Chemical Descriptors for a Large-Scale Study on Drop-Weight Impact Sensitivity of High Explosives

Abstract

The drop-weight impact test is an experiment that has been used for nearly 80 years to evaluate handling sensitivity of high explosives. Although the results of this test are known to have large statistical uncertainties, it is one of the most common tests due to its accessibility and modest material requirements. In this paper, we compile a large data set of drop-weight impact sensitivity test results (mainly performed at Los Alamos National Laboratory), along with a compendium of molecular and chemical descriptors for the explosives under test. These data consist of over 500 unique explosives, over 1000 repeat tests, and over 100 descriptors, for a total of about 1500 observations. We use random forest methods to estimate a model of explosive handling sensitivity as a function of chemical and molecular properties of the explosives under test. Our model predicts well across a wide range of explosive types, spanning a broad range of explosive performance and sensitivity. We find that properties related to explosive performance, such as heat of explosion, oxygen balance, and functional group, are highly predictive of explosive handling sensitivity. Yet, models that omit many of these properties still perform well. Our results suggest that there is not one or even several factors that explain explosive handling sensitivity, but that there are many complex, interrelated effects at play.

Introduction

All high explosives (HE) react violently to external stimuli, although some do so more easily than others. The ability to understand, reliably predict, and manipulate the handling safety of new explosives would have a revolutionary effect on both basic explosives research and applied commercial systems. However, the handling safety of explosives can vary markedly, even between related compounds. The coupling between mechanics and chemistry in explosives is evidently complex and depends on properties that span the molecular scale through the mesoscale. Most explosives development is based on a costly trial-and-error approach, consisting of iterated rounds of synthesis and testing, in part because it has been difficult to predict the sensitivities of new HEs. Theory and simulation struggle to provide consistent guidance for the explosive design process, due to the multitude of factors that influence handling safety.

The handling safety of explosives is currently characterized by a suite of empirical tests that subject explosives to a range of stimuli.¹ The most common method to evaluate material handling safety in the initial design stage is the use of a drop-weight impact test. In a typical drop-weight impact test, about 40 mg of HE is placed between two anvils, one of which is typically covered with fine grit paper. A weight (usually 2.5 kg) is dropped from a specified height onto the anvils and transducers detect the acoustic signature of a thermal explosion. This test is used routinely at the earliest stages of explosive development because of its simplicity and its use of modest quantities of material. The mechanism of initiation is not explicitly known,² but has been debated for many decades, and is most often assumed to be thermal activation through friction.³⁻⁵ Results of the test are reported as an estimate of the H₅₀ (or E₅₀) value, the drop height (or drop energy) at which the HE will undergo a violent explosion in 50% of tests. Though the drop-weight impact test involves a somewhat simplified view of an explosive’s sensitivity, it is widely available and allows for rapid comparison to common explosives with well-defined properties.

Drop-weight impact tests measure, albeit indirectly, the activation enthalpy for an explosive to undergo a thermal explosion.⁶⁻¹⁰ Recent molecular dynamics simulations have confirmed the connection between the thermal stabilities of explosives and their sensitivities.^11,12 Numerous researchers, dating back to the seminal works of Wenograd,⁶ Kamlet and co-workers,^7,8 Bowden and Yoffe,⁹ and Storm, Stine, and Kramer,¹⁰ have sought to identify correlations between the physical and chemical properties of explosives and their impact sensitivities. Correlations between physically intuitive descriptors such as the oxygen balance of the molecule—as well as more esoteric properties like bond charge densities—have been investigated.¹³⁻²⁷ Machine learning methods such as artificial neural networks have been applied to predicting impact sensitivities for well over 20 years.²⁸⁻³³ While these methods have been somewhat successful in predicting trends in impact sensitivities, they are strictly interpolative, and they provide limited insights or physical understanding. In fact, Brill and James³⁴ have specifically cautioned against taking correlations of properties with impact sensitivity as the actual cause of the observed sensitivity trends, or only looking at one correlation and disregarding the importance of other effects. For example, as we see in this work, two neural networks that are trained on the same data by the same authors may give equally good predictions, but use different sets of descriptors with limited overlap.²⁸

The lack of a coherent theory in this area is in part due to the fact that previous research has focused on correlations of a limited number of molecular properties (often less than 5) with a relatively small (<200–300) number of explosives, often of similar types. Analyzing a small number of explosives and descriptors inherently adds bias to the estimated model before the study even begins. Such studies are likely to find trends that are specific to the set of explosives considered and may not generalize outside the class of explosives analyzed.

This field is missing several critical areas of study: (1) analyses of how much variation to expect within a test of a single material at a single site or multiple sites (this has only occurred on single explosives in previous work³⁵⁻³⁷) and (2) data collected from more than a few hundred sources with more than a few molecular descriptors. A key contribution of our work is the compilation of a data set of over 1500 drop-weight impact tests from multiple institutions, including hundreds of repeat measurements collected from common explosives such as PETN, TNT, and RDX. We have collected data from common benchmark data sets¹⁰ and other historical tests at LANL,³⁸ as well as more recent studies of new synthesized molecules from Chavez and co-workers,^39,40, Klapotke et al.,⁴¹ Sabatini et al.,⁴² and Shreeve et al.⁴³ We have measured and calculated over 100 descriptors for molecular properties, including energetic functional groups, heat of explosion, oxygen balance, specific heat, and hydrogen bonding. We hope that our data-driven approach to understanding which molecular properties matter in the drop-weight impact test will inform and direct future studies addressing these critical problems in the HE literature.

Data

We modeled logE₅₀ (“drop energy”) as a function of the molecular descriptors. We used only observations that had complete data for all descriptors. These data consisted of 1533 observations with 156 descriptors. There were 25 molecules with repeat observations (which made up 1032 of the 1533 observations); the remaining 501 molecules had a single observation. The five most common explosives tested, and the number of observations per explosive, are given in Table 1. Unless otherwise noted, we use logE₅₀ to represent sensitivity, where “log” is the base-10 logarithm, as in Kamlet,⁷ Nefati et al.,²⁸ and Wang et al.⁴⁴ The main source of data was legacy LANL tests; however, many of the data were obtained from the published scientific literature.^11,45−170 Particularly sizable data sources are the works of Kamlet,⁷ Marrs et al.,³⁶ and Keshavarz and Jaafari,²⁹ the latter of which incorporates most of the data of Storm et al.¹⁰ More recent works of Rice and Hare¹⁴ and Mathieu¹⁷¹ draw on these same data sources.

Table 1. Five Most Common Explosives and Number of Observations of Each.

PETN	HMX	RDX	NQ	TNT
490	164	130	44	40

We considered both categorical and continuous descriptors; however, the majority of descriptors we considered were continuous or integer valued; see Table 3 for a summary of descriptors. In the simplest case, we included the stoichiometry of the molecule (C, H, N, O, etc.) in the molecular formula of each observation. We computed oxygen balance from the chemical formula of each molecule. Oxygen balance is a measure of the relative weight of oxygen in deficiency, or excess, of what is required to burn all carbon to carbon dioxide and all hydrogen to water.^172,173 Explicitly, the oxygen balance is computed using

1

where MW is the molecular weight of the compound; n_C, n_H, and n_O are the numbers of carbon, hydrogen, and oxygen atoms in the compound; and n_M is the number of metallic atoms that form metallic oxides. The atomic connectivity within each molecule was evaluated using a condensed version of the classification scheme developed originally by Kober et al.⁴ Descriptors relating to energetic and electronic properties of the molecules were evaluated using semiempirical density functional tight binding (DFTB) theory with the lanl31 parametrization;^174−176 see the following subsections for more details of the of the computed descriptors. Lastly, we considered descriptors that were outputs of the CHEETAH software from Lawrence Livermore National Laboratory software that predicts explosive performance properties, including mechanical energy of detonation, detonation velocity, and reaction products for over 100 unique molecules. The correlations observed with the properties calculated by CHEETAH were ultimately found to be encompassed by the correlations with heat of explosion and with oxygen balance.

Table 3. Summary of Descriptors.

Predictor(s)	Description
Lab	Laboratory where drop-weight impact test was performed
Grit	Identifies whether the test series used a bare anvil or an anvil covered with grit paper
Method	Identifies the statistical method used to analyze the drop-weight impact series
Functional group	Most sensitive explosive group
C, . . ., Si	Number of atoms of each element in the molecule
C[C], . . ., C[NNNN]	Atomic connectivity
Q (per g)	Specific heat of explosion computed from the difference in the heat of formation of the reactant and products
N, O, and NO groups	Number of functional groups with an N (N group), O (O group), or an N and O (NO group)
Mol mass	Molecular mass of energetic molecule (amu)
Oxygen balance	Oxygen balance (1)
Heat of formation	Gas phase heat of formation of energetic molecule computed at the DFTB-lanl31 level of theory176
Dipole	Molecular dipole (eA)
Max change	Maximum charge on any atom (e)
Min change	Minimum charge on any atom (e)
Atomization energy	Atomization energy (eV) computed at the DFTB/lanl31 level of theory
Normalized atomization energy	Normalized by number of atoms (eV/atom)
Band energy	DFTB band energy (eV)
Coulomb energy	DFTB Coulombic energy (eV)
HOMO–LUMO gap	DFTB HOMO–LUMO gap (eV)
Moment1, . . ., Moment4	Moments of the DFTB electron densities of states
ZPE (kJ/g)	Vibrational zero point energy per g (kJ/g)
ZPE (kJ/mol)	Vibrational zero point energy per mole (kJ/mol)
Cv (J/mol K)	Heat capacity per mole (J/mol K)
Cv (J/g K)	Heat capacity per g (J/g K)
H donor	Number of hydrogen bond donors per molecule
H acceptor	Number of hydrogen acceptors per molecule

Functional Group and Other Categorical Descriptors

To examine the effect of chemical composition on sensitivity, we sorted measured drop energies (logE₅₀), by the presence of functional groups. The functional groups we considered were peroxides (O₂), azides (N₃), nitrate esters (ONO₂), nitromines (NNO₂), and nitros (CNO₂). In this portion of the study, we allow sensitivities to be double counted; that is, if a sample had both CNO₂ and NNO₂ functional groups, we classified its sensitivity for both groups, CNO₂ and NNO₂. Table 2 shows the number of observations for each functional group, and/or pair of functional groups, as appropriate.

Table 2. Number of Observations for Each Functional Group^a.

	O2	N3	ONO2	NNO2	CNO2
O2	2	5	0	0	11
N3	5	19	1	10	23
ONO2	0	1	538	8	5
NNO2	0	10	8	397	84
CNO2	11	23	5	84	367

^aThe diagonal shows the number with a single functional group, and off diagonal entries count those observations with multiple functional groups.

Previous studies have shown that the test setting, such as the test prescription method, the laboratory where the test was performed, and whether grit was used on the anvil or not, may be predictive of observed drop energies.³⁶ So, in addition to the “functional group” variable, we included the test method,^177,178 the laboratory, and the use of grit as categorical descriptors. The test prescriptions consisted of the Bruceton (746), Neyer (759), and BAM (28) methods. Drop energy measurements from tests with unspecified prescription methods were assumed to be Bruceton (for tests from before 1995) or Neyer (after 1995). We split the laboratories where the tests were performed into the two laboratories with the most tests (LANL and NRL, with 870 and 204 tests each, respectively) and an “other” laboratory category (440 tests). Finally, measurements that did not specify whether grit paper was used were assumed to be have used grit (1249 total tests), compared with 284 tests without grit (see Table 3).

Connectivity

The atomic connectivity within each molecule was evaluated using a condensed version of the classification scheme developed originally by Kober et al.⁴ Here, atoms are considered to be covalently bonded when their interatomic distance is less than 120% of the sum of their covalent radii. We then denote the coordination environments of atom X as X[A], X[AB], X[ABC], and X[ABCD], which count the number of instances where atoms of species X are directly bonded to between one and four atoms of species A–D, respectively. We do not report unphysical coordination environments, such as H[CCCC], which would correspond to a hydrogen atom that forms covalent bonds to four carbon atoms. By limiting the number of atoms in the coordination environments to one for H (i.e., H[A]), two for O (O[A] and O[AB]), three for N (N[A], N[AB], and N[ABC]), and four for C (C[A], C[AB], C[ABC], and C[ABCD]), we produce a set of 115 unique atomic coordination environments for up to four bonded atoms. Finally, we count only the number of unique environments and exclude those associated with trivial permutations of the atoms. Illustrative examples of the coordination environments for a handful of simple molecules are provided below:

• Hydrogen dimer, H₂: H[H] = 1, all other environments = 0

• Water, H₂O: H[O] = 2, O[HH] = 1, all others = 0

• Methane, CH₄: C[H] = 4, C[HH] = 6, C[HHH] = 4, C[HHHH] = 1, all others = 0

• Aminomethanol, NH₂CH₂OH: C[H] = 2, C[N] = 1, C[O] = 1, H[N] = 2, H[O] = 1, C[HH] = 1, C[HN] = 2, C[HO] = 2, C[NO] = 1, N[CH] = 2, N[HH] = 1, O[CH] = 1, C[HHN] = 1, C[HHO] = 1, C[HNO] = 2, N[CHH] = 1, C[HHNO] = 1, all others = 0

DFTB-Calculated Properties

Descriptors relating to energetic and electronic properties of the molecules have been evaluated in the gas phase using semiempirical density functional tight binding (DFTB) theory with the lanl31 parametrization.^174−176 The lanl31 DFTB parametrization for molecules containing C, H, N, and O was demonstrated to exhibit near DFT accuracy at a much smaller computational cost, which makes the combination of model and parameters ideal for rapidly computing the properties of large numbers of molecules containing tens of heavy atoms.

The total energy in DFTB theory is a sum of three contributions,

where E_band = 2 Tr((P – P₀)H₀) describes the cohesion arising from the overlap of valence orbitals on neighboring atoms. Here, Tr(X) denotes the trace of matrix X; H₀ is the two-center, Slater-Koster tight binding Hamiltonian;^179,180P₀ is the density matrix for neutral, noninteracting atoms; and P is the self-consistent density matrix computed from the DFTB Hamiltonian,

where S is the overlap matrix; i labels atoms; and α labels orbitals. V are electrostatic potentials,

arising from the set of atom-centered Mulliken partial charges, where U⁽ⁱ⁾ is the Hubbard U parameter for the species at site i; N is the total number of atoms; and γ_ij is a screened Coulomb potential.¹⁷⁴E_qq is the Coulombic energy arising from the set of Mulliken partial charges, that is,

where

and n_i is the number of valence electrons assigned to neutral atom i. Finally, E_rep is a sum of pairwise terms, Φ(R_ij), that provide strong repulsion at short interatomic distances. We used the three terms in the total DFTB energy, E_band, E_qq, and E_rep, as descriptors for the energetic molecules owing to their clear and unambiguous physical meanings. The atomization energy,

which is equal to the change in energy upon completely separating all of the atoms in a molecule, was also computed using DFTB/lanl31 and used as a descriptor. Here, is the energy of noninteracting atom i in its ground state electronic configuration (which includes spin multiplicity), and u is the total energy of the molecule.

Heat of Formation

The gas-phase heat of formation, ΔH_f°, of each molecule was computed using the total DFTB energy, u, at the self-consistent electronic ground state with the four parameter atom equivalent energy scheme described in Cawkwell et al.,¹⁸¹ that is,

where η_l is the number of atoms of species l in the molecule, and ϵ_l is the corresponding atom equivalent energy. The total energies, u, were obtained following a geometry optimization until the maximum force acting on any atom was less than 0.001 eV/Å.

Charge and Electrostatic Properties

The self-consistent charge DFTB formalism yields a set of atom-centered Mulliken partial charges, {q}. We have used the self-consistent Mulliken charges at the optimized geometry of each molecule to compute (i) the maximum and minimum Mulliken partial charges in the molecule, q_max and q_min, and the molecular dipole moment,

where N is the number of atoms in the molecules, and R_i are the Cartesian coordinates of atom i.

Properties of Electronic Eigenspectra

We have computed a series of descriptors of the ground state electronic structures of the molecules at the DFTB level of theory. The allowable electronic energy levels of a molecule are equal to the eigenvalues, ε, of the DFTB Hamiltonian, H. The band energy is equal to the sum of the occupied energy levels,

where electronic, or molecular orbital, energy levels are occupied sequentially with the available electrons, with two electrons per level, starting from the lowest eigenvalue. The separation in energy between the highest occupied eigenvalue and lowest unoccupied eigenvalue is known as the HOMO–LUMO gap. The HOMO–LUMO gaps of explosive molecules have been investigated heavily with respect to sensitivity because the rupture of bonds typically involves the closure of, or excitation of electrons across, the HOMO–LUMO gap (see, for example, refs (182 and 183)). Therefore, we might expect molecules with small HOMO–LUMO gaps to be relatively sensitive. Hence, the DFTB HOMO–LUMO gap has been included in our set of electronic descriptors.

The distribution of the eigenspectrum controls the structural stabilities of molecules and materials. While binding energy as a function of structure can be obtained numerically from the eigenspectrum, {ε}, directly, the first few moments of the eigenvalue spectrum,

provide deep insights into the origin of structural stability. The moments theorem, which showed how the electronic eigenspectrum can be connected to molecular topology via hopping paths of length p that start and finish on the same orbital, was first developed by Cyrot-Lackmann and Ducastelle^184−186 and became the basis for subsequent major developments in linear scaling electronic structure theory.^{180,187−189} We have used moments of the DFTB eigenspectra up to and including p = 4 to construct descriptors to understand whether explosive sensitivity is connected to electronic structure and molecular topology in more profound ways than just through the relatively simple concept of the HOMO–LUMO gap.

The zeroth moment, μ₀, returns the total number of valence orbitals N_orb. The normalized first moment, , gives the center of mass of the eigenspectrum. The subsequent moments are calculated with respect to , that is, the eigenspectrum is shifted so that . The normalized second moment, , gives the mean square width of the eigenspectrum, and the dimensionless third moment, , measures the skewness of the distribution. Finally, we use the normalized fourth moment, , to build the dimensionless shape parameter,

that measures whether the distribution is bimodal (s < 1) or unimodal (s > 1).¹⁹⁰ We refer to , , , and s as Moment1 through Moment4 in our study. The applications of the moments theorem to the electronic eigenspectra of PETN and 2,4,6-TNT are illustrated in Figure 1, which also serves to highlight that the HOMO–LUMO gap is just one of numerous possible descriptors of the electronic structures of organic molecules. The four moments-based descriptors derived from the eigenspectra of PETN and 2,4,6-TNT are given in Table 4. The eigenspectra of PETN and 2,4,6-TNT do not differ markedly; the mean square width, , of the spectrum of 2,4,6-TNT is slightly larger than that of PETN, and both exhibit positive skewness and a unimodal distribution, s ≈ 1. The vast majority of the explosive molecules in our data set exhibit a unimodal eigenvalue distribution with s ≈ 1, which we attribute to the absence of four-membered rings within their mostly flat 2D topologies that would promote a small s and a strong bimodal distribution.¹⁸⁸ Indeed, the shape parameter, s, (Moment4), was not ranked highly as a descriptor of explosive sensitivity, presumably because of the structural similarities among the small organic molecules sampled by our data set.

Figure 1.

Electronic eigenspectra of PETN and 2,4,6-TNT from the ground state DFTB electronic structure. The energies of the HOMO, LUMO, and center of mass, , are highlighted. 2,4,6-TNT has a slightly wider eigenspectrum than PETN, which is measured by the normalized second moment, . Both eigenspectra exhibit positive skewness, , which is larger for 2,4,6-TNT, and moderately unimodal distributions, s ≈ 1.

Table 4. Moments-Based Descriptors for PETN and 2,4,6-TNT.

	μ̂1 (eV)	μ̂2 (eV2)	μ̂3/μ̂23/2	s
PETN	–6.43	14.01	0.62	1.02
2,4,6-TNT	–3.42	16.48	0.76	0.96

Vibrational Modes

The vibrational normal modes, {ω}, were computed following a geometry optimization until the maximum force acting on any atom was less than 0.001 eV/Å by building and diagonalizing the force constant matrix.¹⁹¹ The vibrational zero point energy,

and heat capacity at T = 300 K,

on a per gram and per mole basis are computed from the sets of vibrational normal modes.

Numbers of H Bond Donors and Acceptors

The numbers of hydrogen bond donors and acceptors were obtained using the Chemical Identifier Resolver tool using the simplified molecular-input line-entry system (SMILES) string in the data, which is available from the U.S. National Institutes of Health at https://cactus.nci.nih.gov/chemical/structure.

Reaction Products

Reaction products were calculated according the oxidation priority established in the literature for detonation reactions.¹⁹² All nitrogen is assumed to be released as N₂. Oxygen is assumed to first convert all hydrogen to water vapor. Further oxygen converts all carbon to carbon monoxide followed by full oxidation to carbon dioxide. Excess oxygen is released as molecular oxygen, O₂. In oxygen deficient compounds, excess hydrogen and carbon are released in molecular form as H₂ and soot (C). Heats of explosion (Q) were calculated through comparison of experimental gas phase heats of formation, ΔH_f°, of the products,¹⁹³ with heats of formation of the reactants, calculated by the DFTB method explained above.

Methods

Correlation Analysis

Before modeling impact sensitivity as a function of all descriptors, we evaluated the relationship of each descriptor with drop energy in isolation (and the relationship of each pair of descriptors). To evaluate the relationship of the continuous- and integer-valued descriptors with drop energy, we used the Spearman correlation. The Spearman correlation quantifies the ordinal correlation between two continuous quantities, allowing for strong correlation when one variable is a monotonic function of another (rather than exclusively linear functions as in the Pearson correlation). We used the coefficient of determination in a one-way analysis of variance (ANOVA) model to evaluate the relationship between continuous measurements and the categorical descriptors. We emphasize that the correlation analysis has no bearing on subsequent variable selection.

Predictive Modeling

We modeled drop energy as linear, and nonlinear, functions of the descriptors. Due to the large number of descriptors and drop-weight impact tests, we wish to employ methods that simultaneously estimate a model for drop energy and subset the descriptors to only the most useful descriptors. Models with fewer descriptors are not only easier to interpret but also are less likely to “overfit” by, for example, incorporating random variation present in the data into the learned model. The methods we employ estimate the following regression model:

2

where i refers to each observation; y_i is the sensitivity of observation i; and x_i is a vector of p descriptors for observation i. The function f(.) is the object of inference; the random errors ϵ_i are assumed to be mean zero and have constant variance. These assumptions imply that variation in drop-weight impact tests outside of the descriptors x_i is not systematic, and departures from these assumptions will degrade performance, for example, if certain explosive materials have less variable drop-weight impact tests than others, or if certain methods of determining “Go” do not correlate with others.

LASSO

The least absolute shrinkage and selection operator (LASSO) is a method for estimating a linear function f(.) to represent the sensitivity function.¹⁹⁴ Explicitly,

3

where β is a vector of coefficients to be estimated. The LASSO selects the coefficients β to trade off the number of nonzero values (and thus the complexity of f(.)) with model fit. It does so by minimizing the criterion

4

where |β|₁ denotes the sum of absolute values of nonzero entries in β. The parameter λ > 0 controls the trade-off between model fit and model complexity and is selected using cross-validation. The λ that predicts best out-of-sample (or nearly the best) is chosen to guard against overfitting. The LASSO optimization problem is known to have a unique solution, so that it chooses the one best β value that minimizes the optimization criterion for a given λ. The LASSO is a fast method and is known (for sufficiently large λ) to set some entries in β to zero, effectively removing descriptors from the model. Drawbacks include that the function f(.) is constrained to be linear (although this type of model is easier to interpret) and, in the canonical setting, does not include interactions between the descriptors. We use the LASSO method as implemented in the R statistical programming language.¹⁹⁵

Random Forest

The random forest method represents f(.) as a sum of decision tree models.¹⁹⁶ Thus, f(.) is a highly flexible piecewise constant function. Each tree is trained on a random subset of the data with a random subset of the number of descriptors. The number of trees and number of variables are tuning parameters to be selected by the user, although the defaults often work well for a range of applications. We explore the choice of various hyperparameters for the random forest method(s) in the Supporting Information,^197,198 finding that the defaults (p/3 descriptors and minimum number size of terminal node of 5) perform as well as any other settings examined. Trees are learned in an optimal way that controls the out-of-sample performance. That is, splits are added to the tree using the variable that reduces the out-of-sample mean-square error the most,

5

where Θ is the set of observations under consideration. The random forest optionally reports the typical decrease in this error as each descriptor is added to each tree, where large decreases in error suggest that the descriptor is useful in predicting the response y_i. For more details on the random forest method, see Hastie et al.,¹⁹⁹ for example.

Random forest models are quite flexible, where f(.) is inherently nonlinear and includes interactions. However, this flexibility makes interpretation of the estimated function f(.) challenging. Further, the random searching for the “best” function f(.) is not guaranteed to have a single solution, such that another function g(.) may perform just as well.

Unlike the LASSO method, canonical random forests do not automatically subset the descriptors to the most useful ones—although the random forest method does offer some notion of variable “importance”—and such subsetting may not be necessary for good performance.²⁰⁰ In hopes of improving interpretability, and possibly to improve predictive performance, we employ a method for reducing the descriptors to the most useful ones for random forests. This method, variable selection using random forests (VSURF),²⁰¹ takes several passes through the data to select the smallest subset of descriptors that effectively represent the relationship between drop energy and the descriptors. We use the importance measure native to random forest methods to interpret the relative contribution of each (selected) descriptor to the prediction of drop energy. We use the random forest and VSURF methods as implemented in the R statistical programming language.^202,203

Model Evaluation

Evaluation of predictive performance on the drop-weight impact data is challenging due to the imbalance of molecules represented in the data, and the stochastic nature of estimating E₅₀ from relatively few binary measurements. In hopes of addressing these challenges, we compared predicted values of drop energy to the mean drop energy across observations of molecule k, for each of the unique molecules.

An additional consideration in evaluating predictive performance is that we are more certain of for molecules which we have large number of observations, such as PETN, where the number of observations m_k = 490. Hence, we should weight errors of molecules for which m_k is large higher than errors for those which have m_k = 1. We introduce a weighted root-mean square error,

6

where w_k is the weight for molecule k, and is the predicted drop energy for molecule k.

When w_k = m_k/∑_km_k, each error is weighted by the number of times molecule k was observed, whereas when m_k = 1/K, we ignore the multiplicity of observations for each molecule. Generally, we seek a compromise between these two extremes. The extreme unbalance in the data makes weighting by w_k = m_k/∑_km_k seem too strong: should roughly 1/3 of the error criterion be generated by a single molecule (PETN)? Or should molecules with two or three observations count double those with a single observation? And yet, completely ignoring the multiplicity of each molecule does not recognize the true structure of the data, nor the fact that we are more certain about some values of than others. Thus, we define η ∈ [0, 1] as a mixing parameter between the extremes w_k = m_k/∑_km_k and m_k = 1/K, so that the desired predictive model should minimize the following RMSE over a broad range of η:

7

We evaluated the fit of each predictive model using 10-fold cross-validation. Here, 10-fold cross-validation is an approach that is meant to evaluate predictive performance of each model on data outside of that used in estimating each model, thus mimicking the use of the model to design new molecules, and is a canonical machine learning technique.²⁰⁴ We randomly partitioned the data set into 10 “folds” of roughly equal numbers of unique molecules (no molecule was allowed to reside in more than one fold). For each fold, we selected variables (LASSO and VSURF) and estimated each predictive model (LASSO, random forest without subsetting descriptors, and VSURF) using the data outside the fold and generated the predictions from each model within the fold. Repeating for each fold, we obtained an out-of-sample prediction for each unique molecule in the data set. We fit models to the complete data set (including repeat observations) and to a data set consisting of each unique molecule and only its mean drop energy across repeat observations. The former approach implicitly weights predictions toward molecules for which we have more observations and are thus more certain about the true drop-weight impact sensitivity; we allow predictive performance to decide which approach is best for these data. We computed the weighted RMSE in (eq 7) for 1000 equally spaced values of η ∈ [0, 1]. We repeated the entire procedure 10 times, with 10 different partitionings of the unique molecules. Averaging across the 10 random partitions and 1000 values of η gave a final error criterion, , which we used to compare predictive models.

Results

Correlation Analysis

We studied five major functional groups: peroxides (O₂), azides (N₃), nitrate esters (ONO₂), nitromines (NNO₂), and nitros (CNO₂). We found that about 86% of observations had exactly one major functional group, about 9% of observations had two or more of the major functional groups (see Table 2), and about 5% had none of the major functional groups. Following Mathieu,¹⁷¹ we labeled each observation with the most reactive functional group (where “reactivity” corresponds to the lowest median drop height by functional group). That is, in a hypothetical molecule with ONO₂ and NNO₂ functional groups, the “group” label for this molecule was “ONO₂”. Figure 2 shows the estimated sensitivities separated by functional group. As expected, the functional group does appear to affect impact sensitivity, explaining about 40% of the variation in sensitivity (as measured using coefficient of determination, commonly denoted R²). Peroxides and azides were the most sensitive, and molecules with aromatic nitro groups (and/or none of the major functional groups) are the least sensitive.

Figure 2.

Drop energy trends with the strongest related categorical and continuous descriptors, functional groups, and oxygen balances, respectively.

The Spearman correlations among continuous predictors are displayed in Figure 3, for the descriptors with the largest correlations with sensitivity. We observe that oxygen balance has the highest correlation with sensitivity, followed by the first moment of the distribution of electron states, and the heat of explosion, Q, has the third largest correlation with sensitivity. We notice that correlations between descriptors are often at least as large as the correlations between the descriptors and drop energy, logE₅₀.

Figure 3.

Spearman correlations of descriptors most strongly correlated with logE₅₀, ordered by the absolute value of this correlation.

In Figure 2, we plot drop energy against oxygen balance (separated by functional group), which is the descriptor with the strongest correlation with drop energy. We observe that the trend is negative for most functional groups and nonlinear in general. The general negative trend indicates that molecules with higher oxygen balance (more performant) have lower drop energies (and thus higher sensitivity). Moreover, the strength of the trend varies by functional group. Drop energies of peroxides have little trend with oxygen balance, while drop energies of nitro compounds have a much stronger trend. This fact suggests that an interaction between functional group and oxygen balance may be appropriate. Finally, we point out that the presence of vertical stacks of points indicates molecules with many repeat observations. These molecules have the same oxygen balance, but different measured drop energies.

Predictive Performance

The estimated performance, measured by discussed in the previous section, of the predictive models is given in Table 5. Both random forest and VSURF models outperform the linear LASSO method by a substantial margin, suggesting that there may be some nonlinearity in the true function relating the descriptors and the observed drop energies. We also notice a slight improvement in the VSURF method over the random forest method, especially when training on all the data, including repeats. VSURF achieves this performance improvement with far fewer descriptors as well (only 19 when selecting on the complete data set, compared to all 156). Finally, in Figure 4, we plot the mean RMSE_η (see eq 7) across random partitions as a function of η and see that the VSURF method trained on all data performs best for the broadest range of η. The value of for the selected method corresponds to an approximate coefficient of determination R² ≈ 0.64 (see Supporting Information for computation of this value from ). The performance of the VSURF method trained on all data, along with the limited number of descriptors, leads us to select this method for further study and interpretation.

Table 5. Performance of Predictive Models As Measured by in 10 10-Fold Cross-Validations^a.

	Means data	All data
LASSO	0.321	0.322
RF	0.305	0.307
VSURF	0.304	0.303

^aThe models were trained on data consisting only of mean drop energies of unique molecules (“Means data”) and on data including all repeat observations (“All data”).

Figure 4.

(Left) Average RMSE_η across random cross validation partitions, as a function of η. Dashed lines denote models trained on data consisting only of mean drop energies of unique molecules (“Means data”), and solid lines denote models trained on data including all repeat observations (“All data”). (Right) Average predicted drop energy (logE₅₀), across 10 random cross-validations, from VSURF method trained all data, compared to observed mean drop energy for all unique molecules. Molecules with five or more repeated observations are highlighted in color.

The observed mean values of drop energy, logE₅₀, are compared to the predicted mean values (for the VSURF method trained on all data) in the right panel of Figure 4. We emphasize that the predictions represent out-of-sample predicted mean drop energies from the cross-validation study (averaged across random partitions). Generally, we see good agreement between the observations and the predictions from the model, especially for the molecules with several repeat observations.

We expect that the of about 0.30 to be roughly representative of the accuracy of the selected VSURF predictive method. Concretely, we expect to predict (mean) drop energy within a multiplicative factor of 10^2(0.3) ≈ 4, where the factor of “2” results from choosing error bounds two standard deviations from the predicted value. Since the measurement error of the drop-weight impact test (for a single test) has been reported to be accurate to a multiplicative factor of 1.5–3 in Marrs et al.,³⁶ the degree of error in the predictive model is not unexpected.

Important Descriptors

To determine which descriptors are most influential in predicting drop-weight impact sensitivity, we trained the best-performing method, VSURF, to the entire data set. In Table 6, we report the descriptors selected when using the VSURF method and their estimated importance. To compute the importance, we used the permutation method described in Fisher et al.²⁰⁵ Briefly, we randomly permuted each descriptor and then recomputed , for 1000 random permutations for each descriptor. The mean increase in (see Table 6) for a given descriptor corresponds to how “important” that descriptor is for the model, where large increases correspond to higher importance.

Table 6. Descriptor Importance of VSURF Regression Model, Ordered by Decreasing Importance^a.

	VSURF	Spearman ρ
Oxygen balance	0.111	–0.668
Moment1	0.100	0.664
Functional group	0.095	0.659
Q (kcal/g)	0.081	–0.668
ZPE (kJ/g)	0.054	0.280
H acceptor	0.054	0.589
Moment3	0.054	0.193
Max charge	0.040	0.493
Moment2	0.039	–0.543
Gas C	0.039	0.085
Band Energy	0.033	–0.582
Lab	0.033	0.560
HOMO–LUMO gap	0.031	0.577
N[CO]	0.024	0.178
Min charge	0.024	0.326
N[CHH]	0.021	0.141
N[N]	0.020	0.196
Method	0.005	–0.432
Grit	0.005	0.415

^aImportance corresponds to the expected decrease in when each descriptor is randomly permuted. The last column gives the Spearman correlation between the descriptor and drop energy (in log J).

Discussion

One caveat of our analysis is that the data analyzed are observational and are not the result of designed experiments to examine the effect of particular descriptors on observed drop energy. Further, as discussed in the Results section, the descriptors are largely correlated. As a simple example, the count of oxygen atoms in a given molecule is related to the oxygen balance of that molecule (since the number of oxygen atoms is used to compute the oxygen balance). Thus, our results can be interpreted as primarily correlational. However, by analyzing a large data set of a broad range of explosives, and by restricting learning to the most parsimonious models, we are able to make conclusions regarding the most important chemical and physical factors (out of those available) for estimating drop energy.

To examine the effects of correlated descriptors on the importance of said descriptors selected by VSURF in Table 6, we removed each descriptor in order of importance and then refit and reevaluated the selected model. Hence, we fit the model corresponding to Table 6 with all descriptors except oxygen balance, then with all descriptors except oxygen balance and Moment1, and so on. We also worked from the other direction, removing Method, then Method and NCO, and so on. We used each submodel in the 10 10-fold cross validations to determine , as in the model selection study. The for the submodels is plotted for each model in Figure 5. We notice immediately that the inclusion of the first four descriptors (oxygen balance, Moment1, Q, and functional group) account for most of the predictive performance of the model. The inclusion of further descriptors does improve performance, but only slightly. On the other hand, the removal of oxygen balance, Moment1, Q, and functional group leaves model performance largely unchanged. Although at first these facts may appear contradictory, they suggest that, although oxygen balance, Moment1, Q, and functional group succinctly describe the influence of chemical structure on impact sensitivity, all the information contained within these descriptors is also contained within other descriptors. This result may have been expected, since oxygen balance, Moment1, Q, and functional group are all computed from descriptors that are present in the set of descriptors considered. However, this study does illustrate the challenge in interpreting importance of the descriptors. For example, it is unclear whether heat of formation is selected for the model because it may be correlated with oxygen balance and Q, or because it affects sensitivity directly.

Figure 5.

Mean cross-validation performance of the selected model when adding each descriptor in turn (“forward”) and removing each descriptor in turn (“backwards”).

The estimated mean drop energy from the selected VSURF model is, in general, a nonlinear function that allows many interactions of the descriptors. One way to envision the average effect of each descriptor on the estimated drop energy, as predicted by the selected model, is to use the accumulated local effects.^205,206 The estimated average effects of oxygen balance and Moment1, separated by functional group, on drop energy are shown in Figure 6 (it is challenging to interpret plots that include the effect of more than two descriptors on drop energy). The general trends across all the data agree with what we expect: drop energy decreases (sensitivity increases) with increasing oxygen balance and with decreasing Moment1. However, these trends are different for each functional group. For example, the peroxides show little trend of drop energy with either oxygen balance or Moment1. The effect of oxygen balance appears stronger for molecules with nitro groups than the trend for molecules with nitrate esters. Finally, we note that both Moment1 and oxygen balance show saturation effects for large drop energies (low sensitivities). We cannot determine whether this is a real physical phenomenon—that insensitive molecules perform the same beyond a certain sensitivity threshold—or an artifact of limitations in drop-weight testing, where there is some maximum drop height (320 cm at LANL), and hence maximum drop energy that can be measured.

Figure 6.

Estimated average effects of oxygen balance (left) and Moment1 (right), separated by functional group.

We examined the residuals to see where the model might predict incorrectly. In general, there is little trend of any of the selected descriptors with residuals. The exception is functional group, where the model estimates peroxide and azide groups as more sensitive than they truly are, and slightly underestimates sensitivity for “other” molecules (see Figure 7). The model most likely favors functional groups with many observations—nitros, nitromines, and nitrate esters—over those with few observations. Although the fact that the model performs less well on azide, peroxide, and other functional groups is not unexpected, one should take care in using predictions for molecules with these functional groups. Performance of the model could be improved by collecting more data for these underrepresented functional groups.

Figure 7.

Residuals (observed drop energy less predicted drop energy) for each functional group.

Conclusions

We have collected and analyzed a large and diverse data set of drop-weight impact tests available in the literature. To mitigate the variability inherent in the drop-weight impact test, we have collected as many replications of drop-weight impact tests on the same molecules as were available. We illustrated that, despite the variability of the drop-weight impact test, we can predict drop energy relatively well across a broad range of explosives. We estimate that the predictions from the model come within a multiplicative factor of 4 of the mean drop energy for a hypothesized, unobserved molecule, corresponding to an out-of-sample coefficient of determination of about R² ≈ 0.64. Our data set includes hundreds of repeat molecules on a variety of explosives, and we find that the variation of values for each individual explosive is substantial. This study also indicates that caution must be taken when drawing conclusions about important molecular properties when analyzing drop-weight impact test results, because many of the important variables appear to be interrelated.

We have illustrated the difficulty in interpreting the structure of the estimated random forest model. However, the model selects several descriptors convincingly: oxygen balance, Moment1, Q, and functional group; oxygen balance and Q are known to be predictive of explosive performance. We feel confident in concluding that, generally, more performant explosives are more sensitive, which confirms reports elsewhere.^{8,10,171,207,208} We postulate that the functional group descriptor likely relates to the ease of initiating reaction in the material.¹² Although it has been hypothesized that the HOMO–LUMO gap is important in determining molecular properties, the moments of the electronic eigenspectrum, especially Moment1, which gives a more complete picture of the electronic structure, appear to correlate much better with drop-weight impact sensitivity.

The interpretation of other descriptors may be even more challenging. For example, band energy (the sum of occupied molecular orbital energies) is selected by the VSURF method. However, band energy is also highly correlated with oxygen balance and Moment1. It is unclear whether band energy is selected because of these correlations, because band energy influences sensitivity in some other way, or because all are calculated from similar inputs. Although we can only learn an interpolative model from this large data set, we know how each descriptor is computed from other data streams. Future work should explicitly encode these relationships. For example, encoding that the number of oxygen atoms feeds into the oxygen balance calculation can allow one to test whether the number of oxygen atoms influences sensitivity only through oxygen balance, or if there is additional influence. A model with the known physical hierarchy encoded should improve our ability to be confident in drawing conclusions regarding the chemical descriptors of sensitivity based on the best predictive model fit to observational data.

Data Availability Statement

The data used in this article are publicly available as Supporting Information. All methods are existing and available in open source software, and thus, we do not provide reproduction software. All statistical analysis was performed using the programming language R, version 4.1.3, available at https://cran.r-project.org/src/base/R-4/. Key packages for reproduction, and their versions, are randomForest (4.6-14), VSURF (1.1.0), glmnet (4.1-3), ALEPlot (1.1), and mltools (0.3.5). All packages are available online at https://cran.r-project.org/. All of the density functional tight binding calculations were performed using the open-source code LATTE, which is available for download from https://github.com/lanl/LATTE. The lanl31 parametrization is supplied with the LATTE source via the GitHub repository and in Cawkwell and Perriot.¹⁷⁶ Open Babel version 3.1.0 was used to convert SMILES to three-dimensional geometries in the standard .xyz format.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jcim.2c01154.

• Supplementary results, simulation studies, and additional details (PDF)

• All data (XLS)

The authors declare no competing financial interest.