A total of 33 stable trajectories of open-shell singlet state were selected from a non-adiabatic trajectory surface-hopping dynamics simulation from our previous work. The B3LYP/6-31G* level of the method (Reiher et al., 2001; Salomon et al., 2002) was used to calculate the ESP atomic charge of each structure in the singlet, triplet, and quintuplet state. We finally achieved 39,368 converged structures owing to the convergence of the calculation. The data preparation was time consuming. By and large, it took 2 weeks to complete all the calculations of the 39,368 structures for each spin state with four computer nodes; each node had dual Intel 2683v3 CPUs. All the electronic structure calculations were implemented with a Gaussian 16 package (Frisch et al., 2016), and the detail charge distribution of each atom in the different spin states were analyzed and shown in the section 3.

The raw dataset was preprocessed firstly to extract appropriate features, such as the descriptors of structures and input of model. Specifically, each RFR model was constructed separately under certain spin states for each atom according to the flow shown in Figure 2. Since there were 14 atoms in the simplified heme model, 14 independent RFR models were constructed by training for each spin state. There were 42 models in total.

Let D = {(x₁, y₁), ⋯⋯, (x_N, y_N)} denote the training data, with N = 39368/2, xi=(xi,1,⋯⋯xi,p)Trepresenting the information relative to atom i in each structure described with p features, and y_i denoting the ESP charge. During the training process, for each decision tree in the forest, a bootstrap sample D_j from the training data of N molecules was drawn first. Starting with all observations (x₁, y₁)⋯⋯(x_N, y_N), of D_j at each node, m predictors were selected at random from the p predictors (m < p), and the node was split into two descendant nodes using the best split among the remaining predictors. This process was repeated until no further splits ere possible to grow a tree, and the steps were repeated again until all the trees were grown.

Although Random Forests can obtain good results using the default parameters in most cases, appropriate parameters can further improve the accuracy for particular situations. There is only one parameter to which random forests is somewhat sensitive—m. This denotes the number of randomly selected predictor variables at each node. The default value of m is often set by p/3. In the RFR model, combined with symmetry functions, different values of m were tested, and, finally, m = 5 was determined by comparing the Pearson correlation coefficient (r) between the predicted charges and the ESP charges of Fe²⁺. Another parameter, B, which represents the number of trees in the forest, can be chosen to be as large as desired; Breiman (2001) showed the generalization error for random forests converges almost surely to a limit as B increases. Here, B was set as 200.

When the training is completed, the prediction charge of a given atom i in a new geometry structure will be given by the average prediction of all individual trees. Thus, the predicted charge is assigned as Equation (1):

The standard deviation of the predicted charge for atom i by the tree T is defined as Equation 2:

where q_i(T_j) is the partial charge predicted by tree T_j. The RFR algorithm was implemented using the scikit-learn module in Python.

To encode the physical features and the mandatory symmetries of the problem, many descriptors have been introduced (Imbalzano et al., 2018). For example, Huan et al. (2017) utilized a d-dimensional vector V_{i, α}, representing the atomic environment of atom i viewed along the Cartesian α direction (Huan et al., 2017). Heid et al. (2019) used the type of each atom and its connectivity as the input for the neural network. Schutt et al. (2017) introduced a vector of nuclear charges and a matrix of atomic distances to describe the molecular structures. In addition, molecules can be represented as Coulomb matrices (Rupp et al., 2012; Lilienfeld, 2015), scattering transforms (Hansen et al., 2015), bags of bonds (Bartók et al., 2010; Bartók et al., 2013), and so on. Among these various descriptors, atomic-based symmetric function, which was first proposed by Behler et al. (2007), has been widely used in machine learning (Behler et al., 2007; Behler, 2011a,b). Here, we adopted this method to describe the molecular structure.

Atom-based symmetric functions describe the chemical environment of atom i in terms of radial and angular terms. Therefore, each atom's Cartesian coordinate R_i = (x_i, y_i, z_i) needs to be converted into the so-called symmetric function form of Equation (3):

where Gi, E1radial represents the total contribution of the distance between all the surrounding atoms, and atom i, and Gi,Ei,Ejangular represents the angular relationship between any two surrounding atoms and itself. All atoms are distinguished according to their element E_i, and the set of symmetric functions of two atoms belonging to the same element are thus the same.

In this study, Equation (4) was used to describe the distance component of each atom, where R_ij represents the distance between atom i and j. The cutoff function f_c(R_ij) was introduced in Equation 5 because the atoms in the molecular dynamic simulation may enter or leave the cutoff distance, which can lead to the number of neighbor atoms to be variable. Here, R_c was thus set to 99 Å to include all the atoms, and R_s and η were both set to 1.0.

Equation (6) is the angular component, which defines the angular distribution centered on each reference atom; here, λ = 1.0, ζ = 1.0.

Therefore, through coordinate transformation, the symmetric functions for each atom can be obtained and combined with the ESP charge to finally form the training set as the input of model.

Meanwhile, in order to compare the effect of descriptor selection on prediction performance, 11 structural parameters were manually selected and used as descriptors to train the model. Specifically, the 11 parameters included eight distance values (Fe-N1, Fe-N2, Fe-N3, Fe-N4, Fe-N11, Fe-O12, Fe-O13, and O12-O13), one angle value (Fe-O12-O13), and two dihedral angles (N2-Fe-N1-C10 and N1-Fe-N2-C5). According to our chemical perception, these 11 parameters reflect the features of molecule structure, so they can well-describe different conformations.

It can be seen in Figure 3 that most variations range from 0.5 to 0.7e; the fluctuation of Fe²⁺ was the most significant, as it was close to 2e. The variation of O12 was larger than that of O13. It can also be found that there was a slight tendency for the mean value of N to decrease and the mean value of C to increase. For Fe and the coordinating O12 and O13, the difference among the mean values under different spin states was relatively more significant. Specifically, the atomic charge of Fe²⁺ in the singlet state was distributed around 1.2 and 1.5e in quintuplet. Further analysis of the charge distribution of different spin states showed that the triplet charge of Fe²⁺ in most structures was greater than the singlet charge (Δ31 > 0, see Figure 4), with the difference being at the highest probability concentrated at 0.1e, while, for the quintuplet and triplet spin state, the difference reached 0.2e. The results confirmed that different spin states in the same structure had distinct charge distributions.

Additionally, it should be noted that the atomic charge of each atom fluctuated within a certain range, among which Fe²⁺ fluctuated the most. Just taking the singlet state as an example, the variation ranged from 0.3 to 2.2e, which implied that the charge distribution of a certain atom in a specific spin state was conformation dependent.

In order to better distinguish between different molecular structures, the atom-based symmetry functions were used to convert atomic coordinates into a series of function values, which embed the atoms in their neighborhood depending on the element type (Schutt et al., 2017). It is an efficient way to consider the chemical environments that the invariances, such as translation, rotation, and permutation, can be guaranteed to be exploited by. By doing so, the RFR model combined with symmetry functions and ESP charge was constructed.

As mentioned above, although complex parameter tuning is not required in the RFR model, it is sensitive to the number of descriptors. To this end, we tested and compared the predicted charge of Fe²⁺ at different values of m (i.e., the number of features selected from p descriptors at random; here p = 19) and then calculated the correlation between the predictions and the ESP charges. The results are shown in Table 1. It can be seen from Table 1 that, when m = 5, the correlation between the predicted value and the fitted value is the largest (0.9784), which indicated that prediction gave the best performance. The parameter m was consequently set to five in the subsequent analysis.

To assess the prediction performance of the charge models, the mean absolute error (MAE) was calculated for each atom in the three spin states, and the standard deviation of the error was given as well (Table 2). According to Table 2, the MAEs of the predicted charges in three spin states are all within 0.015e for all the spin states. There was no obvious difference between two states. For each state, most of the MAEs of the atoms were within 0.02e as well, except for Fe²⁺, which reached a maximum of 0.047e. Moreover, the Pearson correlation coefficient (r) between the predicted charges and the ESP charges of the RFR model in all three states was above 0.96. These data demonstrated that the model had high prediction accuracy, especially for N1 and N2. At the same time, the MAE and error standard deviation were close in the three states, indicating that our RFR model had good stability.

For clarity, we further selected three atoms—Fe²⁺, N11, and O12—to plot their charge distributions for comparison (Figure 5). As shown in Figure 5, the predicted charges of the RFR model are basically gathered around the straight line y = x; they were very close to the high-precision charges calculated by DFT, indicating that our model achieved satisfactory accuracy.

By carefully comparing the distribution of each atom in different spin states, it can be found that the predicted values of Fe²⁺ have a good aggregation and few scattered points. However, the predictions are larger when the corresponding ESP charges are <1.5e and smaller when they are above 1.5e. The aggregation centers in the three states were different but essentially distributed in 0.5–2.25e, which is consistent with the analysis in Figure 5. For N11 atom, the predictions were more concentrated in the singlet and triplet, as there existed a few scattered points in the quintuplet. For O12 atoms, the correlation coefficients in all three spin states exceeded 0.98, and although there were some scattered points, the distribution was uniform. Finally, by comparing the distribution of different atoms in the same spin state, it can be found that the predictions in the triplet state were more concentrated overall. In summary, it was demonstrated that our model can predict the atomic charge of most structures well.

To compare the performance of the RFR models with different descriptors, we manually screened 11 parameters to describe the molecular structure, including eight bond lengths (Fe-N1,Fe-N2, Fe-N3, Fe-N4, Fe-N11, Fe-O12, Fe-O13, and O12-O13), one bond angle (Fe-O12-O13), and two dihedrals angles (N2-Fe-N1-C10 and N1-Fe-N2-C5). The same process and method were used for RFR model training and prediction. A comparison of the prediction performance of selected atoms (Fe²⁺, N11, and O12) is shown in Figure 6, and, for comparison, the root mean square error RMSE of the prediction of each atom in different spin states was further calculated and is shown in Figure 7.

It can be seen clearly from Figure 6 that both models have good prediction performances, and the same model has a similar RMSE of predictions for different spin states. When 11 structural parameters were used as descriptors, however, the prediction values were more concentrated, and the model prediction performance was better than with in the case of symmetry functions. The average RMSE and the RMSE of each atom were reduced. Among these, the RMSE of Fe²⁺ reduced from 0.07 to 0.0035e, which is a maximum 0.06e improvement. We think that this is partially due to the use of a dihedral angle as the descriptor, which is a four-body term and is not included in the symmetry function.

In conclusion, choosing different descriptors will affect the prediction performance of the RFR model; the 11 manually selected parameters can better describe the molecular structure and thus achieved better results. At the same time, however, it should be noted that the difference between the two cases is not significant. As shown in Figure 7, the RMSE of Fe²⁺ is relatively larger, but its fluctuations are still below 0.04e, and the variations of RMSE for other atoms are all below 0.02e. This indicates that, even if there is no empirical experience involved, the RFR model with symmetry functions can achieve satisfactory predictions, and the advantage is that it can be automatized.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.