first-principles study of the four polymorphs of crystalline...

First-Principles Study of the Four Polymorphs of CrystallineOctahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine

Weihua Zhu,*,† Jijuan Xiao,† Guangfu Ji,‡ Feng Zhao,‡ and Heming Xiao*,†

Institute for Computation in Molecular and Materials Science and Department of Chemistry,Nanjing UniVersity of Science and Technology, Nanjing 210094, China, and Laboratory forShock WaVe and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics,Mianyang 621900, China

ReceiVed: June 28, 2007; In Final Form: August 15, 2007

The electronic structure and vibrational properties of the four polymorphs of crystalline octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) have been studied using density functional theory within the local densityapproximation. The results show that the states of N in the ring make more important contributions to thevalence bands than these of C and N of NO2 and so N in the ring acts as an active center. From the lowfrequency to high-frequency region, the molecular motions of the vibrational frequencies for the four HMXpolymorphs present unique features. It is also noted that there is a relationship between the band gap andimpact sensitivity for the four HMX polymorphs. From the cell bond order per unit volume, we may infer thevariation order of crystal bonding for the four polymorphs and so predict their impact sensitivity order asfollows: â-HMX < γ-HMX < R-HMX < δ-HMX, which is in agreement with their experimental order.

1. Introduction

Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine, known asHMX, is an important and commonly used energetic ingredientin various high performance explosives and propellant formula-tions due to its thermal stability and high detonation velocityrelative to other explosives.1,2 HMX is known to exist in fourcrystalline phases, denoted asR, â, δ, and γ.3,4 Althoughâ-HMX is the thermodynamically stable form under ambientconditions, post-mortem analysis5 of samples recovered fromsafety experiments involving low-velocity projectile impacts onthe HMX-based plastic-bonded explosive (PBX)-95016 hasshown the formation ofδ-HMX in the vicinity of damagedregions within the material. It is also known that a layer ofδ-HMX is formed at the solid-melt interphase in deflagratingHMX. These observations present a safety concern sinceδ-HMX is considerably more sensitive thanâ-HMX. Therefore,a detailed and comparative study of the four HMX polymorphsoffers the possibility of determining microscopic properties andunderstanding their explosive properties.

Materials such as propellants and explosives contain tightlybonded groups of atoms that retain their molecular characteruntil a sufficient stimulus is applied to cause exothermicdissociation. This in turn triggers further dissociation leadingto initiation or ignition. The macroscopic behavior is ultimatelycontrolled by microscopic properties such as the electronicstructure and interatomic forces. Thus, a desire to probe morefundamental questions relating to the basic properties of HMXas a solid energetic material is generating significant interest inthe basic solid-state properties of such energetic systems.Although the detailed decomposition mechanisms by whichenergetic materials release energy under mechanical shock are

still not well understood, it has been suggested that thesedecompositions may result from transferring thermal andmechanical energy into the internal degrees of freedom of themolecules in energetic solids.7-9 It is thus very important tounderstand the electronic and vibrational properties of the fourHMX polymorphs. Many experimental studies have beenperformed to investigate the vibrational spectra of the HMXpolymorphs.10-14

The investigation of the microscopic properties of energeticmaterials, which possess a complex chemical behavior, remainsto be a challenging task. Theoretical calculations can play animportant role in investigating the physical and chemicalproperties of complex solids at the atomic level and theestablishment of the relationships between their structure andfunction. Previously, density functional theory (DFT) calcula-tions based on norm-conserving pseudopotentials and a localdensity approximation (LDA) were performed to examine theenergetics of the three pure polymorphic HMX (R, â, δ).15

Afterward, Lewis16 used the same method to investigate theenergetics of HONO formation in the three pure polymorphsof condensed-phase HMX, where intermolecular hydrogentransfer occurs. Recently, Ye and Koshi17 have evaluated theenergy transfer rates of the three pure HMX polymorphs in termsof the density of vibrational states and the unharmonic vibron-phonon coupling term, which were calculated by using a flexiblepotential containing both intra- and intermolecular terms. Asthe electronic structure and vibrational properties of the fourHMX polymorphs are not systematically investigated andcompared, there is a clear need to gain an understanding of thoseat the ab initio level.

In this study, we report a systematic study of the electronicstructure and vibrational properties of the four HMX crystallinephases (R, â, δ, andγ) from DFT. Our main purpose here is toexamine the differences in the microscopic properties of thepolymorphs and to understand their structure-function relation-ships.

* To whom correspondence should be addressed. Fax:+86-25-84303919. E-mail: [email protected]; [email protected].

† Nanjing University of Science and Technology.‡ China Academy of Engineering Physics.

12715J. Phys. Chem. B2007,111,12715-12722

10.1021/jp075056v CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 10/12/2007

The remainder of this paper is organized as follows. A briefdescription of our computational method is given in section 2.The results and discussion are presented in section 3, followedby a summary of our conclusions in section 4.

2. Computational Method

The calculations performed in this study were done withinthe framework of DFT,18 using Vanderbilt-type ultrasoft pseudo-potentials19 and a plane-wave expansion of the wave functions.The self-consistent ground state of the system was determinedby using a band-by-band conjugate gradient technique tominimize the total energy of the system with respect to theplane-wave coefficients. The electronic wave functions wereobtained by a density-mixing scheme20 and the structures wererelaxed by using the Broyden, Fletcher, Goldfarb, and Shannon(BFGS) method.21 The LDA functional proposed by Ceperleyand Alder22 and parametrized by Perdew and Zunder23 namedCA-PZ, was employed. The cutoff energy of plane waves wasset to 300.0 eV. Brillouin zone sampling was performed by usingthe Monkhost-Pack scheme with ak-point grid of 3× 3 × 3,4 × 2 × 3, 3 × 3 × 1, and 2× 3 × 2 for R-, â-, δ-, andγ-HMX, respectively. The values of the kinetic energy cutoffand thek-point grid were determined to ensure the convergenceof total energies.

The R phase of HMX crystallizes in an orthorhombicFdd2space group and contains eight H8C4N8O8 molecules per unitcell.24 Theâ phase contains two H8C4N8O8 molecules per unitcell in a monoclinic lattice with space groupP21/c.25 The δphase crystallizes in a hexagonalP61 space group with sixH8C4N8O8 molecules per unit cell.26 Theγ phase contains two2H8C4N8O8·0.5H2O molecules per unit cell in a monocliniclattice with space groupPn.4 Theγ phase is a hydrate and nota true polymorph. When the explosive decomposition of thisHMX hydrate occurs water does not work. Strictly speakingonly three HMX phases should be named polymorphs. Figure1 displays the unit cell of the four HMX crystalline phases,and conformation and atomic numbering of H8C4N8O8 moleculein the â-HMX phase is shown in Figure 2.

Starting from the above-mentioned experimental structures,the geometry relaxation was performed to allow the ionicconfigurations, cell shape, and volume to change. In thegeometry relaxation, the total energy of the system wasconverged less than 2.0× 10-5 eV, the residual force less than0.05 eV/Å, the displacement of atoms less than 0.002 Å, andthe residual bulk stress less than 0.1 GPa. The Mulliken chargesand bond populations were investigated using a projection ofthe plane wave states onto a linear combination of atomic

orbitals (LCAO) basis set,27,28which is widely used to performcharge transfers and populations analysis. The phonon frequen-cies at the gamma point have been calculated from the responseto small atomic displacements.29

3. Results and Discussion

3.1. Bulk Properties. As a base for studying other HMXphases and as a well-studied benchmark, we apply three differentfunctionals to bulkâ-HMX as a test. The LDA (CA-PZ) andgeneralized gradient approximation (GGA) (PBE30 and PW9131)functionals were selected to fully relax theâ phase without anyconstraint. The calculated lattice parameters are given in Table1 together with their experimental values.25 It is found that theerrors in the LDA (CA-PZ) results are slightly smaller than thatin the GGA (PBE and PW91) results in comparison with theexperimental values. This shows that the accuracy of LDA isbetter than that of the GGA functionals. Table 2 presents thebond lengths and bond angles of theâ phase along with thecorresponding experimental data. It is seen that the bond lengthsand bond angles compare well with experimental values. Thecomparisons confirm that our computational parameters arereasonably satisfactory. We thus used LDA in all subsequentcalculations.

Figure 1. Unit cell of R-HMX (a), â-HMX (b), δ-HMX (c), andγ-HMX (d). Gray, blue, red, and white spheres stand for C, N, O, and H atoms,respectively.

Figure 2. Conformation and atomic numbering of H8C4N8O8 moleculein â-HMX.

TABLE 1: Experimental and Relaxed Lattice Constants (Å)for â-HMX

this work

LDA PBE PW91 experimental25

a 6.539 6.625 6.617 6.540b 11.030 11.202 11.188 11.050c 8.689 8.824 8.810 8.700â 123.9° 124.3° 124.2° 124.3°

12716 J. Phys. Chem. B, Vol. 111, No. 44, 2007 Zhu et al.

3.2. Electronic Structure. The calculated total density ofstates (DOS) and partial DOS (PDOS) for the four HMX phasesare displayed in Figures 3-6, respectively. The DOS of thefour HMX phases are finite at the Fermi energy level. This isbecause the DOS contain some form of broadening effect. Inthe upper valence band, all of the four HMX phases have asharp peak near the Fermi level, which shows that the topvalence bands of their band structures are flat. The top of theDOS valence band shows three main peaks forR- andδ-HMXand two main peaks forâ- and γ-HMX. These peaks arepredominately from the p states. After that, several main peaksin the upper valence band are superimposed by the s and p states.The conduction band of each phase is dominated by the p states.This indicates that the p states for each HMX phase play a veryimportant role in its chemical reaction.

The atom-resolved DOS and PDOS of the four HMX phasesare also shown in Figures 3-6, respectively. The main featurescan be summarized as follows. (i) In the upper valence band,the PDOS of the states of N in the ring are far larger than thatof the states of C and N of NO2. It is expected that the states ofN in the ring make more important contributions to the valencebands than these of C and N of NO2. This shows that N in thering acts as an active center. (ii) Some strong peaks occur atthe same energy in the PDOS of a particular C atom and aparticular N atom in the ring. It can be inferred that the twoatoms are strongly bonded. Similarly, a particular N atom ofNO2 and a particular N atom in the ring are strongly bonded.(iii) There are some differences in the PDOS of the C atomsfor the four HMX phases. This is due to the differences in theirlocal molecular packing. The same is true of the N atoms inthe ring and the N atoms of NO2. (iv) In the conduction bandregion of DOS, the peaks are dominated by the N-p states ofring and the N-p states of NO2. (v) In the energy range from-5.0 to 0 eV, the DOS ofR-, â-, andγ-HMX are superimposedby the states of C and N in the ring, whereas that ofδ-HMXmainly arises from the states of N in the ring. It is inferred thatthe C-N bond fission in the ring may be favorable in thedecomposition of crystallineR-, â-, and γ-HMX. This isconsistent with the previous conclusion32 that in a bulk

condensed phase the C-N bond scission reaction of the ringfor R-HMX may be energetically favorable because of the stericconstrains that will disfavor N-NO2 bond dissociation. Of

TABLE 2: Experimental and Relaxed Bond Lengths (Å)and Bond Angles (deg) forâ-HMX

bond length bond angle

LDA expt25 LDA expt25

N2-C1 1.435 1.448 O1-N1-O2 125.7 125.9N2-C2′ 1.461 1.471 N2-N1-O1 118.5 118.0N3-C1 1.440 1.455 N2-N1-O2 115.8 116.1N3-C2 1.421 1.437 N1-N2-C2′ 115.8 115.2N2-N1 1.358 1.354 N1-N2-C1 118.7 118.2N3-N4 1.374 1.373 C1-N2-C2′ 123.7 122.4N1-O1 1.253 1.233 N2-C1-N3 111.4 113.5N1-O2 1.245 1.222 N2-C1-H1 106.8 108.5N4-O3 1.246 1.210 N2-C1-H2 111.8 109.8N4-O4 1.243 1.204 N3-C1-H1 107.3 106.6C1-H1 1.106 1.111 N3-C1-H2 111.6 111.8C1-H2 1.108 1.091 H1-C1-H2 108.0 106.2C2-H3 1.107 1.101 O3-N4-O4 126.5 126.7C2-H4 1.106 1.095 N3-N4-O3 116.5 115.9

N3-N4-O4 117.0 117.4N4-N3-C1 117.9 117.4N4-N3-C2 118.2 118.2C1-N3-C2 123.5 123.8

N2′-C2-N3 110.6 110.2N3-C2-H4 108.2 107.3N3-C2-H3 109.9 110.5N2′-C2-H4 109.9 110.1N2′-C2-H3 109.0 109.2H3-C2-H4 109.1 109.6

Figure 3. Total and partial density of states (DOS) of C-states, N inring-states, N of NO2-states, andR-HMX. The Fermi energy is shownas a dashed vertical line.

Figure 4. Total and partial density of states (DOS) of C-states, N inring-states, N of NO2-states, andâ-HMX.

Figure 5. Total and partial density of states (DOS) of C-states, N inring-states, N of NO2-states, andδ-HMX.

Four Polymorphs of Crystalline HMX J. Phys. Chem. B, Vol. 111, No. 44, 200712717

course, near areas with large surfaces or voids there is noconfining environmental and N-N bond dissociation wouldlikely be more favorable. This is in agreement with the previousreports33,34 that the N-N bond scission reaction of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) in the gas-phase would befavorable. For the N-NO2 and C-N bond fission reactions ofδ-HMX, it is difficult to judge which one is more favorablesince the contribution of the states of C and of NO2 to the totalDOS from-5.0 to 0 eV is very small. Recently, Feng and Li35

have investigated two polymorphs of flufenamic acid andelucidated the difference in the solid-state reaction of the twopolymorphs based on their electronic structures.

3.3. Vibrational Properties. Here we investigate the vibra-tional properties of the four HMX phases.â-HMX contains aring conformation that the NO2 groups adopt a chairlikearrangement. This gives the entire molecule a center ofsymmetry.24,25The ring conformation ofR-, δ-, andγ-HMX isa “boat” form that the NO2 groups are positioned on one sideof the molecule.24,26 It is interesting to note that theR-, δ-, andγ-HMX phases have similar vibrational spectra, whereas theâphase has a significantly different vibrational spectrum.10-14

Therefore, the calculated vibrational frequencies forâ-HMXare presented in Table 3, whereas these forR-, δ-, andγ-HMXare presented in Table 4. The corresponding experimental valuesare also listed for comparison.

The lattice mode spectra ofâ-, δ-, and γ-HMX displaysunique features in the region 27-80 cm-1, whereas there is onlyone librational vibration mode at 31.2 cm-1 in the R-HMXspectrum.â-HMX has five frequencies in this region, whichare in agreement with the experimental reports.13 The latticemode spectra of the other three phases were not reported inthis region by the experiments because their lattice region spectracould not be routinely obtained. In the region of 80-500 cm-1,the molecular motions of the first four frequencies are thetwisting of NO2 about N-N bonds, denotedγNN(NO2), and themotions corresponding to the following frequencies is concen-trated in the distortions of C4N4 rings.

In the region of 500-730 cm-1, the molecular motions ofthe frequencies are the bending of N-N-O angles, denotedb(NNO), and the stretching of N-N bonds, denotedV(NN). The637.8 cm-1 mode corresponding to the bending of N-N-Oand N-N-C angles and the stretching of C-N bonds appearsin γ-HMX but not in any of the other three polymorphs. Themodes in the range of 730-760 cm-1 are the wagging of N

atoms out of the NO2 plane, denotedσ(NO2). We note thatR,δ, andγ-HMX share similarities in band separation, whereasâhas a different spectral pattern. This is because the crystalstructures ofR, δ, andγ-HMX differ from one another primarilythrough small differences in the relative positioning of the NO2

groups, whileâ-HMX has a different ring conformation fromthe other phases. In the region of 760-1260 cm-1, the molecularmotions of the frequencies are mainly the stretching of C4N4

rings including the symmetric stretch of C-N-C and N-N-C2 bonds, the asymmetric stretch of C-N-C, N-C-N, andN-N-C2 bonds, and the rocking in the CH2 plane. The modesof 1220.4 cm-1 in R, 1234.2 cm-1 in δ, and 1228.2 cm-1 in γcorrespond to the stretching of only one N-O bond in NO2

moieties, denotedV(NO), plus the stretching of N-N bonds andthe bending of C-N-C angles, which do not appear inâ-HMX.The R-HMX phase has the smaller motion modes of thefrequencies in this region than any of the other three polymorphs.

In the region of 1260-1350 cm-1, the molecular motions ofthe frequencies involve the symmetric stretching of O-N-Obonds, denotedVs(NO2), and the twisting about bisector ofH-C-H angles, denotedγ(CH2). The modes of the fourfrequencies in the range of 1350-1500 cm-1 involve thewagging of H atoms out of the CH2 plane, denotedω(CH2),and the bending of H-C-H angles, denoted b(HCH), whichdoes not appear inR-HMX but appears in the other threepolymorphs. In the region of 1500-1750 cm-1, the molecularmotions of the frequencies are the asymmetric stretching ofO-N-O bonds, denotedVas(NO2). The calculated frequenciesfor the â-HMX phase in this region are much larger than thecorresponding experimental values. The motions of thesefrequencies is concentrated in the NO2 moieties; thus, thisdiscrepancies may be due to intermolecular hydrogen bondinginteractions present in the crystal lattice, which are not welldescribed by DFT. In the region of 2900-3100 cm-1, themolecular motions of the frequencies are divided into twoseparate groups. The motions of the first four frequenciesinvolve the symmetric stretching of H-C-H bonds, denotedVs(CH2), whereas these of the next four frequencies areconcentrated in the asymmetric stretching of H-C-H bonds,denotedVas(CH2). These modes do not appear inR-HMX butappear in the other three polymorphs.

Recently, Stevens et al.36 has reported the single-crystalpolarized Raman spectra forâ-HMX. It can be seen from Table3 that our calculated frequencies are in agreement with theirexperimental ones. They have also presented complete symmetryassignments for the Raman-active modes ofâ-HMX. This isuseful for understanding the proposed vibrational mechanismsfor the initiation of detonation.

In a word, theâ-HMX phase has clearly very differentvibrational properties fromR-, δ-, and γ-HMX. This issupported by previous experimental studies.10,12,14From the lowfrequency to high-frequency region, the molecular motions ofthe frequencies for the four HMX polymorphs present uniquefeatures, which could be used to distinguish the polymorphseasily from one another. At lower (below 120 cm-1) or higher(above 1150 cm-1) frequencies, the motions seem very localizedin certain atom groups, whereas elsewhere the motions arediffusely distributed among different atom groups. Since thetorsional motions of the molecule’s functional groups are usuallysupposed to be highly coupled to the other moiety of themolecule,37 it is possible that the torsion motions of the NO2

groups act as doorways through which kinetic energy can flowinto the molecule from its surroundings. These suggestions show

Figure 6. Total and partial density of states (DOS) of C-states, N inring-states, N of NO2-states, andγ-HMX.


that the NO2 groups play a very important role in thedecomposition and detonation of HMX.

3.4. Electronic Structure and Impact Sensitivity. In thissection, an attempt is made to correlate the impact sensitivityof the four HMX phases with their electronic structure. Bandgap is an important parameter to characterize the electronicstructure of solids. Table 5 presents the energy gaps betweenvalence and conduction bands forR-, â-, δ-, andγ-HMX. Thisshows that the band gap decreases in the sequence ofâ-, γ-,R-, δ-HMX, whereas their experimental impact energy (shownin Table 5) increases in the following order:â-HMX < γ-HMX∼ R-HMX < δ-HMX.38 Therefore, there is a relationshipbetween the band gap and impact sensitivity for the four HMXpolymorphs. This indicates that the smaller the band gap is,

the easier the electron transfers from the valence band to theconduction band, and the more the HMX phase becomesdecomposed and exploded. In the previous studies,39,40 a“principle of the easiest transition” (PET) was put forward toinvestigate the relationship between the band gap and impactsensitivity for the metal azides. Although these calculations areperformed at the semiempirical discrete variational XR (DV-XR) and extended Hu¨ckel-crystal orbital (EH-CO) levels, theresults have shown that their band gap could be correlated withtheir explosive characters. Our recent study on the heavy-metalazides within the framework of periodic DFT41 has alsoconfirmed the relationship between the band gap and impactsensitivity. In several papers,42-45 Gilman has emphasized thatthe role of HOMO-LUMO (highest occupied molecular orbital-

TABLE 3: Vibrational Frequencies (cm-1) for â-HMX

experimental experimental

infrared (IR) Raman (R) infrared (IR) Raman (R)

mode assignmentathis

work ref 14 ref 10 ref 13 ref 12 ref 13 mode assignmentathis

work ref 14 ref 10 ref 13 ref 12 ref 13

1 librationalvibration

38.8 36 40 ν(NN), F(CH2) 923.2 950 953

2 translationalvibration

57.3 59 41 νas(CNN), F(CH2) 930.9 965 966

3 rotational latticevibration

63.2 42 νas(CNN), F(CH2) 938.4 967 965 966

4 librationalvibration

65.9 65 43 ν(NN), F(CH2) 1071.2 1088 1090 1089

5 translationalvibration

78.8 79 81 44 νas(NNC2) 1086.1 1090 1088

6 γNN(NO2) 84.0 45 νas(CNN), F(CH2) 1146.1 1146 1146 11457 γNN(NO2) 96.1 90 97 46 νas(CNN), F(CH2) 1150.0 1168 11708 γNN(NO2) 111.9 110 47 νas(NC2) 1196.9 1190 11929 γNN(NO2) 112.9 126 48 νas(NC2) 1213.9 1204 1205 1203

10 σ(CNC) 147.6 140 149 49 νas(NC2) 1228.0 1239 124011 b(NNC) 156.2 152 155 50 νas(NC2) 1228.4 1248 125112 σ(CNC) 217.0 51 νs(NO2) 1275.3 1268 127013 b(NNC), b(CNC) 217.6 224 230 231 52νs(NO2) 1280.6 1279 1280 128114 b(NNC), b(NCN) 256.1 53 νs(NO2) 1293.4 1296 129815 b(NNC), b(NNO) 295.6 281 282 54 γ(CH2) 1309.0 1312 131016 b(NNC) 332.0 312 318 55 νs(NO2) 1329.2 131817 ν(CN), ν(NN),

b(CNC)337.5 56 γ(CH2) 1332.0 1325 1320 1325

18 ν(NN), b(CNC),b(NNC)

369.4 358 364 57 γ(CH2) 1344.9 1349 1348 1348

19 b(NNO), b(NNC) 398.8 385 412 417 58γ(CH2) 1346.4 1350 135120 b(NNO), b(NNC) 404.0 419 420 59 ω(CH2) 1367.8 1368 136921 b(CNC), b(NNC) 424.8 432 437 60 ω(CH2) 1390.8 1385 1397 139522 b(CNC), b(NNO),

b(NNC)426.1 438 441 61 ω(CH2) 1394.5 1395 1401

23 b(NNO) 593.3 600 605 62 ω(CH2) 1423.8 1418 142024 b(NNO) 595.5 597 600 63 b(HCH) 1426.8 1438 143825 b(NNO),ν(NN) 629.0 627 630 64 b(HCH) 1442.8 1433 1433 143426 b(NNO),ν(NN) 630.4 638 640 65 b(HCH) 1446.5 1460 146127 b(NNO),ν(NN) 641.0 660 658 66 b(HCH) 1459.5 1462 1465 146328 b(NNO),ν(NN) 658.6 662 663 67 νas(NO2) 1700.4 1532 153529 b(ONO),ν(CN) 722.4 721 722 68 νas(NO2) 1704.0 1534 1540 153830 σ(ONO) 750.2 754 752 755 759 762 69νas(NO2) 1706.9 1558 156031 σ(ONO) 750.9 763 70 νas(NO2) 1715.2 1563 1570 156532 σ(ONO) 751.0 71 νs(CH2) 2955.433 σ(ONO) 752.7 760 758 762 72 νs(CH2) 2955.5 2985 297834 b(NCN) 774.4 772 769 773 73 νs(CH2) 2955.6 2992 298335 νs(NC2) 837.9 834 834 74 νs(CH2) 2955.7 2992 299436 νs(NC2) 838.2 832 827 833 75 νas(CH2) 3029.9 3028 302837 νs(NNC2) 878.8 872 871 872 76 νas(CH2) 3030.3 3027 302738 νs(NNC2) 879.6 881 884 77 νas(CH2) 3030.8 3037 303739 νas(NNC2) 920.7 947 945 948 78 νas(CH2) 3031.1 3037 3038

a γNN(XY2): twist of XY2 about NN bond;σ(XY2): wag of X atom out of XY2 plane; b(XYZ): bend of X-Y-Z angle;νs(XY2): symmetricstretch of Y-X-Y bonds;νs(XXY 2): symmetric stretch of X-X-Y2 bonds;νas(XY2): asymmetric stretch of Y-X-Y bonds;νas(XXY 2): asymmetricstretch of X-X-Y2 bonds;F(XY2): rocking in XY2 plane;γ(XY2): twist about bisector of Y-X-Y angle;ω(XY2): wag of Y atoms out of XY2plane.


TABLE 4: Vibrational Frequencies (cm-1) for r-HMX, δ-HMX, and γ-HMX

R-HMX δ-HMX γ-HMX

experimental experimental experimental

mode assignmenta this work IR14 IR10 R12 this work IR14 IR10 R12 this work IR10 R12

1 librational vibration 31.2 32.6 27.22 translational vibration 46.5 43.73 rotational lattice vibration 55.6 51.94 librational vibration 56.2 53.35 translational vibration 70.9 68.66 γNN(NO2) 90.1 92.5 89.27 γNN(NO2) 92.0 95.4 93.38 γNN(NO2) 104.9 105.8 101.49 γNN(NO2) 108.8 105.8 105.8

10 σ(CNC) 128.3 124.9 129.211 b(NNC) 175.1 186.9 191.612 σ(CNC) 187.8 191.7 20513 b(CNC) 214.0 216.6 216.6 22414 b(NNC), b(CNC) 248.2 241.8 255.415 b(NNC) 332.5 339.2 329.7 32816 ν(CN), ν(NN), b(NCN) 344.1 359.7 347.1 36517 ν(NN), ν(CN), b(CNC) 374.3 386.918 ν(NN), b(NNO) 395.8 392.719 b(NNC), b(NNO),F(CH2) 406.8 395.9 404.2 40220 b(NNO), b(NNC),F(CH2) 409.9 396.0 400 392 408.921 b(NNO), b(NNC),F(CH2) 450.2 446 450.222 b(CNC),F(CH2) 454.9 451 450 470 473 457.9 45823 b(NNO) 587.0 585.8 590 579.724 b(NNO) 591.1 603 603 594 589.9 599 605 601 592.8 59225 b(NNO),ν(NN) 611.5 622 625 620 610.4 621 625 622 610.9 622 61826 ν(NN), b(ONO) 619.1 637.6 632.2 63627 ν(NN), b(ONO) 649.5 646 648 646 639.1 646 655 654 632.3 658 64828 b(NNO), b(NNC),ν(CN) 637.829 ν(NN), b(ONO) 691.5 714 714 712 696.6 711 713 713 712.6 712 71030 σ(ONO) 732.4 741 742 740 727.8 733 740 735 730.331 σ(ONO) 736.9 751 753 750 733.2 750 750 751 730.8 73532 σ(CNC) 741.9 758 742.5 763 744.733 σ(ONO) 747.7 766 768 742.6 763 765 745.6 751 75334 σ(ONO) 759.4 770 751.4 767 759.0 76835 νs(NC2) 824.1 830 843.8 845 84036 νs(NC2) 856.9 847 848 846 835.8 841 848 846 844.1 84737 b(ONO),νs(NNC2) 867.2 864 862 866.6 867 866 870 863.138 b(ONO),νs(NNC2) 868.8 878 878 869.1 864.639 b(ONO),νs(NNC2) 915 888.3 913 910 885.4 87840 νs(NNC2) 920 928 906.3 930 925 930 905.6 91841 ν(NN), b(ONO) 944 945 945 911.6 941 940 943 910.8 941 92842 ν(CN), F(CH2) 936.3 912.1 920.0 94043 νs(NNC2) 1030 1032 1030 989.4 1016 1040 1013.9 1016 99544 νs(NNC2), F(CH2) 1075.7 1089 1090 1085 1036.6 1087 1088 1092 1081.2 1090 109045 νs(NNC2), F(CH2) 1095.8 1109 1110 1089.5 1109 1110 1111 1097.1 1112 111246 νas(NC2) 1155.8 1148 1154.1 1147 1153.3 1149 116647 νas(NC2) 1208.0 1215 1220 1215 1206.5 1204 1215 1208.5 119048 ν(NO), ν(NN), b(CNC) 1220.4 1234.2 1222 1225 1228.2 1221 122549 νas(NC2) 1247.9 1254.9 1249.4 1255 125950 νas(NC2) 1250.1 1259 1258 1255.3 1249 1249.451 νs(NO2) 1268.0 1268 1257 1256 1274.9 127152 νs(NO2) 1273.0 1280 1280 1280 1281.5 1274 1270 1275 1275.9 128053 νs(NO2) 1310.6 1293.0 1290 1291 1296.7 1320 129454 γ(CH2) 1314.1 1304.3 1321.5 131955 νs(NO2) 1330.5 1319 1320 1318 1305.2 1320 1325 1317 1330.2 133056 γ(CH2) 1335.6 1344.2 1341 1340 1332 1336.757 ω(CH2) 1360.5 1359.858 γ(CH2) 1364 1377.2 1375 137559 γ(CH2) 1370 1370 1368 1385.4 1370 1370 1371 1382.1 1398 138560 ω(CH2) 1386 1385 1390.8 1383 1382 1383.5 139261 ω(CH2) 1393 1394 1391 1400.3 1395 1393 1393 1403.0 141162 b(HCH) 1400.3 1420.8 1422 141963 ω(CH2) 1414 1416 1412 1400.9 1419 1420 1415 1423.464 b(HCH) 1432 1432 1422 1400.9 1422 1442.5 144065 b(HCH) 1451 1448 1422 1442.9 1454 145266 b(HCH) 1441 1468 1451 145567 νas(NO2) 1545.3 1550 155768 νas(NO2) 1559.6 1541 1539 1545 153669 νas(NO2) 1570.3 1561 1550 1561 1562 1560 1562 156370 νas(NO2) 1574.4 1611.8 1586.2 1570 157371 νs(CH2) 2923.7 2918


lowest unoccupied molecular orbital) gap closure in moleculessuffering shear strain. Further reports34,46 on the excitonicmechanism of detonation initiation show that the pressure insidethe impact wave front reduces the band gap between valenceand conducting bands and promotes the HOMO-LUMOtransition within a molecule. Although these studies havesuggested that the HOMO-LUMO gap in molecules sufferingshear strain, impact wave, or distortion relates directly to thesensitivity, they further support our conclusion here that thereis the relationship between the band gap and sensitivity.

It has been suggested that initiation of combustion ordetonation in energetic solids is associated with small regionsof the solid termed “hot spots”.47-49 In general, the initiationof explosion in HMX is concerned with the decomposition ofa small volume of material in the region of the “hot spot” at ahigh temperature and in a short time. The thermal decompositionof the energetic material is prior to its detonation; thus, theinformation of the thermal decomposition seems to provide animportant basis for understanding the explosive characteristics.

For the crystal as a whole, a more meaningful quantity is thetotal cell bond order that is the sum of all bond order values inthe crystal.50 Since the four HMX polymorphs contain differentnumber of atoms and bonds within the unit cell of each phase,a much more useful concept is the cell bond order per unitvolume. These numbers are listed in Table 5. It turns out thatâ-HMX has the highest value of 0.080 for the cell bond orderper unit volume. TheR-, δ-, andγ-HMX phases have the valuesof 0.073, 0.057, and 0.078, respectively. Therefore, we mayconclude that theâ-HMX crystal has the strongest crystalbonding among the four polymorphs. This is consistent withthe fact thatâ-HMX is the thermodynamically most stable formunder ambient conditions among the HMX polymorphs. Thecrystal bonding for the HMX phases weakens in the followingorder: â-HMX > γ-HMX > R-HMX > δ-HMX, so theactivation order for thermal decomposition of the four poly-morphs is as follows: â-HMX < γ-HMX < R-HMX <δ-HMX. From these discussions, we may infer that the impactsensitivity for the HMX phases increases in the followingsequence:â-HMX < γ-HMX < R-HMX < δ-HMX, which isin agreement with their experimental sensitivity order.38

4. Conclusions

In this study, we have performed a detailed density functionaltheory study of the electronic structure and vibrational propertiesof the four HMX polymorphs in the local density approximation.Then an attempt has been made to correlate the impactsensitivity of the four phases with their electronic structure.

From the density of states for the four HMX phases, it isfound that the states of N in the ring make more importantcontributions to the valence bands than these of C and N ofNO2 and so N in the ring acts as an active center.

The â-HMX phase has clearly very different vibrationalproperties fromR-, δ-, andγ-HMX. From the low frequencyto high-frequency region, the molecular motions of the frequen-cies for the four HMX polymorphs present unique features,which could be used to distinguish the polymorphs easily fromone another.

It is also noted that there is a relationship between the bandgap and impact sensitivity for the four HMX polymorphs. Fromthe cell bond order per unit volume, we may infer the variationorder of crystal bonding for the four polymorphs and so predicttheir impact sensitivity order.

Acknowledgment. We thank the funding from Key Labora-tory for Shock Wave and Detonation Physics. This work waspartly supported by National Natural Science Foundation ofChina (Grant No. 10576016) and National “973” Project.

References and Notes

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TABLE 4: Continued

R-HMX δ-HMX γ-HMX

experimental experimental experimental

mode assignmenta this work IR14 IR10 R12 this work IR14 IR10 R12 this work IR10 R12

72 νs(CH2) 2923.7 292873 νs(CH2) 2935.574 νs(CH2) 2970 2975 2963.3 2970 2974 2935.5 298275 νas(CH2) 3047 3061.1 303276 νas(CH2) 3049 3058 3061.2 304577 νas(CH2) 3056 3058 3070.878 νas(CH2) 3053 3058 3071.0

a γNN(XY2): twist of XY2 about NN bond;σ(XY2): wag of X atom out of XY2 plane; b(XYZ): bend of X-Y-Z angle;νs(XY2): symmetricstretch of Y-X-Y bonds;νs(XXY 2): symmetric stretch of X-X-Y2 bonds;νas(XY2): asymmetric stretch of Y-X-Y bonds;νas(XXY 2): asymmetricstretch of X-X-Y2 bonds;F(XY2): rocking in XY2 plane;γ(XY2): twist about bisector of Y-X-Y angle;ω(XY2): wag of Y atoms out of XY2plane.

TABLE 5: Experimental Impact Energy and CalculatedBand Gap and Cell Bond Order per Unit Volume forr-HMX, â-HMX, δ-HMX, and γ-HMX

R-HMX â-HMX δ-HMX γ-HMX

impact energy (Kg/cm2)38 0.20 0.75 0.10 0. 20band gap (eV) 2.42 3.62 0.021 3.38cell bond order per unit volume 0.073 0.080 0.057 0.078


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