Electronic structure of solid nitromethane: Effects of ...dio/JCP-117-02.pdf · Electronic...

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 2 8 JULY 2002

Electronic structure of solid nitromethane: Effects of high pressureand molecular vacancies

Dionisios Margetisa) and Efthimios Kaxirasb)

Division of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138

Marcus Elstnerc)

Department of Physics, Harvard University, Cambridge, Massachusetts 02138

Th. FrauenheimUniversitat-GH, Paderborn, Fachbereich Physik, Theoretische Physik, D-33098 Paderborn, Germany

M. Riad ManaaLawrence Livermore National Laboratory, Chemistry and Materials Science Directorate, Livermore,California 94551

~Received 24 July 2001; accepted 8 February 2002!

The combined effect of pressure and molecular vacancies on the atomic structure and electronicproperties of solid nitromethane, a prototypical energetic material, is studied at zero temperature.The self-consistent charge density-functional tight-binding method is applied in order to investigatechanges induced in the band gap of this system by uniform and uniaxial strain of up to 70%,corresponding to static pressure in the range of up to 200 GPa. The effects of molecular vacancieswith densities ranging from 3% to 25% have also been considered. A surprising finding is thatuniaxial compression of about 25–40 GPa along theb lattice vector causes the C–H bond to behighly stretched and leads to proton dissociation. This event also occurs under isotropic compressionbut at much higher pressure, being indicative of a detonation chemistry which is preferential to thepressure anisotropy. We also find that the band gap, although evidently dependent on the appliedstrain, crystal anisotropy and vacancy density, is not reduced considerably for electronic excitationsto be dominant, in agreement with other recent first-principles studies. ©2002 American Instituteof Physics. @DOI: 10.1063/1.1466830#

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I. INTRODUCTION

A series of experiments have investigated the effectsstress and shocks on energetic materials,1–6 demonstratingstrong correlations among the applied mechanical stressintrinsic solid-state properties and the onset of chemicalactions that may lead to detonation.7 These observationhave in turn stimulated theoretical interest into the micscopic nature of the detonation initiation, in which a comnation of shock parameters such as duration and prescause widespread chemical reactions. Despite concenteffort,8–10 the mechanisms by which the propagating shoaffect atomistic conditions that favor bond-breaking and ssequent detonation are not as yet fully understood. A loterm objective is to construct reliable models that encompnonlinear dynamics, dispersion and chemical~or phase!transformations,11,12 in order to predict aging and other properties of energetic materials pertaining to safety.

Experiments1,4,13 suggest that local heterogeneity adefects in solids, such as dislocations, vacancies, microvo

a!On leave from the Department of Mathematics, Massachusetts InstituTechnology, Cambridge, MA 02139.

b!Also at: Department of Physics, Harvard University, CambridMA 02138.

c!Also at: Universita¨t-GH, Paderborn, Fachbereich Physik, TheoretiscPhysik, D-33098 Paderborn, Germany, and Department of Molecularphysics, German Cancer Research Center, D-69120 Heidelberg, Germ

7880021-9606/2002/117(2)/788/12/$19.00

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impurities, and cracks, play an important role in the sensiity and performance of energetic materials to shockimpact.14 The experimental and theoretical investigationsuch imperfections lends support to the theory of ‘‘hot spoformation, according to which the ignition starts in localizeregions of highly concentrated energy associated wdefects.15–17 The nature of these hot spots is an unresolvissue. Dlott and Fayer16 have proposed that the anharmoncoupling between phonons and low-frequency molecularbrations is strong at such hot spots, and hence causes dmolecules to transiently attain high vibrational temperaturthese temperatures favor chemical reactions that normallnot occur in the bulk. However, no widely agreed upon mcroscopic model exists to describe how the size, morpholoand density of the defects in the crystalline solid affectoverall rate of the reaction energy release.

Under shock propagation or impact, the energetic soexperiences a sudden compression of up to 30% of its onal volume. At least by intuition, one expects that the atomand electronic properties of a heterogeneous molecular scould change dramatically and alter its overall mechanresponse at high pressures. Several proposals currentlyto place this speculation on firm physical grounds. Accordto Williams,18 initiation of detonation in solids can occuthrough changes induced in the Fermi level because odouble electron and positive hole injection followed bchemical reactions. Coffey19 postulated that the initiation

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© 2002 American Institute of Physics

license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

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789J. Chem. Phys., Vol. 117, No. 2, 8 July 2002 Solid nitromethane

process is due to tunneling of dislocations in a solid; whthe shock velocities are sufficiently high, the dislocations chave energy adequate to directly pump the internal vibtional modes of the constituent molecules. More recenDremin20 made an attempt to explain why molecules lyingthe shock front dissociate; his proposed dissociation menism directly involves electronic excitations. In the samspirit, Gilman21–24suggested that the compression due topressure wave front causes local metallization whenbending of certain covalent bonds reduces the energybetween the highest occupied~HOMO! and the lowest unoccupied molecular orbitals~LUMO!. A candidate bond anglefor bending is that between the C–N bond and the planethe almost omnipresent nitro (NO2) group in explosives.This effect in turn favors chemical decomposition, in copliance with a criterion given in the 1920s by Herzfeld25,26

for crystals with dense structures. According to this criteria material becomes metallic if its molar refractivity, whichproportional to polarizability and hence approximately pportional to the inverse of the optical gap, exceeds the mvolume.24

Simulations of band-gap properties of imperfect engetic molecular solids, such as the cyclotrimethylene trinramine ~RDX, @C3H6N6O6#! and pentaetythritol tetranitrat~PETN,@C(CH2ONO2)4#! crystals with vacancies and dislocations, have been attempted recently by Kuklja aKunz.27–31 These authors make use of the first-principHartree–Fock approach with an approximate treatmenelectron correlations via second-order many-body pertution theory~MBPT!.32 Their results have indicated that compression of the RDX crystal in the presence of single29,30anddimer29,31 vacancies reduces the optical gap of this mateappreciably, thus decreasing the excitation energy needethe insulator-metal phase transition. Specifically, these calations showed that the edge dislocations cause a dramreduction of the optical gap due to the splitting of the locelectronic states from both the valence and conducbands. These findings have in turn prompted a mechanbased on electronic excitations induced by the impact wpropagating through the crystal.33,34 According to this hy-pothesis, the pressure exerted by the impact wave fcauses the dramatic reduction of the band gap to nearlyvalues and results in the breakage of the N–NO2 chemicalbond in RDX, thus initiating detonation and chemical chareactions.34

Recently, Reedet al.35 examined the nature of electronexcitations in crystalline solid nitromethane (CH3NO2) un-der static and dynamic compression by using densfunctional theory~DFT! calculations in the context of thgeneralized-gradient approximation~GGA!. These authorsconclude that the band gap is not lowered sufficientlyproduce considerable thermal population of the electroexcited states in the crystal, in apparent contradistinctionthe results of Kuklja and Kunz,29,30 who dealt with differentsystems and larger primitive cells. Reedet al.35 also referredto earlier calculations by Manaa and Fried,36 who consideredthe role of nonradiative transitions in isolated nitromethamolecules and indicated that metallization in the gas phcan be achieved by bending of the nitro group; this gro


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originally lies in the CNOO plane of the singlet ground staChemical reactions and phase transformations for varimaterials and detonation temperatures have also been stuvia molecular-dynamics simulations.35,37–43

In this paper, we study the combined effect of stapressure and molecular vacancies on the optical gap ofnitromethane crystal for volume strains up to 70% by folloing the behavior of the band gap at the singlek50 point ~G!.This gap is known to be related to the HOMO and tLUMO electron states that are localized near the nitro groThere are at least two reasons to choose nitromethaneprototypical energetic material: the first is that the ntromethane molecule provides the simplest model oC– (NO2) bond in energetic materials; the second is that,mentioned previously, recent first-principles calculation36

involving the single molecule of nitromethane have reporthat singlet–triplet minimum energy crossings require a snificant bending of the nitro group off the CNOO planethe ground state. It is therefore expected that the crystalform of nitromethane is a good candidate to check Gilmahypothesis,21–24 or other theories on the hot spot formatioIn particular, a question addressed in this paper is whetheappreciable bending of the NO2 group inside the crystal canbe the primary cause of band-gap reduction within theperimentally accessible pressure range.

Our electronic-structure calculations are based onself-consistent charge density-functional tight binding44

~SCC-DFTB! scheme. This is an extension of the standatight binding~TB! approach45 in the context of DFT,46,47andself-consistently describes total energies, atomic forces,charge transfer. This method has been tested for systsuch as small organic molecules and biomolecules,44,48 andsemiconductors.44,49 An essential element of our analysisthat, for each fixed set of lattice parameters, all interatompositions are determined by total energy minimization, ithe atoms are fully relaxed.

The tight-binding character of the SCC-DFTB makesfeasible to study large supercells, containing up to 32 mecules, with concentrations of molecular vacancies rangfrom 3.125% to 25%. The present study therefore diffwith respect to computational tools and system sizes frrecent first-principles calculations,35 which consideredsmaller supercells with more demanding, first-principcomputations. The two methods give very similar resultsthe band gap of systems accessible to both, as discussSec. III. In particular, our calculations give a small reductiof the HOMO-LUMO gap and no sign of significant distotion of bond angles involving the nitro group for uniformvolume strain up to 70%. For the present study, the tendeof the SCC-DFTB to overestimate band gaps is not a hacap, since only the band gapchangeis of interest, as a func-tion of the applied strain. To corroborate this claim wshowed that, by imposing the bending of the nitro groupthe nitromethane molecule, the SCC-DFTB method indepredicts a significant reduction of the HOMO-LUMO gapremarkable agreement with high-level first-principlcalculations.36

A noteworthy finding of our static calculations is that thC–H bond is highly stretched at a relatively early stage


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790 J. Chem. Phys., Vol. 117, No. 2, 8 July 2002 Margetis et al.

uniaxial compression along the direction of theb lattice vec-tor, a prediction which is made for the first time by simultions to our knowledge. As the pressure increases, this bis stretched further, while its hydrogen atom becomes acand tends to dissociate. Accordingly, a serious reductionthe band gap at high pressures is favored by the formationearly free electrons and protons. Our simulations undertropic compression indicate a similar tendency for prodissociation at a much higher hydrostatic pressure, pointo the preliminary conclusion that the chemistry onsetstrongly affected by pressure anisotropy.

Molecular-dynamics simulations can reveal changesthe atomic and electronic structure that may not be captuby static calculations.35 However, our purpose here is tprobe the atomic and electronic properties in an adiabsense, as a first step toward more accurate, first-princisimulations for large systems. A prerequisite at this stagto understand how the electronic structure depends on mroscopic parameters, such as strain and stress, in the simpossible context. In this connection, we also need to clahow reliable our method actually is.

The remainder of this paper is organized as follows:Sec. II the key ideas of the SCC-DFTB method are outlinalong with a brief description of its computational details aof the first-principles adaptive-coordinate real-spaelectronic-structure50–52 ~HARES! method, used here onlimited scale for comparison purposes. In Sec. III, the SCDFTB method is applied in order to calculate the total engies and the HOMO-LUMO energy gap of the single ntromethane molecule, the band gap atG of the perfectperiodic crystal, as well as of crystals with molecular vcancy densities of 3.125%, 6.25%, 12.5%, and 25%. Asther discussed in Sec. III, these calculations show astrictly monotonic behavior of the gap as a function of tvacancy density for fixed compression, with a distinct depdence on the crystal anisotropy. We find that sufficient uform or uniaxial compression causes the C–H bond tohighly stretched, before any significant bending of the nigroup occurs. Finally, in Sec. IV we conclude the paper wa summary of our findings and comparisons to other calations and experimental data available in the literature.alert the reader that the terms ‘‘band gap’’ and ‘‘HOMOLUMO gap’’ are used interchangeably in the present wor

II. COMPUTATIONAL METHODS

The key idea underlying the SCC-DFTB method is tformulation of the traditional TB approach45,53 in terms of avariational principle; this gives the TB approach as a statiary approximation to DFT.54–57Accordingly, the SCC-DFTBmethod is based on a second-order expansion of theenergy functional from DFT with respect to charge densfluctuationsdn relative to a reference densityn0 . The first~zeroth-order! term amounts to the standard TB approaMore precisely, the total energy is

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whereH0 is a single-particle Hamiltonian operator resultinfrom the input densityn0 , c i are the single-particle electrowave functions corresponding to occupied states of valeelectrons,Erep stands for the combined ion–ion repulsioand double-counting Hartree and exchange-correlation terepresented by an effective short-range pair potential,57 andDE(2) is the correction due to density fluctuationsdn. It isworthwhile noting thatDE(2) accounts for the charge transfer among atoms and renders the proposed scheme aconsistent extension of TB.44 DE(2) is formally defined as44

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is an approximation to Eq.~2! that is found by decomposingthe charge density fluctuations into atom-centered contrtions, treating them within a multipole expansion and retaing monopolar terms only. The first term in Eq.~4! is theband-structure energy, whileDqa5qa2qa

0 are the atom-centered charge density fluctuations in the usual Mullikcharge analysis.58 The Hamiltonian matrix elementsHmn

0 arecalculated in the ‘‘two-center approximation’’55 using aminimal set of atomic orbitals. Equation~5! represents thelong-range Coulomb interactions between point chargedifferent sites under the monopole approximation, andcludes the self-interaction contributions of single atoms.44 Byminimizing the right-hand side of Eq.~4!, one derivesRoothaan-type equations58 for the coefficientscn i which be-come self-consistent through the modification of the Hamtonian matrix elements because of the nonzero termDE(2).Erep is determined by appropriate fitting of data for the dference between the cohesive energy versus distance inlocal density approximation~LDA ! and the correspondingTB band-structure energy for a suitable reference structur44

In order to apply the SCC-DFTB to the nitromethane crysthe TB Hamiltonian and overlap integrals have been evaated as functions of distance between the requisite O, Nand H atom types. In all calculations of this paper, we uminimal basis sets consisting ofs and p atomic orbitals forO, N, C, and only thes orbital for H.

We have also used the adaptive-coordinate real-spelectronic-structure~HARES! code,50 in order to calculatethe HOMO-LUMO energy gap in the case of the singletromethane molecule for comparison purposes. The band


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calculation within this approach was supplemented bylowest-order correction on the basis of a theory originaformulated by Fritsche,59 who generalized the Kohn–Shatheory by considering changes in the electron density ofground state due to electron excitations from the HOMOthe LUMO.

III. RESULTS AND DISCUSSION

A. The nitromethane molecule

The structure of the nitromethane molecule in its singground state is shown in Fig. 1~a!. The molecule exhibitsCS

symmetry, with a dominant single closed-shell configuratcharacter.60,61 The constituent atoms were fully relaxed bSCC-DFTB and the C atom was found to be coplanar wthe NO2 group. Molecular orbitals of importance are thesand s* of the C–N pair, the ‘‘inplane’’ NO2 bonds, thepandp* of the NO2 group, and thes ands* of the C–H pair~for more details see Ref. 36!. The starting point for thecalculations is the equilibrium structure given by Manaa aFried,36,62 which was obtained via the multiconfiguratioself-consistent field63 ~MCSCF! method. The bond lengthand angles determined after atomic relaxation by SCC-DFare within 2–4% of the values given in their Table 1.62 Thebending angleg depicted in Fig. 1~b! is defined as the anglbetween the plane of the NO2 group and the original CNOOplane of the fully relaxed structure.

The HOMO-LUMO gaps corresponding to the planand bent structures, forg553°, are shown in Table I evalu

FIG. 1. Two different configurations of a nitromethane molecule.~a! Themolecule with the atoms fully relaxed; C, N, and O all lie in the same pl(g50). ~b! The molecule with the NO2 group bent out of the originalCNOO plane of the relaxed structure (g'53°); the structure is taken to bvery close to the one given in Ref. 36, which corresponds to a singlet–trminimum energy crossing point found by using a first-principles metho


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ated separately by the SCC-DFTB and HARES methodslowest-order correction has also been supplied~fourth col-umn of Table I! to account for the rearrangement of the eletron occupied states due to excitations, accordingFritsche’s generalization of the Kohn–Sham theory.59 Thiscorrection is calculated using the electron density and sinelectron wave functions from HARES within LDA.52 Thevalue given in the last column of Table I for the undistortmolecule is in very good agreement with the electron-impspectroscopy experiments by Flickeret al.,64 where asinglet–triplet transition was observed to have an onset ateV and an intensity peak at 3.8 eV. Note that the SCC-DFoverestimates the HOMO-LUMO gap by nearly 1 eV whg'0, as expected.44

The geometry corresponding to the second row of TaI has been chosen to be very close to the geometry ofminimum energy crossing determined recently by ManaaFried36 by use of MCSCF, with all atomic positions kepfixed during the calculations. The values reported here arqualitative agreement with their results, according to whthere is a singlet–triplet minimum energy crossing pointg close to 50°. We emphasize that the precise location ofminimum energy crossing of course depends on the metemployed. The essential point is that the SCC-DFTB crectly indicates a tendency toward closure of the HOMLUMO gap wheng deviates significantly from zero.

B. Perfect crystal

1. Crystal structure

Nitromethane is known to be a liquid at room tempeture. Its crystalline form is employed here for the sakesimplicity; it also makes meaningful future comparisonsproperties of other energetic materials, such as RDXPETN, which are known to be solids at room temperatuThe crystal structure of nitromethane at nearly zero pressis orthorhobic, with 4 molecules per unit cell.65 The valuesused here for the initial lattice constants area55.198 Å, b56.266 Å, c58.649 Å ~space group P212121!, which havebeen determined via lattice relaxation by starting with texperimental lattice parameters and using the CASTplane-wave code. The initial volume of the primitive cellapproximatelyV05281.7 Å3. These numerical values arwithin 2% of the experimental ones, determined by x-rsingle crystal diffraction and neutron diffraction data.65,66

The bond lengths and angles found after full atomic relation by SCC-DFTB are within 2–4% of those determinby MCSCF for the nitromethane molecule in the gas phas62

Uniform or uniaxial compression was applied to the ucell by successively decreasing the lattice parametersrelaxing the atomic positions at each step. Use was mad

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TABLE I. HOMO-LUMO energy gap for each of the two single-molecuconfigurations of Fig. 1. The bending of the NO2 group causes a dramatireduction of the gap.

g SCC-DFTB~eV! HARES/LDA ~eV! Fritsche corr.~eV!

0.0 4.75 3.70 3.8753° 0.54 0.13 0.34


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the conjugate gradient technique for the atomic relaxatiowith a threshold of the order of 1024 a.u. for the maximummagnitude of the atomic forces. The resulting change intotal energy per unit cell is shown in Fig. 2 for compressiup to 50% of the original volume. Because the initial lattiparameters of the primitive cell have not been relaxed bySCC-DFTB method, the total-energy curves of Fig. 2found to be monotonically decreasing in the volume str(V02V)/V0 for sufficiently small values of the strain, wherV is the volume of the unit cell at each step of compressiThe pressure, estimated by invoking the formulaP52]Etot /]V whereEtot is the total energy per unit cell, irelatively small and negative for this range of strain.P be-comes zero when the total energy attains its minimumV/V0'0.76 for uniform compression, andV/V0'0.83, 0.72,and 0.91 for uniaxial compression inx, y, andz, respectively.For higher values of the strain~smaller values ofV/V0!, thetotal energy is of course found to increase with compress

FIG. 2. Total energy per unit cell of the nitromethane perfect crystal ununiform ~hydrostatic! and uniaxial compression of the unit cell alongx ~forthe lattice parametera!, y ~for b!, or z ~for c!. V is the volume of thecompressed unit cell. The geometry of the initial unit cell of volumeV0 isvery close to the experimental one. Since no lattice relaxation is appeach energy curve is monotonically increasing inV/V0 when this is suffi-ciently close to 1. The corresponding pressure is negative but small.


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and P is positive and increasing, expressing the strengththe intermolecular interactions which progressively deveas the molecules approach each other. The unit cell ofnitromethane crystal is shown in Fig. 3~a!. Before compres-sion, the HOMO-LUMO gap is approximately equal to 4eV, as seen in Fig. 3~b!. This value is close to that given inTable I for the single nitromethane molecule, because ofweak intermolecular forces in the initial primitive cell.

2. Uniform compression

To simulate the effect of uniform strain on solid ntromethane, the initial lattice parameters were successidecreased by keeping their ratio fixed. Throughout this pathe pressureP is estimated by using the low-temperatuformulaP52]Etot /]V. The initial pressure is found to be omagnitude about 1.6 GPa, close to the value given by refirst-principles calculations.35 For hydrostatic compressioup to 50% of the original volumeV0 , the pressure rises toabout 50 GPa, while the HOMO-LUMO gap dropped by 0eV, i.e., by 13% of its original value. This compression rsults in a simultaneous increase of the HOMO andLUMO energies~Kohn–Sham eigenvalues!, while the de-crease of the HOMO-LUMO gap is almost monotonic in tvolume strain, as depicted in Fig. 3~b!. The change inducedin the band gap by very high hydrostatic compressionshown in Fig. 4 for strain equal up to 70%.

It is of interest to note that the mutual orientations of tC–N axes vary smoothly with the strain, (V02V)/V0 , whenthis is less than 40%, with the corresponding pressureexceeding 20 GPa. WhenV/V0 is between 58% and 60% anP is estimated to lie between 15 and 25 GPa, the atoconfiguration undergoes an abrupt change accompanierotations of the methyl (CH3) groups, in agreement withfirst-principles calculations;35 a similar transition was ob-served in that work forV/V0 between 59% and 77% andP inthe range 10–30 GPa. This transition is expected becausbarrier for the rotation of the methyl group is known to bvery low.66

The following changes of the bond lengths and boangles were observed as the strain varied from 0 to 5

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FIG. 3. ~a! Unit cell of the ni-tromethane molecular crystal. Atomare relaxed and the nitro group lies ithe CNOO plane in each molecule.~b!HOMO-LUMO energy gap~in eV! ofthe perfect crystal under uniform anuniaxial compression of the primitiveunit cell.


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While practically no bending of the nitro group in the moecules took place, the C–N bond lengths were invariashortened at most by 6%. In contrast, the eight N–O blengths in the unit cell exhibited different variations, withmaximum deviation of61% from their equilibrium valuesThe bond angles involving the nitro group in each molecremained practically intact. A few of the C–H bonds westretched up to 2% of their initial values. No significachange in the bond angles involving the nitro group wnoticed even as the uniform strain was increased from 5to 70%. However, two of the C–H bonds were highstretched when the strain was extended to values above 6The same effect on the C–H bond in all molecules of the ucell was also observed under uniaxial stress, yet the emated pressure is appreciably lower. This case is discussmore detail below.

3. Uniaxial compression

Compared to uniform compression, uniaxial strain aloone of the lattice vectors is more likely to lead to detonatiDick has proposed67 that detonation initiation in ni-tromethane is favored by shock-wave propagation in spedirections related to the orientation-dependent steric hdrance to the shear flow. This proposal is based on a maccording to which the sterically hindered shear proccauses preferential excitation of optical phonons stroncoupled with vibrons.

We applied uniaxial compression along each of thex, y,andz axes, and tracked the HOMO-LUMO gap as a functiof the volume change, for strains not exceeding 50%corresponding estimated pressures as high as 100 GPa@seeFig. 3~b!#. Among the three types of compression, the onex andy yielded the highest estimated pressure. The gapdecreased at most by 1.4 eV~a 30% reduction of the originavalue! under stress iny. This drop is higher than in the caswith uniform strain, while the gap behavior here is nstrictly monotonic, i.e., the gap does not continuously

FIG. 4. HOMO-LUMO gap change for the nitromethane crystal under uform compression down to 30% of the original volume. The gap valuesshown relative to the initial configuration of the~unstrained! unit cell. Thesudden drop of the gap to lower values corresponds to the reportedbond high stretching.


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crease as the uniaxial strain increases. The strains inx andyseem to cause a similar overall behavior of the gap. Tcompression iny, however, causes a steeper reduction ofgap asV/V0 approaches 50%. In contrast, the compressionz reduces the energy gap more drastically for intermedvalues ofV/V0 , roughly between 65% and 80%. This laeffect can be visualized as follows. The molecules approeach other inz by bringing the methyl group of one close tthe nitro group of the next, while the cell lattice parametecapproaches the intermolecular distance. Consequently,electron densities of the HOMO and the LUMO, which alocalized near the nitro group, are distorted more than inother type of compression of similar magnitude and the bgap is reduced more rapidly for this type of uniaxial strai

A closer look at the atomic configurations reveals strutural transitions which are triggered preferentially by presure anisotropy. The most frequent transition in thex com-pression involves of course molecule translations, rotatiof the methyl groups~which occur even forV/V0 close to 1!and reorientations of the molecules, primarily via relatitranslations and rotations around their C–N axes. Two sabrupt transitions are observed whenV/V0567– 70% andthe estimated pressure is 25–30 GPa, andV/V0559– 62%with an estimated pressure of 45–55 GPa. In thez compres-sion the methyl groups seem to undergo smoother transitas each molecule’s center-of-mass is translated inz. In con-trast, the y compression causes abrupt transitions whV/V0576– 81% and 63–66% and the pressure is estimato lie in the ranges 2–5 and 13–16 GPa, respectively.

Notably, the dramatic stretching of four C–H bonds, oin each nitromethane molecule of the unit cell, occurs unstress iny when V/V0 is between 59% and 62%, with aestimated pressure of 25–40 GPa; more precisely, theselengths are stretched by more than 10–12% of their origvalues in this case. The increase of strain iny above 50%causes these bonds to be stretched further and leads tabstraction of their protons. This indication of proton dissciation corresponds to the steep part of the curve in Fig. 3~b!,and renders they andx compressions qualitatively differen

The C–N bond lengths are again invariably shortenedmost by 6%, 9%, and 2–3% of their initial values underx, y,andz compression, respectively. In contrast, the N–O bolengths exhibit positive or negative deviations from thequilibrium values that depend both on the location ofcorresponding molecules and the direction of compressiois worthwhile noting that only the compression iny causesthe N–O bond length to be continuously stretched in all mecules, at most by 3% of the initial value. The bond angdo not exhibit any change worthy of reporting for this ranof compression.

4. C–H bond high stretching under uniaxialcompression

As mentioned previously, the SCC-DFTB calculatiopredict a high stretching of the C–H bond that leads to pton abstraction undery compression when the strain becomof the order of 40% or higher; for uniform compression, tcorresponding strain takes the significantly higher valuesthe range 60–62%, at estimated pressure of 150–180 G

-re

H


e

, b

e

d


FIG. 5. Atomic configurations of theunit cell for uniaxial compressionalong b5by, with the nitromethanemolecules numbered from 1 to 4 in thclockwise sense~top left molecule isnumbered 1!. In each molecule, the Hatoms are designated by the letters aand c~not to be confused with the lat-tice parameters!. Thex axis is perpen-dicular to the plane of the page.~a!Original unit cell of volume V0

5281.7 Å3. ~b! Configuration forV/V050.79; the cell undergoes phastransformations ~molecule reorienta-tions and methyl-group rotations!. ~c!Configuration for V/V050.60; theC–H bond lengths have been stretcheby more that 12% of their original val-ues. ~d! Configuration for V/V0

50.51. In ~c! and ~d! the dissociatedacidic H atoms~protons! are shown aslarger light gray circles.

thiH

ha

d

e-

nC–ta

dtsins

ithighe

l-s

as

si-on

r tot thetherederithce

theidithna-e

eticri-

One of the intermediate atomic configurations leading toproton dissociation in the former case of uniaxial strainshown in Fig. 5. Note that the high stretching of the C–bond is visualized graphically by enlarging the symbols tcorrespond to each abstracted acidic hydrogen atom~theatomic sizes in these figures are proportional to the vanWaals radius of the ionized form!.

A physical picture of the C–H bond stretching is thfollowing: upon contraction of the initially single C–N covalent bond, electronic charge is transferred to the regiospace between the C and N atoms. Consequently, thebond becomes stronger and its character approaches thadouble covalent bond. The adjacent C–H bonds actsources of the electronic charge; at least one of these Homs is deprived of some electronic charge and its bonweakened. On the other hand, the orbitals localized nearNO2 group are not significantly distorted by the compresion. The Mulliken charge changes given in Table II areagreement with this picture. The total Mulliken charge loof the H atoms is found to increase monotonically wuniaxial strain in each molecule, being dominated for hstrain by the charge pertaining to the markedly stretc


es

t

er

ofN

of as

at-ishe-

s

hd

C–H bonds~shown in Fig. 5!, as expected. While these vaues follow closely the Mulliken charge gain of the C atomfor low uniaxial strain, they start to deviate significantlythe strain approaches 50%.

These findings imply that chemical reactions and trantions from covalent to ionic C–H bonds prior to detonatiin nitromethane are more likely to startbeforethe closure ofthe optical gap. Roughly speaking, an actual precursodetonation appears to be the proton dissociation and nohypothesized metallization induced by drastic changes inbond angles.22 Interestingly, energetic materials can therefobe considered as more sensitive to their chemistry ununiaxial rather than uniform compression, in agreement wDick’s67 elaborate results that invoke a steric hindranmodel for nitromethane.

There have been numerous experimental studies ofreaction kinetics that may lead to detonation of liqunitromethane.68–70 Most of these studies are concerned wthe nature of chemical products that may speed up detotion. Shawet al.,68 for example, measured the effect of thconcentration of protonated compounds within the energmaterial on the time needed until explosion. Their expe

ndive

are

TABLE II. Mulliken charge changes~in units ofe! of the C and H atoms of the nitromethane molecules 1 a2 of the unit cell under uniaxial compression alongb, in close correspondence to Fig. 5. All values are relatto the Mulliken charges in the primitive cell of volumeV05281.7 Å3. Each Hic atom (i 51,2) is found tobecome acidic and start dissociating from the corresponding C atom whenV/V050.60. The Mulliken chargesfor the other two molecules~3 and 4! of the unit cell show an essentially identical behavior and their valuestherefore omitted from this table.

V/V0 C1 H1a H1b H1c C2 H2a H2b H2c

0.79 0.012 20.010 0.016 20.019 0.012 20.010 0.016 20.0200.60 0.126 20.016 20.026 20.152 0.108 20.019 20.008 20.1550.51 0.294 20.014 20.074 20.258 0.168 20.051 20.030 20.245


-


FIG. 6. ~a! Unit cell of the ni-tromethane molecular crystal with 1molecular vacancy, identified by alarge circle ~25% vacancy concentration!. ~b! The HOMO-LUMO gap~ineV! for uniform and uniaxial compres-sion of the unit cell with 1 molecularvacancy.

oro

ss.ilailalyn

ith

ofnepsu

scs

ehe%,ne

er-elli-

g.m

inlesmea31ith0%

ments indicated that a reaction involving hydrogen atomsprotons is involved in the first steps of this fast-reaction pcess. This suggestion was later corroborated by Blaiset al.69

by more direct measurements, who also referred to aquence of previous investigations by Engelke and other70

Our zero-temperature, static calculations point to simconclusions for the nitromethane molecular crystal. A simprediction for this material was also reached concurrent71

via ongoing first-principles, molecular-dynamics simulatioat high densities~1.5–2.5 gr/cm3! and high temperatures (T52000– 4000 K). We plan to supplement our results wfirst-principles static calculations in the near future.72

C. Effect of molecular vacancies

In view of Dick’s67 proposal and the persistenceuniaxial strain following shock propagation, the cases of uform and uniaxial compression are herein presented srately. In order to simulate vacancies of desired densitiethe nitromethane crystal, we first superimposed two, foand eight unit cells in order to create 23131, 23231, and23232 supercells where the numbers express multiplethe primitive unit cell dimensions. More precisely, a vacanconcentration of 25% was formed in two different way


r-

e-

rr

s

i-a-inr,

ofy:

first, by removal of one entire molecule from the primitivunit cell and, second, by removal of two molecules from t23131 supercell. Vacancy concentrations of 12.56.25%, and 3.125% were also modeled by removal of omolecule from the 23131, 23231, and 23232 super-cells, respectively. In the following, unless it is stated othwise, there is no lattice relaxation and the initial supercstructure merely stems from a multiple of the starting primtive cell along thex, y, andz axis.

The unit cell with a molecular vacancy is shown in Fi6~a! and the corresponding HOMO-LUMO gaps for uniforand uniaxial compression are given in Fig. 6~b!. The 23131 supercell models are shown in Figs. 7~a! and 8~a! withtheir HOMO-LUMO gaps depicted in Figs. 7~b! and 8~b!.The larger supercells are not shown. A pair of vacanciesthe 23231 supercell is created by removing the molecunearest to the endpoints of the cell diagonal. To realize soof the underlying complexity, it suffices to mention that23232 supercell with one molecular vacancy containsnitromethane molecules, which amounts to 217 atoms w744 valence electrons. Each supercell is strained up to 5of its original volume.

r,-

n-

FIG. 7. ~a! 23131 supercell of thenitromethane crystal with 1 moleculavacancy, identified by a large circleand relaxed atomic positions, corresponding to a 12.5% vacancy concetration. ~b! HOMO-LUMO gap ~ineV! for uniform and uniaxial compres-sion of this supercell.


r,-

ntn


FIG. 8. ~a! 23131 supercell of thenitromethane crystal with 2 moleculavacancies, identified by large circlescorresponding to a 25% vacancy concentration. Note that the arrangemeof vacancies in the periodic extensioof the supercell is different from theone of Fig. 6.~b! HOMO-LUMO gap~in eV! for uniform and uniaxial com-pression of this supercell.

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1. Uniform compression

The HOMO-LUMO gap with one molecular vacancy~aconcentration of 25%! in the primitive unit cell is shown inFig. 6~b!. As V/V0 is changed from 100% to 50%, the presure is estimated to vary from 1 GPa to 15 GPa. These vaare lower than in the case of uniform compression ofperfect crystal, as expected. A comparison with Fig. 3~b!shows that the band gap exhibits a tendency to remain flaa wider range of the volume strain in the presence of acancy, but the overall drop is about the same as in the cwith no vacancy, i.e., 0.6 eV. This resistance of the gapchanges induced by stress stems from the increase in theof the effective free space that each molecule experienbecause of the vacancy. The gap reduction is so small thcannot allow for considerable electron excitations fromHOMO to the LUMO. The molecules reorient themselvesthat the structure relieves the applied stress; roughly sping, the molecules tend to take positions that are closesthose of minimum pressure for the given lattice parametThe most frequent structural change observed is againrotation of the methyl group, yet the variation in the atompositions appears to be smoother in the presence of acancy.

The effect of vacancy concentrations of 12.5% and 2on the HOMO-LUMO gap are shown in Figs. 7~b! and 8~b!on the basis of a 23131 supercell. WithV/V0 varying from100% to 50%, the estimated pressure reaches almost 30for concentration of 12.5% and 10 GPa for concentration


ese

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seoizeest iteok-tos.he

a-

Paf

25%. Note the differences from the values in Fig. 6~b!,which may reflect the anisotropy of the crystal since the retive positions of the vacancy are different in the two casAgain, no appreciable narrowing of the band gap is fouThe small jumps appearing in the simulation data forband gap correspond to abrupt structural transitions invoing relative reorientations of the C–N axes and rotationsthe methyl groups, especially of the molecules that lieclose proximity to the vacancies.

In Fig. 9~a! the HOMO-LUMO gap of the 23131 su-percell is depicted for vacancy concentrations of 12.5% a25%, with the inclusion of partial optimization of the initialattice parameters. Accordingly,V0 is now the volume thatcorresponds to the relaxed supercell in each case. Notaany two of these curves intersect. In other words, for afixed V/V0 , the gap is not a strictly monotonic function othe vacancy concentration. The pressures at the intersepoints for the perfect crystal are estimated to be 5 GPa18 GPa~from right to left!. In Fig. 9~b! the gaps are shownfor vacancy concentrations of 12.5% and 6.25% from23231 supercell, and 3.125% from the 23232 supercell,where the starting point (V/V051) is that of the minimumpressure for the given vacancy concentration and hydroscompression. The curves are well-separated now. Forfixed volume strain, the gap is roughly a monotonically dcreasing function of the density. In all these cases, the gareduced at most by 1.3 eV, dropping to 3.25 eV. The smal

t

of-

FIG. 9. HOMO-LUMO gap ~in eV!for uniform compression and differenvacancy concentrations:~a! Perfectcrystal, and vacancy concentrations25% and 12.5% from an approximately optimized 23131 supercellwith defects.~b! Vacancy concentra-tions of 3%, 6.25% and 12.5% fromthe 23231 and 23232 supercellswith defects.


te-

f-


FIG. 10. HOMO-LUMO gap~in eV!for uniaxial compression and differenvacancy concentrations, in close corrspondence to Fig. 9.~a! Perfect crys-tal, and vacancy concentrations o25% and 12.5% from an approximately optimized 23131 supercellwith defects.~b! Vacancy concentra-tions of 3%, 6.25%, and 12.5% from23231 and 23232 supercells withdefects.

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decrease in the gap occurs when the vacancy concentratithe highest, i.e., equal to 25%.

2. Uniaxial compression

Application of uniaxial strain does not alter the ordermagnitude of the HOMO-LUMO gap reduction, as shownFigs. 6~b!, 7~b!, 8~b!, and 10 where only strain in thex di-rection is considered. In fact, for this uniaxial compressthe gap is reduced at most by 1 eV. The unstrained strucof Fig. 10 corresponds to the minimum pressure for the gicompression inx by starting with the nonoptimized latticparameters of the unit cell. In each set of curves in F10~a! and 10~b! the smallest change in the gap is again oserved when the vacancy density is the highest, 25%. In10~a! one sees that the gap decreases with the vacancysity, and thus the curves appear to be more distinct, cpared to the case with uniform strain.

IV. SUMMARY AND CONCLUSIONS

We studied the combined effect of static pressuremolecular vacancies on the crystal geometry, atomic confiration and electronic structure of the prototypical solidtromethane. We found that exerting pressure on the crydoes not cause a dramatic band-gap reduction assocwith the bending of the nitro group.

As a step toward simulating the effect of shocks,applied uniform and uniaxial compression to thetromethane crystal, with and without molecular vacancOur calculations demonstrated only a slight reduction ofHOMO-LUMO gap for uniform compression down to 30%of the original volume and hydrostatic pressures up to 1GPa or higher. This small reduction essentially prohibits aconsideration of electron excitations, and hence of nonrative transitions due to the crossing of adjacent energyfaces of molecules.36 The band-gap drop is most likely atributed to the relatively weak intermolecular interactions,the molecules tend to maintain positions that yield the lowpossible pressure in the crystal. Accordingly, no appreciabending of the nitro group is observed for this range of vume strains.

In a physical picture that can explain our results, tconstituent molecules undergo changes in their bond lengrelative orientations of the C–N axes, and rotations of th


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methyl groups in order to relieve the applied stress. Wconfiguration prevails of course depends on how the enelandscape evolves under adiabatic compression, and thfore on how high the barriers are for transitions to neighbing local minima of other configurations.

The most common structural transition observed in osimulations is the rotation of the methyl group. We emphsize that it is crucial to be able to describe accuratelyatomic forces and to include full atomic relaxation, an esstial feature of the SCC-DFTB method, which makes it posible to capture this effect. This aspect of the calculatiorenders our analysis distinctly different from that by Kuklet al.,34 who applied a ‘‘rigid molecule approximation’’ andthus calculated a large decrease of the band gap in the Rcrystal. Direct comparisons of our results to theirs arecourse not compelling because of the different systemsvolved. However, we expect that allowing for full atomrelaxation necessarily leads to a band-gap behavior thatcludes dominant electron excitations in either system.

Because of the crystal anisotropy, on the other hand,sequence and the frequency of phase transformations deon the character of compression. For example, the compsion in z forces the molecules to develop more rapidly eletrostatic interactions, which in turn affect the electroncharge of the nitro group and therefore result in a steedecrease of the band gap with applied pressure. Since chtransfer is important in the description of atomic forces,expect that its realistic description within the SCC-DFTmethod captures the general features of the molecularsponse to external stress.

It follows from the physical picture outlined previouslthat the stress relief is in general facilitated by the preseof vacancies, since the effective size of free space in thecell increases. Indeed, our results show a higher overallsistence of the band gap of the crystal with vacancy defewith applied pressure. It should be pointed out, however, tthe present model of periodically distributed vacancies isstricted in its applicability; in real materials the vacancifollow an inhomogeneous, and even random, distribution

We also paid attention to the distortion of bonds undcompression, even above 100 GPa, of the unit cell or sucell. The C–H bond in particular was found to be highstretched, and its proton most likely dissociates, in the p


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fect crystal under both uniform and uniaxial strain in theydirection, but in the latter case the requisite pressure is emated to be significantly lower, lying in the accessiblegime of 20–40 GPa. We interpreted this effect as beingsociated with the induced electron-charge transfer fromhydrogen to the region between the carbon and the nitrosince the C–N bond is noticeably contracted and its charais progressively modified from a single to a double covalbond.

These results point to the preliminary conclusion thatchemistry prior to detonation is strongly dependent onpressure anisotropy, and starts before any band-gap cloi.e., metallization which would be caused by bending of cvalent bonds involving the nitro group. Accordingly, furthdecrease of the band gap seems to be favored by the fotion of protons in the crystal. It remains to check this hypoesis by applying a more accurate, first-principles methBecause the formation of a free-electron gas is favored unvery high pressure, it can be argued that LDA within DFshould suffice to produce reliable results for this regime. Iremarkable that our results, being derived under adiabcompression and for zero temperature, are in qualitaagreement with ongoing first-principles, molecular-dynamsimulations at both high density and temperature.71 Specifi-cally, these last simulations also predict that the C–H bonhighly stretched at an early stage of the compression.

It is of course premature to make any firm conjectubased on the present results alone. Yet, it is tempting to pour discussion in the context of detonation experimentsliquid nitromethane.68–70It has not escaped our attention ththe prediction of a C–H bond high-stretching here suggethat a proton abstraction may be indeed involved in the fiactivation step, thus reducing the observed time to sexplosion under high pressure.68 This type of proton disso-ciation has a long history in high pressure studies. Whileprediction is not rigorous, the reasoning that implies it wrants some attention; we believe that more theoretical wis needed in this direction.

It is expected that our results for the band-gap behawill be modified with inclusion of dynamical effectsMolecular-dynamics simulations are only suggestive atmoment.35,71 Although restricted to the ground state, thpresent analysis has indicated properties in agreementmore expensive, first-principles calculations that are tycally limited to smaller systems.35 It appears therefore promising to apply the SCC-DFTB method to more complex eergetic materials such as RDX and PETN.

ACKNOWLEDGMENTS

Part of the research in this work was performed unthe auspices of the U.S. Department of Energy by LawreLivermore National Laboratory under Contract No. W-740Eng-48. One of the authors~M.R.M.! also acknowledgessupport from the Advanced Strategic Computing Initiatiprogram. The authors wish to thank Umesh WaghmaIoannis Remediakis, Olivier Politano, and Daniel Orlikowsfor useful discussions.


ti--s-en,ert

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