First principles reactive simulation for equation of state prediction

First principles reactive simulation for equation of state prediction
 Ryan B. Jadrich,1, 2, a) Christopher Ticknor,1 and Jeffery A. Leiding1, b)
 1)
 Theoretical Division, Los Alamos National Laboratory, NM, 87545
 2)
 Center for Nonlinear Studies, Los Alamos National Laboratory, NM, 87545
 (Dated: 11 May 2021)
 The high cost of density functional theory has hitherto limited the ab initio prediction of equation of state
 (EOS). In this article, we employ a combination of large scale computing, advanced simulation techniques, and
 smart data science strategies to provide an unprecedented, ab initio performance analysis of the high explosive
 pentaerythritol tetranitrate (PETN). Comparison to both experiment and thermochemical predictions reveals
arXiv:2103.09849v2 [physics.chem-ph] 10 May 2021

 important quantitative limitations of DFT for EOS prediction, and thus the assessment of high explosives.
 In particular, we find DFT predicts the energy of PETN detonation products to be systematically too high
 relative to the unreacted neat crystalline material, resulting in an underprediction of the detonation velocity,
 pressure, and temperature at the Chapman-Jouguet (CJ) state. The energetic bias can be partially accounted
 for by high-level electronic structure calculations of the product molecules. We also demonstrate a modeling
 strategy for mapping chemical composition across a wide parameter space with limited numerical data, the
 results of which suggest additional molecular species to consider in thermochemical modeling.

 I. INTRODUCTION

 Predictive modeling of high explosive (HE) materials
 is a complex multi-scale physics problem whereby atom-
 istic physics informs mesoscale modeling, enabling the
 prediction of macroscopic explosive performance proper-
 ties (e.g., the detonation velocity and explosive work)1–5 .
 Bridging the atomistic and mescoscopic scales is the HE
 equation of state (EOS), which is frequently decomposed
 into an EOS for the unreacted material and for the det-
 onation products.1–5 The former state is typically an or-
 ganic crystal or liquid at standard conditions bound by
 dispersion forces.6–8 The latter state is generally a su- FIG. 1. Qualitative graphic of the atomic configuration
 changes that occur upon detonation of a typical molecular
 percritical fluid of small molecules (e.g., H2 O, CO, CO2 ,
 crystalline HE (PETN in this case) to that of a supercritical
 N2 ).1–5 A qualitative graphic depicting the detonation fluid of small molecule products (ex., carbon dioxide, water,
 of pentaerythritol tetranitrate (PETN) from its molecu- molecular nitrogen).
 lar crystalline form to supercritical products is shown in
 Fig. 1.
 Predicting the thermodynamic properties for both the
 reactants and products requires condensed phase simu- and D3 correction schemes,12,13 are sufficient.6–8,11 Suc-
 lations. Ab initio molecular simulation is the current cesses of dispersion corrected DFT (often termed DFT-
 state of the art in this respect. In terms of the trade-off D) include the accurate prediction of crystal cell param-
 between computational feasibility and accuracy, Kohn- eters (and thus density), equation of state, and even vi-
 Sham Density Functional Theory9,10 (denoted as DFT brational modes.6–8,11 Accurate prediction of vibrational
 in this work) is a popular choice for the calculation of modes (phonons),14,15 is critical to accounting for the
 forces and energies in these atomistic simulations. zero-point energy14,15 missed in a standard ab initio sim-
 It is relatively straightforward to apply DFT to the ulation where nuclei are propagated classically.6–8,11,15
 reactant HE because the relevant range of thermody- Quantum corrections can be important near the standard
 namic conditions is often relatively limited (by compar- state (298 K and 1 bar), which unreacted HEs often are.
 ison to product states) and chemical reactions are not The overall success of DFT for characterizing unreacted
 occurring. Many studies have shown that DFT has great HEs motivates its application to study detonation.
 predictive capability.6–8,11 Inclusion of dispersion inter- Simulation of the products requires vastly more com-
 actions is generally necessary for condensed phase calcu- putational time to chemically equilibrate small molecule
 lations to properly capture the cohesive energy. Post- products at a vast array of potentially relevant state
 hoc DFT corrective interactions, like the Grimme D2 points. Roughly speaking, detonation conditions can
 span temperatures from 1,000-10,000 K and densities
 from 0.8 to 2.5 g/cc.3–5 Within these ranges, many
 state points will require long simulation times to estab-
 a) Electronic mail: rjadrich@lanl.gov lish chemical equilibrium, which is a slow activated pro-
 b) Electronic mail: jal@lanl.gov cess dominated by the breaking and re-forming of strong

2

chemical bonds.16–18 High density is also likely to com- the simulation of reactive atomic systems while exactly
plicate matters by slowing the diffusion of atoms by geo- retaining DFT level results. The ML potential is used in
metric crowding. a nested Monte Carlo (NMC)45–52 framework to propose
 To circumvent the computational challenges of a com- a long chain of small moves that are then accepted or
prehensive detonation products study, researchers have rejected in toto according to how well the ML potential
successfully leveraged alternative (more approximate) predicts the energy change. This scheme avoids having
methods than DFT. The well known ReaxFF force field to compute the expensive DFT potential for every small
is a popular choice among such computationally expedi- update. In addition, we fit noisy simulation data via a
ent approaches. ReaxFF leverages a many-body, bond- machine learning based EOS that (1) enforces thermody-
order type interaction to furnish a fully reactive poten- namic consistency and (2) simultaneously utilizes energy
tial.19,20 Large simulation sizes and long run times are and pressure measurements to fit the model. Machine
easily achieved. Studied materials include an impres- learning is also leveraged to maximize the information
sive array of high explosives.15,21–26 While successful, it available in chemical composition data.
is reasonable to assume that classical models will have In addition to getting unprecedented accuracy for the
a more restricted range of applicability than DFT given product states, reactive ab initio detonation products
the coarse-grained electronic degrees of freedom.13 . simulations are uniquely suited to validating and im-
 Modern machine learning (ML) force fields are pro- proving the thermochemical models of high explosives.
viding additional options for computationally tractable Thermochemical techniques leverage approximate sta-
chemically reactive simulations.27–40 . ChiMES is one tistical mechanical theories to model chemical equilib-
such approach that has been applied to chemically re- rium in a coarse grained, computationally expedient
active fluids. Successful applications include the mod- manner. Two common thermochemical codes include
eling of carbon clustering in C/O systems at extreme Cheetah53 , of Lawrence Livermore National Laboratory,
conditions30 and the detonation state of HN3 .29 A major and Magpie54,55 , of Los Alamos National Laboratory. A
advantage of ML potentials are their ability to rapidly key limitation of both aforementioned models is the need
iterate and train on new data in a mostly automated to a priori specify a list of possible molecular products,
fashion, a power that we leverage in this work. Like their classical interaction potentials, and internal parti-
ReaxFF though, these models also integrate over elec- tion functions. First-principles reactive simulations are
tronic degrees of freedom to produce an effective classical uniquely suited to test the assumptions regarding which
potential. molecular species are relevant to detonation performance
 The very diverse range of conditions relevant to deto- calculations, as well as the resultant EOS. Fully account-
nation products coupled with a desire to perform max- ing for the most relevant species in a thermochemical
imally predictive and minimally parameterized simula- modeling approach is integral to getting the right answer
tions, strongly motivates the use of finite-temperature for the right reasons; only then can one have some faith
DFT based approaches.41,42 For example, thermal effects in the reliability of predictions coming from thermochem-
on the electronic populations (excited states) are natu- cial models at new, untested conditions.
rally accounted for in finite-temperature DFT by mod- Before outlining the manuscript, we briefly discuss
ification of the functional to include the electronic en- the formation of solid carbon in HE detonation prod-
tropy.41,42 Accounting for thermal electronic excitations ucts. The formation of solid carbon (soot) in the det-
is certainly important at higher temperatures as, in gen- onation products of carbon-rich/oxygen-poor HEs has
eral, excitations will lead to a general softening of bonds been the subject of much previous research56–59 includ-
and increased reactivity, more so than would be expected ing relatively recent work.60,61 Solid carbon (as diamond
from enhanced kinetic energy of the atoms alone. High or graphite for example) is often included in the list of
densities are likely to yield complications too. The very possible products of thermochemical codes like Magpie;
notion of well-defined molecular entities is challenged at however, PETN is a relatively oxygen-rich HE, and thus
highly compressed thermodynamic states, often of rel- solid carbon is a minor consideration.62 Indeed PETN
evance to shocks or detonations. Extreme conditions has been accurately modeled without the inclusion of
can lead to non-standard behavior, such as the forma- solid carbon.63 In this work, we also neglect the forma-
tion of electrically conductive states, which often possess tion of solid carbon in the detonation products calcula-
unique extended structures16,43,44 . Classical force fields tions via Magpie and obtain good agreement with exper-
that “integrate over” the electronic degrees of freedom iment. Indeed, modeling solid carbon in a first-principles
are unlikely to be predictive at such state points unless context would be very difficult, as one would need to
they were specifically calibrated for these regions. simulate with very large simulation cells to model the
 In this paper we employ novel simulation and data- phase separation directly, or perform a Gibbs-ensemble
analysis techniques to overcome the computational bur- simulation where the solid phases are modeled with sep-
den of applying DFT to HE products. We employ arate cells; however, the former is becoming within reach
a Monte Carlo (MC) simulation scheme that uses a with the advent of accurate machine-learning interac-
machine-learned (ML) reactive force field, akin to those tion potentials.64,65 This difficulty motivated the choice
described above, as a reference potential to accelerate of PETN as our first real-world HE application of the

3

new algorithms outlined herein. B. Gas-phase electronic structure
 In this article, we discuss the results of a fully ab
initio simulation study of the high explosive pentaery- In Section III A, we found it necessary to estimate the
thritol tetranitrate (PETN) covering an unprecedented errors of the relative energies of BLYP66,67 with respect
range of thermodynamic conditions. DFT simulation is to high-accuracy CCSD(T)73–77 calculations. For these
used to construct a products EOS, from which detona- calculations, we optimized isolated molecules (unreacted
tion properties are calculated and compared to experi- PETN and several molecules found in the products) at
mental and thermochemical results spanning a range of the MP2/6-31G78,79 level of theory and calculated single-
initial unreacted HE densities. The combination of ac- point energies at the CCSD(T)/6-31G level of theory.
celerated simulations and data-driven analysis, provides All gas-phase electronic structure calculations were per-
an unprecedented, ab initio EOS for high explosive prod- formed in the MOLPRO80–82 electronic structure package.
ucts. Sect. II discusses the various ML methods for sim- Further details of how these energies were used are pro-
ulating, EOS construction, and modeling of molecular vided Section III A.
populations in addition to the electronic structure, sim-
ulation, and thermochemical modeling details. Sect. III
presents our HE performance predictions in comparison C. Monte Carlo simulation
to experiment and thermochemical modeling. We discuss
the quantitative limitations of DFT and present a possi- In this work, we perform Monte Carlo (MC)83–86
ble remedy. We conclude in Sect. IV and put forward the simulations at the DFT(BLYP) level of theory as dis-
need for future studies of more high explosives to further cussed above. MC, as opposed to molecular dynamics
assess thermochemical modeling and to test the general- (MD),85,86 was chosen for this work so as to leverage non-
ity of our DFT correction for quantitative predictions. physical moves that (1) accelerate kinetically slow bond
 rearrangements and (2) execute large-scale translations
 and rotations of molecules. MC has the added bene-
 fit of sampling directly from the equilibrium distribution
 without approximation.83–86 While standard MC imple-
 mentation suffers from smaller moves/updates in between
II. METHODS costly QM calculation than MD, often displacing only a
 handful of atoms at a time, we use the following strate-
A. Density functional theory gies to circumvent this difficulty.85,86
 To mimic the all-atom updates afforded by MD in our
 MC simulations, we use force-biased moves.87–89 Force
 Simulations and single-point calculations utilized biasing uses both the magnitude and the direction of the
Kohn-Sham DFT9,10 with the BLYP66,67 functional and QM forces to move every atom in the simulation. In par-
Grimme D3 dispersion corrections12,13 (with the C9 con- ticular, a displacement (δx) for each Cartesian direction
tribution). We chose the BLYP functional for its gener- is drawn from a probability ∝ exp [βF δx/2] with a max-
ally good performance in regards to organic molecules. imum allowed displacement of δxmax where F is the cor-
All calculations used the CP2K software package68,69 responding Cartesian force component. The dominant
with triple-ζ double polarization basis sets,70 GTH pseu- factor governing the acceptance rate of atom displace-
dopotentials,71,72 and were spin-unrestricted with a mul- ments is the temperature, not the density, as the short
tiplicity of unity. Energy cutoffs for the plain wave and and strong chemical bonds (not collisions between atoms
Gaussian contributions to the basis sets were 600 and 60 of separate molecules) are the limiting factor. The domi-
Ry respectively. Convergence was further aided by the nant temperature dependence is captured via the empir-
NN50 density and derivative smoothing implemented in ical relation: δxmax ≡ a + b × T where a = 0.014 Å and
CP2K. With these choices, increasing the respective cut- b = 6 × 10−6 Å/K
offs to 1000 and 100 Ry yielded numerically insignificant Two additional MC moves are useful in chemically re-
differences for an array of representative configurations. active mixtures, namely, swap and cluster moves. Swap
Excited electronic effects were accounted for via Fermi- moves are straightforward to implement and involve pick-
Dirac smearing;41,42 as such, the “energies” output by ing two random atoms and attempting to interchange
CP2K are electronic free energies (and the forces are them.18,45,90 The move is accepted or rejected according
electronic potentials of mean force). Atomic configura- to the standard Metropolis acceptance criterion. Cluster
tions were sampled according to the electronic free en- moves, on the other hand, have greater flexibility in the
ergy to correctly account for the electronic contribution exact implementation.85,86 In this work, we define a clus-
to the partition function. For equation of state calcu- ter as a group of atoms connected via the cutoff distance
lations though, we need to access the total internal en- di,j = ci + cj + δB where ci is the covalent radius of atom
ergy (not free energy), amounting to a simple removal of i and δB = 0.2 Å. By randomly picking a “seed” atom,
the electronic entropy contribution according to standard a cluster is recursively built and randomly displaced and
thermodynamics. rotated about the center of position (COP). The max-

4

imum allowed magnitude for a random displacement in (in that order). An insertion attempt was rejected if the
each Cartesian direction takes into account the density distance between the inserted atom and any other previ-
via δxCOP = 0.31 × (1.0/ρ)1/3 , which is the dominant ously inserted atom was smaller than the corresponding
factor in determining acceptance. Empirically, we found pairs mean Van der Waals radius. Skipping 20 frames
that setting the maximum allowed angular displacement at a time, a total of six snapshots were extracted from
in proportion to that of the positional displacement was the end of each simulation on which to calculate statis-
effective: δαCOP = κδxCOP where κ ≡ 100 ◦ /Å. Ap- tics. While six snapshots is insufficient for calculating
pendix A details the specific implementation and how physical quantities at any one state, by utilizing all 225
these parameters affect the motion. simulations to fit regularized models for equation of state
 Alone, swap and cluster moves only update a handful and molecular population (see Sect. II D and Sect. II G),
of atoms in between each costly QM calculation; to in- we are able to maximize the information content. In this
crease the efficiency we perform a long sequence of such manner, predictions at a state-point leverage the infor-
moves via Nested MC (NMC).45–52 Specifically, NMC mation from not one, but a whole array of neighboring
simulates the system on an approximate reference poten- simulations, leading to more confident estimates of phys-
tial (U0 ) for a fixed number of moves. The “full” or DFT ical quantities than would be expected from an isolated
potential energy (U ) is then calculated at the end points simulation.
of the chain. The whole chain of moves is accepted or In addition to reducing estimation error by fitting
rejected in toto according to min[1, exp[−β(δU − δU0 )]]. physically constrained models, our “scattershot” simu-
The acceptance probability is determined by the length lation approach in concert with the enhanced rare-event
of the nested chain, and the accuracy of the reference po- reactive sampling (relative to MD) helps erase sampling
tential with respect to the full potential. To maximize the biases that might be present in the simulations due to
acceptance rate for the nested chain of moves, we leverage a rugged energy landscape as more relevant energy min-
a machine learned (ML) many-body potential for U0 .45 ima will be sampled. More simulations must be run, but
Details regarding the ML potential are discussed in Ap- each can be run for shorter lengths of time as only a small
pendix B along with the training details. We note that amount of data at apparent equilibrium needs to be used.
any ML potential27–40 that can handle chemical reactiv- The bias reduction from our scattershot approach is not
ity can in principle be used instead, with the correspond- unlike that achieved in “bagging”91 (as used in machine
ing acceptance probability being quantitative measures learning) whereby a “community of models” yields an
of their agreement with DFT. Force-biased moves and overall less biased result. Regardless, we note that it is
nested moves are performed randomly in a 50/50 ratio. always possible that the limited run times and system
Within the nested chain, swap and cluster moves are ran- sizes affordable in ab initio simulations could yield a bias
domly performed in a 50/50 ratio. that our approach cannot fully deal with. If such a bias
 An important aspect of our NMC approach is that it is present, it will not be included in the uncertainty esti-
preserves exact sampling at the target level of theory mates which only probe the estimation error–a fact that
(DFT, in this work) despite the approximate nature of is true of virtually any uncertainty analysis.
the underlying reference force field. NMC can be con-
sidered an on-the-fly statistical re-weighting of configu-
 D. Equation of state modeling
rations generated by the reference potential.45–52 A poor
overlap of the reference potential distribution with the
exact distribution will result in a poor acceptance rate To maximize the information content from our sim-
(α) for nested moves. In this manner, α provides a real- ulations, we leverage energy and pressure data simul-
time metric of how well the reference potential ensemble taneously to train a Helmholtz free energy model with
replicates the exact ensemble. For our reactive simula- physically reasonable behavior. Specifically, we break the
tions an α ≈ 50% is achievable for nested chains of ≈ 30 Helmholtz free energy per volume (a) into ideal (aid ) and
steps, with some variation due to the temperature. For excess (aex ) contributions
reference, this is roughly an order of magnitude lower
than what was realizable in a He/Ar mixture45 where a ≡ aid + aex (1)
the reference potential could be made virtually exact.
 where
For our current reactive system, the reference is less ac-
curate than our previous work, but this is to be expected 
 aid ≡ kB T n ln(n) − 1 − 3 ln(kB T )/2
 
 (2)
given the much greater complexity of super-critical reac-
tive mixtures. Overall, our reference potential is suffi- and n is the number of atoms per unit volume. In Eqn. 2,
cient for fast NMC sampling. constant terms have been dropped for numerical fitting
 In total, 225 separate simulations were run on an convenience–these have no bearing on the modeling and
evenly spaced 15 × 15 grid of ρ and T with respective serve only to scale out the units. The ideal entropy of
ranges [0.7, 2.5] g/cc and [1000, 8000] K. Each simulation mixing is also not explicitly in Eqn. 2 as the atomic com-
started from randomly generated configuration by insert- position is (1) fixed throughout this work and (2) is irrel-
ing 20 carbons, 16 nitrogens, 48 oxygen and 32 hydrogens evant to the thermodynamic quantities of interest. The

5

energy per volume and pressure are obtained by the stan- ture, density) as well as a sudden change from unre-
dard relations: e = a−T (∂a/∂T ) and P = n(∂a/∂n)−a. acted high explosive (before the front) to equilibrated
Flexibility is embedded in the excess free energy contri- small molecule products (behind the front).1–5 The locus
bution, which is described in Appendix C. Uncertainty of possible states that the products can be in is readily
estimates are also calculated via bootstrap re-sampling determined from the Rankine-Hugoniot jump conditions
of the data and the fitting procedure (see Appendix C). for mass, momentum, and energy conservation
 To assist in training the EOS model, a baseline refer-
ence energy was subtracted from the raw simulation data ρ D
 = (3)
to yield more manageable numbers. We define this ref- ρ0 D−u
erence energy (E) e as the sum of the heats of formation
(HF ) for isolated CO2 , H2 , N2 and O2 molecules at stan-
dard conditions (1 bar, 298.15 K) in PETN relative abun- P = ρ0 uD (4)
dances: E e = 5HF (CO2 )+4HF (H2 )+2HF (N2 )+HF (O2 ).
This choice is convenient as accurate experimental data !
is available and analogous DFT calculations are inexpen- 1 1 1
 E − E0 = (P + P0 ) − (5)
sive and simple to perform. Specifically, DFT predic- 2 ρ0 ρ
tions leveraged standard statistical mechanical results for
molecules whereby uncoupled translation, rotation and where D and u are the detonation and material veloc-
vibration is assumed.92,93 ; the only required computa- ities, respectively, ρ is the density (mass per volume),
tion is that of a cheap small-molecule DFT normal mode E is the energy per mass of material, P is the pres-
analysis to parameterize the vibration partition function. sure (force per area), and the naught indicates the ini-
The values for experiment94 and simulation, respectively, tial unreacted material state. Chapman-Jouguet theory
are EeEXP = −6.223 kJ/g and E eDFT = −2198.654 kJ/g. then states that the single unique point with minimum D
All energies in this work, whether derived from experi- (i.e., the CJ state) corresponds to the steady state det-
ment or DFT, will have the corresponding reference en- onation.1–5 This unique state point is demarcated by a
ergy subtracted out. subscript with “CJ” throughout the text.
 Both a DFT and experimental estimate for E0 are cal-
 culated in reference to the baseline E. e Using the ex-
 96
E. Thermochemical equation of state perimental HF for crystalline PETN we find E0,EXP =
 4.52 kJ/g. The analogous DFT value is derived from
 Thermochemical modeling is performed with the a standard phonon analysis of crystalline PETN. Specifi-
Magpie software package developed at Los Alamos na- cally, two PETN molecules were placed in a periodic sim-
tional laboratory.54 Magpie approximates the free energy ulation cell with experimentally determined relative ar-
for a mixture of molecules via a decomposition into intra- rangements. The cell was scaled to the experimental den-
and inter-molecular contributions. The former is approx- sity of 1.778 g/cc observed at 298 K.97 . Atomic positions
imated by splitting into independent electronic, vibra- were relaxed by an energy minimization and a phonon
tion, rotation and translation contributions that are de- analysis of the energy minimized configuration was per-
scribed by standard statistical mechanical forms .92,93 formed, the results of which parameterized the standard
The electronic and vibration contributions are parame- harmonic statistical mechanical model14,92,93 . Again
terized via electronic structure and/or experimental re- with the reference energy removed, we find E0,DFT =
sults. Inter-molecular interactions between molecules of 4.33 kJ/g, which is close to the experimental estimate.
the same type are approximated by Ross perturbation As mentioned in the Introduction (Sect. I), previous
theory95 in which molecules are coarse-grained to radi- DFT-D modeling of crystalline high explosives found
ally isotropic exponential-6 pair potentials. The thermo- near quantitative performance. For the remainder of the
dynamics of mixtures are then obtained by ideally mixing text, all results are shown using E0,EXP , though the det-
the pure molecular fluids. This may seem to be a crude onation results are visually indistinguishable for the two.
approximation; however, it is very accurate at the condi-
tions of relevance to high explosives products. This has
been explicitly shown for the high explosive considered G. Molecular populations
in this work (PETN).55 Our choice of Magpie product
molecules are provided in Table D1 of Appendix D. To approximately extract molecular species in our
 ab initio simulation, we developed a graph-isomorphism
 software tool to label atomic clusters according to their
F. Jump conditions and Chapman-Jouguet theory topology of atomic connections. Clusters of atoms are
 defined by cutoff distances between atomic species and
 The detonation front in a high explosive is well ap- graphs are constructed using atoms as nodes and bonds
proximated as a near discontinuous change in all ther- as edges (edges are unweighted and only imply the cut-
modynamic variables (e.g., energy, pressure, tempera- toff criteria is met). The bond cutoff distance for all atom

6

 (Sect. II D). In doing so, we maximized the information
 ≡ , = Energy
 Initial samples
 ≡ , = Pressure
 content obtained from the many noisy simulation results
 Ex., fast simulation, existing data and obtained a smooth, representation of the EOS with
 ≡ , = Moles
 ≡ Nested acceptance rate uncertainties. From our EOS, the performance of PETN
 is assessed over a large range of initial packing densities.
 (Re)-train ML Quantitative limitations of DFT simulation are found in
 potential
 ML Potential DFT Potential
 respect to experiment. Similar to the EOS data, a model
 is trained to the noisy molecular composition data and a
 Low High NMC simulations spanning , direct comparison of molecular populations predicted via
 Perform N steps… DFT simulation and the thermochemical code Magpie
 New samples is carried out for a limited set of species. Our results
 suggest the inclusion of another important molecular en-
 Equilibrated tity in Magpie. The overall workflow of our approach
 Compute equilibrium measurables
 is summarized in Fig. 2. Specific details about each of
 , , , … the components can be found in Sect. II and the relevant
 subsections.

 Bootstrap resample Fit physically
 physical measurables informed models A. Detonation
 !, !, !, …
 As shown in Fig. 3a-b, our ab initio (DFT simulation)
 …

 Calculate EOS metrics predictions exhibit a systematic underprediction of DCJ ,
 " , " , " , … and PCJ relative to a diverse set of experimental results.
 Ensemble = uncertainty
 Fig. 3c suggests that TCJ may be underpredicted relative
 to experiment as well, but far less data are available and
 large (or unavailable) uncertainties obfuscate the com-
FIG. 2. The overall workflow employed in this study. Details
 parison. It is unlikely that the differences between our
about each stage/process are elaborated upon in Sect. II
 ab initio results and experiment can be attributed to ex-
 perimental biases or uncertainties as the experimental
 measurements come from a variety of separate sources
pairs was set to the sum of the covalent radii plus a an ex- and are mutually consistent with one another in regards
tra threshold of 0.2 Å except for Oxygen-Nitrogen which to DCJ and PCJ . Magpie predictions are in better agree-
was set to 0.4 Å. Graphs were constructed using the Net- ment with the experimental data, but again an overall
workX python package which provides easy graph ma- underprediction is observed in DCJ , and PCJ . We note
nipulation, characterization, and comparison abilities.98 that the Magpie TCJ seems more reasonable, but the lack
Two clusters (molecules) are considered identical if their of definitive data makes it hard to come to any strong
respective graphs are isomorphic. Our approach does not conclusions.
account for charge, and cannot distinguish between iso-
 In contrast to the experiments, our DFT simulation re-
mers or enantiomers, but it is sufficient for the purposes
 sults are in reasonable agreement with two independent
of this work. Furthermore, our approach does not take
 simulation studies, as shown in Fig 3. Both studies uti-
into account the relative stability of a “bonded” pair as
 lized the ReaxFF force field, however, only one refined
measured by the (1) bond order or (2) time stability. The
 the configurations with short ab initio molecular dynam-
former requires a complex analysis beyond the scope of
 ics runs. A similar degree of underprediction in DCJ to
this work, and the latter is better suited to molecular
 that found in this work was observed in each study. For
dynamics99,100 with real time. Our analysis provides a
 PCJ , the prior work brackets our prediction, though the
coarse-grained structural overview of the chemistry only.
 uncertainties are fairly large. Furthermore, the prior sim-
 ulation studies have CJ state temperatures that closely
 match our results, as seen in Fig. 3c.
III. RESULTS AND DISCUSSION Assuming no other complications related to chemical
 equilibrium or a breakdown of CJ theory, any discrep-
 By leveraging the advanced simulation and modeling ancies between experiment, theory and simulation must
efforts, outlined above, we ran 225 DFT simulations, cov- arise from differences in the underlying equilibrium equa-
ering a large range in state space–yielding a first prin- tion of state. To help probe for issues in the DFT sim-
ciples EOS and some complementary molecular popula- ulation predictions, it is useful to compare the underly-
tions. Each simulation leveraged the ML enhanced NMC ing energy and pressure predictions between Magpie and
simulation approach of Sect. II C. Individual, noisy mea- simulation, especially since the former agrees better with
surements from each simulation were not employed di- experiment. As shown in Fig. 4a, the pressures from our
rectly, but rather, were used to train our ML EOS model DFT simulations and Magpie are in quite good agree-

7

 DFT sim. Exp. 4 Magpie (a) PDFT sim. PMP
 Exp. 1 Exp. 5 Sim. 1 8 4
 Exp. 2 Exp. 6 Sim. 2
 Exp. 3 Exp. 7
 6 3

 T (1000K)
 8 (a) 2
 7 4
 DCJ (km/s)

 6 1
 2
 5 0
 4 1.0 1.5 2.0 2.5
 (g/cc)
 (b) (b)
 30 EDFT sim. EMP
 8 6
 PCJ (GPa)

 20 5
 6 4
 10
 T (1000K) 4
 3
 2
 (c)
 4000 2 1
 0
TCJ (K)

 1.0 1.5 2.0 2.5
 3000 (g/cc)
 FIG. 4. Difference between the DFT simulation and Magpie
 0.8 1.0 1.2 1.4 1.6 1.8 pressure (a) and energy (b) predictions. The average of the
 0 (g/cc) DFT simulation results is over the uncertainty obtained by
 the model bootstrapping procedure outlined in Sect. II D

FIG. 3. Detonation velocity (a), Pressure (b) and Tempera-
ture (c) at the Chapman-Jouguet state. Uncertainty bounds
are provided if available in the reference data sources. Thus, a pressure, density and temperature are either all avail-
lack of error bounds does not imply that the error is insignif- able or can be supplemented with the the jump condi-
icant). Non-indexed entries correspond to this work whereas tions (Eqn. 3-5) to obtain any missing information. Ex-
index values, Exp. 1101 , 2-5102 , 6-7103 and Sim. 122 , 224 , perimental data with sufficient information are in lim-
come from the literature. ited supply due to the challenges with measuring a CJ
 temperature. While there is no difficulty in calculating
 properties from simulation, there are minimal data with
ment. Simulation tends to yield slightly higher pressures which to compare. In total, we found four points with
with increasing temperature, relative to Magpie, (can ap- which compare our results, two simulations and two ex-
proach ≈ 3 GPa); however, both simulation and Magpie periments. The four state-points are indicated on the
converge at lower temperatures and there appears to be x-axis of Fig. 5, each with either a distinct experiment
no systematic bias. In comparison, the simulation en- or simulation result from the literature; along with each
ergy is systematically elevated with respect to Magpie are corresponding predictions from DFT simulation and
by ≥ 2 kJ/g, as shown in Fig. 4b. The systemati- Magpie. As shown in Fig. 5a, the pressures are all in good
cally higher DFT simulation product state energies cer- agreement, though the uncertainties on some points are
tainly contribute to the systematic underprediction in large. In contrast, the energies in Fig. 5b generically dis-
DCJ , PCJ , and TCJ relative to Magpie. agree. First, DFT simulation energies are again seen to
 A comparison, similar to that in Fig. 4, can be carried be systematically elevated relative to those of Magpie.
out for select literature CJ state-points where energy, Second, the two experimental results are in conflict: one

8

 DFT sim. Magpie DFT sim. Exp. 4 Magpie
 Exp. 2 Sim. 1 Exp. 1 Exp. 5 Sim. 1
 Exp. 3 Sim. 2 Exp. 2 Exp. 6 Sim. 2
 Exp. 3 Exp. 7

 40
 8 (a)
 P (GPa)

 30

 DCJ (km/s)
 7
 20 (a) 6
 5
 5.0
E E0 (kJ/g)

 (b)
 2.5 30

 PCJ (GPa)
 0.0 20
 (b)
 2.5 10
 2.40g/cc 2.33g/cc 2.25g/cc 2.51g/cc
 4200K 3400K 2460K 2960K

FIG. 5. Comparison of DFT simulation and Magpie pre- (c)
dictions at four state points for which either experimental or
simulated pressure (a) or energy (b) measurements exist. The 4000
 TCJ (K)

x-axis provides the state-points of relevance taken from the
various literature results, the citations of which are provided
in the figure legend. All uncertainty bounds are at the 95%
confidence interval. DFT simulation and Magpie uncertainties
 3000
take into account any quoted uncertainty in the state points
(ρ, T ) obtained from the literature citations provided in the
figure legend. A lack of uncertainty on literature data means 0.8 1.0 1.2 1.4 1.6 1.8
only that it was not reported. Non-indexed entries correspond 0 (g/cc)
to this work whereas index values, Exp. 2-3102 and Sim. 122
and 224 , come from the literature.
 FIG. 6. Detonation velocity (a), Pressure (b) and Tempera-
 ture (c) at the Chapman-Jouguet state. Uncertainty bounds
 are provided if available in the reference data sources. Thus, a
agrees with Magpie and the other is in between DFT sim- lack of error bounds does not imply that the error is insignif-
ulation and Magpie. Third, both previous simulations icant. Non-indexed entries correspond to this work whereas
agree with our DFT simulation predictions as apposed index values, Exp. 1101 , 2-5102 , 6-7103 and Sim. 122 , 224 ,
to Magpie. Further experimental studies with CJ tem- come from the literature. (Same as Fig. 3 except a post hoc
perature measurements will be critical to resolving these energy correction as described in the text has been applied to
disparities. the DFT simulation results.)
 Application of a post hoc energy correction to our DFT
simulations, derived from high-level electronic structure
calculations of the detonation products, is enough to PETN in the product and reactant states of BLYP
bring our predictions into substantially better agreement with respect to CCSD(T). Compositions for the prod-
with experiment. Specifically, CCSD(T) energy calcula- ucts states are required to calculate ∆∆E at various
tions of the detonation products may provide a full ab ini- densities and temperatures. We find that DFT composi-
tio alternative to calibrating the energy to known experi- tional ambiguities, due to the sometimes vague notion of
mental measurements. We estimated an inherent error in what constitutes a molecule in a reactive system, com-
the energy of the products due to the BLYP functional plicates the definition of a bulk molecular composition;
with the following procedure. We calculated ∆∆E 
 = thus, we choose the well-defined chemical compositions
(Ep,BLYP − Er,BLYP )− Ep,CCSD(T) − Er,CCSD(T) , which predicted by the Magpie thermochemical code. We re-
represents the shift in the energy difference between strict ourselves to the gas phase in calculating ∆∆E for

9

obvious computational reasons. This is reasonable, as a
large portion of the energy in a molecular fluid comes
 DFT sim. H Magpie H
from the chemical bonding itself. Along the entire CJ
 DFT sim. HO Magpie H2O
locus we find a correction of about 1.8 kJ/g; this is 75 DFT sim. H2O
% of the value shown in Figure 4 near the same locus of
states, suggesting that the bulk of the inherent error in
 = 2.25 (g/cc)
the BLYP energy of the CJ products can be determined
by this recipe. Once we’ve calculated the ∆∆E (ρ, T )
 4

 mol X / mol PETN
surface, we add this to the energy surface used in our CJ
calculations. This produces a corrected CJ locus, shown 3
in Figure 6, which is in better agreement with the exper-
imental data and Magpie calculations. 2
 To be clear, we do not assert that this process will work
as well every time and for every HE. We merely present 1
it here as a possible recipe to approximate the error in
one’s choice of DFT functional for the condensed phase
calculation. This, very importantly, allows the method to
 0
persist as a predictive capability, and thus, it remains free = 1.00 (g/cc)
from requiring experimental data on which to calibrate 4
an energy shift of the DFT-products energy surface. We

 mol X / mol PETN
are currently performing simulations of HMX, DAAF,
and TATB. These future studies will help determine the
 3
generality of the energy correction procedure described
here. 2
 A second potential source of error is the lack of
quantum-nuclear molecular vibrations; however, we ex- 1
plored this possibility by re-calculating the Magpie CJ
locus in Figure 3 wherein the quantum harmonic os- 0
cillators were replaced with classical ones, and found
very small differences in Us (0.25-0.5%), P (1-1.5%) and
 2000 4000 6000 8000
T (1.5-1.8%), implying that this is not a significant source T (K)
of error.
 FIG. 7. Composition of water (H2 O), hydroxyl (HO), and
 atomic hydrogen (H) along two isochores as predicted by DFT
B. Molecular composition simulation and Magpie.

 First-principles simulation provides a unique window
into microscopic chemical details that can be used
to assess and improve thermochemical modeling. As
mentioned in Sect. II E, thermochemical models like
Magpie54,55 and Cheetah53 require a selection of possi- has some precedence in the literature,22,24,108 much less
ble product molecules. For high explosives composed of is known regarding the general importance of including
C, H, N, and O, typical entities are shown in Table D1. HO for a typical HE. In contrast, Fig. 8 shows a qualita-
The choice of species to include is generally motivated by tive agreement between the simulation and Magpie pop-
many previous theoretical works,62,104–106 as well as ex- ulations of CO2 , CO, and O; the agreement is better
periment107 . Optimizing intermolecular parameters for at lower density where many body crowding effects are
new species is a non-trivial effort, as such, the prospect minimal and the “mean field” assumptions in Magpie are
of narrowing the scope for new molecules via ab initio more accurate. The importance of other carbon/oxygen
simulation, is enticing, and will be addressed in future species is not totally ruled out by this analysis, but it is
works. suggestive that the chemical description is at least qual-
 As demonstrated in Fig. 7, our DFT simulations itatively adequate for carbon-oxygen species. A more
strongly support the inclusion of HO (hydroxyl) with H complete assessment of simulated molecular populations
and H2 O for thermochemical modeling. The lack of HO and how they compare to thermochemical codes neces-
in Magpie is undoubtedly a major driving force behind sities further simulation. Specifically, data on multiple
the observed discrepancies in the populations of H2 O, high explosives will provide suggested improvements and
and H. In general Magpie has much more water than molecular species for inclusion that are relevant across
the simulations and the inclusion of HO would provide a the board, and not just possibly for one HE, which is the
mechanism for its reduction. While the presence of HO ultimate goal.

10

 calculations, CCSD(T), on the product molecules. Using
 DFT sim. CO Magpie CO product compositions obtained from Magpie, CCSD(T)
 DFT sim. CO2 Magpie CO2 calculations suggest that BLYP actually overestimates
 DFT sim. O Magpie O the energies of isolated product molecules–which in the
 relative abundances for PETN detonation products–
 = 2.25 (g/cc) roughly amounts to ≈ 1.8 kJ/g along the CJ locus. For
 a select set of DFT configurations where molecule defini-
 4 tions are clear-cut, similar energy corrections are found.
mol X / mol PETN

 Important future extensions of this work are three-
 3 fold. First, new high explosives must be examined to
 see if the post hoc energy correction, described above,
 2 is robust. If yes, this implies that a quantitatively ac-
 curate first-principles performance analysis of a HEs is
 1 within reach, and importantly, without a need for cali-
 brating to a small number of experimental measurements.
 0 Work along these lines is already underway for HMX,
 DAAF, and TATB. Second, the product molecules pro-
 = 1.00 (g/cc) duced by ab initio DFT simulation warrant a closer ex-
 amination for species of possible importance to thermo-
 4 chemical modeling. Such information has the potential to
mol X / mol PETN

 validate and improve thermochemical modeling efforts in
 3 a manner that is complementary to experiments. Key in-
 formation beyond what we explored in this work includes
 2 the electronic environment of the molecular aggregates to
 differentiate between charged and radical species. Third,
 1 our approach can be used as a validation tool for ML
 force fields. Specifically, by using ML force fields directly,
 0 much greater system sizes and time scales are accessible.
 2000 4000 6000 8000 The reliability of these predictions can be probed by com-
 T (K) paring the results of small-scale ML + NMC simulations
 with direct ML simulations.

FIG. 8. Composition of carbon dioxide (CO2 ), carbon monox-
ide (CO), and atomic oxygen (O) along two isochores as pre-
dicted by DFT simulation and Magpie.
 ACKNOWLEDGMENTS

IV. CONCLUSIONS
 R.B.J. acknowledges funding from the Nicholas C.
 Metropolis Postdoctoral Fellowship and the ASC-PEM-
 We have demonstrated a modern computational strat-
 HE program at Los Alamos National Laboratory. J.A.L.
egy that enables the ab initio performance analysis of a
 and C.T. acknowledge funding from the ASC-PEM-HE
high explosive over an unparalleled range of conditions.
 program at Los Alamos National Laboratory. This work
Crucial to the success of our approach is the combina-
 was supported by the US Department of Energy through
tion of (1) an efficient Monte Carlo simulation framework
 the Los Alamos National Laboratory. Los Alamos Na-
catered to reactive systems and (2) a machine learning
 tional Laboratory is operated by Triad National Secu-
based equation of state model to maximize the informa-
 rity, LLC, for the National Nuclear Security Admin-
tion content available from the many noisy simulations.
 istration of U.S. Department of Energy (Contract No.
CJ state predictions for the detonation velocity, pressure
 89233218NCA000001).
and temperature were obtained over an unprecedented
range of initial unreacted HE densities.
 DFT(BLYP) simulation systematically under-predicts
the CJ detonation velocity, pressure and temperature
relative to both experiment and thermochemical predic- DATA AVAILABILITY STATEMENT
tions. This discrepancy is attributable to an overesti-
mated energy gap between the product molecule state-
points and the unreacted crystalline high explosive. In- The data that support the findings of this study are
terestingly, the energy discrepancy may be mostly ac- available from the corresponding authors upon reason-
counted for by post hoc high-level electronic structure able request.

11

Appendix A: Monte Carlo cluster moves Appendix B: Machine learned reference potential

 A cluster move is a composed from the following se- Our machine learning model assumes a decomposition
quence: of the total system energy into individual atom energies
 (the locality assumption). Furthermore, the energy of an
 1. Randomly pick a seed atom. atom is decomposed into six unique contributions. With-
 2. Build a cluster from a set of atom-atom cutoff dis- out specifying the exact functional forms yet, the first two
 tances based on the atom types. contributions correspond to explicit two- and three- body
 correlations via
 3. Unwrap the coordinates from periodic imaging to N
 ensure a contiguous representation of the cluster is
 X
 E2 (i) ≡ f2 (i, j) (B1)
 obtained. j

 4. Randomly displace the cluster center of position and
 (COP).
 N
 (κ) (κ)
 X
 5. Randomly rotate the cluster about the COP. E3 (i) ≡ f3 (i, j, k) (B2)
 j,k,
 6. Reapply periodic boundary conditions to the clus- j6=k
 ter. (κ)
 where f2 and f3 with κ ∈ [1, 2] are yet to be speci-
 7. Using the same seed atom, attempt to build the fied functions of two and three atoms (denoted by their
 cluster again and reject the move if not possible indices) respectively. Analagous to the explicit forms
 due to the formation of new connections (required above, three different “infinite” body forms are used as
 for preserving detailed balance).85,86 well
 8. If not already rejected, perform a standard, unbi- N
 Y
 ased, metropolis acceptance rejection check based e2 (i) ≡
 E [1 + f2 (i, j)] (B3)
 on the potential energy change.83–86 j

The COP displacement (Step 4) is of the standard form and
used in MC: each Cartesian component is displaced by N
 e (κ) (i) ≡ (κ)
 Y
a uniformly randomly sampled amount in the interval E3 [1 + f3 (i, j, k)] (B4)
[0, δxmax ]. The random rotation (Step 5) is more involved j,k,
and elaborated on below. j6=k

 A random 3D rotation about the COP is realized via In practice, the “infinite body” forms never involve all of
successive rotation of the Euler angles (φ, θ, ψ) to the the atoms in a system as a radial cutoff (described below)
COP removed Cartesian coordinate vector x. The first is imposed.
and last rotations are about the z-axis, which we denote As originally used by Jadrich and Leiding 45 , the spe-
by the rotation matrix Rz (γ) where γ is the rotation cific two and three body terms are built from a radial (R)
angle. The middle rotation is about the x-axis and is function and an angular (A) function as
denoted as Rx (γ). The rotated coordinate is obtained
via f2 (i, j) ≡ fR (ri,j ) (B5)

 x̃ ≡ Rz (ψ)Rx (θ)Rz (φ)x (A1) and
 (κ)
The reverse rotation is obtained by negating the angles f3 (i, j, k) ≡ fR (ri,j )fR (ri,k )[δ1,κ + δ2,κ fA (θi,j,k )] (B6)
and swapping ψ and φ, yielding where ri,j is the radial distance between particles i and
 x ≡ Rz (−φ)Rx (−θ)Rz (−ψ)x̃ (A2) j, θi,j,k is the angle formed between a vector pointing
 from particle i → j and from particle i → k. The radial
The most straightforward way to ensure detailed balance component is
(actually, super-detailed balance) is to guarantee that X e−r/Zi
the reverse rotation is equally likely to be proposed as fR (r) ≡ Ai cos[ki r − Qi ] (B7)
the forward rotation. We realize this by uniformly ran- i=1
 r
domly sampling each angle from the symmetric range
 where {Ai , Zi , Qi } is the set of flexible parameters with
[−αmax , αmax ] where αmax is the maximum allowed an-
 associated wave-vectors {ki }. The corresponding angular
gular rotation mentioned in Sect. II C. Technically, θ can
 term is
be sampled from a different symmetric range than φ and X 2 2
ψ, but setting all of them equal was effective and conve- fA (θ) ≡ Bi e−(cos[θ]−Ci ) /(2σi ) (B8)
nient. i=0

12

where {Bi , σi } is the set of flexible parameters with fixed Appendix C: Equation of state model
{Ci } sampled from the range [-1, 1] at 200 evenly spaced
points. Independent parameter sets for pairs and triplets As mentioned in Sect. II D, we decompose the
of species were employed for each of the five energy Helmholtz free energy per volume (a) into ideal (aid ) and
contributions (Eq. B1-B4). Further simplification was excess (aex ) contributions. Model flexibility is embedded
achieved by enforcing invariance to permutation of atoms in the excess free energy contribution
in the two body terms of Eqn. B1 and to permuting the
neighbor (non-central j and k) atoms in any angular func- O m−1
 X X
tion. For the wave-vectors, we used 400 evenly spaced aex ≡ ne0 + λi Γin Γm−i
 T (C1)
wave-vectors in the range (0, 4π). m=2 i=1

 In an effort to regularize our ML potential, we added in where e0 is an optimized baseline reference energy per
a core repulsion of the Weeks-Chandler-Anderson (WCA) atom, O is the order of the multi-variable polynomial
form at close atom-atom separations as well as a finite expansion,
range cutoff between atoms. For a pair of atom types,
the hard core was set to values slightly (∼ 0.2 Å) below (n/n)2
the minimum inter-atomic distance between those atom Γn ≡ (C2)
 δn + n/n
types encountered in the training data. The cutoff dis-
tance was set to rC = 3.0 Å. Before the hard cutoff, and
a smooth cutoff using the standard Sigmoid form, S(r),
with a switching distance of δr = 0.02 Å is applied to all δT,1 + T /T
 ΓT ≡ (C3)
radial functions of the form S((rC − 4δr − ri,j )/δr) to en- δT,2 + T /T
sure negligibly small forces at the hard cutoff. An inner
smooth cutoff is also applied to every radial function via and n is the number of atoms per unit volume (num-
S((ri,j − rI )/δr) where rI = 0.3 Å. ber density) and λi are optimized weights. The density
 and temperature shift factors (δn , δT,1 , δT,2 ) are also op-
 Through an iterative process, we created an initial
 timized while the reference density (n) and temperature
ML model that was periodically retrained to newly ac-
 (T ) are set to the mean of the density and temperature
quired simulation data. The initial data was obtained via
 in the data set.
Density Functional Theory Tight Binding simulations in
 The choice of the above form enforces a few key fea-
CP2K using the UFF force field with UFF dispersion cor-
 tures. The first, is the limit to ideality with vanishing
rections. 1000 configurations evenly spaced in the den-
 density. While a better limiting view at low density is an
sity range of 0.7 − 2.5 g/cc at simulation temperatures of
 ideal gas of molecules, this is more challenging to capture
4000 K, 6000 K, 8000 K and 10000 K were collected. En-
 in a fully reactive context where molecules are emergent
ergy and forces were recalculated within the Kohn-Sham
 entities. Second, the choice of Γn provides a low density
DFT formalism, outlined in Sect. II A, on the entire ag-
 crossover from quadratic to linear with increasing den-
gregated data-set of configurations using electronic tem-
 sity. This provides the correct low density quadratic cor-
peratures of 1000 K, 4500 K, and 8000 K. The initial ML
 rection on top of the reference energy (i.e., corrections
model was fit to this mixed electronic temperature data-
 must come from 2-body and higher order correlations)
set with the goal of realizing a model that performs well
 while capturing the empirically observed linear-like be-
across all conditions. Retraining with additional DFT
 havior at higher densities. Third, the form of ΓT yields a
simulation configurations took place at unspecified inter-
 finite “cold curve” (ground state) contribution as T → 0
vals whenever the acceptance rate for NMC (using chain
 as well as a finite contribution as T → ∞, corresponding
lengths of 30-40) was observed to drop below 40%. Re-
 to an unweighted average of the potential energy over
training ceases upon approach to equilibrium.
 configuration space.
 Overall, our NMC approach provided a significant ac- For a fixed set of {n, T } points with corresponding e, P
celeration over an equivalent (non-nested) MC approach measurements,
 P 2we fit 2the model according to the loss
for a comparatively small upfront investment comprised function i [∆e,i + ∆P,i ]/(2M ) where ∆e and ∆P are
of generating training data and training the ML mod- the respective energy and pressure differences between
els. The acceleration of our NMC scheme for the nested the model and the DFT simulation data and M is the
moves (swap and cluster), relative to bare MC, was number of measurements. 3-fold cross validation is used
O(10).45–48 Even with the acceleration of swap and clus- to identify the optimal model order (O) for training to
ter moves, the 225 simulations totaled O(1M) CPU hours the whole data set. Our model is implemented in the
to reach the final set of equilibrium samples. The total PyTorch machine learning package.109
cost of the simulations far outweighs the initial training For extracting uncertainties in all EOS derived quan-
data generation cost of O(10K) CPU hours. Occasional tities, we employ bootstrap re-sampling. Raw simulation
training/retraining of the ML reference potential is even derived e, P data points are re-sampled with replacement
less burdensome; it was performed on a standard desktop to create 100 new data sets of the same size as the origi-
in a few hours. nal. A cross-validated model is fit to each of the data-sets

13

to yield an ensemble of EOS models, from which an en- The density and temperature shift factors
semble of any derived parameter can be arrived at. From (∆ρ,1 , ∆ρ,2 , ∆T,1 , ∆T,2 ) are all optimized along with
the ensemble we calculate the 50th percentile (median) to the weights κi while the reference density (ρ) and
indicate the predicted value as well as the 2.5th (lower) temperature (T ) are set to the mean of the density and
and 97.5th (upper) percentile bounds to account for the temperature in the data set. The squared form ensures
95% confidence intervals. the predictions are positive definite and the form of the
 density and temperature variables yield asymptotically
 finite contributions in the cold and low density limits.
Appendix D: Magpie products Models
 P 2 were fit according to the following loss function
 i ∆fM ,i /M where ∆fM is the population difference
 between the prediction and DFT simulation and M is
 Magpie products the number of measurements. Population uncertainties
 were estimated using the same protocol outlined in
 Common Name Formula
 Sect. C for the equation of state model.
 atomic hydrogen H
 atomic carbon C
 atomic nitrogen N 1 P. W. Cooper, Explosives engineering (John Wiley & Sons,
 atomic oxygen O 2018).
 2 J. A. Zukas, W. Walters, and W. P. Walters, Explosive effects
 water H2 O
 and applications (Springer Science & Business Media, 2002).
 molecular hydrogen H2 3 W. C. Davis, “High explosives: The interaction of chemistry
 molecular nitrogen N2 and mechanics,” Los Alamos Science 2, 48–75 (1981).
 4 J. B. Bdzil, T. Aslam, R. Henninger, and J. Quirk, “High-
 molecular oxygen O2
 explosives performance,” Los Alamos Sci 28, 96 (2003).
 carbon dioxide CO2 5 C. A. Handley, B. D. Lambourn, N. J. Whitworth, H. R.

 carbon monoxide CO James, and W. J. Belfield, “Understanding the shock and
 nitric oxide NO detonation response of high explosives at the continuum and
 meso scales,” Applied Physics Reviews 5, 011303 (2018),
 nitrogen dioxide NO2 https://doi.org/10.1063/1.5005997.
 6 A. C. Landerville and I. I. Oleynik, “Vibrational and thermal
 nitrous oxide N2 O
 properties of β-hmx and tatb from dispersion corrected density
 methane CH4
 functional theory,” AIP Conference Proceedings 1793, 050007
 ammonia NH3 (2017), https://aip.scitation.org/doi/pdf/10.1063/1.4971541.
 7 A. C. Landerville, M. W. Conroy, M. M. Budzevich, Y. Lin,
 isocyanic acid HNCO
 C. T. White, and I. I. Oleynik, “Equations of state for energetic
 formic acid CH2 O2
 materials from density functional theory with van der waals,
 thermal, and zero-point energy corrections,” Applied Physics
Table D1. Molecules used in the Magpie thermochemical Letters 97, 251908 (2010), https://doi.org/10.1063/1.3526754.
calculation of the equilibrium product states. All species are 8 Z. Zheng and J. Zhao, “Unreacted equation of states of typ-
uncharged within Magpie; however, the chemical formulas are ical energetic materials under static compression: A review,”
more generally used in this article to denote a molecular ag- Chinese Physics B 25, 076202 (2016).
 9 W. Kohn and L. J. Sham, “Self-consistent equations includ-
gregate with the bonding topology specified by the common
name. ing exchange and correlation effects,” Phys. Rev. 140, A1133–
 A1138 (1965).
 10 J. Kohanoff, Electronic Structure Calculations for Solids and
Appendix E: Molecular population model Molecules: Theory and Computational Methods (Cambridge
 University Press, Cambridge, U.K., 2006).
 11 W. Perger, J. Zhao, J. Winey, and Y. Gupta, “First-principles
 Molecular populations were modeled using a custom study of pentaerythritol tetranitrate single crystals under high
analytical form implemented in PyTorch109 pressure: Vibrational properties,” Chemical Physics Letters
 #2 428, 394 – 399 (2006).
 12 S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, “A consistent
 " O m−1
 X X
 fM ≡ κi Λin Λm−i
 T (E1) and accurate ab initio parametrization of density functional dis-
 m=2 i=1
 persion correction (dft-d) for the 94 elements h-pu,” J. Chem.
 Phys. 132, 154104 (2010), https://doi.org/10.1063/1.3382344.
 13 J. G. Brandenburg and S. Grimme, “Accurate modeling
where fM is the number of molecules of a given type
 of organic molecular crystals by dispersion-corrected den-
yielded per molecule of high explosive (PETN) and sity functional tight binding (dftb),” The Journal of Physi-
 cal Chemistry Letters 5, 1785–1789 (2014), pMID: 26273854,
 ∆ρ,1 + ρ/ρ https://doi.org/10.1021/jz500755u.
 ΛT ≡ (E2) 14 N. W. Ashcroft, N. D. Mermin, et al., “Solid state physics [by]
 ∆ρ,2 + ρ/ρ
 neil w. ashcroft [and] n. david mermin.” (1976).
 15 B. W. Hamilton, M. P. Kroonblawd, M. M. Islam, and A. Stra-
and chan, “Sensitivity of the shock initiation threshold of 1,3,5-
 triamino-2,4,6-trinitrobenzene (tatb) to nuclear quantum ef-
 ∆T,1 + T /T fects,” The Journal of Physical Chemistry C 123, 21969–21981
 ΛT ≡ (E3)
 ∆T,2 + T /T (2019), https://doi.org/10.1021/acs.jpcc.9b05409.