial Conditions Sampled on
RC
(fs)
pA1D1D3A1)kA1D1D3 (s–1)
R = A1 ; P = A2A3A4D1D3D2D4
A1-vs-all-SOAP-SVM110 ± 5(1.79 ± 1.86) 10–3(1.63 ± 1.70) 1010
A1-vs-D1-SOAP-SVM105 ± 3(1.81 ± 1.98) 10–5(1.72 ± 1.88) 108
interpolated SOAP-PCV104 ± 4(1.95 ± 2.26) 10–4(1.87 ± 2.17) 109
R = A1A2A3A4D2D4 ; P = D1D3
A1-vs-D1-SOAP-SVM105 ± 2(3.31 ± 2.97) 10–4(3.15 ± 2.83) 109
interpolated SOAP-PCV108 ± 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
TransitionkTransition–AMS (s–1)kTransition–hTST (s–1)
Water rotations
A1A2A3(3.08 ± 1.43) 10107.55 × 1010
A2A3A1(1.49 ± 0.46) 10112.06 × 1012
A2A3A4(4.33 ± 2.20) 10103.64 × 1010
A4A2A3(2.35 ± 0.87) 10115.66 × 1011
A1A4(3.34 ± 6.56) 1062.04 × 108
A4A1(1.34 ± 0.68) 10108.65 × 1010
Hydroxyl rotation
D1D3D2D4Ø2.38 × 109
D2D4D1D3(2.86 ± 4.71) 1084.15 × 109
Formation and dissociation of water
A1D1D3(1.64 ± 1.59) 1093.37 × 1011
D1D3A1(2.32 ± 1.59) 10101.13 × 1012
A2A3D2D4(7.86 ± 7.53) 1095.45 × 1013
D2D4A2A3(1.28 ± 0.54) 10111.17 × 1013
A2A3D1D3ØØ
D1D3A2A3(2.33 ± 3.14) 108Ø
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 A2A3A4 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 A2A3D2D4 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 A1A4 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:
KRP=kRPkPR
(9)
and
ΔGRP(T)=NAkBTln(KRP)
(10)
where NA is the Avogadro number and KRP 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 ± 2.665.50
2.81 ± 2.834.56
13.8 ± 4.4310.1
Hydroxyl rotations  
Ø0.93
Water dissociations  
4.41 ± 3.88–2.56
4.64 ± 3.542.01

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 A4A1. 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 A4A2A3A1 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 D1D3A2A3 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 A1D1D3 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 A4A1 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

Figure 7. Schematic representation of the two types of paths for the A4A1 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 pRP(q) = 0.5, where pRP(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 pRP(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 pRP(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 Σzkilln+1 starting on the surface Σzkilln is 1ηkilledn+1Nrep. By identifying the level n0.5 such that
n=n0.5nmax(1ηkillednNrep)=0.5
(11)
one can define the iso-level Σ0.5 of the reaction coordinate. The configurations corresponding to reactive trajectories crossing this surface are such that p^RP(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 Σ0.5, 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 A1D1D3, the interpolated SOAP-PCV reaction coordinate with the reactant and product states defined as R = A1A2A3A4D2D4 and P = D1D3 leads to a mean structure of configurations such that pRP(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

Figure 8. Ball and sticks representation of a) saddle point on the PES and b) mean structures such that pRP(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 = ∪ A2A3A4D1D3D2D4, the definition of the stochastic TS is of poor quality. This comes from the fact that this AMS mostly samples A1A2A3 trajectories (and only rarely A1D1D3 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 Σ0.5 iso-level of the reaction coordinate, while the red one is the Σ0.5 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

Figure 9. Schematic representation of poor match of the Σ0.5 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

ARTICLE SECTIONS
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

ARTICLE SECTIONS
Jump To

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 A1D1 dissociation trajectory (AVI)

  • Video 2 showing an A4A1 “top” rotation trajectory (AVI)

  • Video 3 showing an A4A1 “side” rotation trajectory (AVI)

Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

ARTICLE SECTIONS
Jump To

  • Corresponding Authors
    • Thomas Pigeon - MATHERIALS team-project, Inria Paris, 2 Rue Simone Iff, 75012 Paris, FranceCERMICS, École des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, FranceIFP Energies Nouvelles, Rond-Point de l’Echangeur de Solaize, BP 3, 69360 Solaize, FranceOrcidhttps://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, FranceMATHERIALS 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, FranceOrcidhttps://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, FranceMATHERIALS 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, FranceOrcidhttps://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, FranceOrcidhttps://orcid.org/0000-0002-4908-401X
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

ARTICLE SECTIONS
Jump To

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).

References

ARTICLE SECTIONS
Jump To

This article references 71 other publications.

  1. 1
    Broadbelt, L. J.; Snurr, R. Q. Applications of molecular modeling in heterogeneous catalysis research. Appl. Catal. A: Gen. 2000, 200, 2346,  DOI: 10.1016/S0926-860X(00)00648-7
  2. 2
    Chizallet, C.; Raybaud, P. Density functional theory simulations of complex catalytic materials in reactive environments: beyond the ideal surface at low coverage. Catal. Sci. Technol. 2014, 4, 27972813,  DOI: 10.1039/C3CY00965C
  3. 3
    Chen, B. W. J.; Xu, L.; Mavrikakis, M. Computational methods in heterogeneous catalysis. Chem. Rev. 2021, 121, 10071048,  DOI: 10.1021/acs.chemrev.0c01060
  4. 4
    Piccini, G.; Lee, M.-S.; Yuk, S. F.; Zhang, D.; Collinge, G.; Kollias, L.; Nguyen, M.-T.; Glezakou, V.-A.; Rousseau, R. Ab initio molecular dynamics with enhanced sampling in heterogeneous catalysis. Catal. Sci. Technol. 2022, 12, 1237,  DOI: 10.1039/D1CY01329G
  5. 5
    Feynman, R. P. Forces in molecules. Phys. Rev. 1939, 56, 340343,  DOI: 10.1103/PhysRev.56.340
  6. 6
    Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 1935, 3, 107115,  DOI: 10.1063/1.1749604
  7. 7
    Bennett, C. H. Algorithms for Chemical Computations; American Chemical Society: Washington, DC, USA, 1977; Chapter 4, pp 6367.
  8. 8
    The IUPAC Compendium of Chemical Terminology; International Union of Pure and Applied Chemistry (IUPAC), 2014;  DOI: 10.1351/goldbook.t06468 .
  9. 9
    Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 1990, 62, 251341,  DOI: 10.1103/RevModPhys.62.251
  10. 10
    Evans, M. G.; Polanyi, M. Some applications of the transition state method to the calculation of reaction velocities, especially in solution. Trans. Faraday Soc. 1935, 31, 875,  DOI: 10.1039/tf9353100875
  11. 11
    Wigner, E. The transition state method. Trans. Faraday Soc. 1938, 34, 29,  DOI: 10.1039/tf9383400029
  12. 12
    Collinge, G.; Yuk, S. F.; Nguyen, M.-T.; Lee, M.-S.; Glezakou, V.-A.; Rousseau, R. Effect of collective dynamics and anharmonicity on entropy in heterogenous catalysis: Building the case for advanced molecular simulations. ACS Catal. 2020, 10, 92369260,  DOI: 10.1021/acscatal.0c01501
  13. 13
    Chandler, D. Statistical mechanics of isomerization dynamics in liquids and the transition state approximation. J. Chem. Phys. 1978, 68, 2959,  DOI: 10.1063/1.436049
  14. 14
    Free energy calculations; Chipot, C., Pohorille, A., Eds.; Springer Berlin Heidelberg, 2007.
  15. 15
    Rousset, M.; Stoltz, G.; Lelièvre, T. Free Energy Computations; Imperial College Press: London, 2010.
  16. 16
    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
  17. 17
    Horiuti, J. On the statistical mechanical treatment of the absolute rate of chemical reaction. Bull. Chem. Soc. Jpn. 1938, 13, 210216,  DOI: 10.1246/bcsj.13.210
  18. 18
    Keck, J. Statistical investigation of dissociation cross-sections for diatoms. Faraday Discuss. 1962, 33, 173,  DOI: 10.1039/df9623300173
  19. 19
    Vanden-Eijnden, E.; Tal, F. A. Transition state theory: Variational formulation, dynamical corrections, and error estimates. J. Chem. Phys. 2005, 123, 184103,  DOI: 10.1063/1.2102898
  20. 20
    Miller, W. H.; Schwartz, S. D.; Tromp, J. W. Quantum mechanical rate constants for bimolecular reactions. J. Chem. Phys. 1983, 79, 48894898,  DOI: 10.1063/1.445581
  21. 21
    Dellago, C.; Bolhuis, P. G.; Geissler, P. L. Advances in Chemical Physics; John Wiley & Sons, Inc., 2003; pp 178.
  22. 22
    Mandelli, D.; Hirshberg, B.; Parrinello, M. Metadynamics of paths. Phys. Rev. Lett. 2020, 125, 026001,  DOI: 10.1103/PhysRevLett.125.026001
  23. 23
    Hill, T. Free Energy Transduction in Biology: The Steady-State Kinetic and Thermodynamic Formalism; Elsevier Science and Technology Books, 2012.
  24. 24
    Baudel, M.; Guyader, A.; Lelièvre, T. On the Hill relation and the mean reaction time for metastable processes. Stoch Process Their Appl. 2023, 155, 393436,  DOI: 10.1016/j.spa.2022.10.014
  25. 25
    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
  26. 26
    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, 77627774,  DOI: 10.1063/1.1562614
  27. 27
    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
  28. 28
    Huber, G.; Kim, S. Weighted-ensemble Brownian dynamics simulations for protein association reactions. Biophys. J. 1996, 70, 97110,  DOI: 10.1016/S0006-3495(96)79552-8
  29. 29
    Cérou, F.; Guyader, A. Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 2007, 25, 417443,  DOI: 10.1080/07362990601139628
  30. 30
    Glielmo, A.; Husic, B. E.; Rodriguez, A.; Clementi, C.; Noé, F.; Laio, A. Unsupervised learning methods for molecular simulation data. Chem. Rev. 2021, 121, 97229758,  DOI: 10.1021/acs.chemrev.0c01195
  31. 31
    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
  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, 47574775,  DOI: 10.1021/acs.jctc.0c00355
  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
  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
  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
  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), 19651975,  DOI: 10.1021/cs4002833
  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, 19051920,  DOI: 10.1021/acscatal.6b00080
  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, 265268,  DOI: 10.1126/science.282.5387.265
  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, 15,  DOI: 10.1006/jcat.2002.3741
  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, 5468,  DOI: 10.1016/j.jcat.2004.04.020
  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, 1443014449,  DOI: 10.1021/ja3042383
  42. 42
    Pigeon, T.; Chizallet, C.; Raybaud, P. Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces. J. Catal. 2022, 405, 140151,  DOI: 10.1016/j.jcat.2021.11.011
  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, 2156121570,  DOI: 10.1021/acs.jpcc.6b07191
  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, 1241012419,  DOI: 10.1021/la802295x
  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, 1035110363,  DOI: 10.1021/acs.jpcc.7b00101
  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, 191199,  DOI: 10.1021/acsanm.7b00100
  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, 93105,  DOI: 10.1016/j.cattod.2005.04.005
  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
  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, 1893818942,  DOI: 10.1002/anie.202006065
  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, 61936202,  DOI: 10.1021/acs.jctc.1c00458
  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, 11981208,  DOI: 10.1002/jcc.25778
  52. 52
    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, 29832989,  DOI: 10.1021/acs.jctc.6b00277
  53. 53
    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
  54. 54
    Cérou, F.; Delyon, B.; Guyader, A.; Rousset, M. On the Asymptotic Normality of Adaptive Multilevel Splitting. SIAM-ASA J. Uncertain. Quantif. 2019, 7, 130,  DOI: 10.1137/18M1187477
  55. 55
    Bréhier, C.-E.; Gazeau, M.; Goudenège, L.; Lelièvre, T.; Rousset, M. Unbiasedness of some generalized adaptive multilevel splitting algorithms. J. Appl. Probab. 2016, 26, 35593601,  DOI: 10.1214/16-AAP1185
  56. 56
    Binder, A.; Lelièvre, T.; Simpson, G. A generalized parallel replica dynamics. J. Comput. Phys. 2015, 284, 595616,  DOI: 10.1016/j.jcp.2015.01.002
  57. 57
    Kresse, G.; Hafner, J. Ab-initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558561,  DOI: 10.1103/PhysRevB.47.558
  58. 58
    Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999, 59, 17581775,  DOI: 10.1103/PhysRevB.59.1758
  59. 59
    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
  60. 60
    Bartók, A. P.; Kondor, R.; Csányi, G. On representing chemical environments. Phys. Rev. B 2013, 87, 184115,  DOI: 10.1103/PhysRevB.87.184115
  61. 61
    Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials. Phys. Rev. B 2019, 99, 014104,  DOI: 10.1103/PhysRevB.99.014104
  62. 62
    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
  63. 63
    Bartók-Pártay, A. The Gaussian Approximation Potential; Springer: Berlin/Heidelberg, 2010.
  64. 64
    Bartók, A. P.; Kermode, J.; Bernstein, N.; Csányi, G. Machine learning a general-purpose interatomic potential for silicon. Phys. Rev. X 2018, 8, 041048,  DOI: 10.1103/PhysRevX.8.041048
  65. 65
    Himanen, L.; Jäger, M. O.; Morooka, E. V.; Canova, F. F.; Ranawat, Y. S.; Gao, D. Z.; Rinke, P.; Foster, A. S. DScribe: Library of descriptors for machine learning in materials science. Comput. Phys. Commun. 2020, 247, 106949,  DOI: 10.1016/j.cpc.2019.106949
  66. 66
    Murphy, K. P. Probabilistic Machine Learning: An introduction; MIT Press, 2022.
  67. 67
    Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res. 2011, 12, 28252830
  68. 68
    Fleurat-Lessard, P. http://pfleurat.free.fr/ReactionPath.php.
  69. 69
    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
  70. 70
    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
  71. 71
    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, 69466955,  DOI: 10.1021/acs.jpclett.0c01061

Cited By

This article has not yet been cited by other publications.

  • Abstract

    Figure 1

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

    Figure 2

    Figure 2. First iteration of the AMS algorithm with kmin = 1 and Nrep = 3. Purple points represent the initial conditions on ΣR. a) Identify the kill level zkill1=zmaxkmin,0 and kill the replicas such that zmaxi,0zkill1, 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 zkill1 and continue the trajectory of the replica until it reaches either the state R or the state P.

    Figure 3

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

    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.

    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.

    Figure 6

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

    Figure 7

    Figure 7. Schematic representation of the two types of paths for the A4A1 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.

    Figure 8

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

    Figure 9

    Figure 9. Schematic representation of poor match of the Σ0.5 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.

  • References

    ARTICLE SECTIONS
    Jump To

    This article references 71 other publications.

    1. 1
      Broadbelt, L. J.; Snurr, R. Q. Applications of molecular modeling in heterogeneous catalysis research. Appl. Catal. A: Gen. 2000, 200, 2346,  DOI: 10.1016/S0926-860X(00)00648-7
    2. 2
      Chizallet, C.; Raybaud, P. Density functional theory simulations of complex catalytic materials in reactive environments: beyond the ideal surface at low coverage. Catal. Sci. Technol. 2014, 4, 27972813,  DOI: 10.1039/C3CY00965C
    3. 3
      Chen, B. W. J.; Xu, L.; Mavrikakis, M. Computational methods in heterogeneous catalysis. Chem. Rev. 2021, 121, 10071048,  DOI: 10.1021/acs.chemrev.0c01060
    4. 4
      Piccini, G.; Lee, M.-S.; Yuk, S. F.; Zhang, D.; Collinge, G.; Kollias, L.; Nguyen, M.-T.; Glezakou, V.-A.; Rousseau, R. Ab initio molecular dynamics with enhanced sampling in heterogeneous catalysis. Catal. Sci. Technol. 2022, 12, 1237,  DOI: 10.1039/D1CY01329G
    5. 5
      Feynman, R. P. Forces in molecules. Phys. Rev. 1939, 56, 340343,  DOI: 10.1103/PhysRev.56.340
    6. 6
      Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 1935, 3, 107115,  DOI: 10.1063/1.1749604
    7. 7
      Bennett, C. H. Algorithms for Chemical Computations; American Chemical Society: Washington, DC, USA, 1977; Chapter 4, pp 6367.
    8. 8
      The IUPAC Compendium of Chemical Terminology; International Union of Pure and Applied Chemistry (IUPAC), 2014;  DOI: 10.1351/goldbook.t06468 .
    9. 9
      Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 1990, 62, 251341,  DOI: 10.1103/RevModPhys.62.251
    10. 10
      Evans, M. G.; Polanyi, M. Some applications of the transition state method to the calculation of reaction velocities, especially in solution. Trans. Faraday Soc. 1935, 31, 875,  DOI: 10.1039/tf9353100875
    11. 11
      Wigner, E. The transition state method. Trans. Faraday Soc. 1938, 34, 29,  DOI: 10.1039/tf9383400029
    12. 12
      Collinge, G.; Yuk, S. F.; Nguyen, M.-T.; Lee, M.-S.; Glezakou, V.-A.; Rousseau, R. Effect of collective dynamics and anharmonicity on entropy in heterogenous catalysis: Building the case for advanced molecular simulations. ACS Catal. 2020, 10, 92369260,  DOI: 10.1021/acscatal.0c01501
    13. 13
      Chandler, D. Statistical mechanics of isomerization dynamics in liquids and the transition state approximation. J. Chem. Phys. 1978, 68, 2959,  DOI: 10.1063/1.436049
    14. 14
      Free energy calculations; Chipot, C., Pohorille, A., Eds.; Springer Berlin Heidelberg, 2007.
    15. 15
      Rousset, M.; Stoltz, G.; Lelièvre, T. Free Energy Computations; Imperial College Press: London, 2010.
    16. 16
      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
    17. 17
      Horiuti, J. On the statistical mechanical treatment of the absolute rate of chemical reaction. Bull. Chem. Soc. Jpn. 1938, 13, 210216,  DOI: 10.1246/bcsj.13.210
    18. 18
      Keck, J. Statistical investigation of dissociation cross-sections for diatoms. Faraday Discuss. 1962, 33, 173,  DOI: 10.1039/df9623300173
    19. 19
      Vanden-Eijnden, E.; Tal, F. A. Transition state theory: Variational formulation, dynamical corrections, and error estimates. J. Chem. Phys. 2005, 123, 184103,  DOI: 10.1063/1.2102898
    20. 20
      Miller, W. H.; Schwartz, S. D.; Tromp, J. W. Quantum mechanical rate constants for bimolecular reactions. J. Chem. Phys. 1983, 79, 48894898,  DOI: 10.1063/1.445581
    21. 21
      Dellago, C.; Bolhuis, P. G.; Geissler, P. L. Advances in Chemical Physics; John Wiley & Sons, Inc., 2003; pp 178.
    22. 22
      Mandelli, D.; Hirshberg, B.; Parrinello, M. Metadynamics of paths. Phys. Rev. Lett. 2020, 125, 026001,  DOI: 10.1103/PhysRevLett.125.026001
    23. 23
      Hill, T. Free Energy Transduction in Biology: The Steady-State Kinetic and Thermodynamic Formalism; Elsevier Science and Technology Books, 2012.
    24. 24
      Baudel, M.; Guyader, A.; Lelièvre, T. On the Hill relation and the mean reaction time for metastable processes. Stoch Process Their Appl. 2023, 155, 393436,  DOI: 10.1016/j.spa.2022.10.014
    25. 25
      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
    26. 26
      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, 77627774,  DOI: 10.1063/1.1562614
    27. 27
      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
    28. 28
      Huber, G.; Kim, S. Weighted-ensemble Brownian dynamics simulations for protein association reactions. Biophys. J. 1996, 70, 97110,  DOI: 10.1016/S0006-3495(96)79552-8
    29. 29
      Cérou, F.; Guyader, A. Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 2007, 25, 417443,  DOI: 10.1080/07362990601139628
    30. 30
      Glielmo, A.; Husic, B. E.; Rodriguez, A.; Clementi, C.; Noé, F.; Laio, A. Unsupervised learning methods for molecular simulation data. Chem. Rev. 2021, 121, 97229758,  DOI: 10.1021/acs.chemrev.0c01195
    31. 31
      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
    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, 47574775,  DOI: 10.1021/acs.jctc.0c00355
    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
    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
    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
    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), 19651975,  DOI: 10.1021/cs4002833
    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, 19051920,  DOI: 10.1021/acscatal.6b00080
    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, 265268,  DOI: 10.1126/science.282.5387.265
    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, 15,  DOI: 10.1006/jcat.2002.3741
    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, 5468,  DOI: 10.1016/j.jcat.2004.04.020
    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, 1443014449,  DOI: 10.1021/ja3042383
    42. 42
      Pigeon, T.; Chizallet, C.; Raybaud, P. Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces. J. Catal. 2022, 405, 140151,  DOI: 10.1016/j.jcat.2021.11.011
    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, 2156121570,  DOI: 10.1021/acs.jpcc.6b07191
    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, 1241012419,  DOI: 10.1021/la802295x
    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, 1035110363,  DOI: 10.1021/acs.jpcc.7b00101
    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, 191199,  DOI: 10.1021/acsanm.7b00100
    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, 93105,  DOI: 10.1016/j.cattod.2005.04.005
    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
    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, 1893818942,  DOI: 10.1002/anie.202006065
    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, 61936202,  DOI: 10.1021/acs.jctc.1c00458
    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, 11981208,  DOI: 10.1002/jcc.25778
    52. 52
      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, 29832989,  DOI: 10.1021/acs.jctc.6b00277
    53. 53
      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
    54. 54
      Cérou, F.; Delyon, B.; Guyader, A.; Rousset, M. On the Asymptotic Normality of Adaptive Multilevel Splitting. SIAM-ASA J. Uncertain. Quantif. 2019, 7, 130,  DOI: 10.1137/18M1187477
    55. 55
      Bréhier, C.-E.; Gazeau, M.; Goudenège, L.; Lelièvre, T.; Rousset, M. Unbiasedness of some generalized adaptive multilevel splitting algorithms. J. Appl. Probab. 2016, 26, 35593601,  DOI: 10.1214/16-AAP1185
    56. 56
      Binder, A.; Lelièvre, T.; Simpson, G. A generalized parallel replica dynamics. J. Comput. Phys. 2015, 284, 595616,  DOI: 10.1016/j.jcp.2015.01.002
    57. 57
      Kresse, G.; Hafner, J. Ab-initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558561,  DOI: 10.1103/PhysRevB.47.558
    58. 58
      Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999, 59, 17581775,  DOI: 10.1103/PhysRevB.59.1758
    59. 59
      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
    60. 60
      Bartók, A. P.; Kondor, R.; Csányi, G. On representing chemical environments. Phys. Rev. B 2013, 87, 184115,  DOI: 10.1103/PhysRevB.87.184115
    61. 61
      Drautz, R. Atomic cluster expansion for accurate and transferable interatomic potentials. Phys. Rev. B 2019, 99, 014104,  DOI: 10.1103/PhysRevB.99.014104
    62. 62
      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
    63. 63
      Bartók-Pártay, A. The Gaussian Approximation Potential; Springer: Berlin/Heidelberg, 2010.
    64. 64
      Bartók, A. P.; Kermode, J.; Bernstein, N.; Csányi, G. Machine learning a general-purpose interatomic potential for silicon. Phys. Rev. X 2018, 8, 041048,  DOI: 10.1103/PhysRevX.8.041048
    65. 65
      Himanen, L.; Jäger, M. O.; Morooka, E. V.; Canova, F. F.; Ranawat, Y. S.; Gao, D. Z.; Rinke, P.; Foster, A. S. DScribe: Library of descriptors for machine learning in materials science. Comput. Phys. Commun. 2020, 247, 106949,  DOI: 10.1016/j.cpc.2019.106949
    66. 66
      Murphy, K. P. Probabilistic Machine Learning: An introduction; MIT Press, 2022.
    67. 67
      Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E. Scikit-learn: Machine Learning in Python. J. Mach. Learn. Res. 2011, 12, 28252830
    68. 68
      Fleurat-Lessard, P. http://pfleurat.free.fr/ReactionPath.php.
    69. 69
      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
    70. 70
      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
    71. 71
      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, 69466955,  DOI: 10.1021/acs.jpclett.0c01061
  • Supporting Information

    Supporting Information

    ARTICLE SECTIONS
    Jump To

    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 A1D1 dissociation trajectory (AVI)

    • Video 2 showing an A4A1 “top” rotation trajectory (AVI)

    • Video 3 showing an A4A1 “side” rotation trajectory (AVI)


    Terms & Conditions

    Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Pair your accounts.

Export articles to Mendeley

Get article recommendations from ACS based on references in your Mendeley library.

Pair your accounts.

Export articles to Mendeley

Get article recommendations from ACS based on references in your Mendeley library.

You’ve supercharged your research process with ACS and Mendeley!

STEP 1:
Click to create an ACS ID

Please note: If you switch to a different device, you may be asked to login again with only your ACS ID.

Please note: If you switch to a different device, you may be asked to login again with only your ACS ID.

Please note: If you switch to a different device, you may be asked to login again with only your ACS ID.

MENDELEY PAIRING EXPIRED
Your Mendeley pairing has expired. Please reconnect