Development of Prediction Models for the Self-Accelerating Decomposition Temperature of Organic Peroxides

Abstract

Thermal risk assessment is very important in the primary stages of chemical compound development. In this study, a model to estimate the self-accelerated decomposition temperature of organic peroxides was developed. The structural information of compounds was used to calculate descriptors, on which partial least-squares (PLS) regression and support vector regression were applied for temperature prediction. Molecular mechanics and density functional theory calculations were performed before descriptor calculations, for structure optimization, using a genetic algorithm for variable selection. Structure optimization and variable selection immensely improved the prediction accuracy. Thus, a PLS model, with R² = 0.95, root mean square error = 5.1 °C, and mean absolute error = 4.0 °C, exhibiting higher accuracy than existing self-accelerating decomposition temperature prediction models, was constructed.

Introduction

Thermal risk assessment is extremely crucial in the development of chemical compounds. Self-accelerating decomposition temperature (SADT) is a key parameter characterizing the thermal risk of organic peroxides. It is the lowest temperature for self-accelerating decomposition in organic peroxides and self-reactive substances (used in transportation packaging). Thus, it determines the optimum temperature-control to avoid thermal hazards during material storage and transport.¹ Several experimental methods measure SADT for thermal risk assessment; however, the associated cost, risk, and chemicals make early-stage evaluation very difficult. Therefore, it is beneficial to develop a simple and high-accuracy SADT prediction method.

Previous studies have proposed quantitative structure–property relationship models to predict SADT. Wang et al.² performed density functional theory (DFT) calculations [6-31G(d)/B3LYP] in Gaussian 09 to obtain descriptors. Geometrical descriptors (bond length, bond angle, and dipole moment) and quantum chemical descriptors [highest occupied molecular orbital (HOMO)/ lowest unoccupied MO (LUMO), bond dissociation energy] were used to construct prediction models with multiple linear regression (MLR) and support vector regression (SVR) to estimate SADT. He et al.³ used the semiempirical molecular orbital technique (AM1) for preprocessing before geometry optimization and frequency calculations. Descriptors (excluding quantum chemical descriptors) were calculated in DRAGON 6.0, and the genetic algorithm (GA) was applied for variable selection, followed by MLR and SVR construction, to estimate SADT. The first and second methods suffered from the limitations of high computational load and low accuracy, respectively.

In this study, a model with high accuracy and low calculation load was developed to estimate the SADT of organic peroxides. Descriptors were calculated using the optimal molecular conformation, determined by molecular mechanics (MM) and DFT calculations. GA was used for variable selection, followed by the application of partial least-squares (PLS) regression, as a linear regression method, and SVR, as a nonlinear regression method, to predict SADT. Prediction models, including and excluding structural optimization and variable selection, were developed and analyzed.

Methods

Data Set

The data set included 65 organic compounds, with 90.14–571.00 molecular weights, and −5.0 to 196.5 °C SADTs, obtained from the literature,²⁻⁴ determined using different calorimetric methods (TG-DSC and C80). However, as previously reported, SADT is independent of the determination method used.⁴ Compounds included commonly used organic peroxides, such as dialkyl peroxide, diacyl peroxide, hydroperoxide, peroxyester, ketone peroxide, peroxy carbonate, and diperoxide. Organic peroxides used here, and their experimental SADTs, are listed in Table 1. The data set was divided into two subsets: a training set (52 samples) and a test set (13 samples), following the method reported in a previous publication.³

Table 1. Compounds and Their Experimental SADTs.

no.	compound name	CAS no.	MW	SADT [°C]
1	tert-butyl hydroperoxide	75-91-2	90.14	120.4
2	cumyl hydroperoxide	80-15-9	152.21	79.0
3	dicumyl peroxide	80-43-3	270.40	77.8
4	p-menthane hydroperoxide	80-47-7	172.30	73.5
5	dibenzoyl peroxide	94-36-0	242.24	80.0
6	diisopropyl peroxydicarbonate	105-64-6	206.22	5.0
7	tert-butyl peroxyacetate	107-71-1	132.18	65.0
8	tert-butyl peroxyisobutyrate	109-13-7	160.24	30.0
9	di-tertbutyl peroxide	110-05-4	146.26	80.9
10	diacetyl peroxide	110-22-5	118.10	35.0
11	disuccinic acid peroxide	123-23-9	234.18	25.0
12	bis(2,4-dichlorobenzoyl)peroxide	133-14-2	380.00	60.0
13	tert-butyl peroxybenzoate	614-45-9	194.25	65.8
14	bis(1-oxononyl)peroxide	762-13-0	314.52	20.0
15	butyl 4,4-bis(tert-butylperoxy)pentanoate	995-33-5	334.51	55.0
16	2,5-bis-(t-butylperoxy)-2,5-dimethyl-3-hexyne	1068-27-5	286.46	84.8
17	methyl ethyl ketone peroxide	1338-23-4	210.26	60.0
18	dicyclohexyl peroxydicarbonate	1561-49-5	286.36	25.0
19	2,3-dimethyl-2,3-diphenylbutane	1889-67-4	238.40	196.5
20	tert-butyl peroxy isopropylcarbonate	2372-21-6	176.24	62.2
21	tert-butyl peroxy diethylacetate	2550-33-6	188.30	35.0
22	2,5-dimethyl-2,5-di-(benzoylperoxy)hexane	2618-77-1	386.48	69.0
23	tert-butyl peroxy-2-ethylhexanoate	3006-82-4	216.36	35.0
24	1,1-bis-(tertbutylperoxy)cyclohexane	3006-86-8	260.42	60.0
25	2,5-dimethyl-2,5-bis(hydroperoxy)hexane	3025-88-5	178.26	105.0
26	bis(2-chlorobenzoyl)peroxide	3033-73-6	311.12	51.3
27	dipropionyl peroxide	3248-28-0	146.16	30.0
28	diisobutyryl peroxide	3437-84-1	174.22	0.0
29	tert-butyl cumyl peroxide	3457-61-2	208.33	77.1
30	1,1-bis(tert-butylperoxy)-3,3,5-trimethylcyclohexane	6731-36-8	302.51	60.0
31	3,4-dimethyl-3,4-diphenylhexane	10192-93-5	266.46	158.6
32	cyclohexanone peroxide	12262-58-7	246.34	80.0
33	tert-butyl 3,5,5-trimethylperoxyhexanoate	13122-18-4	230.39	24.0
34	bis(4-tert-butylcyclohexyl)peroxydicarbonate	15520-11-3	398.60	40.0
35	di(n-propyl)peroxydicarbonate	16066-38-9	206.22	–5.0
36	di-n-butyl peroxydicarbonate	16215-49-9	234.28	5.0
37	di-sec-butyl peroxydicarbonate	19910-65-7	234.28	0.0
38	α,α-dimethylbenzyl peroxypivalate	23383-59-7	236.34	15.0
39	bis(tert-butyl peroxyisopropyl)benzene	25155-25-3	338.54	80.8
40	dihexadecyl peroxodicarbonate	26322-14-5	571.00	37.5
41	cumyl peroxyneodecanoate	26748-47-0	306.49	7.8
42	di-isopropylbenzene hydroperoxide	26762-93-6	196.32	65.0
43	acetylacetone peroxide	37187-22-7	230.24	64.7
44	tert-butyl peroxy-2-ethylhexyl carbonate	34443-12-4	246.39	51.0
45	2,4,4-trimethylpentyl-2-peroxyneodecanoate	51240-95-0	314.52	18.1
46	bis(3-methoxybutyl)peroxydicarbonate	52238-68-3	294.34	15.0
47	di(2-ethoxyethyl)peroxydicarbonate	52373-74-7	266.28	10.0
48	diacetone alcohol peroxide	54693-46-8	230.34	50.0
49	tert-amyl peroxyneodecanoate	68299-16-1	258.45	10.8
50	tert-amyl peroxy 2-ethylhexyl carbonate	70833-40-8	260.42	55.0
51	t-hexyl peroxide benzoate	124350-67-0	222.31	62.2
52	cumyl peroxyneoheptanoate	130097-36-8	278.38	10.0
53	dioctanoyl peroxide	762-16-3	286.46	25.9
54	bis(3,5,5-trimethylhexanoyl)peroxide	3851-87-4	314.52	20.0
55	tert-butyl peroxyneodecanoate	26748-41-4	258.40	15.0
56	tert-amyl peroxypivalate	29240-17-3	188.30	21.6
57	bis-(2-ethylhexyl)peroxydicarbonate	16111-62-9	346.52	15.4
58	dimyristyl peroxydicarbonate	53220-22-7	514.88	19.2
59	tert-butyl peroxypivalate	927-07-1	174.27	27.0
60	di-lauroyl peroxide	105-74-8	398.70	46.0
61	di-decanoyl peroxide	762-12-9	342.58	31.0
62	2,5-bis-(2-ethylhexanoylperoxy)-2,5-dimethylhexane	13052-09-0	430.70	38.6
63	tert-amyl peroxy-2-ethylhexanoate	686-31-7	230.39	35.0
64	2,2-bis(tert-butylperoxy)butane	2167-23-9	234.38	70.0
65	di-tert-amyl peroxide	10508-09-5	174.32	70.4

Geometry Optimization

Molecular structures of the 65 compounds were prepared in the molfile format. Some descriptors were molecular-structure-dependent, making geometry optimization very important. Auto geometry optimization on Avogadro,⁵ a molecular modeling software for quantum chemical calculations, was performed for each mol file. The basic MM potential energy function includes bonded terms (for covalently bonded atomic interactions) and nonbonded terms (for long-range electrostatic and van der Waals forces). Here, UFF (universal force field) was used to improve bond lengths and bond angles to obtain a minimum-energy conformation. Prior MM calculations improved DFT convergence, with less computational load. DFT calculations using Firefly (PC GAMESS),⁶ via MoCalc2012,⁷ optimized the molecular structures. “Geometry optimization” job type, using hybrid density functional B3LYP with 6-31G basis set, was used. B3LYP hybrid functional incorporates approximations to the exchange–correlation energy functional in DFT (combination of exact exchange from Hartree–Fock theory, and from other sources). Polarization-incorporated 6-31G, 6-31G(d), is commonly used for organic compounds. Here, 6-31G was chosen to reduce the computational load and verify the effect of d-orbitals.

Descriptor Calculation and Selection

After structural optimization, 5666 and 552 molecular descriptors of organic peroxides were calculated by alvaDesc 2.0.8⁸ and CODESSA 3,⁹ respectively, most of which were irrelevant to this study. These methods calculate molecular descriptors and fingerprints from structural information. Descriptors with small standard deviations and strong multicollinearity were eliminated. CODESSA was used to calculate the quantum chemical descriptors, such as HOMO/LUMO, and GA was used to find a descriptor set for model construction.

GA, applied as a variable selection method, is an iterative procedure that continuously improves the fitness function, from which the fitness score (indicating probability of descriptor set selection) is calculated. An initial population of descriptors (a few hundred sets) was selected at random or heuristically. Each iteration step calculated and assigned a fitness value to the descriptor sets, and proportional probabilities were used to select a new descriptor population. This selection procedure cannot independently generate a new point in the search space; thus, crossover and mutation were additionally used by GA to generate new descriptor sets. Here, the coefficient of determination (R²), after fivefold cross validation in PLS and SVR modeling, labeled GA-PLS and GA-SVR, respectively, was used as the fitness function. Number of components in the population was 100, crossover probability was 0.5, mutation probability was 0.2, and maximum number of generations was 200.

Regression and Validation

Here, PLS and SVR were used to develop the prediction models. PLS is a statistical linear regression method used to find fundamental relations between explanatory variables (X) and response variables (Y). It is widely used when the number of explanatory variables is significantly larger than the number of samples. Decompositions of X and Y, as shown below, were constructed to maximize the covariance between the latent variables (T) and response variables (Y).

1

2

where A is the number of latent variables, t_a is the ath latent variable, p_a is the ath loading, q_a is the weight on ath latent variable, and E and f are the error terms that cannot be explained by X and Y. p_a and q_a were calculated to minimize the sum of squares of errors. The number of latent variables with the highest R², obtained via fivefold cross validation, was used in the prediction model.

Variable importance in projection (VIP)¹⁰ scores of each descriptor were calculated to identify variables that contributed significantly to the prediction. VIP scores, as shown below, are defined for each X variable and j, as the sum of latent variables of its PLS-weight value (w_ij), weighted by the percentage of explained Y variance.

3

where h is the number of latent variables, w_i is the weight vector, and R²(y, t_a) is the percentage of explained Y variance.

SVR is also a regression method, performing linear and nonlinear regression using the kernel trick, implicitly mapping inputs into high-dimensional feature spaces. If x⁽ⁱ⁾ is the explanatory variable for the ith sample, the response variable, f(x⁽ⁱ⁾), is expressed as follows

4

5

where b is a constant term, w is the weight vector, and k is the number of dimensions. The error function is expressed as follows

6

where C is the regularization parameter, n is the number of training samples, and ε specifies the epsilon tube within which no penalty is associated with the loss function.

The RBF kernel, a kernel function used in machine learning, shown below, was used.

7

where γ is the RBF kernel parameter. SVR includes three hyperparameters (C, ε, and γ) to be provided before model construction.

Model development and descriptor selection were performed using Python 3.7. Scikit-learn,¹¹ a machine learning library in Python, was used for the PLS and SVR calculations. GridSearchCV,¹² a parameter estimator in Python, was used for hyperparameter optimization. Fast optimization of hyperparameters was implemented following the procedure adopted by a previous publication.¹³

Hyperparameter optimization and descriptor selection were performed only on the training data set and then evaluated the performance of the model for the test data set. Model accuracy was evaluated on the common statistical parameters: root mean square error (RMSE), mean absolute error (MAE), and R². These parameters were calculated as follows

8

9

10

Results and Discussion

Prediction models, including and excluding structural optimization and variable selection, were developed and analyzed. Additionally, prediction accuracies of existing models were compared (Table 2).

Table 2. Comparison of Predictive Performance of Existing Models for Test Data.

	Wang2		He3
geometry optimization	DFT B3LYP/6-31G(d)		MM+/MO PM1
frequency calculation
variable selection			GA-MLR
number of descriptors	8		9
modeling method	MLR	SVR	MLR	SVR
number of training data	40		57
number of test data	10		14
RMSE	12.0	6.43	9.91	9.79

Case 1 (Geometry Optimization: MM)

In this case, only the MM calculation was performed before model development. After preprocessing, the model was developed by PLS, with five latent variables, and SVR (C = 8.0, ε = 0.00098, and γ = 0.00024). A parity plot of actual values versus calculated values is shown in Figure 1. Statistical parameters of the model for the training set were as follows: R² = 0.90, RMSE = 12.35, and MAE = 9.45 for PLS, and R² = 0.99, RMSE = 3.92, and MAE = 0.67 for SVR and those for the test set were as follows: R² = 0.26, RMSE = 22.40, and MAE = 17.46 for PLS, and R² = 0.10, RMSE = 24.69, and MAE = 19.95 for SVR. Prediction performance was lower than those of the existing models. Changing the number of latent variables in PLS, and the hyperparameter values in the SVR, did not improve the prediction accuracy.

Figure 1.

Actual vs calculated values of SADT in case 1 (left: PLS, right: SVR).

Case 2 (Geometry Optimization: MM/DFT)

In this case, the MM and DFT calculations were performed before model development. After preprocessing, the model was developed by PLS using 13 latent variables and SVR (C = 4.0, ε = 0.00098, and γ = 0.00049). A parity plot of actual values versus calculated values is shown in Figure 2. The statistical parameters of the model for the training set were as follows: R² = 0.99, RMSE = 0.98, and MAE = 0.76 for PLS, and R² = 0.99, RMSE = 2.15, and MAE = 0.34 for SVR, and those for the test set were as follows: R² = 0.82, RMSE = 9.80, and MAE = 7.70 for PLS, and R² = 0.77, RMSE = 11.07, and MAE = 8.68 for SVR. Prediction performance was significantly higher than case 1 and comparable to existing models. Prediction accuracy of both the models could be improved by DFT calculations, and the addition of quantum chemical descriptors, before model building. Changing the number of latent variables in PLS, and hyperparameter values in SVR, did not improve the prediction accuracy, similar to case 1.

Figure 2.

Actual vs calculated values of SADT in case 2 (left: PLS, right: SVR).

Case 3 (Geometry Optimization: MM/DFT, Variable Selection: GA)

In this case, in addition to preprocessing (similar to case 2), the descriptors were selected using GA-PLS and GA-SVR, before model building. After variable selection, the fitness function of GA was improved from R² = 0.69 to R² = 0.91 for PLS and from R² = 0.38, to R² = 0.90 for SVR. The model was developed by PLS with 11 latent variables and SVR (C = 466, ε = 0.02343, and γ = 0.00000714). A parity plot of actual values versus calculated values is shown in Figure 3. The statistical parameters of the model for the training set were as follows: R² = 0.99, RMSE = 1.57, and MAE = 1.14 for PLS, and R² = 0.99, RMSE = 3.69, and MAE = 1.78 for SVR, and those for the test set were as follows: R² = 0.95, RMSE = 5.11, and MAE = 4.03 for PLS, and R² = 0.91, RMSE = 6.87, and MAE = 5.15 for SVR. Prediction performance dramatically improved compared to case 2, and the prediction accuracy was also better than the existing models. Thus, it can be said that the prediction accuracy could be improved by the appropriate selection of variables before model building.

Figure 3.

Actual vs calculated values of SADT in case 3 (left: PLS, right: SVR).

Comparison of Prediction Accuracy between PLS and SVR

Comparison between PLS and SVR prediction accuracies are listed in Table 3. “Descriptors” line shows the change in the number of descriptors due to preprocessing, and “variable selection” line shows the number of descriptors after applying GA. The prediction accuracy was good overall, with a higher value for PLS than SVR, for the data set used in this study. Geometry optimization using MM/DFT calculations, addition of quantum chemical descriptors, and variable selection using GA influenced prediction accuracy.

Table 3. Model Development Condition and Validation Results.

	case 1		case 2		case 3
descriptors	5889 to >2659		1216 + 553 to >1586		1216 + 553 to >1586
calculated by	alvaDesc 2		alvaDesc 2 + CODESSA 3		alvaDesc 2 + CODESSA 3
geometry optimization	MM (UFF)		MM (UFF), DFT (6-31G/B3LYP)		MM (UFF), DFT (6-31G/B3LYP)
variable selection					GA-PLS	GA-SVR
					1586 to >559	1586 to >524
modeling method	PLS	SVR	PLS	SVR	PLS	SVR
RMSE	22.4	24.7	9.8	11.1	5.1	6.9
MAE	17.5	20.0	7.7	8.7	4.0	5.2
R2	0.26	0.23	0.82	0.77	0.95	0.91

Comparison of Prediction Accuracy with Existing Models

The model in the literature² had small RMSE and high prediction accuracy; however, the computational load was very high due to DFT calculation. On the other hand, the model in the literature,³ which used the semiempirical molecular orbital method (AM1), had relatively larger RMSE, although the computational load was low.

The prediction accuracy of the proposed model was significantly improved by the addition of quantum chemical descriptors and variable selection using GA. Changing the basis set for DFT calculation to 6-31G could reduce the computational load while maintaining the prediction accuracy.

The addition of quantum chemical descriptors and appropriate optimization of molecular conformation before calculating the descriptors were important for a model with high accuracy. However, improvement in prediction accuracy reached a ceiling at some point, so improvement in prediction accuracy should be balanced with a computational load to create an effective model (Table 4).

Table 4. Comparison with Predictive Performance of Existing Models for Test Data.

	proposed method		Wang2		He3
geometry optimization	MM/DFT 6-31G/B3LYP		DFT 6-31G(d)/B3LYP		MM+/MO PM1
frequency calculation
variable selection	GA-PLS	GA-SVR			GA
number of descriptors	559	521	8		9
modeling method	PLS	SVR	MLR	SVR	MLR	SVR
number of training data	52		40		57
number of test data	13		10		14
RMSE	5.11	6.87	12.0	6.43	9.91	9.79

Descriptors with High Impact on Prediction Accuracy

Top 15 descriptors with highest VIP scores are shown in Table 5. Various descriptors related to oxygen bonding (bond order, valence, and charge), and quantum chemical descriptors (LUMO and repulsion/attraction energy), are included in the table. The result shows that these descriptors obviously influenced the prediction accuracy of SADT.

Table 5. Top 15 Descriptors with Highest VIP Scores.

no.	VIP	descriptor	calculated by	explanation
1	2.248	AvgBondOrd_O	CODESSA	average bond order for all atoms of O type
2	2.231	MaxOneCent-ElecElecRepEn	CODESSA	maximum one-center electron–electron repulsion energy
3	2.182	SM02_EA(dm)	alvaDesc	spectral moment of order 2 from edge adjacency mat. weighted by dipole moment
4	2.176	SM08_EA(dm)	alvaDesc	spectral moment of order 8 from edge adjacency mat. weighted by dipole moment
5	2.133	MinOneCent-CoreElecAttrEn	CODESSA	minimum one-center core-electron attraction energy
6	2.082	SM07_EA(dm)	alvaDesc	spectral moment of order 7 from edge adjacency mat. weighted by dipole moment
7	2.040	MaxTwoCent-TotEn_AB	CODESSA	maximum two-center total energy, all bonds
8	2.021	SpMax_B(s)	alvaDesc	leading eigenvalue from Burden matrix weighted by I-state
9	2.004	AvgVal_O	CODESSA	average valence for atoms of O type.
10	1.978	MaxTwoCent-CoreElecResEn_AP	CODESSA	maximum two-center core-electron resonance energy, all pairs
11	1.971	B02[O–O]	alvaDesc	presence/absence of O–O at topological distance 2
12	1.958	AvgBondOrd_O_O	CODESSA	average bond order among all bonds between atoms of type O and O
13	1.949	qpmax	alvaDesc	maximum positive charge
14	1.907	MinTwoCent-CoreElecAttrEn_AP	CODESSA	minimum two-center core-electron attraction energy, all pairs
15	1.861	LUMOEn	CODESSA	energy of lowest energy unoccupied molecular orbital

Analyzing Effects of Preprocessing Including and Excluding DFT/GA

In no. 53, 54, 55, 59, 64, and 65, the prediction accuracy was improved by performing the DFT calculations before descriptor calculations, but in no. 57, 58, and 60, preprocessing by DFT calculations did not significantly improve the prediction accuracy. Additionally, no. 56, 62, and 63 showed high prediction accuracy from the beginning. In all the cases, except no. 62, variable selection via GA improved the prediction accuracy, but its overall influence on prediction-accuracy improvement was smaller than the effect of DFT optimization (Figures 4).

Figure 4.

Comparison of prediction accuracy between PLS models.

Figure 6.

Molecular structure of no. 54, bis(3,5,5-trimethylhexanoyl)peroxide.

Figure 7.

Molecular structure of no. 56, tert-amyl peroxipivalate.

No. 53, 54, 56, and 58 were compared as representatives (Figures 5–8). Conformation (bond length and angle) near the O–O bond of each molecule significantly changed after DFT calculation, as presented in Table 6. No. 53 and 54 exhibited relatively larger bond length changes than no. 56 and 58 due to the greater difference between the initial and optimal states for no. 53 and 54 than between no. 56 and 58. No. 56 exhibited a low prediction error for case 1, despite only 18 iterations, which could be due to the optimal initial conformation, where only MM calculations were performed. A large number of iterations yielded a high prediction accuracy, with some exceptions. Prediction error for no. 58 could not be reduced significantly. This could be due to the DFT calculation errors because 6-31G, instead of 6-31G(d), was used as the basis function.

Figure 5.

Molecular structure of no. 53, dioctanoyl peroxide.

Figure 8.

Molecular structure of no. 58, dimyristyl peroxydicarbonate.

Table 6. Comparison Results for Four Representative Molecules.

element	unit	no. 53	no. 54	no. 56	no. 58
improvement of prediction accuracy	°C	19.9	52.9	1.0	3.1
	%	79.6	86.2	17.3	16.6
iterations	times	21	51	18	25
length (O1–O2)	angstrom	+0.263	+0.237	+0.162	+0.174
length (C1–O1)	angstrom	+0.059	+0.064	+0.020	+0.024
length (C2–O2)	angstrom	+0.059	+0.063	+0.065	+0.046
angle(∠C1O1O2)	degree	–9.9	–12.5	–7.6	–13.5
angle(∠O1O2C2)	degree	–9.9	–9.9	–2.9	–12.5

Thus, appropriate optimization of molecular conformation and addition of quantum chemical descriptors immensely influence SADT prediction (Tables 7–10).

Table 7. Representative Bond Lengths and Angles before/after Optimization for No. 53.

element	unit	before optimization	after optimization	difference
O1–O2	angstrom	1.268	1.531	+0.263
C1–O1	angstrom	1.359	1.418	+0.059
C2–O2	angstrom	1.359	1.418	+0.059
∠C1O1O2	degree	124.5	114.6	–9.9
∠O1O2C2	degree	124.5	114.6	–9.9

Table 10. Representative Bond Lengths and Angles before/after Optimization for No. 58.

element	unit	before optimization	after optimization	difference
O1–O2	angstrom	1.272	1.446	+0.174
C1–O1	angstrom	1.354	1.378	+0.024
C2–O2	angstrom	1.352	1.398	+0.046
∠C1O1O2	degree	121.7	108.2	–13.5
∠O1O2C2	degree	121.1	108.6	–12.5

Table 8. Representative Bond Lengths and Angles before/after Optimization for No. 54.

element	unit	before optimization	after optimization	difference
O1–O2	angstrom	1.273	1.510	+0.237
C1–O1	angstrom	1.352	1.416	+0.064
C2–O2	angstrom	1.353	1.416	+0.063
∠C1O1O2	degree	122.3	109.8	–12.5
∠O1O2C2	degree	120.5	110.4	–9.9

Table 9. Representative Bond Lengths and Angles before/after Optimization for No. 56.

element	unit	before optimization	after optimization	difference
O1–O2	angstrom	1.297	1.459	+0.162
C1–O1	angstrom	1.360	1.380	+0.020
C2–O2	angstrom	1.417	1.482	+0.065
∠C1O1O2	degree	125.0	117.4	–7.6
∠O1O2C2	degree	108.2	105.3	–2.9

Double Cross-Validation

Here, following a previous study, the data set was divided into training and test data to conduct holdout validation of the prediction accuracy. The temperature range of the test data was 15–70 °C, and it was unclear whether the model was applicable to a wider temperature range. To verify the extrapolation (generalization performance) for a high temperature range, using a small data set, double cross-validation was conducted using case 3 data, variables, and model building conditions. In inner cross-validation, hyperparameters were determined via fivefold cross-validation, whereas in outer cross-validation, leave-one-out cross-validation was performed. Statistical parameters of the PLS model were as follows: R² = 0.76, RMSE = 17.74, and MAE = 13.29, and those of the SVR model were as follows: R² = 0.74, RMSE = 18.11, and MAE = 13.50. No significant difference in the prediction error at low and high temperatures was observed, and both the models evenly predicted a wide range of SADTs, with good accuracy (Figure 9).

Figure 9.

Actual vs calculated values of SADT (left: PLS, right: SVR).

Conclusions

In this study, we constructed a model to estimate the SADT of organic peroxides. PLS regression and SVR were applied on the descriptors calculated using the structural information of the compounds, to predict the SADTs. MM and DFT calculations were performed before calculating the descriptors, and GA was used for variable selection. In DFT calculation, B3LYP with the 6-31G basis set was used instead of 6-31G(d), significantly improving prediction accuracy and reducing computational load. Thus, a model with higher accuracy than the existing SADT prediction models was developed.

Appropriate preprocessing and variable selection were important for a model with high accuracy, and optimizing compound conformation before descriptor calculations improved SADT prediction accuracy. However, the improvement in prediction accuracy should be balanced with a computational load to create an effective model.

In the future, application of machine learning models other than SVR, preprocessing methods for descriptor calculations, and descriptor selection with high contribution to SADT prediction, to further improve prediction accuracy, will be investigated.

Glossary

Abbreviations

SADT

self-accelerating decomposition temperature

QSPR

quantitative structure–property relationship

DFT

density functional theory

MLR

multiple linear regression

SVR

support vector regression

GA

genetic algorithm

MM

molecular mechanics

PLS

partial least squares

HOMO

highest occupied molecular orbital

LUMO

lowest unoccupied molecular orbital

VIP

variable importance in projection

RMSE

root mean square error

MAE

mean absolute error

The authors declare no competing financial interest.

Notes

Data and Software Availability: structures of chemical compounds were downloaded from the Chemical Book database.¹⁴ Some of the software used to calculate optimal conformation of molecules can be freely downloaded from the linked sites.⁵⁻⁷ Python 3.7 was used to build the models and the source code was referenced to the linked site.¹⁵ All data underlying the results are available as part of the article and no additional source data are required.