Theoretical studies on triplet-state driven dissociation of formaldehyde by quasi-classical molecular dynamics simulation on machine-learning potential energy surface

Abstract

The H-atom dissociation of formaldehyde on the lowest triplet state (T₁) is studied by quasi-classical molecular dynamic simulations on the high-dimensional machine-learning potential energy surface (PES) model. An atomic-energy based deep-learning neural network (NN) is used to represent the PES function, and the weighted atom-centered symmetry functions are employed as inputs of the NN model to satisfy the translational, rotational, and permutational symmetries, and to capture the geometry features of each atom and its individual chemical environment. Several standard technical tricks are used in the construction of NN-PES, which includes the application of clustering algorithm in the formation of the training dataset, the examination of the reliability of the NN-PES model by different fitted NN models, and the detection of the out-of-confidence region by the confidence interval of the training dataset. The accuracy of the full-dimensional NN-PES model is examined by two benchmark calculations with respect to ab initio data. Both the NN and electronic-structure calculations give a similar H-atom dissociation reaction pathway on the T₁ state in the intrinsic reaction coordinate analysis. The small-scaled trial dynamics simulations based on NN-PES and ab initio PES give highly consistent results. After confirming the accuracy of the NN-PES, a large number of trajectories are calculated in the quasi-classical dynamics, which allows us to get a better understanding of the T₁-driven H-atom dissociation dynamics efficiently. Particularly, the dynamics simulations from different initial conditions can be easily simulated with a rather low computational cost. The influence of the mode-specific vibrational excitations on the H-atom dissociation dynamics driven by the T₁ state is explored. The results show that the vibrational excitations on symmetric C–H stretching, asymmetric C–H stretching, and C=O stretching motions always enhance the H-atom dissociation probability obviously.

I. INTRODUCTION

Studies on the dissociation dynamics of volatile organic compounds (VOCs), such as alkanes, alkenes, and carbonyls, are always an essential topic in physical chemistry, since the volatility, toxicity, and diffusivity of VOCs are great threats to human health and eco-environment.^1–7 Theoretically, various dynamics approaches, ranging from quantum to quasi-classical dynamics, were developed to provide the atomic-level simulations of molecular dissociation of these VOCs.^8–14 No matter which dynamics approach was taken, it is necessary to build the accurate potential energy surface (PES) with high efficiency, while this is not a trivial work for polyatomic systems. One widely used approach is to build the analytical expressions to represent the potential energy as a function of nuclear coordinates. After the construction of an analytical PES, the evaluation of the potential energy (and gradients) only requires very minor computational efforts in the dynamics calculations. However, the selection of suitable fitting functions becomes extremely difficult for high-dimensional systems. Alternatively, it is also possible to run on-the-fly Born–Oppenheimer dynamics,^15–19 in which the electronic-structure calculations are performed at each time step. The on-the-fly approach, in principle, is very attractive for the study of the dynamics of high-dimensional polyatomic systems, because all degrees of freedom are included, and it is not necessary to build the completed analytical PES in advance. However, the computational cost of the on-the-fly calculation is generally high because the time-consuming electronic-structure calculations have to be conducted at each time step. Therefore, many on-the-fly molecular dynamics calculations only provided a limited number of trajectories, and it is very difficult to achieve the calculations with high statistical accuracy. Moreover, when several initial conditions are considered, each requires a large number of trajectories in dynamics calculations. Within the framework of on-the-fly simulation, the computational cost may become extremely high and even unaffordable.

Recently, the machine learning (ML) approaches received huge developments, which became popular in many scientific fields,²⁰ such as image recognition,²¹ speech recognition,²² and reconstructing brain circuits.²³ In theoretical chemistry, great progress was made in the employment of ML approaches, such as the neural network (NN) approaches, to build the PES. In the construction of the ML potential model, it is preferable to make sure that the system symmetry properties are considered, i.e., the input of the ML potential model should take into account the transitional, rotational, and permutational invariance.^24–28 Several groups suggested different approaches to satisfy such symmetry properties.^27–32 One of the promising approaches was proposed by Behler,³¹ in which the total energy was decomposed as the summation of individual atomic energy contributions from each atom. The atomic energy contribution of each atom was predicted by its own individual deep neural networks (NNs), and a set of atom-centered symmetry functions (ACSFs) built from original Cartesian coordinates were used as the network inputs. Several efforts were made to improve the performance of the model by employing new or updated descriptors or model expressions. For instance, Liu and co-workers suggested to include the four-body interaction terms in the Behler Parrinello Neural Network (BP-NN) framework by introducing a new group of descriptors, power-type structural descriptors (PTSDs),³³ which improved its performance in the surface reactions. Roitberg and co-workers developed the so-called ANAKIN-ME (Accurate NeurAl networK engINe for Molecular Energies) model that modified the original version of the BP-NN potential.³⁴ This gives the better description of the local chemical environment of each single atom and improves the transferability problems of the NN-PES model. Unke and Meuwly developed the “PhyNet” approach, in which atomic interactions were modeled by learnable distance-based attention masks biased toward a physically meaningful (exponential) decay behavior.³⁵ Based on PhyNet, Schütt et al. successfully constructed the highly accurate global PES of acetaldehyde.³⁶ Schütt et al. developed the so-called “SchNet” approach, which used the machine learning approach to directly generate the descriptors for the inputs of the network.³⁶ The updated version of the Deep Potential²⁹ also employed the descriptors generated by the neural network.³⁷

Recently, the ML-PES approaches were also employed in the simulation of the excited-state dynamics. For more details, please see the recent reviews.^38–44 Hu and Lan studied the non-adiabatic dynamics of 6-aminopyrimidine by applying Machine Learning Kernel Ridge Regression (KRR) Potential Energy Surfaces in the Zhu–Nakamura surface-hopping algorithm.⁴⁵ In this work, the clustering approaches were employed to reduce the training dataset size, and the confidence interval was determined by a statistical way. Dral, Barbatti, and Thiel used the kernel ridge regression (KRR) approach to conduct the surface-hopping dynamics for the spin-boson systems.⁴⁶ Dral and Barbatti also provided a broad overview on machine learning applications in excited-state research studies⁴⁴ and evaluated the performance of popular machine learning potentials in terms of accuracy and computational cost.⁴⁷ Using the Deep Potential Molecular Dynamics (DPMD) approach,²⁹ Cui and his co-workers provided an accurate representation of the complete active space self-consistent field (CASSCF)-level ground- and excited-state potential energy surfaces of CH₂NH.⁴⁸ Marquetand and co-workers made the systematical contribution,^49–52 who applied various ML approaches in excited-state simulations; see their recent reviews for details.^41,53 For instance, they combined the SchNet approach and their surface hopping including arbitrary couplings (SHARC) approach to define the SchNarc model (atomic-energy deep-learning NN-based approach).⁴⁹ They also tried to use another NN model to run long-time photodynamics on nanosecond time scales,⁵⁰ in which they also used the active learning approach suggested by Behler⁵⁴ and took several NN models to detect the out-of-confidence region,^{45,50,51,55–62} where the accuracy of the NN-PES model is very low due to the missing of the enough training data points. In addition, employing the NN-PES model and the same active learning approach, Li et al. developed the Python Rapid Artificial Intelligence Ab Initio Molecular Dynamics (PyRAI2 MD) software for the cis–trans isomerization of trans-hexafluoro-2-butene and the 4π-electrocyclic ring-closing of a norbornyl hexacyclodiene.⁶³

One of the challenging problems in the NN-PES model construction is that the prediction and training examples may not lie on the same configuration. As an interpolated method, the NN-PES model only works when the new geometries are inside the geometrical space composed by the samplings. When a geometry is in the out-of-confidence areas, the prediction is generally not reliable. Thus, it is necessary to develop the suitable approach to pick up such points. Thus, many trajectory-based iterative sampling algorithms, which can be called active learning,⁶⁴ were developed to pick up such points.^55,65–71 For instance, Behler proposed an adaptive sampling approach,⁷² which used different NN models to predict the physical observables, such as energy, at the same time. When different NN models give quite different values, the data are believed to fall in the out-of-confidence region.

Among all ML-PES models, the Behler and Parrinello (BP) model was applied in the study of different systems, such as crystal, metal, and alloy.^72–74 To improve the accuracy of the BP model in the treatment of the central atom and its surrounding environment in molecular systems, the modified descriptors, weighted ACSFs (wACSFs), were proposed by Gastegger et al.,⁷⁵ in which the nuclear charge number was used as a weight to distinguish different types of atoms. As the BP NN-PES based on the wACSF descriptors provided a feasible way to build the accurate potential surfaces of the high-dimensional systems,⁶¹ it is certainly interesting to apply such an idea to study the dissociation dynamics of small VOCs, such as formaldehyde.

As a typical organic VOC, formaldehyde widely exists in Earth’s present stratosphere⁷⁶ and its polluted troposphere,^77–79 and thus, its thermal and photo-induced reactions received great attention. It is well-known that the photodissociation of formaldehyde plays an important role in its removal processes in atmosphere. Therefore, the photodissociation of formaldehyde was extensively studied experimentally^79–81 and theoretically.^82–95 Several early studies proposed that the direct S₁ → S₀ non-adiabatic decay may be responsible for the photolysis of formaldehyde,^96,97 while the studies on the S₁–S₀ conical intersections showed that this internal conversion is only possible in the high-energy excitations.⁹⁷ Therefore, some studies^98,99 suggested that the lowest triplet state (T₁) is deeply involved in the photo-induced dynamics of formaldehyde. After the photo-excitation of the formaldehyde into the (S₁) state, the inter-system crossing may take place and the T₁ state is formed, giving the possible explanation on the radical products in photolysis.^81,100 In theoretical calculations, considerable efforts were devoted to understand the dissociation dynamics on the ground state of the formaldehyde,^{86,88,90,101–122} including isomerization, dissociation, and roaming H-atom pathways. The possible dissociation mechanism on the T₁ state was also studied by both electronic-structure calculations and dynamics simulations.^{87,95,109,119} For instance, the very early work by Hayes and Morokuma⁸⁸ and by Fink⁸⁷ provided the initial view on the reaction pathways on the T₁ state. Yates et al.⁹⁵ used the high-level the complete active space self-consistent field (CASSCF), multi-reference configuration interaction with single and double excitations (MR-CISD) approaches to study in the T₁-driven dissociation pathways, in which the reaction barrier height (28.9 kcal/mol) and the exit barrier height at 0 K (7.8 kcal/mol with the zero-point energy correction) were given. Yamaguchi et al. studied the T₁ energies and all critical geometries at the Coupled Cluster method with single and double excitations and perturbative treatment of triple excitation level,¹⁰⁹ giving the estimation of the dissociation barrier height to be 20.1 kcal/mol with respect to the T₁ minimum. Bowman and co-workers explored the role of the T₁ state in the photolysis reaction of formaldehyde in detail using the quasi-classical molecular dynamics simulation.^123,124 They confirmed the importance of the T₁-driven dissociation of formaldehyde, particularly under high photolysis energy. Zhang et al. built the T₁ PES function based on the highly accurate ab initio data,¹⁰³ in which the permutation symmetry was properly taken into account. On the T₁ state, the barrier height of the H-atom dissociation channel is much lower than that of the isomerization channel. The photolysis of formaldehyde was studied by Lasorne, Worth, Robb, and co-workers using the direct quantum dynamics approach based on the variational multiconfiguration Gaussian wavepacket at the CASSCF level.¹⁰ The quantum evolution on the T₁ state starting from different initial conditions was also simulated, which demonstrates the important role of the H-atom dissociation channel. In a few of these theoretical studies,^{10,121,125,126} the possible involvement of the S₀ − T₁ crossings in the T₁ dissociation dynamics was also suggested. Recently, Guan et al. tried to construct the diabatic models including S₀, S₁, and T₁ PESs¹²⁷ and their couplings by using the neural networks with proper consideration of the symmetry properties. This opens the possibility to run the full-dimensional dynamical simulations with all the three states included.

In this work, we study the T₁-driven dissociation of formaldehyde by using molecular dynamics simulation on the NN-PES. The NN-PES of T₁ state is successfully constructed with the BP model based on wASCF descriptors. In fact, we need to emphasize that the NN-PES construction is not trivial, and several techniques are employed to improve the fitting efficiency and increase the accuracy of the NN-PES model. In particular, the training dataset is built by the clustering analysis of the original sampled data. The mini-batch approach is used in the NN model construction. When the NN-PES model is used in the quasi-classical dynamics simulation, the trajectory may enter the out-of-sampling regions. To overcome such a critical problem, two useful standard approaches are applied to detect the out-of-sampling regions in the dynamics simulation, which are based on the confidence interval of the training dataset and the multiple-fit validation^50,72,128 by different NN-PES models. In addition, the NN-PES model is built in an iterative manner by adding more relevant data, until the reasonable model is obtained.

The resulted NN-PES model gives a rather accurate description of the reaction pathway along the intrinsic reaction coordinate (IRC). In addition, the quasi-classical dynamics obtained at both NN-PES and on-the-fly electronic-structure calculations are highly consistent. After the accuracy of the NN-PES model is confirmed, a large number of trajectories are propagated on the NN-PES in the quasi-classical dynamics, allowing us to get the full description of the T₁-driven formaldehyde dissociation dynamics with high efficiency. This NN-PES model also allows us to perform the effective simulation of the full-dimensional dissociation dynamics from different initial conditions with a very low computational cost. For instance, the influence of the mode-specific vibrational excitation of symmetric C–H stretching, asymmetric C–H stretching, and C=O stretching motions on the dissociation dynamics are explored. The current work not only demonstrates that the NN-PES model is a powerful approach to speed up the full-dimensional dynamics simulation, but also provides some valuable information to understand the T₁-driven dissociation of formaldehyde.

II. METHODOLOGY

A. T₁-driven dissociation model

Although the photo-induced dynamics of formaldehyde starts from the singlet excited state created by the photo-excitation, the theoretical description on the whole excited-state photodissociation dynamics is rather challenging, when all degrees of freedom are taken into account. For instance, the intersystem crossing dynamics from the S₁ to T₁ does not take place in the ultrafast femtosecond or picosecond time scales,⁹⁹ due to the lack of heavy chemical elements. The direct simulations of such a “slow” dynamics approach at the atomic level are extremely difficult, if all involved electronic states and nuclear coordinates are treated explicitly. If the photodissociation dynamics on the T₁ state is simulated directly, it is also not clear on how to set up the proper initial conditions of nuclear coordinates and velocities. In addition, previous work also indicated that the non-adiabatic decay from T₁ to S₀ may take place in the dissociation pathways, and it is not easy to simulate such intersystem crossing processes directly by the dynamics simulations. Particularly, if one wants to obtain the comprehensive theoretical understanding of the full-dimensional photodissociation dynamics of formaldehyde at the atomic level, the treatment of all degrees of freedom is also not trivial. In this sense, nowadays, it is still very challenging to conduct the atomic simulations of the full-dimensional photo-induced quantum dynamics of formaldehyde.

The quasi-classical dynamics simulations of the T₁-driven dynamics certainly provide useful information to understand the dissociation mechanism of formaldehyde at the atomic level.^15–19 Thus, the H-atom dissociation dynamics on the T₁ state is simulated in this work, which includes all nuclear degrees of freedom. As it is not easy to give a clear definition of the initial conditions, several initial conditions are taken into account. First, the samplings of initial nuclear coordinates and velocities are performed at the T₁ minimum, giving the boundary condition representing the lowest-energy limit situation of the photodissociation dynamics on the T₁ state. Second, the sampling of initial nuclear coordinates and velocities is also conducted at the S₀ minimum, and placing them vertically into the T₁ state defines the initial conditions of quasi-classical dynamics. This gives the upper-energy limit of the photodissociation dynamics on the T₁ state.

Since the H-atom dissociation is relevant to the C–H stretching and C=O stretching motions of formaldehyde, the possible impact of the mode-specific vibrational excitations of the C–H stretching and C=O stretching motions on the H-atom dissociation dynamics is also studied. The above initial conditions provide us a useful view to understand the dynamical features of the H-atom dissociation of formaldehyde on the T₁ state.

In the current quasi-classical dynamics simulations, the full-dimensional PES model is employed in order to properly take all atomic interactions into account. Due to the high dimensionality of the PES, the NN-PES model is employed here.

B. Structure of the NN-PES

The T₁ PES of formaldehyde is built by using the NN model proposed by Behler and Parrinello²⁴ (BP), and the NN structure is shown in Fig. 1. The total energy of the molecular system is estimated by the summation of the contribution of each atom, and the atomic energy contribution of each atom is represented by a feed forward neural network (FFNN) with full connectivity. To keep the rotational, translational, and permutation symmetries, the input vector of each individual fully connected FFNN is composed of a set of atom-centered symmetry functions (ACSFs) constructed from the nonlinear transformation of atomic Cartesian coordinates, and the same FFNN is employed for the same element.

FIG. 1.

The structure of the whole NN-PES. The total energy of the molecular system is given by the summation of the contribution of each atom, and the atomic energy contribution of each atom is represented by a feed forward neural network (FFNN) with full connectivity. The input vector of each individual fully connected FFNN is composed of a set of atom-centered symmetry functions (ACSFs) constructed from the nonlinear transformation of atomic Cartesian coordinates, and the same FFNN is employed for the same element (H). The red and blue lines show that the symmetry functions of two different H atoms are given separately as the input of the same FFNN. This way gives the individual atomic energy contribution for each H atom.

In this work, an FFNN with three hidden layers connecting the input layer and the output layer, denoted as n − m − m − m − 1, is employed for all the atoms. The structure of the FFNN is shown in Fig. 2. It has n nodes in the input layer, which are equal to the number of symmetry functions (15 in this work), and one node in the output layer corresponding to the atomic energy contribution of the particular type element with the input vector. The three hidden layers have m neurons, respectively. The output of the ith neuron in the first hidden layer is

$$H_i^{(1)}=f\left(\sum _{r=1}^n{w}_{ri}^{(1)}{G}_r+b_i^{(1)}\right),$$ (1)

and the output of the ith neuron in the second and third hidden layers is

$$H_i^{(2)}=f\left(\sum _{p=1}^m{w}_{pi}^{(2)}{H}_p^{(1)}+b_i^{(2)}\right),$$ (2)

$$H_i^{(3)}=f\left(\sum _{q=1}^m{w}_{qi}^{(3)}{H}_q^{(2)}+b_i^{(3)}\right),$$ (3)

and the final output is given by

$$E=\sum _{l=1}^m{w}_l^{(4)}{H}_l^{(3)}+b^{(4)},$$ (4)

where G_r is the rth input symmetry function, the weight $w_{ri}^{(l)}$ connects the rth neuron of the (l − 1)^th layer and the ith neuron of the lth hidden layer ($w_{ri}^{(1)}$ connects the input layer and the first hidden layer and $w_r^{(4)}$ connects the output layer and the third hidden layer), and the bias $b_i^{(l)}$ determines the threshold of the ith neuron of the lth hidden layer (b⁽⁴⁾ determines the threshold of the output layer). f is the Sigmoid function, which is used as the activate function in this work.

FIG. 2.

The structure of the single FFNN with the input layer (n nodes), three hidden layers (m nodes), and an output layer (1 node). G_n represents the nth node in the input layer. $H_i^l$ represents the ith node of the lth hidden layer. E represents the node in the output layer (in this model, only one node in the output layer).

In the BP model, ACSFs are defined as inputs of FFNNs, which describes the coordinates and chemical environment of each atom. The original ACSFs in the BP model do not distinguish the element types of the neighbor atoms adjacent to the center atom. Specifically speaking, when the center atom is fixed, the values of ACSFs remain the same, when a neighbor atom is replaced by other elements. To improve the description ability of ACSFs, weighted ACSFs (wACSFs), proposed by Gastegger et al.⁷⁵ are used in the current work, which distinguish different atom types by their nuclear charge numbers Z. Thus, wACSFs are employed as the input of the NN potential. The radial wACSF G^(r) is shown as

$$G_i^{(r)}=\sum _{j\ne i}^{N_{atom}}{Z}_j\exp [-\eta {(r_{ij}-r_s)}^2]f_c(r_{ij})=\sum _{j\ne i}^{N_{atom}}{Z}_{j}g(r_{ij})f_c(r_{ij}),$$ (5)

which includes the two-body interaction terms g(r_ij). The angular wACSF G^(a) is given as

$$G_i^{(a)}=2^{1-\zeta }\sum _{j\ne i}^{N_{atom}}\sum _{k\ne i,k>j}^{N_{atom}}{Z}_j{Z}_k{F}_{ijk}^1{F}_{ijk}^2{F}_{ijk}^3,$$ (6)

with three-body interaction terms, i.e., $F_{ijk}^{(1)}$, $F_{ijk}^{(2)}$, and $F_{ijk}^{(3)}$, given as follows:

$$F_{ijk}^{(1)}={(1+\lambda \cos {\theta }_{jik})}^{\zeta },$$ (7)

$$F_{ijk}^{(2)}=e^{-\eta [{(r_{ij}-{\mu }_s)}^2+{(r_{ik}-{\mu }_s)}^2+{(r_{jk}-{\mu }_s)}^2]},$$ (8)

$$F_{ijk}^{(3)}=f_c(r_{ij})f_c(r_{ik})f_c(r_{jk}).$$ (9)

In above equations, i is the index of the center atom and j, k are its neighbors. r_ij is the interatomic distance between atoms i and j, and θ_jik is the angle formed by the center atom i and its neighbors j, k. Constants such as η, ζ, r_s, μ_s, and λ are adjustable parameters to make wACSFs capable to cover the dynamically relevant configuration space. To completely describe chemical environments, a set of wACSFs are used for each atom. f_c(r) is the cutoff function,

$$f_c(r)=\left\{\begin{array}{cc}0,\hfill & r>r_c,\hfill \\ 0.5\left[\cos \left(\frac{r}{r_c}\pi \right)+1\right],\hfill & r\le r_c,\hfill \end{array}\right.$$ (10)

where the cutoff radius r_c was set to 6 Å as usual.

C. NN-PES construction

To construct a reliable NN-PES model, an iterative process is designed, and the working flow chart is given in Fig. 3. All steps are explained as below, and more details are given in Appendixes A–C.

FIG. 3.

Flow chart of the iterative NN-PES building process.

1. Preliminary training/validation dataset

The primitive training set contains a huge number of data points generated by quasi-classical ab initio molecular dynamics (AIMD) simulations on the T₁ state from different initial conditions. In principle, the AIMD-generated configurations may not be equally distributed in the whole dynamics-relevant configuration space. For instance, more snapshots may exist in the low-energy regions, and thus, the resulting NN-PES should be accurate enough in these areas. However, much less sampled geometries exist in the vicinity of reaction barriers, leading to the strong deviation of the NN-PES in such regions, while the precise description of the PES in these areas should be extremely important for the proper treatment of chemical reactions.

Therefore, one important task in the fitting procedure is to make all training data points evenly distributed in the whole sampled space. This way treats all relevant PES regions with equal importance in the NN-PES fitting. Then, at least the error of the NN-PES model in different nuclear configuration regions should be similar. In practice, this treatment is realized by the clustering algorithm, which is widely used in different ML approaches.^{45,46,51,56,71,129–131} All sampled geometries are divided into several groups by the clustering algorithm, and each cluster represents an individual PES area. Next, the preliminary training/validation dataset is built by selecting the same number of geometries from each cluster. At the same time, this approach also reduces the number of training data points, largely saving the computational efforts in the NN training step.

2. Training

In the training step, all original data are divided into several mini-batches (N sets of mini-batches), after they are randomly shuffled. Next, one mini-batch is randomly chosen as the validation set label as sub-set V. In all remaining N − 1 sets, we pick up one set as the training set, labeled sub-set K. The early-stop algorithm^132,133 is employed to train the NN model by using two sub-sets (K and V). When the fitting error starts to increases over several successive iterations, we stop the training based on K and V, choose the other training sub-set K′ from N − 1 sets, and repeat the early-stop training procedure based on K′ and V. After the traversal of all N − 1 sets, all original data are mixed again, and all the above procedure is repeated. At the end, when both the training and validation errors are smaller than the pre-defined threshold, the NN model is constructed.

3. Preliminary dynamics simulations based on the NN-PES

After the construction of the preliminary NN-PES, the quasi-classical dynamics is run again, in which several trajectories starting from different initial conditions are propagated. In the simulation process, trajectories may enter the out-of-sampling areas that are not well sampled by the training dataset. It is worth noting that the NN is a fitting tool with reliable interpolation results and inaccurate extrapolation performances. As a consequence, the NN-PES cannot provide the correct description of the PES in the out-of-sampling regions. In this sense, it is rather important to judge whether the trajectories enter such out-of-sampling regions in the dynamics simulation. Several technical tricks are employed to detect such regions:

• (a)We trained two independent NN-PES models at the same time, and the difference between their predicted energies provides an indicator to identify the out-of-sampling area. According to the work by Behler,¹²⁸ the fitting accuracy of the NN-PES decreases dramatically in the out-of-sampling areas. Therefore, different NN-PES models may not provide similar energies in these areas. In the current work, two NN-PES models with rather different network structures are built based on the training dataset. One NN-PES is set as the main model on which the quasi-classical dynamics is run. The second NN-PES model is set as the assistant one, which is also used for the validation of the prediction energies by the first NN-PES in the dynamics simulations. When the difference of potential energies predicted by two NN-PES models is above 0.2 kcal mol⁻¹, we assume that the out-of-sampling regions are reached. We took the more conservative criteria to detect out-of-sampling regions and, thus, ensure the quality of the NN-PES. This process is the so-called multiple-fit validation.^50,72,128
• (b)The out-of-sampling areas can also be detected by the confidence interval of the training dataset. In principle, when a geometry is located in the dense-sampled regions, it is easy to obtain the accurate energy in the NN prediction. When a geometry is in the marginal regions or even in the out-of-sampling region of the training dataset, it is not possible to predict the reasonable energy of this geometry by using the NN-PES model. Here, we define the confidence interval to detect the out-of-sampling regions, by modifying the essential idea proposed by previous work.⁴⁵ More details are given in Appendix B.

When the confidence interval for each dimension of molecular descriptors is known, a configuration is assumed to be in the out-of-sampling areas if the value of any dimension goes out of the corresponding interval. In such situations, we switch back to electronic-structure calculations instead.^45,134,135

As a short summary, it is necessary to use all of the above tricks. This largely reduces abnormal reaction channels in the quasi-classical dynamics simulations.

4. Iterative process for NN-PES construction and validation

After the above preliminary dynamics simulations, the data points coming from out-of-sampling areas are collected. After the clustering analysis, we randomly select a certain amount of points from each cluster as representative data and add them to the training set. The new NN-PES model is built again, and the new sets of dynamics simulations are re-run. In each iteration, we compare the dynamics results based on the NN-PES and ab initio PES, particularly the dissociation probability (P) of the H-atom dissociation reaction, to access the reliability of the NN-PES. If both give consistent results, we believe that the NN-PES is reliable. If not, the above procedure is repeated until consistent results are achieved.

After the NN-PES construction, the IRC reaction pathway is built to test the fitting accuracy of the NN-PES. In addition, for checking the performance of the NN-PES model, the molecular motions on the T₁ state of formaldehyde are simulated by the small-scaled quasi-classical dynamics based on the NN-PES model and the on-the-fly AIMD simulations. Two different initial nuclear conditions are sampled, i.e., either at the T₁ minimum or S₀ minimum of formaldehyde.

D. NN-PES MD simulations

After the validation of the accuracy of the NN-PES model, a large number of trajectories are simulated in the quasi-classical dynamics. The Wigner sampling at the S₀ minimum is performed to give the initial nuclear conditions.

We also examine the influence of the vibrational mode-specific excitation on the reactive dynamics. Particularly, the roles of the symmetric C–H bond stretching, asymmetric C–H bond stretching, and C=O bond stretching motions are explored. Here, the action-angle sampling approach is better because the vibrational excited states are involved.

III. RESULT

A. Validation of the NN-PES model

After the NN-PES model is built, it is necessary to examine its accuracy.

The first step is to examine the fitting errors. We generate an independent testing dataset composed of 11 000 configurations obtained from the quasi-classical dynamics, which totally has no relationship with our training data and is taken to examine the performance of the NN-PES model. As seen in Fig. 4, the high accuracy of the NN-PES model is achieved with the Root Mean Square Errors (RMSEs) as 0.58 kcal mol⁻¹ (main NN-PES) and 0.59 kcal mol⁻¹ (assistant NN-PES), which reach to the level of chemical accuracy. Overall, this accuracy is enough for our next step of dynamics calculations.

FIG. 4.

Training errors of two NN-PES models: the main one (left panel) and the assistant one (right panel). The energy of the S₀ minimum is used as the zero value of potential energy in the figure (same in the following figures).

The second step is to test the accuracy of the NN-PES model along the reaction pathway on the T₁ state of formaldehyde. Here, the IRC reaction pathway toward the H-atom dissociation reaction (the major reaction on the T₁ state) channel is constructed with electronic-structure calculations, and all involved geometries are re-calculated using the NN-PES model (Fig. 5). A similar height of the reaction barrier is given by electronic-structure calculations and the NN-PES predictions, and both of them are highly consistent with previous results obtained at accurate high-level electronic-structure calculations.⁹⁵ Overall, both the methods give very similar PES profiles along the reaction pathway, and this provides additional evidence on the accuracy of the NN-PES model.

FIG. 5.

IRC pathway. X-axis is the bond length of the broken C–H bond in the H-atom dissociation reaction.

The third step is to evaluate whether the current NN-PES model gives the reliable description near the T₁ minimum region. In some sense, the NN-PES model in this region must be correct if we wish to use it for future calculations. Starting from the same initial conditions (the lowest vibrational level of the T₁ state), both AIMD and the NN-PES MD are employed to propagate the trajectories on the T₁ surface for comparison. In this case, the ratio of ab initio calculations (R_ab) is 0.9% in the NN-PES MD, and this shows that the NN-PES model describes the T₁ minimum region well. In this set of initial conditions, no trajectory shows a reaction in NN-PES MD, and this is consistent with AIMD results. For benchmark, we examine the distribution of three key chemical bonds over all trajectories. The distribution of these critical internal coordinates is shown in Fig. 6. The NN-PES MD and AIMD give highly consistent results. This indicates that the NN-PES model gives the correct description of the PES near the T₁ minimum.

FIG. 6.

Bond length distribution of key bond lengths in NN-PES MD and AIMD. (a) Summation of two C–H bond lengths, (b) difference between two C–H bond lengths, and (c) C=O bond length.

The last validation step is to examine the reactive dynamics features on the T₁ state. After all initial conditions are generated using Wigner sampling of the lowest vibrational level at the S₀ minimum, we directly put them into the T₁ PES to run both AIMD and the NN-PES MD. In the NN-PES MD, the ratio of ab initio calculations (R_ab) is 6.3%. The H-atom dissociation probability in the NN-PES MD is 14.3%, consistent with the value of 15% in the AIMD. This basically indicates that the NN-PES model provides the reliable description on the H-atom dissociation dynamics for the current initial conditions, while the computational cost is much smaller. The dissociation probability is given at 1.5 ps in this step.

For a representative trajectory that gives H-atom dissociation, we re-compute energies of all configurations along the trajectory propagation with electronic-structure calculations. As shown in Fig. 7, two approaches predict very similar time-dependent PESs with a very minor deviation. This gives us another evidence to confirm the accuracy of the NN-PES model.

FIG. 7.

Time-dependent potential energy evolution of the representative trajectory in the NN-PES MD calculations. The energies of all snapshots are re-computed using the electronic-structure calculations. To clearly show the result, NN energies and ab initio energies are showed for only the first and last 100 fs in this trajectory before dissociation.

B. NN-PES MD simulations

When the reliability of the NN-PES model is confirmed, it is preferable to use it for the large-scaled dynamics simulations. These simulations allow us to obtain a better understanding of the T₁-driven H-atom dissociation dynamics with small computational efforts. Particularly, since the NN-PES model is available, it is rather easy to run the large-scaled quasi-classical dynamics simulation starting from different initial conditions with high efficiency. In contrast, the same calculations based on the pure AIMD may require the extremely huge computational cost that is far beyond the current computational facility.

The NN-PES MD simulation gives the H-atom dissociation probability of 16.3%, close to the dynamics with a much smaller number of trajectories.

Next, we try to discuss the role of the mode-specific excitations in the H-dissociation dynamics. Particularly, the symmetric and asymmetric C–H stretching motions, and the C=O stretching motions are considered here. Four initial vibrational levels are simulated, including (0_s, 0_a, 0_co), (1_s, 0_a, 0_co), (0_s, 1_a, 0_co), and (0_s, 0_a, 1_co) at the S₀ minimum, in which the labels of s and a represent the symmetric and asymmetric C–H stretching motions and co represents the C=O stretching motions, respectively. The action-angle sampling approach is taken here because it is more easily applied in the case with vibrationally excited levels.

Starting from the lowest vibrational level (0_s, 0_a, 0_co), the H-atom dissociation probability is around 11.6%, which is smaller than the result starting from initial conditions generated by the Wigner sampling near the S₀ minimum. The deviation can be understood by the different energy distribution of two initial conditions given by Wigner and action-angle samplings. As shown in Fig. 8, the Wigner sampling gives a broader distribution in the energy domain, i.e., more samples appear in both low-energy and high-energy domains. Since the quasi-classical dynamics employs the classical Hamilton equation to propagate the nuclear motions, only the trajectories with enough initial energy may overcome the barrier on the T₁ state and move toward the H-atom dissociation channel. In the Wigner sampling, more trajectories start from the high-energy domains, and this results in the larger H-atom dissociation probability.

FIG. 8.

Total energy distribution in two sets of initial conditions sampled by the Wigner and action-angle approaches.

Next, let us consider the other initial cases with mode-specific excitations. Starting from the (1_s, 0_a, 0_co), (0_s, 1_a, 0_co), and (0_s, 0_a, 1_co) cases, the H-atom dissociation probability reaches to around 18.5%, 14.5%, and 31.7%, respectively. This means that the one-quantum vibrational excitations of these vibrational modes significantly improve the H-atom dissociation channel. At the first glance, it is easy to figure out that the excitations of the symmetrical and asymmetrical C–H modes may enhance the H-atom dissociation probability since more energies are added in the reaction coordinate, while it is not fully transparent to understand the dependence of this channel on the excitations on the C=O stretching motions.

In fact, the impact of the mode-specific vibrational excitations on the H-atom dissociation channel is simply attributed by different initial energies in the quasi-classical dynamics simulations. When the action-angle sampling is taken to define all initial conditions, their corresponding energy distribution is displayed in Fig. 9. Please note that here the potential energy part is defined by the T₁ potential energy obtained at each sampled geometry. Here, only the high-energy tails of all initial conditions may contribute to the H-atom dissociation probability, because only the trajectories starting from the high-energy domains can go over the barrier on the T₁ state. If we closely examine the high-energy part of the energy distribution in Fig. 9, the ratio of the high-energy samples rises in the following order: (0_s, 0_a, 0_co) < (0_s, 1_a, 0_co) < (1_s, 0_a, 0_co) < (0_s, 0_a, 1_co), when we consider the cutoff energy as the barrier height. This tendency clearly explains that the H-atom dissociation probability increases according to the same order.

FIG. 9.

Total energy distribution of different initial conditions, including (0_s, 0_a, 0_co), (1_s, 0_a, 0_co), (0_s, 1_a, 0_co), and (0_s, 0_a, 1_co) cases.

We wish to point out that the classical vibrational energy transfer is generally much faster. Thus, in the current system, no matter which modes are excited, the additional energies may quickly flow to the reaction coordinate, and this simply drives the H-atom dissociation dynamics. When more additional energies are added, the H-atom dissociation probability increases as well. Such ultrafast vibrational energy transfer is the outcoming of the classical dynamics simulations, while the same feature may not be observed in the full quantum dynamics.

IV. DISCUSSION

In this work, the dissociation of formaldehyde on the T₁ state is studied by the quasi-classical dynamics simulations on the NN-PES model, instead of building an analytical model of the PES. The analytical PES model has been widely used in the field of quantum dynamics for decades.^136–138 Normally, the analytical PES function model is assumed according to physical inspirations, and all parameters inside this model are fitted by using ab initio data. This way has many advantages, for instance, the physical insight can be directly viewed by the function form; the asymptotical limit may be satisfied. Therefore, this approach received great success in the treatment of the reaction of small molecular systems. However, when the degrees of freedom increase, it is rather difficult to write down the proper analytical function to include all types of interaction terms for high-dimensional systems. In contrast, the ML model, such as the NN-PES model, provides an extremely flexible function form that serves as a strong fitting tool to represent the dependence of potential energies on the geometries. With proper training and test protocol, it provides a reasonable PES model for complex systems. However, the problem is that such an NN-PES model may not provide enough information of physical insight. In addition, as the simple FFNN model is an interpolation method, the NN-PES model used here may not perform correctly in the out-of-sampling region, and thus, it cannot give the correct asymptotical limit.

Our purpose is to build a reasonable NN-PES that can be used to examine the T₁-driven dynamics of formaldehyde. The first sets of trajectories in the quasi-classical dynamics are only used for sampling purpose. The generation of the dataset by using such quasi-classical dynamics itself may be very expensive. However, after the construction of such an NN-PES, we may run further quasi-classical dynamics easily. This approach has several advantages. If we examine the computational work in the reactive scattering dynamics field, many studies of the quasi-classical dynamics on the analytical PES employed a huge number of trajectories (10 000 even more) to achieve convergence.^139,140 However, the normal on-the-fly simulation cannot reach such number of trajectories due to high computational cost. A large number of trajectories are also necessary, when the dissociation probability is rather low. For instance, in this work, the dynamics from the initial conditions created by vertically placing the lowest vibrational state of the S₀ state to the T₁ state only gives around 11.6% of reactive trajectories. With more trajectories, more precise reactive probability can be achieved. In the NN-PES construction process, totally about 800 k ab initio calculations (813 973) are performed and finally 130 990 geometries are taken to build the NN-PES model. In this case, when 1000 trajectories are simulated for each initial condition (corresponding to a vibrational level), we may not save the considerable computational cost in the NN-PES MD, with respect to the pure AIMD. The efficiency of the former approach may be more pronounced when a much larger number of trajectories are considered. In this work, different initial conditions refer to different vibrational levels or different temperatures. For a chosen initial condition with a fixed vibrational level or a fixed temperature, we may perform sampling (by Wigner distribution or action-angle) to generate a set of coordinates and momenta. This defines the initial conditions of the quasi-classical dynamics.

No matter how many geometries are included in the NN-PES model training step, we never can make sure that all dynamical relevant regions are covered by these data due to below reasons. The Wigner distribution of the lowest vibrational level is a multi-dimensional Gaussian function. When more and more data are sampled, it may always be possible that the sampled geometries/velocities are far away from the distribution center. If so, this initial condition may give a trajectory propagating beyond the confidence interval regions. In addition, the current dynamics is the nonlinear dynamics due to the existence of anharmonicity. This means that two slightly different initial conditions may give totally different trajectory propagations. Thus, we always find that some trajectories may enter the out-of-sampling region with the trajectory propagation, even after more and more points are added to the training dataset. Certainly, if such a trajectory re-enters the confidence interval regions, we use the NN-PES model again. Both previous work^50,58 and our work used an iterative training and discuss about the out-of-confidence problems. Some implementation details are different. In this reference work,⁵⁰ several properties predicted by two different NNs are compared (which is called multiple-fit validation^50,72,128). If the outcomes of both NNs are not sufficiently similar, the nuclear configuration was recomputed with quantum chemistry and added into the training set. In our current detection of the out-of-sampling region, we apply multiple-fit validation on the potential energy and use the additional criteria defined by confidence interval of input features. To solve these problems, we switch back to the electronic-structure calculations if the out-of-sampling geometries are found in the trajectory propagation. Here, we wish to build a reliable NN-PES model with reasonable computational cost in the NN-PES model construction step. Thus, when a set of trajectories show only less than 10% of geometries falling in out-of-sampling regions, we stop the iterative procedure in the NN-PES construction. In all large-scaled trajectory calculations, we switch back to the density functional theory (DFT) calculations when the trajectory enters the out-of-sampling regions. Since the NN-PES model is accurate enough, for each initial condition, the ratio of ab initio calculations (R_ab) is less than 9.8%. This still largely saves the computational cost compared with the pure on-the-fly calculations.

In our current ML approaches, we only include the potential energies in our NN model construction. Therefore, a large number of configurations are used in the training dataset. The inclusion of the gradient information in the NN model construction may largely reduce the number of configurations in the NN model training, as demonstrated by recent work.^50,63,141 However, our purpose here is to build a reliable NN-PES model, and this purpose is achieved here when we examine the accuracy of the current PES model. In this sense, the current manner of building the NN-PES model is acceptable. In addition, the current approach provides some useful ideas on the NN-PES construction, when the analytical gradients are still not available in the employed electronic structure methods, such as quantum Monte Carlo. Certainly, the inclusion of the gradient information is important, and this will be the research interest of our future work.

V. SUMMARY

The NN-PES model is built within the BP model and wACSF descriptors. The iterative approach is taken in the NN-PES construction, and in each iteration, new points are added to the training dataset to build a new NN-PES model. Several standard technical tricks are employed in this work to improve the performance of the NN-PES model. In the setup of the training dataset, the clustering approach is taken to process sampled data points, giving the training data points equally distributed in the configuration space relevant to the dynamics propagation. This way allows the NN-PES model to show the similar prediction ability in all dynamically relevant PES regions. To detect the out-of-sampling regions in the dynamics, various standard approaches are employed. For example, two NN-PES models are built. The first one is used for the dynamics simulations, and the second one is taken for the validation of the results provided by the first network. When two NN-PES models give different energies, we assume that the out-of-sampling regions are reached. In addition, the confidence interval of the overall distribution of the training dataset is defined from the statistical view. When the geometry during the dynamics propagation is not located within such an interval, we assume that the trajectory enters the out-of-sampling regions. All these tricks significantly improve the performance of the NN-PES model.

The reliability of the NN potential energy surface is examined by several benchmark calculations against ab initio data. Both NN-PES and electronic-structure calculations give the similar H-atom dissociation pathway. Starting from the initial nuclear geometries and velocities sampled near the T₁ and S₀ minimum, the trial dynamics simulations based on NN-PES and ab initio PES give consistent results. This confirms the reliability of the NN-PES model.

Based on the well-established NN-PES model, a large number of trajectories can be efficiently calculated in the quasi-classical dynamics. Particularly, the influence of the mode-specific excitations on the H-atom dissociation dynamics of formaldehyde is explored, which include (0_s, 0_a, 0_co), (1_s, 0_a, 0_co), (0_s, 1_a, 0_co), and (0_s, 0_a, 1_co) initial conditions. The result shows that the excitations of symmetric C–H stretching, asymmetric C–H stretching, and C=O stretching motions always improve the dissociation probability, while the excitations of symmetric C–H stretching motions give a stronger impact on the enhancement of the dissociation channel than asymmetric C–H stretching motions. In fact, all the above large-scale dynamics simulations may require a huge amount of computational cost, when the pure AIMD is taken. In addition, after the construction of the NN-PES model, the dynamics starting from different initial conditions can also be simulated with high efficiency. In this sense, the employment of the NN-PES model here can significantly reduce the computational cost with respect to AIMD.

Overall, we apply the NN-PES MD approach in exploration of an important chemical reaction relevant to environmental science, i.e., the H-atom dissociation reaction of formaldehyde on its T₁ PES. There are mainly two key outcomes in our current work: (1) We built an accurate NN-PES for the description of the H-atom dissociation reaction of formaldehyde on its T₁ PES. Several technical tricks employed here may provide useful guidelines on similar research studies. (2) With a reliable NN-PES model, we study a few of important dynamical features in the T₁-driven H-atom dissociation of formaldehyde. For instance, the dependence of the H-atom dissociation on the mode-specific vibrational excitations is addressed. This finding may provide some useful ideas to invoke the further theoretical and experimental work in formaldehyde. In summary, this work demonstrates that the employment of the NN-PES model in the quasi-classical dynamics is an effective tool to simulate the reactive dynamics of the polyatomic system, particularly the VOCs. This opens many interesting new research topics in the future.

APPENDIX A: DETAILS OF CONSTRUCTION AND TRAINING

For inputs of the NN-PES, three radial wACSFs and 12 angular wACSFs are used to describe the molecular features of each atom, and all relevant parameters are shown in Tables I and II.

TABLE I.

Parameters in radial symmetry functions.

Function number	η	r s
1	0.8	0.5
2	0.8	1.75
3	0.8	3.0

TABLE II.

Parameters in angular symmetry functions.

Function number	ζ	λ	μ s
1	1.0	1.0	0.50
2	1.0	1.0	1.00
3	1.0	1.0	1.50
4	1.0	1.0	2.00
5	1.0	1.0	2.50
6	1.0	1.0	3.00
7	1.0	−1.0	0.50
8	1.0	−1.0	1.00
9	1.0	−1.0	1.50
10	1.0	−1.0	2.00
11	1.0	−1.0	2.50
12	1.0	−1.0	3.00

The main NN-PES and assistant NN-PES model are trained individually. In this work, the structures of FFNNs for each type of atom are the same, and the hyper-parameters in our current work are determined via tests. In practices, we start from using FFNNs with very simple structures, and then, we increase the number of hidden layers, neurons per hidden layers, and the size of mini-batches, until the reliable NN-PES models are obtained. Several parameters, such as different learning rates and various activate functions [Rectified Linear Unit (ReLU) function, hyperbolic tangent (tanh) function, and Sigmoid function] are tested. We finally choose Sigmoid function for its best performance. Since our NN model gives a reasonable description of the training and testing data, we considered that the NN architectures are suitable. The details of the NN properties are shown in Table III.

TABLE III.

Details of the NN properties.

Property	Main network	Assistant network
Number of hidden layers	3	3
Number of neurons per hidden layer (m)	500	510
Activate function	Sigmoid	Sigmoid
Optimizer	Adam	Adam
Learning rate	0.000 01	0.000 01
Mini-batch size	204 8	204 8

In the NN training tasks, the ADAM optimizer¹⁴² is employed, and the Root Mean Square Error (RMSE) is used as the loss function during the training process, as given by

$$RMSE(E)=\sqrt{\frac{1}{N}\sum _i^N{(E_i^{\mathrm{ref}}-E_i^{nn})}^2},$$ (A1)

where i is the index of the configuration in the training (or validation) dataset and N is the total number of configurations. For configuration i, $E_i^{\mathrm{ref}}$ and $E_i^{nn}$ are its energy calculated by ab initio calculation and NN-PES.

In the training process, all hyper-parameters and the input symmetry functions are initialized by the methods suggested by Behler and Stende.^128,143 Our NN training and prediction programs are built based on Google’s open-source toolkit Tensorflow.¹⁴⁴

The raw training data are initially generated by the AIMD simulation.

In the very beginning, the preliminary PES model is built by running the ground state ab initio molecular dynamics (AIMD) with different nuclear initial conditions sampled by the Wigner distribution function (at the lowest vibrational level of the S₀ minimum). The AIMD run lasts 2.0 ps, and the time step is 0.5 fs. The simulation is terminated if any bond length increases to above 2.5 Å, which indicates that the corresponding chemical bond is broken. This step totally generates 352 931 points in the electronic-structure calculations, and then, these points will be refined using the clustering algorithm, which was introduced in previous work.⁴⁵ After the clustering, 109 075 geometries are picked up to build the preliminary NN-PES model.

Next, the iterative process mentioned in Appendix A is conducted, which gets converged very quickly, within only three loops. In all three loops, both AIMD and NN-PES MD are run up to 1.5 ps:

• (1)In the first iterative step, 100 initial nuclear samples (coordinates and velocities) are generated by Wigner distribution (at the lowest vibrational level of the S₀ minimum). Both AIMD and NN-PES based dynamics are run. The out-of-sampling data points in the NN-PES MD are 13.6% (35 148 points) that are treated by the DFT calculations (R_ab is 13.6%), and these points are all collected and added to the training set.

  To extend the area of training data in the configuration space, we also run AIMD simulations with 100 initial conditions sampled by the Wigner distribution function (at the lowest vibrational level of the T₁ minimum). All geometries are collected as well. After removing few noise structures and the clustering, 105 148 geometries are used to rebuild training data.

• (2)In the second iterative step, 100 NN-PES MD trajectories are run again from the lowest vibrational level of the S₀ minimum. The out-of-sampling data points in the NN-PES MD are 8.4% (22 019 points) that are treated by the DFT calculations (R_ab is 8.4%), and these points are all collected and added to the training set.

  We also run 100 NN-PES MD trajectories run from the lowest vibrational level of the T₁ minimum. The out-of-sampling data points in the NN-PES MD are 1.3% (3875 points) that are treated by the DFT calculations (R_ab is 1.3%), and these points are all collected and added to the training set. After removing few noise structures, 130 990 geometries are used to rebuild training data.

• (3)In the third iterative step, 100 NN-PES MD trajectories are run from the lowest vibrational level of the S₀ minimum. The out-of-sampling data points in the NN-PES MD are only 6.3% (16 758) that are treated by the DFT calculations (R_ab is 6.3%). The NN-PES MD and AIMD basically get the similar dissociation probability. Thus, the iterative process converges in this step, which proves that our NN-PES model gives a reliable description in the region near to the S₀ minimum.

  To further validate the accuracy of the converged NN-PES model, both AIMD and the NN-PES MD are employed to propagate the trajectories on the T₁ surface, starting from the lowest vibrational level of the T₁ minimum. Then, the distribution of three chemical bond lengths, which is key information during the propagation, is compared. Both the models give highly consistent results.

At the end, totally 130 990 ab initio data points are included in the training data to build the PES model. The details of initial conditions are shown in Table IV.

TABLE IV.

Initial conditions in the construction of NNPES.

Step	References	Method	Sampling conditions	Numbers
Building preliminary training set	S0 minimum	Wigner	300 K	100
First iteration	S0 minimum	Wigner	Lowest vibrational level	100
Second iteration
Third iteration
First iteration	T1 minimum	Wigner	Lowest vibrational level	100
Second iteration
Third iteration

All ab initio calculations are processed with Gaussian16 software.¹⁴⁵ The S₀ minimum of formaldehyde is optimized at the B3LYP/DEF2-QZVP level, and the T₁ minimum is obtained at the UB3LYP/DEF2-QZVP level. The AIMD calculations are based on the electronic-structure calculation at the UB3LYP/6-31G(d) level. There are two ways to treat the lowest triplet state T₁ within the framework of DFT. We may start from the un-restricted Kohn–Sham equation, give the total multiplicity of the system, and run simple DFT calculations. In this case, the T₁ state is basically the ground state when we set the multiplicity to three. The second approach is that we start from the standard DFT of the single state (S₀) and try to perform the spin–flip excitation to get the linear response transition of the triplet state (T₁). The first approach is widely used. It is possible to discuss that the lowest triplet state T₁-driven dynamics of the current system employed such an approach. We also wish to point that some work also used unrestricted coupled-cluster method using both single and double excitations (UCCSD) to treat a similar problem.^86,109,121 In this sense, we use UB3LYP here to treat the T₁ state here.

APPENDIX B: CONFIDENCE INTERVAL

In statistical theory, it is possible to define the confidence interval of the descriptors by checking the distribution of input features. As different molecular descriptors may be converted to each other by the nonlinear transformations, the shape of the sampling region should be also changed substantially under these transformations. For completeness, it is highly recommended to select different sets of molecular descriptors to represent the distribution of the training dataset. The procedure to calculate the confidence interval is given as below.

Two descriptors are considered to detect the out-of-sampling region in the AIMD run. The first one is composed of all elements of symmetry functions G_i. The second set of molecular descriptors is composed of all interatomic distances r_ij.

Let us assume that an adjustable ratio k% is set for each dimension of the chosen molecular descriptors, such as G_i. After sorting all the values of G_i from the smallest to the largest, two values, G_L and G_H, are selected to define an interval of [G_L, G_H], which satisfy that both [G_min, G_L] and [G_H, G_max] contain k% of the G_i data number in G_i ∈ [G_min, G_max]. Next, we go through all the data points (with the total number N_tot) in the whole training dataset and examine whether each individual data point is inside [G_L, G_H] or not. The numbers of the data points inside and outside the [G_L, G_H] interval are recorded as N_in and N_out (N_in + N_out = N_t), respectively. After going over all dimensions, the total number of the data points outside all intervals is given as $N_{out}^{tot}$. Please note that $N_{out}^{tot}$ does not correspond to the simple summation of all N_out, because one data point may be located out of the pre-defined intervals in several dimensions at the same time. Following the above procedure, the ratio of data outside all intervals is simply $N_{out}^{tot}$/N_t. Each k% value should correspond to its own ratio $N_{out}^{tot}$/N_t. Given such a pre-defined ratio, an iterative procedure can be employed to determine the k% value. In current work, 3% is taken, and this process can be easily generated by the Scikit Learn package.¹⁴⁶

APPENDIX C: DETAILS OF NN-PES MD SIMULATIONS

After the reliability of the NN-PES model is confirmed, we run a large number (1000) of trajectories to explore the T₁-driven reaction dynamics of formaldehyde. The initial coordinates and velocities are also generated by the Wigner sampling of the lowest vibrational level on the S₀ minimum.

Next, the impact of the mode-specific excitations on the (anti-)symmetric C–H stretching motions and C=O stretching motions on the reaction processes are explored. Here, four initial levels (0_s, 0_a, 0_co), (1_s, 0_a, 0_co), (0_s, 1_a, 0_co), and (0_s, 0_a, 1_co) at the S₀ minimum are considered, and the action-angle sampling is conducted to generate initial coordinates and velocities.

For all the above cases, the initial nuclear densities are vertically placed in the T₁ surface to give the initial conditions. In the NN-PES MD simulations, trajectories are propagated up to 2.0 ps with a time step of 0.5 fs. The trajectory simulation is terminated if any bond length is larger than 2.5 Å. When the trajectory enters the out-of-sampling regions, we switch back to electronic-structure calculations. Certainly, the number of the data points obtained from electronic-structure calculations should be small enough, if the well-established NN-PES model is used in the final quasi-classical dynamics. At the end, the dissociation probability is given at 2.0 ps for all initial conditions.

All the initial conditions of NN-PES MD simulations are shown in Tabel V. Here, we use the high-temperature cases in the initial sampling. This effectively broadens the distribution width of the sampled coordinates and momenta in the phase space. In this way, the initial conditions sampled by the Wigner distribution at 0 K should be covered by the training data. For action angle sampling in the (0_s, 0_a, 0_co) situation, we may more-or-less make sure that initial conditions maybe fall into the sampling regions of the training data, by considering that the Wigner distribution function is more de-local in the phase space. For other situations, the inclusion of the high-temperature sampling in the phase space should be very helpful. At the end, all initial conditions are located in the region of the confidence interval, and this feature is confirmed by the examination of out-of-sampling regions in the dynamics calculations. Starting from some initial coordinates and velocities, the trajectory may enter the out-of-sampling regions. In this case, we switch back to the DFT calculations in the large-scale dynamics work. By doing so, we do not need to worry that the NN-PES model cannot cover all dynamical-relevant regions.

TABLE V.

Initial conditions of large-scale NN-PES MD simulation. 0_s, 0_a, 1_s, and 1_a represent the vibrational state of C–H (anti-)symmetric stretching mode. 0_co and 1_co represent the C=O stretching mode.

Group	References	Method	Sampling conditions	Numbers
1	S0 minimum	Wigner	Lowest vibrational level	1000
2	S0 minimum	Action-angle	(0 s, 0a, 0co)	1000
3	S0 minimum	Action-angle	(1s, 0a, 0co)	1000
4	S0 minimum	Action-angle	(0s, 1a, 0co)	1000
5	S0 minimum	Action-angle	(0s, 0a, 1co)	1000

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts of interest to declare.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding authors upon reasonable request. The code and dataset in this work are available from the website: https://github.com/Loadstarc/scmd