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Research Article  |  Open Access  |  21 May 2024

Searching new cocrystal structures of CL-20 and HMX via evolutionary algorithm and machine learning potential

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J Mater Inf 2024;4:5.
10.20517/jmi.2023.37 |  © The Author(s) 2024.
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In this work, we report the discovery of energy cocrystals using an efficient iterative workflow combining an evolutionary algorithm and a machine learning potential (MLP). The compound 2,4,6,8,10,12-Hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane (CL-20) has attracted significant attention owing to its higher energy density than traditional energetic materials. However, the higher sensitivity has limited its applications. An important way to reduce its sensitivity involves cocrystal engineering with traditional explosives. Many cocrystal structures are expected to be composed of these two components. We developed an efficient iterative workflow to explore the phase space of CL-20 and 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane (HMX) cocrystals using an evolutionary algorithm and an MLP. The algorithm was based on the Universal Structure Predictor: Evolutionary Xtallography (USPEX) software, and the MLP was the reactive force field with neural networks (ReaxFF-nn) model. A set of high-density cocrystal structures was produced through this workflow; these structures were further checked via first-principles geometry optimizations. After careful screening, we identified several high-density cocrystal structures with densities of up to 1.937 g/cm3 and HMX: CL-20 ratios of 1:1 and 1:2. The influence of hydrogen bonds on the formation of high-density cocrystals was also discussed, and a roughly linear relationship was found between energy and density.


Crystal search, machine learning, ReaxFF-nn, evolutionary algorithm


Crystal structure prediction (CSP) techniques enable the design of new crystal structures with a given chemical composition. However, the high computational costs of density functional theory (DFT) calculations limit their applications in molecular systems, whose unit cells usually contain dozens or even hundreds of atoms. Recently developed machine learning potentials (MLPs), such as the Gaussian approximation potential (GAP)[16], moment tensor potential (MTP)[79], and DeepMD-kit package[1013], can perform DFT-level calculations at the computational cost of classical force fields. CSP methods, such as Universal Structure Predictor: Evolutionary Xtallography (USPEX)[14,15] and Crystal structure AnaLYsis by Particle Swarm Optimization (CALPSO)[16], combining MLPs[1723] will undoubtedly become the main approaches in the search for new materials. The work by Podryabinkin[24] on CSP combining an MLP and an evolutionary algorithm showed that an accuracy of 11 meV/atom could be achieved in the known structure predictions. It is clear that MLPs[25] can accelerate CSP by replacing most DFT calculations. Although attempts[26,27] have been made to use the classical force field as a local optimizer for structure minimization, the parameter set used by the force field cannot be dynamically adjusted according to the search results and, therefore, cannot guarantee the predicted structure to be the best one. In contrast, machine learning-based methods can provide predictions with quality close to that of first-principles calculations at the computational costs of classical techniques and thus can deal with large-scale[28] systems. In addition, the parameter set can be dynamically updated through a search–check–train–search workflow developed by our group. A similar workflow is used in the active learning procedure used to train deep-learning potential models, such as the Deep Potential GENerator (DP-GEN)[29] and Fast Learning of Atomistic Rare Events (FLARE)[30] software, which contains three Exploration–Labeling–Training iterative steps. In the Exploration step, a sample of trajectories is generated using methods such as molecular dynamics (MD) simulations and randomly sampling the crystal phase space. The search step in our workflow is similar to the Exploration of the crystal space group by the USPEX software. In the Labeling step, and correspondingly in our check step, the ab initio energies are calculated for the configurations generated in the previous step, and the configurations with a large variance are added to the training dataset. In the last step, the model is trained, and a new parameter set is generated. In this work, we combined MLP and evolutionary algorithm to identify new cocrystal structures of 2,4,6,8,10,12-hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane (CL-20)[31] and 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane (HMX)[32], aiming to find an optimal balance between their energy and safety. CL-20 is known for its high detonation power; however, significantly reducing its sensitivity is a great challenge. Effectively addressing this problem involves combining CL-20 with other low-sensitivity materials, producing more attractive explosive cocrystal materials[3337]. A highly promising CL-20:HMX (2:1) cocrystal was synthesized by Bolton et al.[38]; the cocrystal exhibits superior energetics to HMX while maintaining comparable impact sensitivity. Many cocrystal structures composed of CL-20 and HMX are expected to exist because of the large number of crystal structures the single components can form.

As thousands of crystal structures will need to be optimized in one search, using the DFT method as the local optimizer will require years to complete. The machine learning-based approach will speed up the search and provide similar accuracy, as the parameter set is dynamically updated after the search is corrected by DFT calculations. The MLP will "learn" the interactions between molecules and give more accurate predictions. A workflow combining the USPEX evolutionary algorithm and the ReaxFF-nn MLP was developed to explore the configuration space of CL-20 and HMX. Using this approach, we report the discovery of high-density cocrystal structures of HMX and CL-20 with 1:1 and 1:2 ratios. Furthermore, there are discussions on whether C–H...O hydrogen bond interactions[39] exist in crystal structures; hence, the role of C–H...O hydrogen bond interactions in the cocrystal structures is discussed by analyzing hydrogen bond energies.


ReaxFF-nn machine learning potential

The ReaxFF reactive force field, originally introduced by van Duin et al.[40], has been extensively used[41,42] in reactive MD simulations[4351]. We developed[52] an MLP model labeled ReaxFF-nn based on ReaxFF with neural networks for bond orders in previous work[53]. This significantly improved various calculations, such as equations of states, reaction energies, and potential energy surfaces. The total energies for ReaxFF and ReaxFF-nn are respectively given in:

$$ \begin{aligned} E_{system}^{ReaxFF} = E_{bond} + E_{lp} + E_{over} + E_{under} + E_{val} + E_{pen} + E_{coa} + \\ E_{tors} + E_{conj} + E_{Hbond} + E_{vdWaals} + E_{Coulomb}, \end{aligned} $$

$$ E_{system}^{ReaxFF-nn} = E_{bond}^{nn} + E_{val} + E_{tors} + E_{Hbond} + E_{vdWaals} + E_{Coulomb}. $$

where $$ E_{bond} $$, $$ E_{lp} $$, $$ E_{over} $$, $$ E_{under} $$, $$ E_{val} $$, $$ E_{pen} $$, $$ E_{coa} $$, $$ E_{tors} $$, $$ E_{conj} $$, $$ E_{Hbond} $$, $$ E_{vdWaals} $$ and $$ E_{Coulomb} $$ correspond to the bond, lone pair, over coordinate, under coordinate, valence angle, penalty, three-body conjugation, torsion rotation barrier, four-body conjugation, hydrogen bond, van der Waals and Coulomb energy terms, respectively[40].

In the current form, as given in Equation (2), neural networks are employed to compute short-range interactions, while long-range interactions, such as Coulomb and van der Waals terms, still use the same formula as ReaxFF. Because most MLPs use prespecified static point charges, they focus on short-range interactions[5456] and have difficulties dealing with long-range Coulomb interactions, which remain a major challenge for these potentials at present. An exception is represented by ReaxFF, and, similarly, ReaxFF-nn, which use charges dynamically calculated by the electronegativity equilibrium[57] method, and further related approaches are expected to be developed in the future[58].

The corrected bond order $$ BO $$ReaxFF and ReaxFF-nn is respectively calculated by:

$$ BO^{ReaxFF} = BO' \cdot f_1(\Delta_i', \Delta_j') \cdot f_4(\Delta'_i, BO'_{ij}) \cdot f_5(\Delta'_j, BO'_{ij}), $$

where the $$ f_1 $$, $$ f_4 $$, and $$ f_5 $$ terms are given by the ReaxFF formula in Ref.[40],

$$ BO^{ReaxFF-nn} = BO' \cdot f_{nn}(\Delta_i', BO'_{ij}, \Delta_j'), $$

$$ f_{nn} $$ used by a hidden-layer neural network is calculated by:

$$ f_{nn}(x) = \sigma(w_o\cdot \sigma(w_h \cdot \sigma(w_i \cdot x + b_i) + b_h) + b_o). $$

where $$ x = (\Delta_i', BO'_{ij}, \Delta_j') $$ represents the input vector, $$ \Delta_i' $$ is the sum of the $$ BO_{ij}' $$ terms of $$ atom_i $$, and $$ BO_{ij}' $$ is the uncorrected bond order, as defined in Ref.[40]; moreover, $$ \sigma $$ is the logistic activation function $$ \sigma(x) = (1+\exp(-x))^{-1} $$, while $$ w $$ and $$ b $$ are the weights and biases of the neural networks, respectively.

The bond energies employed by ReaxFF and ReaxFF-nn are respectively given in:

$$ E_{bond_{ij}}^{ReaxFF} = D^{\sigma}_e \cdot BO_{ij}^{\sigma} \cdot \exp[p_{be_1}(1- BO_{ij}^{\sigma})^{p_{be_2}}] - D^{\pi}_e \cdot BO_{ij}^{\pi} - D^{\pi\pi}_e \cdot BO_{ij}^{\pi\pi}, $$

$$ E_{bond_{ij}}^{ReaxFF-nn} = D^{\sigma}_e \cdot f_{nn}^{e}(BO_{ij}^{\sigma}, BO_{ij}^{\pi}, BO_{ij}^{\pi\pi}). $$

where $$ D^{\sigma}_e $$, $$ D^{\pi}_e $$, $$ D^{\pi\pi}_e $$, $$ p_{be_1} $$ and $$ p_{be_2} $$ are adjusting parameters, and $$ BO_{ij}^{\sigma} $$, $$ BO_{ij}^{\pi} $$, and $$ BO_{ij}^{\pi\pi} $$ are bond order components.

In this case, $$ f_{nn}^{e} $$ represents a hidden-layer neural network that calculates the bond energy. The input vector for this network is the uncorrected bond order $$ BO_{ij}=(BO_{ij}^{\sigma}, BO_{ij}^{\pi}, BO_{ij}^{\pi\pi}) $$. The network output, multiplied by a scalable parameter $$ D_{e}^{\sigma} $$, gives the bond energy of the pair of atoms $$ atom_i $$ and $$ atom_j $$.

Training and testing the potential model

The quality of the dataset used for the MD interaction potential plays a crucial role in determining its accuracy. To ensure the complete sampling of the phase space of molecular structures such as CL-20,2,4,6-trinitrotoluene (TNT), 1,1-diamino-2,2-dinitroethylene (FOX-7), cyclotrimethylene trinitramine (RDX) and HMX, we manually collected a large amount of data using methods including ab initio MD simulations, stretching of specific chemical bonds, scanning covalent bond angles, and equations of state of a crystal structure. After DFT calculations, these data were stored in an atomic simulation environment (ASE)[59]Trajectory object, which can be interpreted as a time series containing many ASE Atoms objects moving in the configurational phase space. This data format is easy to prepare because the ASE package includes many DFT calculators, such as Vienna ab initio simulation package (VASP), Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA), and Quantum Espresso (QE). After performing calculations through these calculators, the Trajectory file can be generated automatically. One can also perform single-point energy calculations and gather the trajectory file after the calculations. The training function accepts a dataset variable, which is a Python dict object that contains all trajectory file names; moreover, the I-ReaxFF package can generate input information for the training function, such as collecting all bonds and all valence angles of the same type into variables that store this information. More details on the data preparation can be found in the documentation and examples of our I-ReaxFF package[53,60].

Figure 1A-C compares the lattice energies calculated by DFT and ReaxFF-nn as a function of the density for the $$ \beta $$-HMX, $$ \varepsilon $$-CL-20, and 1:1 HMX: CL-20 cocrystal. Scaling the box size in three directions changed the density from 1.4 to 2.4 g/cm3. Figure 1D and E shows the energies vs. the N–NO$$ _2 $$ bond distance calculated by DFT and ReaxFF-nn. Figure 1F compares the lattice energies of a batch of 100 cocrystal structures of 1:1 HMX: CL-20 generated by the USPEX software. The results demonstrate a good agreement between our ReaxFF-nn potential model and the DFT calculations. Although active learning algorithms[29,61,62] to collect data automatically are available, the application to molecular systems requires further tests, and we also plan to develop an uncertainty-driven active learning algorithm based on the Z-matrix.

Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 1. Lattice energy as a function of density of (A) $$ \beta $$-HMX, (B) $$ \varepsilon $$-CL-20, and (C) 1:1 HMX: CL-20 cocrystal; Energy of molecular (D) HMX and (E) CL-20 as a function of N–NO2 bond distance; (F) Comparison of DFT- and ReaxFF-nn-calculated lattice energies of 1:1 cocrystal structures randomly generated by USPEX. HMX: 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane; CL-20: 2,4,6,8,10,12-Hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane; DFT: density functional theory; ReaxFF-nn: reactive force field with neural networks; USPEX: Universal Structure Predictor: Evolutionary Xtallography.

After data collection, the ReaxFF-nn model was trained using the I-ReaxFF software package, and the overall precision of this dataset was approximately 94.0%, as calculated by:

$$ Accu = 1.0 - \frac{\sum{\left | E_{DFT} - E_{ReaxFF-nn} \right |}}{\sum(E_{DFT} - E_{ZPE})}, $$

where $$ E_{ZPE} $$ represents the zero-point energy.

To further evaluate the reliability of the ReaxFF-nn potential model, we optimized the crystal structures of $$ \beta $$-HMX (chair conformation[63]) and $$ \varepsilon $$-CL-20 (the most stable crystal form of CL-20[27,38,64]). A comparison of the calculated lattice constants with the experimental data and the first-principles values is presented in Table 1. The analysis shows that the calculated values obtained using ReaxFF-nn agree well with the experimental and first-principles data.

Table 1

The comparison of lattice constants of $$ \varepsilon $$-CL-20 and $$ \beta $$-HMX from experiment, DFT calculated values, and GULP geometry optimization with potential ReaxFF-nn

$$ \varepsilon $$-CL-20, Exp.[27,65]8.85212.55613.386106.82
$$ \varepsilon $$-CL-20, Exp.[66,67]8.83712.55413.289106.87
$$ \varepsilon $$-CL-20, Exp.[68]8.86312.59313.395106.92
$$ \varepsilon $$-CL-20, DFT-D[69]8.87112.55713.477106.62
$$ \varepsilon $$-CL-20, ReaxFF-nn8.98912.40113.428105.90
$$ \beta $$-HMX, Exp.[70]6.52611.0377.364102.67
$$ \beta $$-HMX, Exp.[71]6.53011.0307.350102.69
$$ \beta $$-HMX, Exp.[72]6.54011.0507.371102.83
$$ \beta $$-HMX, DFT-D[69]6.59310.8607.406102.84
$$ \beta $$-HMX, ReaxFF-nn6.43410.5827.876103.40


Molecular CSPs were carried out using the USPEX package[14,15,73,74]. This package is based on the evolutionary algorithm developed by Oganov, Glass, Lyakhov, and Zhu. Initially, a group of 100 structures was randomly generated from the appropriate crystal space group. In subsequent generations, the total population of 100 structures was maintained after operations including heredity, randomization, permutation, rotation, and softmutation. The optimal structures of the previous generation were kept as keptbest[15] structures. Each structure underwent two local relaxation steps using the General Utility Lattice Program (GULP) package[75] and the ReaxFF-nn potential. The fitness of each structure was determined based on the corresponding energy calculated by GULP. Following the principles of biological evolution, structures with lower fitness were then eliminated. Figure 2 illustrates the main search workflow comprising four stages. First, the initial parameter set is generated by training the model against the existing dataset. Second, the USPEX algorithm is combined with the ReaxFF-nn MLP (structural relaxation) to search for cocrystals for specific needs, generating a set of crystal structures. Third, these crystal structures are checked by comparing the corresponding ReaxFF-nn- and DFT-calculated energies. Finally, structures with a very large difference between the calculated energies are added to the training dataset. These steps are repeated until a small difference between the ReaxFF-nn and DFT results or new cocrystal structures meeting the selected requirements have been found.

Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 2. Schematic illustration of four-step iterative workflow combining USPEX and ReaxFF-nn for searching the appropriate cocrystal structures and training the MLP. USPEX: Universal Structure Predictor: Evolutionary Xtallography; ReaxFF-nn: reactive force field with neural networks; MLP: machine learning potential.

Prediction of 1:2 HMX: CL-20 cocrystal configurations

After several searches, we obtained some cocrystal structures with lower enthalpy (potential energy) and higher density. However, to acquire densely packed cocrystals, we set a small pressure in the search process ($$ p $$ = 0.1 GPa). The obtained structures had to be optimized (relaxed) to zero pressure. The optimization results using the ReaxFF-nn and DFT (SIESTA[76]) method with the Perdew-Burke-Ernzerhof (PBE) functional and a double-$$ \zeta $$ basis set at zero pressure are shown in Supplementary Table 1. Furthermore, an experimentally synthesized cocrystal structure was optimized by DFT (SIESTA) to serve as a reference, as shown in the first row of Supplementary Table 1.

Supplementary Table 1 reveals a good agreement between the cocrystal structures optimized by ReaxFF-nn and DFT (SIESTA), demonstrating the high performance of our MLP model. Combined with the data from the structures in Supplementary Table 1, the enthalpy and density of the cocrystal structures are shown as bar graphs in Figure 3. The enthalpy and density values calculated by SIESTA and ReaxFF-nn are shown in red and turquoise, respectively.

Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 3. (A) Enthalpy and (B) density vs. crystal index from USPEX of 1:2 HMX: CL-20 optimized by DFT (SIESTA) and MLP (ReaxFF-nn). USPEX: Universal Structure Predictor: Evolutionary Xtallography; HMX: 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane; CL-20: 2,4,6,8,10,12-Hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane; DFT: density functional theory; SIESTA: Spanish Initiative for Electronic Simulations with Thousands of Atoms; MLP: machine learning potential; ReaxFF-nn: reactive force field with neural networks.

In order to confirm the findings of the search workflow, we chose the five most significant cocrystal configurations with a 1:2 ratio and further optimized them using the van der Waals Density Functional (vdW-DF)[77] method and DFT-D3 method with the PBE functional and an energy cutoff of 600 eV. These DFT-D3 calculations were carried out using the VASP[78] software. The results are detailed in Table 2.

Table 2

The comparison of the enthalpy, density, and lattice constants from DFT (SIESTA), vdW-DF, and DFT-D3 (VASP) calculations of the top predicted 1:2 HMX: CL-20 cocrystal structures, respectively

IDMethodEnthalpy (eV)$$ \rho (g \cdot cm^{-3}) $$$$ a $$(Å)$$ b $$(Å)$$ c $$(Å)$$ \alpha $$(°)$$ \beta $$(°)$$ \gamma $$(°)

Based on the information on the DFT-D3-optimized cocrystal structures from Table 2, the five cocrystal structures with lower energy and higher density and the structure experimentally synthesized by Bolton et al.[38] are shown in Figure 4. The optimized cocrystal structures in this picture show the characteristic alternating arrangements of monolayer HMX and bilayer CL-20, consistent with the cocrystal arrangement of the experimentally synthesized structure.

Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 4. (A) Experimentally synthesized cocrystal structure[38], and (B–F) 1:2 HMX: CL-20 cocrystal structures after DFT-D3 structure optimization at zero pressure. The (B–F) structures correspond to IDs C415, C420, D477, E400, and F111 in Table 2, respectively. HMX: 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane; CL-20: 2,4,6,8,10,12-Hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane; DFT: density functional theory.

Among the structures with the predicted 1:2 HMX: CL-20 ratio obtained from USPEX and MLP, the structure F111 shown in Figure 4F had the highest density of 1.914 g/cm3 after DFT optimization and 1.937 g/cm3 after DFT-D3 optimization. Furthermore, the F111 cocrystal also exhibited the lowest lattice energy.

Prediction of 1:1 HMX: CL-20 cocrystal configurations

A search of 1:1 HMX: CL-20 cocrystal structures was carried out using USPEX and ReaxFF-nn through the search–check–train–search iterative loop, as illustrated in Figure 2. A low pressure was applied to the candidate cocrystal cells; in the next step, the generated structures were optimized using the DFT (SIESTA) method with the PBE functional and a double-$$ \zeta $$ basis set, as well as with the ReaxFF-nn potential to zero pressure. Supplementary Table 1 shows the structural information on the optimized cocrystals with lower enthalpy and higher density for the 1:1 ratio at zero pressure. The enthalpy and density of the cocrystal structures after optimization, obtained using ReaxFF-nn and DFT, are shown in Figure 5, which enables us to identify structures with lower enthalpy and higher density.

Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 5. Comparison of DFT (SIESTA)- and ReaxFF-nn-calculated (A) enthalpy and (B) density values vs. crystal index of cocrystal structures with 1:1 HMX: CL-20 ratio generated by USPEX. DFT: Density functional theory; SIESTA: Spanish Initiative for Electronic Simulations with Thousands of Atoms; ReaxFF-nn: reactive force field with neural networks; USPEX: Universal Structure Predictor: Evolutionary Xtallography.

Similarly, to confirm the findings of the search workflow, we chose the six most significant cocrystal configurations with a 1:1 ratio and further optimized them using the vdW-DF method and DFT-D3 (VASP) method with the PBE functional and an energy cutoff of 600 eV. The results are detailed in Table 3. We selected the better cocrystal structures calculated by DFT-D3, as shown in Figure 6. The optimized structures in Figure 6 show characteristic cocrystal structures with alternating arrangements of HMX and CL-20 monolayers. Theoretically, this is consistent with the characteristic arrangement of cocrystal structures.

Table 3

Comparison of enthalpy, density, and lattice constants from DFT (SIESTA), vdW-DF, and DFT-D3 (VASP) calculations of the top predicted cocrystal structures with 1:1 ratio, respectively

IDMethodEnthalpy (eV)$$ \rho (g \cdot cm^{-3}) $$$$ a $$(Å)$$ b $$(Å)$$ c $$(Å)$$ \alpha $$(°)$$ \beta $$(°)$$ \gamma $$(°)
Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 6. Cocrystal structures with 1:1 HMX: CL-20 ratio after DFT-D3 (VASP) optimization. Structures (A–F) correspond to IDs D3, D558, D559, F377, G1766 and H223 in Table 3, respectively. HMX: 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane; CL-20: 2,4,6,8,10,12-Hexanitro-2,4,6,8,10,12-hexaazaisowurtzitane; DFT: density functional theory; VASP: Vienna ab initio simulation package.

The results show the D558 and D559 structures had the highest density of 1.925 g/cm3 after DFT (SIESTA) optimization. However, the DFT-D3 calculations with the VASP software gave a different result: the D3 structure shown in Figure 6A exhibited the highest density of 1.937 g/cm3 and the lowest lattice energy. To our knowledge, there have been no previous theoretical or experimental reports of 1:1 HMX: CL-20 cocrystal structures, and this is the first report of 1:1 HMX: CL-20 cocrystal structures with a density higher than that of the $$ \beta $$-HMX ($$ d $$ = 1.901 g/cm3)[70] crystal.

Hydrogen bond interactions

Hydrogen bond interactions play an important role in the design of molecular crystals; in practice, we found that increasing the C–H...O hydrogen bond contribution to the total energy can apparently raise the rate of high-density cocrystals. This leads to the conclusion[79] that the C–H...O interaction can be employed to design molecular crystals. Figure 7 shows the relationship between hydrogen bond energy (calculated by the ReaxFF-nn potential) and crystal density. The figure shows that the C–H...O hydrogen bond energy represents the main part of the total hydrogen bond energy; moreover, an approximately linear relationship is observed between the hydrogen bond energy and the density of the cocrystals, with a correlation factor $$ \eta=0.67 $$, which indicates that the C–H...O hydrogen bond energy is the key factor in the design of HMX/CL-20 cocrystals.

Searching new cocrystal structures of CL-20 and HMX <i>via</i> evolutionary algorithm and machine learning potential

Figure 7. Relationship between hydrogen bond energy and density of predicted cocrystals. The hydrogen bond energy was calculated using the ReaxFF-nn potential. ReaxFF-nn: Reactive force field with neural networks.


In this work, we report an evolutionary algorithm and machine learning workflow denoted as search–check–train–search to explore new cocrystal structures of HMX and CL-20, which are expected to have higher energy than $$ \beta $$-HMX and lower sensitivity than $$ \varepsilon $$-CL-20. Using the present workflow, several high-density HMX/CL-20 energetic cocrystal structures with 1:1 and 1:2 HMX: CL-20 ratios were found and validated via DFT and DFT-D3 calculations. The most valuable finding was the D3 cocrystal structure, which had the highest density and lowest lattice energy among the 1:1 cocrystal structures. The density of D3 (1.937 g/cm3) was only slightly lower than that of 1:2 HMX: CL-20 (1.945 g/cm3) reported by Bolton et al.[38] at room temperature. Another high-density cocrystal with a 1:1 ratio was H223, which had a similar lattice energy and a density of 1.921 g/cm3. Among cocrystals with 1:2 HMX: CL-20 ratio, F111 had the lowest lattice energy and the highest density of 1.937 g/cm3. These high-density cocrystal structures can be accessed on the website [80]. In practice, we found that the C–H...O hydrogen bond interactions play an important role in the search for high-density cocrystals. Although these interactions are weak, with an equilibrium hydrogen bond length of 2.254 $$ Å $$ estimated through EOS calculations, they account for the vast majority of the total hydrogen bond energy of HMX/CL-20 cocrystals. An approximately linear relationship was found between the hydrogen bond energy and the density of cocrystal structures.


Authors' contributions

Designed this project: Guo F

Complete all the calculations: Ye ZH

Provided technical and material support: Wen YS, Chai CG, Zhang ZR

Offered writing suggestions: Wang XC, Li HS, Zhang GQ, Cui SX

Availability of data and materials

The raw data (high-density cocrystals and parameter set for LAMMPS) are available from the website[80].

Financial support and sponsorship

This work was supported by the National Natural Science Foundation of China (NSAF. 12002325 and 11504153) and the Natural Science Foundation of Gansu Province (Grant No. I23JRRA1173). We greatly appreciate the assistance.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.


© The Author(s) 2024.

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OAE Style

Ye ZH, Guo F, Chai CG, Wen YS, Zhang ZR, Li HS, Cui SX, Zhang GQ, Wang XC. Searching new cocrystal structures of CL-20 and HMX via evolutionary algorithm and machine learning potential. J Mater Inf 2024;4:5.

AMA Style

Ye ZH, Guo F, Chai CG, Wen YS, Zhang ZR, Li HS, Cui SX, Zhang GQ, Wang XC. Searching new cocrystal structures of CL-20 and HMX via evolutionary algorithm and machine learning potential. Journal of Materials Informatics. 2024; 4(2): 5.

Chicago/Turabian Style

Zhong-Hao Ye, Feng Guo, Chuan-Guo Chai, Yu-Shi Wen, Zheng-Rong Zhang, Heng-Shuai Li, Shou-Xin Cui, Gui-Qing Zhang, Xiao-Chun Wang. 2024. "Searching new cocrystal structures of CL-20 and HMX via evolutionary algorithm and machine learning potential" Journal of Materials Informatics. 4, no.2: 5.

ACS Style

Ye, Z.H.; Guo F.; Chai C.G.; Wen Y.S.; Zhang Z.R.; Li H.S.; Cui S.X.; Zhang G.Q.; Wang X.C. Searching new cocrystal structures of CL-20 and HMX via evolutionary algorithm and machine learning potential. J. Mater. Inf. 2024, 4, 5.

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