Ab Initio Equation of State of the Organic Molecular Crystal:�-Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine

Frank J. Zerilli† and Maija M. Kuklja*,‡

Research and Technology Department, NaVal Surface Warfare Center, Indian Head, Maryland 20640, andDepartment of Materials Science and Engineering, UniVersity of Maryland, College Park, Maryland 20742

ReceiVed: December 12, 2009; ReVised Manuscript ReceiVed: March 9, 2010

We apply a simple strategy for calculating from first principles a thermodynamically complete equation ofstate for molecular crystals using readily available quantum chemistry techniques. The strategy involves acombination of separate methods for the temperature-independent mechanical compression and the thermalvibrational contributions to the free energy. A first principles equation of state for �-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (�-HMX) has been calculated for temperatures between 0 and 400 K and for specificvolumes from 0.42 to 0.55 cm3/g, corresponding to relative volumes from 0.8 to 1.03. The calculated 300 Kisotherm agrees very well with the experimentally measured pressure-volume relation. We also discussthermodynamic properties of the material such as the volumetric thermal expansion coefficient, the Gruneisenparameter, and the specific heat (1.0 kJ/kg/K at 300 K and atmospheric pressure). The developed computationalapproach exhibits a reliable predictive power and is easily transferable to other molecular materials.

I. Introduction

The accurate and reliable modeling of the physical andchemical properties of molecular materials represents a con-siderable challenge because of both the complexity of themolecular and crystalline structures and the inability of firstprinciples techniques to handle a rich variety of externalconditions. An absence of the dispersion forces in computationalschemes and an uncertainty of how to treat, for example, thetemperature dependence and the compressibility within a singlemethod additionally hamper the understanding and the mostefficient use of such materials.

The material selected for this study, �-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (C4H8N8O8), also known as �-HMX,is an organic molecular crystal, which is well-known for itspropensity for rapid explosive decomposition under externalstimuli. The material, therefore, is often called an energeticmaterial and used as an explosive and fuel (propellant) for rocketengines. �-HMX is one of four known polymorphic forms ofHMX, designated as R, �, γ, and δ.1 The high-temperature δphase exhibits the most violent explosive behavior, but it doesnot exist under normal conditions and requires a phase transitiondirectly from the � phase (which is stable up to about 375 Kunder atmospheric pressure) or via the R phase (which is stablebetween 375 and 435 K).1 The γ-phase is not, strictly speaking,a polymorph, but a hydrate that can form from �-HMX in thepresence of water.

Despite the extensive efforts of many research groups,atomistic details of the phase transition and mechanisms thatgovern the ease of initiation of chemistry (sensitivity) and theextent of heat released during a violent event (performance) arefar from being understood. Even the simpler task of obtaininga complete equation of state for four phases of HMX isproblematic because experimental data is difficult, dangerous,or impossible to obtain. From a theoretical point of view, while

compression computation is normally straightforward in ab initiocalculations,2-7 it is not trivial to combine it with the temperaturecomponent.8 We are not aware of any other reported attemptsto obtain a thermodynamically complete EoS for large molecularcrystals from first principles except for the work cited in ref 8.

In this article, we extend a relatively simple approach, recentlyused for obtaining the thermodynamically complete ab initioequation of state (EoS) for the organic molecular crystal, 1,1-diamino-2,2-dinitroethylene (also known as FOX-7),8 to anotherenergetic material, �-HMX. �-HMX has a monoclinic P21/cstructure with two molecules per unit cell.9 The cell parametersfor the P21/c structure, a ) 6.54 Å, b ) 11.05 Å, c ) 8.70 Å,and � ) 124.3°, were reported by Cady, Larson, and Cromer,10

who transformed the measured values of Eiland and Pepinsky11

from the P21/n structure. The unit cell and molecular unit aredepicted in Figure 1.

Our computational strategy is based on readily availablequantum chemical computational methods. The key here is totreat the temperature-independent mechanical compression andthe thermal vibrational contributions to the free energy separatelyand only later to combine them into a complete ab initiothermodynamic equation of state. This recipe allowed us toobtain an EoS for �-HMX over a pressure range from 0 to 10

* To whom correspondence should be addressed. E-mail: mkukla@nsf.gov.† Naval Surface Warfare Center (Retired).‡ University of Maryland.

Figure 1. Structure of the �-HMX crystalline unit cell (a) and themolecular unit (b). The irreducible asymmetric unit consists of atoms1-14.

J. Phys. Chem. A 2010, 114, 5372–53765372

10.1021/jp911767q 2010 American Chemical SocietyPublished on Web 04/05/2010

GPa and a temperature range of 0 to 400 K. The cold mechanicalcompressibility was obtained using the Hartree-Fock methodwith a computational scheme utilizing a basis set consisting oflinear combinations of atomic orbitals as implemented in thecomputer program CRYSTAL.12 The T > 0 vibrational contribu-tions to the Helmholtz free energy were obtained in the quasi-harmonic approximation using density functional theory methodsas implemented in the plane wave pseudopotential computerprogram ABINIT.13 An agreement with available experimentaldata reinforces a reliability and a predictive power of thepreviously reported EoS of 1,1-diamino-2,2-dinitroethylene8 andmakes us believe that the developed computational strategy isgenerally applicable to other materials over a wide range ofconditions.

II. Details of the Calculations

The thermodynamic properties of the system are obtained bycalculating the Helmholtz free energy, which may be written

where U0(V) is the temperature-independent energy of mechan-ical compression and Fvib is the vibrational contribution to thefree energy, written in the harmonic approximation as

and Felect is the electronic contribution. In the quasi-harmonicapproximation used in this work, ω(i,qb), the frequency of theith vibrational mode at a point qb in the Brillouin zone is afunction of the geometry and, hence, the specific volume ofthe crystal. For the temperature range considered here, theelectronic contribution is negligible and we need only considerthe vibrational contribution.

A. Mechanical Compression. The calculation of the me-chanical compression energyU0(V) was reported previously andonly a brief summary is given here.2-4,8 The basic quantumcalculations for a periodic structure were performed with theCRYSTAL computer program, using the Hartree-Fock methodand a linear combination of atomic orbitals basis set (6-21G inPople’s notation). Scaling factors were applied to the outervalence orbitals, reducing their range, to adapt what wouldnormally be a basis set optimized for isolated clusters of atomsfor use in a periodic structure computation. Optimizations ofthe atomic coordinates and lattice parameters were doneseparately and iteratively. The atomic coordinates were firstoptimized using Zicovich-Wilson’s LoptCG script,12 which callsCRYSTAL to calculate the energy for each configuration. Thelattice parameters then were optimized under a fixed volumeconstraint for a set of volumes representing a number ofhydrodstatic compressions using an in-house written programbased on the downhill simplex method of Nelder and Mead.14,15

This was repeated until satisfactory convergence was achieved.All optimizations preserved the space group of the crystal.

An effect of accounting for the basis set superposition error(bsse) using the counterpoise method16 has been analyzed inour recent calculations of the mechanical compression curvefor solid nitromethane with CRYSTAL code using both

Hartree-Fock and density functional theory (DFT) methods.4

Two different 6-21G and 6-31G** basis sets were tested. The6-31G** basis set was optimized by scaling the outer valenceand polarization orbitals. It was found that Hartree-Fockcalculations with a 6-21G basis set, uncorrected for basis setsuperposition error, gave the best agreement with experiment.The DFT methods (Perdew-Wang generalized gradient ap-proximation17) calculations gave nearly the same results. As onemoves toward higher compressions away from the zero pressureequilibrium state, the 6-21G and 6-31G** basis sets gavecomparable results, whether or not corrected for bsse.

Because it gives a result most closely corresponding toexperiment, we choose here to rely on the results of theHartree-Fock calculation with a 6-21G basis set, uncorrectedfor basis set superposition error, for the mechanical compressioncurve of �-HMX.

B. Vibrational Contribution. The dynamical matrix wascalculated with the code ABINIT, which utilizes a plane wavebasis set with pseudopotentials to describe the core electronconfiguration. In this work, the local density approximation(LDA) was used together with pseudopotentials built by D. C.Allan and A. Khein,18 generated with the Troullier-Martinstechnique.19 In these calculations, we used a plane wave energycutoff of 32 hartree (871 eV) with a 2 × 2 × 2 Monkhorst-Pack20 grid in reciprocal space. The importance of the carefultreatment of convergence in DFT calculations was well il-lustrated for a series of energetic crystals by Byrd and Rice.7

In the calculations for �-HMX, we rely on our previousconvergence studies for 1,1-diamino-2,2-dinitroethylene, whichhas the same space group and the same number of atoms in theasymmetric unit.2,3,8 Those calculations showed that the energyfor a typical self-consistent field calculation was 0.05 H (1.4eV) lower (0.01%), with a cutoff of 64 H as opposed to 32 H,and the highest frequency zero wave vector mode was calculatedto be 3036 cm-1 at 32 H and 3038 cm-1 at 64 H.

The optimized lattice parameters and atomic positionsobtained as a result of the calculations for the mechanicalcompression were used as a starting point for the phononspectrum calculations. These results were again optimized atfixed volume with ABINIT in the local density approximationand the dynamical matrices for the eight wave vectors (0 0 0),(1/2 0 0), (0 1/2 0), (0 0 1/2), (0 1/2

1/2), (1/2 0 1/2), (1/21/2 0), and

(1/21/2

1/2) were then obtained from calculations using the newlyoptimized structures. Thermodynamic properties and the phonondensity of states are calculated on a fine grid in the dual spaceby Fourier transforming the dynamical matrices on this regulareight point grid to obtain interatomic force constants and thenusing the interatomic force constants to obtain the dynamicalmatrix at an arbitrary point in the dual space.

III. Results

The mechanical compression curve for �-HMX is shown inFigure 2. Both atomic coordinates and lattice parameters wereoptimized under a fixed volume constraint for each value ofspecific volume. The energy in Figure 2 is shown as a functionof effective linear expansion λ defined as (V/V0)1/3, where thereference volume V0 corresponds to the experimental10 unit cellvolume 519 Å3. The energy is shown for a range of λ from0.93 to 1.01, which corresponds to a range of relative volumefrom 0.80 to 1.03 and a range of specific volume from 0.42 to0.54 cm3/g.

The phonon density of states obtained by sampling theBrillouin zone is shown in Figure 3. The cumulative distributionis shown for each compression in the range from 0.42 to 0.54

F(V, T) ) U0(V) + Fvib(V, T) + Felect(V, T) (1)

Fvib(V, T) ) 12 ∑

i)1

N

∑qbpω(i, qb) +

kT ∑i)1

N

∑qb

ln(1 - e-pω(i,qb)/kT) (2)

�-Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine J. Phys. Chem. A, Vol. 114, No. 16, 2010 5373

cm3/g as a histogram with bin width of 10 cm-1. The mostsignificant changes in the distribution occur for the lowfrequency modes (those below ∼200 cm-1) and the 12 highestfrequency modes, which move higher in frequency withcompression. Modes 153-156 have the largest increase (from∼2505 to 2675 cm-1). The modes in the 1100-1400 cm-1 rangetend to decrease in frequency with compression. This contrastswith the behavior of 1,1-diamino-2,2-dinitroethylene in whichthe 16 highest frequency modes decrease in frequency withcompression and are separated by a gap in the density of stateextending from approximately 1650 up to about 2900 cm-1.Also, in FOX-7, only the low frequency modes have a significantvariation in frequency with compression.

Experimental measurements of the 300 K isotherm usingX-ray crystallography of the material compressed in a diamondanvil cell have been reported by Yoo and Cynn21 and Gumpand Peiris,22 and we compare the ab initio calculated 300 Kisotherm with the experimental measurements in Figure 4. Wenote very good agreement between the calculated and themeasured isotherms. The calculated unit cell volume at 300 Kis 531 Å3 as compared with the experimentally determined valueof 519 Å3. Figures 5-11 summarize the calculated thermody-namic properties (bulk modulus, specific heat, thermal expansioncoefficient, and Gruneisen parameter) of �-HMX over thetemperature range from 0 to 400 K and pressures from 0 to 10GPa. The data in these figures have been linearly interpolatedfrom the data calculated at individual temperatures and volumesto display the thermodynamic properties as functions of tem-perature and pressure.

The isothermal bulk modulus is relatively insensitive tovariation in temperature at constant volume. However, it is a

fairly sensitive function of temperature at constant pressure (seeFigure 5). It is calculated to be approximately 14.2 GPa at 300K and atmospheric pressure. Reported experimentally deter-mined values of the bulk modulus range from 10.6 to 21 GPaat normal temperature and pressure.21-23 In Figure 6, thecalculated isothermal bulk modulus is compared with thatdetermined from the experimental data of Gump and Peiris.22

The calculated specific heat at constant volume is shown inFigure 7 as a function of temperature and pressure. The specificheat is calculated to be 1 kJ/kg/K at 300 K and atmosphericpressure with only a very small variation with respect tocompression. In Figure 8, the calculated specific heat atatmospheric pressure is compared to experimental data of Smith,Ramsburg, and Harrison,24 Shoemaker, Stark, and Taylor,25 and

Figure 2. Energy along the temperature-independent mechanicalcompression curve for �-HMX as a function of effective linearexpansion λ ) (V/V0)1/3.

Figure 3. Phonon density of states for �-HMX for various specificvolumes from 122 to 166 cm3/mol (λ ) 93-101%). Inset shows thedetail of the region from 1100 to 1400 wave numbers.

Figure 4. Calculated 300 K isotherm of �-HMX compared toexperimental data of Gump and Peiris (ref 22) and Yoo and Cynn (ref21).

Figure 5. Calculated bulk modulus of �-HMX as a function oftemperature at constant pressure for pressures from 0 to 10 GPa.

Figure 6. Calculated isothermal bulk modulus of �-HMX comparedwith that derived from the experimental data of Gump and Peiris (ref22).

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Hanson-Parr and Parr.26 Shoemaker et al. and Hanson-Parr andParr also measured the specific heat of δ-HMX and disagreesubstantially, as seen in the figure. Preliminary calculations (notpresented here) of the specific heat of δ-HMX tend to supportShoemaker’s results.

The volumetric thermal expansion coefficient is calculatedto be 210 ppm/K at 300 K and atmospheric pressure and variesconsiderably with compression as well as temperature, fallingto 51 ppm/K at 300 K and pressure of the order of 5 GPa, asshown in Figure 9. The apparent kink in the atmosphericpressure (0 GPa) curve appears to be an artifact of the linear

interpolation from temperature/volume data. At atmosphericpressure and 350 K, the expansion rises to 233 ppm/K and, at400 K, it is 256 ppm/K. Gump and Peiris22 determined thevolumetric thermal expansion coefficient to be 270 ppm/K overthe temperature range from 303 to 413 K and atmosphericpressure. They also noted hardly any expansion in this temper-ature range above 3 GPa, consistent with the calculated valueof the order of 50 ppm/K. Weese and Burnham27 determinedthe coefficient of thermal expansion of � and δ HMX experi-mentally. Their results for temperatures below 413 K, whereHMX is presumably in the � phase, are also shown in Figure9. There appears to be considerable scatter in their data and thevariation with temperature appears not to be physically realistic.

The Gruneisen parameter γ (shown in Figure 10) varies from1.58 at 300 K and atmospheric pressure to about 3.4 at 75 K.At temperatures greater than about 75 K, its variation withpressure is small. At 300 K, it ranges from 1.58 at atmosphericpressure to 1.37 at 10 GPa. Unlike FOX-7 (see ref 8), γ/V hasonly a small variation with pressure over the entire temperaturerange from 0 to 400 K (see Figure 11).

IV. Summary and Conclusions

The theoretical prediction of equations of state is importantin many fundamental and practical problems, in fields as farranging as planetary science and military applications, whereexperimental data may be difficult, expensive, or, in some cases,impossible to obtain. In this regard, there is great interest incomputing the equation of state for a class of organic molecularcrystals that are the ingredients of materials often referred to asenergetic materials because of their propensity for rapid

Figure 7. Calculated specific heat at constant volume of �-HMX as afunction of temperature at constant pressure for pressures from 0 to 10GPa.

Figure 8. Calculated specific heat of �-HMX compared to experimentaldata of Smith et al. (ref 24), Shoemaker et al. (ref 25), and Hanson-Parr and Parr (ref 26).

Figure 9. Calculated thermal expansion of �-HMX as a function oftemperature at constant pressure for pressures from 0 to 10 GPa. Solidcircles are experimental data of Weese and Burnham (ref 27). The solidline terminated with the diamond symbols represents the averagethermal expansion over the temperature range 303-413 K determinedby Gump and Peiris (ref 22).

Figure 10. Calculated Gruneisen parameter of �-HMX as a functionof temperature and pressure for pressures from 0 to 10 GPa.

Figure 11. The ratio γ/V as a function of pressure and temperature.The solid line shows the average and the dotted lines show the minimumand maximum values over the pressure range from 0 to 10 GPa. At300 K, the maximum is 33% larger than the minimum.

�-Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine J. Phys. Chem. A, Vol. 114, No. 16, 2010 5375

explosive decomposition under shock conditions. While fits ofexperimental data to analytical equations of state have achievedsome success, ab initio calculations hold the promise forcalculating the equation of state as well as phase diagrams anda number of other chemical and physical properties of thesematerials in cases in which experimental data is not availableand not easy to obtain.

In this article, a thermodynamically complete equation of statefor a temperature range of 0 to 400 K, and a range of pressuresfrom 0 to 10 GPa, corresponding to a range of density from1.8 to 2.4 g/cm3 was calculated from first principles for theorganic molecular crystal �-HMX. This was achieved by usinga combination of first principles methods for the temperatureindependent mechanical compression and the thermal vibrationalcontributions to the free energy. The mechanical compressionwas obtained by means of solutions of the Schrodinger equationutilizing the Hartree-Fock method. The vibrational contribu-tions to the free energy were obtained from the phononfrequency spectrum as calculated utilizing a density functionaltheory method with the local density approximation. Fromcomparison with experimental data, it appears that the calcula-tion is reasonably accurate. For some properties, for example,thermal expansion, the calculation appears to be more reliablethan the experimental measurements.

The strategy suggested here for calculating a thermodynami-cally complete equation of state for molecular crystals from firstprinciples is relatively simple as it is based on readily availablequantum chemistry techniques. The strategy involves a com-bination of separate methods for the temperature-independentmechanical compression and the thermal vibrational contribu-tions to the free energy. Although these calculations are lengthyand require significant computer resources, they prove to bepractical, as the computational approach exhibits a reliablepredictive power and is easily transferable to other molecularmaterials.

Acknowledgment. This work was supported by the NSWCCore Research Program and by the Office of Naval Research.Computational resources were provided by the AeronauticalSystems Center Major Shared Resource Center, Wright-Patter-son Air Force Base, Ohio, under the Department of DefenseHigh Performance Computing Initiative. M.M.K. is grateful tothe Office of the Director of the National Science Foundationfor support under the Independent Research and DevelopmentProgram. Any appearance of findings, conclusions, or recom-mendations expressed in this material are those of the authorsand do not necessarily reflect views of the National ScienceFoundation.

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