A Physical Organic Approach towards Statistical Modeling of Tetrazole and Azide Decomposition

Abstract

N-rich heterocycles and organic azides have found extensive use in many sectors of modern chemistry from drug discovery to energetic materials. The prediction and understanding of their energetic properties are thus key to the safe and effective application of these compounds. In this work, we disclose the use of multivariate linear regression modeling for the prediction of the decomposition temperature and impact sensitivity of structurally diverse tetrazoles and organic azides. We report a data-driven approach for property prediction featuring a collection of quantum mechanical parameters and computational workflows. The statistical models reported herein carry predictive accuracy as well as chemical interpretability. Model validation was successfully accomplished via tetrazole test sets with parameters generated exclusively in silico. Mechanistic analysis of the statistical models indicated distinct divergent pathways of thermal and impact-initiated decomposition.

Graphical Abstract

Tetrazoles and azides find broad application in pharmaceuticals, chemical biology, and defense. In this work multivariant linear regression and classification are leveraged to develop predictive statistical models for their decomposition temperature and impact sensitivity, which are key in evaluation of the safety of a given molecule. Furthermore, interpretation of the statistical models provided rich insight into the decomposition mechanisms.

Introduction

N-rich organic compounds have found broad applications spanning chemical synthesis, medicine, biochemistry, pharmacology, imaging, and defense.^[1–7] For example, tetrazoles are present in numerous active pharmaceutical ingredients and are frequently used as bioisosteres for carboxylic acids and amides with improved metabolic stability and physicochemical properties.^[8] Organic azides are ubiquitous from materials chemistry to chemical biology as efficient handles for click-chemistry or as versatile synthons in numerous synthetic methods.^[9–11] Tetrazoles and azides have also emerged as high energy materials (HEMs) that rival traditional nitro- and lead-based explosives owing to their energetic properties, relatively mild syntheses, and reduced toxicity.^[6,7,12] Nevertheless, the energetic nature of these N-rich compounds can pose hazards during their preparation, storage, and handling; predicting their sensitivity properties and understanding decomposition mechanisms is thus integral to these applications.^[13–17]

In this context, the in silico study of molecular decomposition has been a common practice in the field of high energy materials and in process hazard risk assessments.^[13,14] While considerable computational work has been devoted to the prediction of HEMs performance properties (detonation velocity, detonation pressure, etc.),^[16,18–21] methods for the prediction of sensitivity properties^[18] (impact sensitivity (IS),^[23–26] friction sensitivity, electric spark sensitivity^[27,28]) and decomposition temperature (T_dec)^[29,30] are less established in comparison. While a number of predictive quantitative structure-property relationships (QSPR) have been reported for HEMs based on nitro groups,^[31–33] nitramines,^[34,35] azides,^[36] and peroxides^[37], prediction of T_dec or IS for nitrogen-containing heterocycles remains underexplored.^[38] In a pioneering example reported by Kamlet, a one-term QSPR was constructed between the oxygen balance of polynitroaromatics and their IS.^[39] This simple model provided insight into the decomposition mechanisms of these HEMs, which inspired subsequent density-functional theory (DFT) studies.^[40] The use of constitutional descriptors in multivariate linear regression (MLR) analysis of HEMs was more recently refined by Keshavarz.^{[27,30,31,36,41,42]} In 1990, Storm published an experimental IS dataset with over 200 nitro-based HEMs including nitroaromatics, nitramines, and nitrates,^[43] which prompted subsequent application of 2D/3D and DFT-calculated descriptors to model this dataset.^{[24,25,33,41,42,44,45]} While satisfactory statistical performance was obtained from these efforts, limited mechanistic information regarding decomposition was achieved due to the wide range of energetic functional groups (i.e., explosophores) in the dataset, which necessitated the use of general descriptors. Focusing on a common explosophore can improve both the mechanistic interpretability and predictive accuracy of statistical models for large HEM datasets, as explosophores are broadly acknowledged to be responsible for the initial steps leading to macroscopic detonation or deflagration. For example, Rice reported analysis of different classes of nitro-based energetics by correlating the electrostatic potentials of reactive C/N/O–NO₂ bonds to high sensitivity.^[23] Similarly, Rotureau developed a QSPR for peroxide T_dec that supported homolytic cleavage of the O–O bond as the initiation step.^[37] Herein, we describe the development of a data-driven workflow to study the decomposition of tetrazoles and organic azides using fundamental principles of physical organic chemistry along with modern data science tools. Leveraging the abundant and readily available data from the energetic materials literature, we obtained statistical models for the decomposition temperature and impact sensitivity of tetrazole-containing HEMs. The dataset contains a diverse range of structures and substitution patterns that are also frequently found in medicinal chemistry. This workflow carries both predictive power and mechanistic interpretability, which could be generalized to the study of other types of high energy compounds and synthetic intermediates across various sectors of chemical synthesis.^[13–15]

Results and Discussion

We focused our initial study on tetrazoles and advanced a data-driven workflow to construct QSPRs for tetrazole-containing HEMs (Figure 1A). Parallel strategies were undertaken to parameterize tetrazoles from a newly-compiled database containing crystal structures and sensitivity properties of over 400 HEMs synthesized by the research groups of Klapötke and Shreeve.^[48] Our first approach utilized solid state structures to extract Hirshfeld surface parameters to provide a framework for QSPR development based on X-ray structures. A second ab initio approach employed molecular mechanics calculations to obtain sets of energetically accessible conformers for each HEM. These conformers were submitted to DFT calculations allowing for the extraction of NBO descriptors with all bonding, anti-bonding, and lone pair NBOs, which outperformed more traditional descriptors based on natural bond orbital (NBO) or electrostatic potential (ESP) charges and NMR shifts. These workflows provided mechanistic insight as well as accurate T_dec and IS prediction for a diverse set of tetrazole-containing HEMs.

Figure 1.

An explosophore-based approach to tetrazole sensitivity property prediction. (A) Schematic workflow for mechanistically oriented property prediction. (B) Decomposition mechanisms of 1,5- and 2,5-tetrazoles with a unifying numbering scheme for featurization. The intermediates and resonance structures shown were based on literature precedents for tetrazole decomposition intermediates identified through trapping studies.^[46] 1,5-tetrazoles decompose via a nitrene that can participate in traditional reactivities such as the insertion into a proximal C–H bond, while 2,5-tetrazoles decompose via nitrile imines, which are a traditional 1,3-dipole and readily participate in [3+2] cycloadditions.^[47]

Explosophore Featurization:

The decomposition of tetrazoles has been investigated in a number of experimental and computational studies, with the prevailing proposed mechanism being a stepwise extrusion of molecular nitrogen for both 1,5- and 2,5-substituted tetrazoles.^[49–53] Among these studies, a number of isomerization events are observed involving substituents on the tetrazole moiety, leading to a variety of decomposition pathways.^[43] Several considerations were made for explosophore-based featurization of tetrazoles: 1) tetrazoles have two different substitution patterns, 2) tetrazoles can bear varied substituents that influence their energetic properties, and 3) many HEMs contain multiple tetrazole explosophores. To adequately describe tetrazoles of both substitution patterns, we constructed descriptors parameterizing 1,5- and 2,5-tetrazoles and devised a numbering scheme unifying tetrazoles of either substitution pattern (Figure 1B).^[46,51–53] Under this numbering scheme, several attributes are maintained regardless of substitution: in the starting tetrazole, double bonds connect N¹ and N² as well as C⁴ and N⁵, while N³ is substituted with R¹ and bears a lone pair. Further, during decomposition via a N₂ extrusion pathway proposed in the literature, the N²–N³ single bond would be broken and a triple bond is formed between N¹ and N². For constitutionally symmetric ditetrazoles, both tetrazole units were treated as equivalent for modeling. This assumption of symmetry is validated by the analysis of the crystal structures: 10 of the 11 constitutionally symmetric ditetrazoles were electronically symmetric in the solid state.

With the crystal structures of our HEMs as input geometries, we conducted DFT geometry optimization (B3LYP/6–31+G(d,p)) and single point calculations (M06–2X/def2-TZVP) in the gas phase. A collection of geometric and electronic parameters (Figure 1A) was then extracted using automated tools disclosed in the SI. Key to our parameter suite was the extraction of occupancies and energies of every NBO bonding, antibonding, and lone pair orbital within each tetrazole motif. Parameters calculated from Hirshfeld isosurfaces were also obtained to explore correlations between crystal packing and HEM sensitivity.^[54,55] Upon closer inspection of our dataset, it became clear that several of the tetrazoles in this study possessed natural Lewis structures containing formal charges (Figure 1B). Feature engineering was performed on calculated NBO orbitals to account for these resonance structures.^[56]

Decomposition temperature:

T_dec measurements are typically obtained through differential scanning calorimetry (DSC) or differential thermal analysis (DTA) measurements at a set temperature ramping rate. To ensure consistency in modeling, T_dec measurements were binned by acquisition method. The majority of the T_dec measurements in our dataset were measured using DSC using a ramping rate of 5 °C/min. We constructed a 3-term statistical model for T_dec via forward stepwise regression using 24 tetrazoles measured with DSC (5°C/min). The descriptors used in this model were obtained from crystal structures and all tetrazoles were well-predicted using a model comprised of two NBO descriptors and one Sterimol descriptor (see SI1 for more details).

Motivated by our initial success in modeling T_dec using descriptor sets constructed from crystallographic data, we sought to conduct ab initio property screening. We implemented a second workflow wherein 2D structural formulae were converted into 3D conformers using molecular mechanics with MacroModel 11.8. Using a Scifinderⁿ search, we added 36 tetrazoles synthesized by Klapötke and Shreeve lacking associated crystal structures to our dataset. All calculated conformers within a 5.0 kcal/mol energy difference of the lowest energy conformer were subjected to DFT calculations for parameter acquisition. With multiple conformations for each compound in hand, we generated two descriptor sets containing parameters extracted from either the lowest energy conformer of each HEM or the Boltzmann average of descriptor values across all conformers.

Using the Boltzmann-averaged descriptor set, we constructed a second model for T_dec (Figure 2A), which offered improved performance relative to our first T_dec model based on crystal structures. This result is consistent with the observation that many of the tetrazoles in our experimental dataset have a melting point below their T_dec. Consequently, these compounds have a high degree of conformational flexibility prior to decomposition, which is more suitably approximated by Boltzmann-averaged descriptors. This model used the same 24-tetrazole training set from the crystal structure model and was validated with an external test set of 5 monotetrazoles and 8 nonsymmetric ditetrazoles (Figure 2A, solid red cycles), all of which were subject to a 5 °C/min DSC procedure for T_dec measurement. Predictions for nonsymmetric ditetrazoles were based on the trigger hypothesis, which states that the bulk decomposition of an energetic material is dominated by the initial decomposition of the explosophore with the lowest activation energy.^[57] Thereby, the lower predicted T_dec between the two tetrazole explosophores was defined as the ditetrazole’s T_dec. Overall, our model presented satisfactory T_dec prediction with a mean absolute error (MAE) of 22 °C for the entire test set and an MAE of 18 °C for the 8 nonsymmetric ditetrazoles, providing validation of our original model and supporting the applicability of the trigger hypothesis for prediction of tetrazole T_dec. The test set contained a single outlier representing an out-of-sample prediction with an experimental T_dec of 347 °C When 13 additional tetrazoles with T_dec measured using DTA (5 °C/min) were included as a test set, an MAE (40 °C) twice as large as that of the training set (19 °C) was obtained (Figure 2A, green triangles). Compounds for which T_dec was acquired using DSC with a ramping rate of 10 °C/min were consistently underpredicted (purple diamonds). It is reported that the apparent T_dec increases at higher heating rates,^[58] which explains underprediction for these samples and highlights the importance of uniform experimental methods for linear modeling T_dec or sensitivity properties.

Figure 2.

Model for T_dec using ab initio featurization and corresponding mechanistic interpretation. (A) A MLR model for prediction of tetrazole T_dec. 95% confidence bands are colored blue and 95% prediction bands are colored light blue. (B) The mechanistic interpretation of the MLR equation based on the coefficients of the parameters.

The statistical model for T_dec contains two NBO parameters and one geometric parameter (Figure 2B). The parameter with the largest coefficient is the averaged bond order (BO) of N¹–N² and N²–N³. Although the feature (σ(N²–N³) BO + σ(N¹–N²) BO)/2 correlates well with the N²–N³ σ bond order, factoring in the N¹–N² σ bond order allowed us to derive a general model, which accounts for charged NBO structures (Figure 1B) that feature an N¹–N² single bond. The positive sign of this term suggests that tetrazoles with a lower N²–N³ bond order are more prone to decomposition, consistent with a rate-determining N²–N³ bond breakage for 1,5- and 2,5- tetrazoles.^[59]

In addition to traditional manual selection of an appropriate MLR model from a list of MLR equations obtained through forward stepwise selection, we also aimed to validate the proposed mechanism in an automated human-on-the-loop workflow. This was implemented through a fully automated statistical analysis of the prevalence and coefficients of features in all MLR models that meet a minimum threshold of statistical performance (here, N = 817, each with a combined training and test set R² ≥ 0.65). Typically, MLR equations have limited numbers of features to control against overfitting and random correlation. This statistical analysis not only removes possible mechanistic bias of the manual selection but also allows for the analysis of more features than included in any single MLR equation. Through this approach, we found that 5 out of the 15 most prevalent features characterized the bond order or σ* occupancy of the reactive N²–N³ bond, substantiating the mechanistic conclusion (see SI for details).

Impact sensitivity:

IS is modulated by factors including solid-state crystal packing, grain size, and sample preparation and is thus not solely determined by the structure of the tetrazole functional group.^[60,61] Given standardized sample preparation and data acquisition procedures, QSPRs may be constructed, which has been demonstrated in previous reports to elucidate the mechanistic features governing IS of other types of HEMs.^{[23,33,39,44,62]} Notably, IS is poorly correlated with other sensitivity properties, suggesting that impact-initiated decomposition proceeds through disparate pathways. Previous work by Klapötke, Zhang, and Shreeve demonstrated the use of Hirshfeld surface analysis to correlate the IS of nitro-based energetics with the percentage of O···O contacts and hydrogen bonds found in the crystal structures.^[62–66] We explored a similar approach by performing Hirshfeld surface analysis on the crystal structures in the tetrazole dataset. While the percentages of O···O, N···N, and N···O did not provide a satisfactory QSPR, we were able to classify impact sensitivity (high or low using TNT as the threshold value) as a function of the sum of N···N, O···O, and N···O intermolecular contacts with high accuracy (0.86) with only a single false negative (Figure 3A, left).

Figure 3.

(A) Comparison of Hirshfeld parameters and constitutional descriptors for threshold analysis of impact sensitivity. 95% confidence bands are colored blue and 95% prediction bands are colored light blue. (B) Model for log(IS) using ab initio featurization.

This parameter quantifies strongly repulsive interactions that destabilize the crystal lattice and provided a better classification than the sum of H···N and H···O contacts (e.g., attractive H-bonding interactions). For comparison, a simple constitutional descriptor measuring combined oxygen and nitrogen content provided virtually no threshold (accuracy = 0.55; Figure 3A, right). This QSPR relates tetrazole IS to the solid-state structure, mirroring the results of the reported qualitative analysis, which showed that repulsive O┄O contacts in nitro-based energetics results in higher sensitivity.^[62–66] This type of classification can be used prospectively when considering the preparation of new analogs.

Aiming to apply our methods to ab initio prediction for IS, we evaluated MLR models for log(IS) based on structures and parameters generated from molecular mechanics calculations.^[22] To avoid systematic errors, tetrazoles that crystallized as hydrates were not modeled due to well-documented lowering of sensitivity upon solvent co-crystallization.^{[62,68,66–68]} The minimum energy conformer descriptor set proved most effective for correlating log(IS). Unlike T_dec measurements, decomposition initiated by impact occurs in the solid state, which is more suitably described using minimum energy conformers than the collective Boltzmann average. Using a training set of 22 tetrazoles, we constructed a 3-term MLR model for impact sensitivity (Figure 3B), which was then used to predict the log(IS) of the remaining 22 monotetrazoles and symmetric ditetrazoles in our dataset.^[69] The model is consistent with decomposition via cleavage of the N²–N³ bond wherein a lower N¹=N² π* energy allows for more efficient stabilization of the partial negative charge in the azide (1,5-tetrazoles) or diazo (2,5-tetrazoles) intermediate (Figure 1B). Statistical mechanistic analysis of all MLR of all MLR models as outlined for decomposition temperature yielded analogous conclusions.

A limitation of this model is its failure to accurately predict the impact sensitivities of nonsymmetric ditetrazoles using the trigger hypothesis. We reason that owing to differing initiation mechanisms, IS may be better described by the features of the entire molecule rather than by a single explosophore. This is supported by the reliance on the %nitrogen parameter in the ab initio model and the successful classification of IS with Hirshfeld parameters.

Principal Component Analysis: During the analyses outlined above, we observed that IS prediction relied on features of the entire molecule and its solid-state structure, whereas T_dec prediction was achieved using parameters localized at the functional group. These findings prompted us to analyze the tetrazoles (minimum energy conformers) using unsupervised principal component analysis (PCA). This resulted in a good correlation of each property with unique orthogonal principal components. Principal component 1 (explained 18.3% of the variance) correlates with log(IS) (R = 0.77, R = 0.32 with T_dec), while principal component 2 (explained 15.1 % of the variance) correlates with T_dec (R = 0.74, R = 0.11 with log(IS) (see SI1 for full correlation analysis and composition of the principal components). Further, the PCA analysis paired with k-means clustering (4 clusters) provided a direct link between substitution patterns and reactivity. For instance, we found that amino substituted tetrazoles were less impact sensitive than their electron-poor counterparts and that tetrazoles bearing a hydrogen substituent tended to have a higher T_dec (Figure 4 and Figure 5).

Figure 4.

Experimental and predicted values of T_dec and log(IS) for tetrazole HEMs investigated in this work. Listed predictions correspond to the models reported in Figures 2 & 3.

Figure 5.

Principal component analysis biplot with four color coded clusters based on k-means clustering and their corresponding common substructures.

Generalization to Azides:

With compelling mechanistic insights and predictive models for tetrazoles in hand, we aimed to further validate our newly developed workflows and features with an additional class of energetic materials. Organic azides have found broad applications in materials chemistry and chemical biology as versatile handles for click-chemistry or as precursors for numerous synthetic methods.^[9–11] As such, they are frequently prepared and purified on scale, yet the prediction of their key safety properties is underdeveloped (T_dec and IS), making them the ideal target for our statistical modeling efforts.^[9,70] Limited previous precedents relied on hardcoded molecular structural fragments and empirically fitted parameters, which are thus unsuitable for ab initio prediction and provided little mechanistic insights.^[36]

Using azides from our HEM database and conformers extracted from their respective crystal structures, we compiled a descriptor set containing quantum mechanical, geometric, and Hirshfeld features, analogous to the one used for tetrazole modeling. The training set consisted of azides from the Klapötke group, while azides prepared by Shreeve and co-workers served as the test set (test set 1 in Figure 6A). Using forward stepwise regression, we constructed a robust three-term model for T_dec (onset T_dec, with DSC (5°C/min)) with a MAE of 12 °C for the training set and a comparable 15 °C MAE for the test set.

Figure 6.

95% confidence bands are colored blue and 95% prediction bands are colored light blue. (A) MLR model for the T_dec of azides. (B) Automated statistical analysis of azide T_dec models. (C) MLR model for the log(IS) for azides. (D) Automated statistical analysis of azide log(IS) models.

Given our observations that predictions for T_dec were sensitive to the heating rate of the measurements, we investigated whether a MLR model trained on onset T_dec could be applied to the prediction of initial T_dec trends. Onset T_dec are extrapolated based on the intercept of the tangents of the baseline and an exothermic peak in a given DSC trace, while initial T_dec are the temperature at which a deviation from the baseline is first observed.^[71] Initial T_dec are frequently reported in synthetic chemistry applications and are typically 10–30 °C lower than the corresponding onset temperatures. We collected a test set of initial T_dec (DSC (5°C/min)) of organic azides from non-energetics applications such as biorthogonal protein labeling^[70,72] (test set 2 in Figure 6A) and extracted ab initio features using their minimum energy conformers. The MLR model successfully predicted the trend of initial T_dec with a test set R² of 0.94 with all samples being consistently overpredicted by 20 °C on average. These results illustrate the utility of our methods beyond the study of energetic materials.

Analysis of the T_dec MLR model, which entirely consisted of terms localized at the azide functional group, suggested that azides bound to electron poor carbons (higher C NBO charge e.g., in acyl azides) have lower decomposition temperatures. Further, statistical analysis of the prevalence of features in all MLR equations for decomposition temperature (Figure 6B) revealed that all five of the most common descriptors directly characterized the azide explosophore, echoing the result for the T_dec of tetrazoles.

In contrast, when the identical descriptor set was used for MLR modeling of azide impact sensitivity (log(IS)), a model consisting entirely of Hirshfeld descriptors was found to most suitably predict the test and training set (Figure 6C). The MLR equation includes the sum of N···N, O···O, and N···O intermolecular contacts, which quantifies the strongly repulsive interactions within the crystal lattice and provided a good classification for tetrazole IS. In analogy to our data for the tetrazole, it suggests that that azides with repulsive interactions in the crystal structure are more sensitive to impact. Notably, this result was further supported by the automated statistical analysis of all MLR equations (Figure 6D), showing that azides with more attractive intermolecular H-bonds (%(H···N, H···O contacts) are less sensitive towards impact. Additionally, we found that azides with denser packing (smaller intermolecular distances (ave. d_i)) were more sensitive towards impact.

Conclusion

In summary, we report a series of computational and statistical methods that enabled prediction of decomposition temperatures and impact sensitivities across a structurally diverse scope of tetrazoles and organic azides. Models were internally and externally validated and were consistent with proposed reported decomposition mechanisms. Moreover, the study of IS and T_dec with consistent datasets provide a clear mechanistic divergence, wherein decomposition temperature can be characterized by the local environment of the reactive functional group, while impact sensitivity is a feature of the entire molecule and its bulk solid-state structure. Clear quantitative structure-property relationships could be constructed based on the repulsive interactions in the crystal lattices using Hirshfeld descriptors, highlighting the importance of crystal engineering towards modulating impact sensitivity. We anticipate that this workflow will be applicable to the analysis of other families of compounds in both high energy materials and other sectors of chemical research and that the critical comparison of T_dec and IS will guide future mechanistic and synthetic efforts.