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-- Empirical Potential for Hydrocarbons for Use in C: NQ0014-89-WX-24146S Simulating the Chemical Vapor Deposition of Diamond Films 1N00014-90-WX-24103

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D.W.BrenerTA: 412n006 --- 02D .W . B ennerWU: 61-3299-00

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13. ABSTRACT (Maximum 200lvo'd_)

An empirical many-body potential-energy expression is developed for hydrocarbons that can modelintramolecular chemical bonding in a variety of small hydrocarbon molecules as well as graphite and diamondlattices. The potential function is based on Tersoff's covalent-bonding formalism with additional terms thatcorrect for an inherent overbinding of radicals and that include nonlocal effects. Atomization energies for a widerange of hydrocarbon molecules predicted by the potential compare well to experimental values. The potential

cretly predicts that the 3-bonded chain reconstruction is the most stable reconstruction on the diamond 111l1}surface, and that hydrogen adsorption on a bulk-terminated surface is more stable than the reconstruction.Predicted energetics for the dimer reconstructed diamond {l100} surface as well as hydrogen abstraction andchemisorption of small molecules on the diamond { 111 } surface are also given. The potential function is shortlyranged 9nd quickly evaluated so it should be very useful for large-scale molecular dynamics simulations ofreacting hydrocarbon molecules.

14. SUBJECT TERMS 15 cj.2rqO PAGES

many-body potentials, hydrocarbons, molecular dynamics'o6 PRICE CODE

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TECHNICAL REPORT # 11

EMPIRICAL POTENTIAL FOR HYDROCARBONSFOR USE IN

SIMULATING THE CHEMICAL VAPOR DEPOSITION OF DIAMOND FILMS*

D.W. Brenner

Code 6119Naval Research Laboratory

Washington, D.C. 20375-5000

Accepted for publication in the

Phys. Rev. B 42, xxxx (1990).

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Reprinted from: Physical Review B. Vol. 42, No. 15, 15 November 1990-l1

Empirical potential for hydrocarbons for use in simulating the chemical vapordeposition of diamond films

Donald W. BrennerTheoretical Chemistry Section, Code 6119, Naval Research Laboratory, Washington, D.C. 203 75-5000

42 EMPIRICAL POTENTIAL FOR HYDROCARBONS FOR USE IN ... 9459

potential-enery expressions have recently been proposed 37) and III-V compounds, 38 as a fitting function for thefor carbon, no potential-energy functions are reactions H+H 2 and 0+02,29 and as a model forcurrently available that can be used in molecular- describing reactive collisions in molecular solids.39

dynamics simulations to model the CVD of diamond The binding energy in the Abell-Tersoff formalism isfilms from hydrocarbons. written as a sum over atomic sites i,

As a first step toward using molecular dynamics tomodel the CVD of diamond films, we have developed an Eb = Ei , (1)empirical potential-energy function that captures manyof the essential features of chemical bonding in hydrocar- where each contribution E, is written asbons. In particular, the goal of this work has been to de-velop a simple expression that (1) can reproduce the in- Ei= , [VR(rIJ)-BqJVA(riJ)] . (2)tramolecular energetics and bonding in solid diamond p*nand graphite as well as a number of essential hydrocar- in Eq. (2), the sum is over nearest neighbors j of atom i,bon molecules; (2) yields realistic energetics and bonding VR(r) and VAr) are pair-additive repulsive and attrac-for structures not included in the data base; (3) allows for tive interactions, respectively, and BY1 represents abond breaking and forming (i.e., chemistry); and (4) is not many-body coupling between the bond from atom i tocomputationally intensive. While a number of classical atom j and the local environment of atom i. As discussedexpressions have been developed for studying the dynam- by Abell and Tersoff,33 . if Morse-type functions are usedics of hydrocarbon molecules, none are able to meet these for the repulsive and attractive pair terms, then B1 cancriteria. For example, valence force fields and related ap- be considered a normalized bond order because the Paul-.proaches such as the molecular mechanics formalism ing relationship between bond order and bond length isdeveloped by Allinger and co-workers32 do not incorpo- realized. 40

rate bond dissociation, and so cannot be used to model Abell has suggested that to a first approximation B,the chemical addition of molecules to diamond surfaces. can be given as a function of local coordination Z in theThe approach taken here is to use a highly parametrized formTersoff bond-order expression 33 as a fitting function for anumber of small molecules as well as graphite and dia- Bi cc Z , (3)mond lattices. Since a bond-order function is fit rather where 8 may depend on the particular system. .Thisthan, for example, a many-body expansion of the poten- analysis yields a relationship between bond length, bind-tial energy, the function should be reasonably transfer- ing energy, and coordination that is realized for a numberable to structures not included in the fitting data set. of systems, including carbon-carbon bonds in diamond,

The organization of this paper is as follows. In the graphite, and in hydrocarbon molecules. 2

next section the Abell-Tersoff bonding formalism is out- The procedure used by Tersoff to develop classical po-lined, and several problems associated with describing tentials for silicon, carbon, and germanium has been to fitcarbon and hydrocarbons are discussed. Two parame- the pair terms and an analytic expression for By to atrizations of a Tersoff-type expression are then givenwhich have been fit to the energetics of a relatively large suctur (g onergies and sllk statenumber of hydrocarbon molecules. The first parametri- structures (e.g., bond energies and lengths, bulk moduli,zatonumber ofohydrcbo mlescripes.n Th carsaraoneton- vacancy formation energies, etc.). His expressionzation provides a good description of carbon-carbon bond developed in this way appears to be relatively transferable

lengths but has stretching force constants that are too tovothe solisa strs no ue inlte ttngfro-to other solid-state structures not used in the fitting pro-

small, while the second yields force constants that are in cedure such as surface reconstructions on silicon 33 andbetter agreement with experiment but produces bond interstitial defects in carbon. 2 s As discussed below, how-lengths for double and triple bonds that are too long. ever, further analysis shows that this expression is unableTransferability of the expressions is tested by calculating to reproduce a number of properties of carbon such as aenergetics for diamond surfaces and hydrocarbons notused in the fitting data base. The energetics of a limited proper description of radicals and conjugated versus non-number of atomic configurations that are of direct conjugated double bonds.relevance to the CVD of diamond films is then given. A. r-bonding in carbon

II. POTENTIAL-ENERGY EXPRESSION Since the Abell-Tersoff expression uses Pauling bond-order ideas, it can realistically describe carbon-carbon

In an attempt to explain a universality relation ob- single, double, and triple bond lengths and energies in hy-served in binding-energy curves,34 Abell derived a general drocarbons and in solid graphite and diamond. In inter-expression for binding energy that is a sum of near- mediate bonding situations, however, the assumption ofneighbor pair interactions that are moderated by the lo- near-neighbor interactions combined with the sum overcal atomic environment. 35 Tersoff subsequently intro- atomic sites [Eq. ()] results in nonphysical behavior. Forduced an analytic potential-energy function based on the example, if a carbon atom with three nearest neighbors isAbell expression that realistically describes bonding in bonded to a carbon atom with four neighbors, Eqs.silicon for a large number of solid-state structures.33 ()-(3) interpolate the bond so that it is intermediate be-Similar expressions have since been used for a range of tween a single and double bond. The formation of a dou-applications, including other IV-IV (Refs. 28, 29, 36, and ble bond, however, results from the overlap of unbonded

9460 DONALD W. BRENNER 42

2p orbitals. Since the atom with a coordination of 4 does B. Empirical potential-energy expression for hydrocarbonsnot have a free orbital, r overlap cannot occur and the Following the discussion given above, the binding ener-bond is better described as a single bond plus a radical or- fol the dicuson given a t bni oerbital. This overbinding of radicals results in a nonphysi- gy for the hydrocarbon potential is given as a sum overcal description of bonding for carbon in a number of bonds ascommon situations. For example, the formation of a va- Eb=y , [VR(rjj)-BijVA(ri/)J , (6)cancy in diamond results in four radicals, and so this I J(>doverbinding of radicals makes fitting the vacancy forma-tion energy in diamond while maintaining a fit to graph- where the repulsive and attractive pair terms are given by

ite difficult if not impossible.2 8 V - - - i1) (7)Similar nonphysical behavior results when conjugated it (rj )f1 /rij)D('f/Sj - )e

and nonconjugated double bonds are examined. For ex- andample, in graphite each atom has a local coordination of3, and Tersoff-type potentials fori carbon have been fit to VA(rij)=fij(rij)D' 1eS/(Si.l)e (8)yield a bond strength apropriate for graphite for this iJ jj j(

atomic environment. 23. 2 A simple analysis of kekule' respectively. The function fj(r), which restricts the pairstructures for graphite shows that due to the conjugation potential to nearest neighbors, is given byeach bond has approximately one-third double-bond andtwo-thirds single bond character. In the molecule I, r<R(1](CH3 )2C = C(CH 3)2 the two carbon atoms connected by rrR /I) (D)the central bond have the r*ame local environment as in fij W) I +Cos it)a( 2, R'j r .... igraphite, but beca .-e the bond is not conjugated it has an fI(r I +cos ",(j I Y

almost entirely do, ' le-bond character. Hence the poten- (0, )> Jtial cannot describe both situations unless nonlocal effects ,rare included. (9)

One way to correct both of these problems while main-taining the fit to diamond and graphite is to rewrite Eqs.(D-03) as a sum over bonds in the form Ile pair terms are of the same form as that used by Ter-

sot,2s.33'36 but rewritten to make the correspondence to a

Eb=X, I [VR(riJ)-BJVA(riJ)J, (4) Morse function more apparent. If S0----2, then the pairi J( >i) terms reduce to the usual Morse potential. Furthermore,

where the well depth DV, equilibrium distance R(e) and 01iareequal to the usual Morse parameters independent of the

By =(By +Bl)/2. (5) value of Si,.

The overbinding of radicals can now be fixed by adding The empirical bond-order function is given by the aver-

corrections to Eq. (5) for bonds between pairs of atoms age of terms associated with each atom in a bond plus a

that have different coordinations. As described below, correction as discussed above:

nonlocal effects can also be added to a first approxima- 1. B.- + B) (i+1( ), NJ) N1,tion to account for conjugated versus nonconjugatedbonding. where

B= i+ Gi(D,)fik(rik)e IJ A -I R ii kA )l+HI (N:H),NiC)) 1'

k ( i~j)

The quantities Nic) and NH) are the number of carbon ters could not be found that could adequately describe

and hydrogen atoms, respectively, bonded to atom i, N"l 1 bond energies for a large number of hydrocarbon mole-

is the total number of neighbors (Ni(C)+NH)) of atom i, cules. Furthermore, as discussed above, radicals andNu" j depends on whether a bond between carbon atoms i nonconjugated double b-tads are not well described.and j is part of a conjugated system, G(Wyk ) is a function To make the potermial function continuous, the func-of the angle between bonds i-j and i-k, and the functions fij(r) are used to define bonding connectivity in the

tions Hij and Fij are described below. The latter func- system, and are therefore used to define values for Nti

tion, which is used for carbon-carbon bonds only, is the N , and . , values for NiH), NC), and

correction discussed above. Forms similar to that used Ni1) for each of the carbon atoms i are given byby Tersoff to describe group-IV alloys were tried,36 but Nil 1H= , fj(ri), (12)because of the large difference in bonding characteristics j( -hydrogen)between hydrogen and carbon (hydrogen is monovalent N!C)= 7" f1j(rj), (13)while carbon has a valency of up to 4), a set of parame- J(-carbon)

.42 EMPIRICAL POTENTIAL FOR HYDROCARBONS FOR USE IN... 9461

and energies." For methane, the heat of formation at 0 K ofNNHI) t)..AC 0.693 eV (Ref. 41) was used to determine carbon-

I-,i --,'+ • (14) hydrogen bond energies of 4.393 eV. Bond dissociation

Values of N('I for neighbors of the two carbon atoms in- energies of 4.861 eV for methane4 ' and 4.944 eV for

volved in a bond can be used to determine whether the methyl radical" were then used to determine bond ener-

bond is part of a conjugated system. For example, if any gies of 4.237 eV for CH3 and 3.833 eV for CH2.neighbors are carbon atoms that have a coordination of Two different parametrizations for Eqs. (6)-18) for

less than 4 (Nil ) <4), the bond is defined as being part of carbon and hydrocarbons have been determined. A simi-

a conjugated system. The value of Nr"J for a bond be- lar function (without the corrections discussed above) has

tween carbon atoms i and j is given by been fit to the binding energy of the C2 diatomic mole-cule, and the binding energies and lattice constants of

1i'j = I + j, fik(rit )F(xik) graphite, diamond, simple cubic, and face-centered-cubiccabons ki.*) structures,29 and so the first parametrization was chosen

+ fj1 (rj,)F(xj,) (15) to reduce to that function. The function Gc(0) is given

carbons /(*l.J) bywhere IcO=o /d22 2 +(I+OO)]I(g

Gc()=a +co/do-co/[do+(1+cosO) 2 ]j (18)

, xik <5 2 and the carbon-carbon parameters are given in Table I.

F(xik )- l+cos[r(xik-2)]/2, 2<x,,<3 (16) The potential yields binding energies and nearest-

0, xik R> 3 neighbor bond lengths of 7.3768 eV/atom and 1.42 A, re-spectively, for graphite, and 7.3464 eV/atom and 1.54 A

and for diamond. It also reproduces calculated values of 7.2

XIk= N -fik (rik). (17) and 7.6 eV for the vacancy formation energies in dia-mond and graphite, respectively." The predicted

This function yields a continuous value of N j as bondls carbon-carbon bond lengths and stretching force con-break and form and as second-neighbor coordinations stants given by this parametrization (denoted as potentialchange. For N7

° J= 1, a bond is not part of a conjugated I) for the bond energies derived above are given in Tablesystem and the function yields appropriate values, and 11. The bond lengths agree very well with the expeiimen-for Nl' j > ? 2 the bond is part of a conjugated system and tal values, the largest difference being 0.02 1 for a singleparameters fit to conjugated bonds are used. As a final bond. The stretching force constants, however, differstep for making the potential continuous, two- and from experimental values by as much as 60%. Thethree-dimensional cubic splines are used for the functions carbon-hydrogen bond length given by the potential isHi and F1 , respectively, to interpolate between values at 1.071 A, which is slightly short of typical experime. 'aldiscrete numbers of neighbors. values of 1.09 A. 40

The procedure used to determine parameters for Eqs. Given in Table III are a second set of hydrocarbon pa-(6)-(17) was to first fit to systems consisting of only car- rameters for Eqs. (6)-(18). This potential was again fit tobon and only hydrogen. Parameters were then chosen for the binding energies and lattice constants of graphite, di-the mixed hydrocarbon system that reproduced additive amond, simple cubic, and face-centered-cubic structuresbond energies. Since the pair terms are first fit to solid- for pure carbon, 42 and the vacancy formation energies forstate carbon structures, the equilibrium carbon-carbon diamond and graphite.4" It yields binding energies anddistances and stretching force constants for hydrocarbons nearest-neighbor bond lengths of 7.3756 eV/atom andare completely determined by fitting to bond energies. 1.45 A, respectively, for graphite, and 7.3232 eV/atom

To determine appropriate energies for hydrocarbons and 1.54 A for diamond. The predicted carbon-carbonwith carbon-carbon bonds, additive bond energies for sin- bond lengths and force constants for hydrocarbons givengle, double, conjugated double and triple carbon-carbon by this parametrization (denoted by potential II) arebonds, and carbon-hydrogen bonds were determined given in Table II. The stretching force constants arefrom molecular atomization energies. Heats of formation closer to the experimental values than potential I, withat 0 K for acetylene, ethylene, ethane, benzene, and cy- the differences being reduced to 11%, 8%, and 26% forclohexane are 2.356, 0.629, -0.717, 1.041, and -0.868 single, double, and triple bonds, respectively. The bondeV, respectively. 41 Combining these values with binding lengths for multiple bonds, however, are not as well de-energies of 7.3768 eV for graphite42 and 2.375 eV for scribed as with potential I, with the double and triplemolecular hydrogen,' 3 and neglecting corrections for zero bonds being larger than the experimental values by 3.7%point energy in the hydrocarbons, yields total atomiza- and 7.5%, respectively. A parametrization for the func-tion energies of 17.149, 23.626, 29.723, 57.472, and tions used for the pair terms could not be found that fit73.634 eV for acetylene, ethylene, ethane, benzene, and carbon-carbon stretching force constants and bondcyclohexane, respectively. Assuming constant carbon- lengths simultaneously, although other forms may repro-hydrogen and carbon-carbon single bond energies, energy duce both sets of properties.values of 4.362, 8.424, 6.175, 5.216, and 3.547 eV can be For hydrogen, the pair terms were fit to properties ofderived for carbon-hydrogen and carbon-carbon triple, the gas-phase diatomic." The parameter 8 H was setdouble, conjugated double (in benzene), and single bond equal to 8

c , and the remaining parameters were fit to


TABLE I. Parameters for Eqs. (4-018) for potential 1. All parameters not given are equal to zero, F(ij,k)-F(j,ik),F(i,j,k > 2)=F(iJ,2), and aF(i,j,k)/ai=aF(jI k)1ai. The partial derivatives are used in the multidimensional cubic splines

Parameter Value Fit to

Carbon R c 1.315 A Lattice constants of diamond and graphite

D 6.325 eV Reference 29

Pgcc 1.5 A-' Reference 29$CC 1.29 Reference 29

SCC 0.80469 Reference 29accc 0.0

R 1.7 A Reference 29

cc 2.0 A Reference 29ao 0.011 304 Reference 29

0 19 Reference 29

0 2.52 Reference 29F(2,3, 1),F(2,3,2) -0.0465 E,, for diamondF(1,2,2) -0.0355 E.. for graphite

Hydrogen R (A 0.74144 A Gas-phase diatomic

DH 4.7509 eV Gas-phase diatomic

PHH 1.9436 A-' Gas-phase diatomicSHH 2.3432 Barrier for reaction (19)

SHH 0.80469 Set equal to carbon valuealHH 3.0 Remove spurious wells from (19)GHH 4.0 Barrier for reaction (19)

R 1.1 A o Near-neighbor interactions( 2)

H 1.7 A Near-neighbor interactions

Hydrocarbons R 1.1199 A Gas-phase diatomic

D 3.6422 eV Gas-phase diatomic

P0, 1.9583 A-' Gas-phase diatomic

SC, 1.7386 (SHHS CC )l2

cR 1.3 A Reaction (19)

CH 1.8 A Reaction (19)axHc,aCHx,ancHaHcc 3.0 A-' H3 value

Hcc( 1,1) -0.0175 CC bond energy in benzeneHcc(2,0) -0.0070 CC double bond in ethyleneHcc(3,O) 0.0119 CC single bond in ethaneHcc(1,2) 0.0115 CC single bond in isobutane

Hcc(2, 1) 0.0118 CC single bond in cyclohexaneHcn(1,0) -0.0760 Atomization energy of CH2

HcH(2,0) -0.2163 Atomization energy of CH3

HcH(3,0) -0.3375 Atomization energy of methaneHcH(0, 1) -0.1792 CH bond energy in acetyleneHcH(0,2) -0.2407 CH bond energy in benzeneHc( 1,I) -0.2477 CH bond energy in ethylene

42 EMPIRICAL POTENTIAL FOR HYDROCARBONS FOR USE IN... 9463

TABLE I. (Continued).


Hc(2,1) -0.3320 CH bond energy in ethane

HcH(O,3) -0.3323 Tertiary-HC bond energy in isobutane

Hc{ !,2) -0.3321 CH bond energy in cyclohexane

aHc( 1,1 0.12805 Centered difference8C

ale(2,0) -0.07655 Centered difference

aCa8cli(O2) -0.13075 Centered difference

8H

aHlc( 1,1) -0.0764 Centered difference8H

F1I,1,) 0.1511 CC triple bond in acetylene

FR2,2,1) 0.075 Average energy of bonds in (CH3 )ZC=C(CH3).

and (CH3 )HC=CH(CH) equal double bond

F91,2,1) 0.0126 Atomization energy of HC=CH2

FMl,3,1),FI,3,2) -0.1130 Single bond in H3C-CH

R0,3,l),F0,3,2) -0.1220 Single bond in H 3C--C

F[0,2,2) -0.0445 Conjugated double bond in C=CH(CH2 )

P10,2,1) 0.0320 Double bond in C=CH2

F10,1.1) 0.1100 Atomization energy of C2H

P11,1,2) 0.0074 Atomization energy of CH2CCH

F(3,1,1) -0.1160 Centered differenceai

aF(3,2, 1) -0.132 05 Centered differenceai"

aF(3,l,2) -0.0610 Centered differenceai

aF(2,3,2) 0.02225 Centered difference8i

aF(2,4,2) -0.03775 Centered difference8i

aF(3,4,2) 0.0565 Centered differenceai

aF(3,4, 1) 0.0565 Centered differenceai

aF(3,2,2) -0.1065 Centered differenceai

reproduce the barrier of 0.425 eV for the ground-state hydrocarbon potential I are given in Table I, and thoselinear exchange reaction for potential II are given in Table III.

Some important points about the function, fitting pro-H+H 2--+H 2 +H (19) cedure, and resulting structures need to be addressed.

First, the angular function G(M) associated with the car-at the hydrogen-hydrogen distance of 0.93 ,1 as calculat- bon centers favors 180 bond angles and hence opened by Liu and Seigbahn 4 9 and to remove spurious wells structures. This is physically motivated by valence shell

from this surface. For simplicity, angular dependence of electron pair repulsion (VSEPR) theory,50 which assumes

the potential for hydrogen centers via the function GH(O) that rqpuisions between pairs of valence electrons tend to

was replaced by the (constant) value of GH used to fit re- maximize bond angles. This assumption, combined with

action (19). The resulting parameters corresponding to the tendency to maximize the binding energy (i.e., num-


TABLE I1. Carbon-carbon bond lengths R, (in A) and force constants F, (in 10 dyn/cm) given bythe two potentials based on bond energies of 3.547, 6.175, and 8.424 eV for single, double, and triplebonds, respectively, as derived in the text.

Single bonds Double bonds Triple bondsF , R e F , R e F , R e

Potential I 2.6 1.56 4.5 1.33 6.1 1.20Potential II 5.0 1.55 8.7 1.38 11.9 1.29Experiment 4.5' 1.54b 9.5a 1.33b 16.0' 1.20 b

'From Ref. 40.bFrom Ref. 48.

TABLE 11. Parameters for Eqs. (4)-(18) for potential Il. All parameters not given are equal to zero, F(i,j,k)=F(jik),F(i,j,k > 2)=F(i,j,2), and aF(i,j,k)/ai=aF(j,i,k)/ai. The partial derivatives are used in the multidimensional cubic splines.


Carbon R e 1.39 A Lattice constants of diamond and graphiteDe 6.0 eV Fit to data in Ref. 42f8cc 2.1 ,-' Fit to data in Ref. 42

,cc 1.22 Fit to data in Ref. 42

8 cc 0.5 Fit to data in Ref. 42

accc 0.0

Ri

o)

cc 1.7 X Fit to data in Ref. 42cc 2.0 ,A Fit to data in Ref. 42

ao 0.00020813 Fit to data in Ref. 42c 3302 Fit to data in Re. 42d0A 3.52 Fit to data in Ref. 42

[2,3,1),FU2,3,2) -0.0363 E.. for diamond

F,22). -0.0243 E, for graphite

Hydrogen R (HeH 0.74144 A Gas-phase diatomic

H 4.7509 eV Gas-phase diatomic

16H 1.9436 ,- Gas-phase diatomic

Snn 2.3432 Barrier for reaction (19)

8HH 0.5 Set equal to carbon value

aHHiI 4.0 Remove spurious wells from (19)

GHH 12.33 Barrier for reaction (19)

R t 1.1 . Near-neighbor interactionsRIZI

H 1.7 , Near-neighbor interactions

Hydrocarbons RnfI 1.1199 A Gas-phase diatomic

D 3.6422 eV Gas-phase diatomic

Pc" 1.9583 A-' Gas-phase diatomic

SCH 1.69077 (SHHSCC) I'

RC1H 1.3 A Reaction (19'C 1.8 Reaction (19)

aHmcaCHl,aHCH,aHCC 4.0 A-' H3 value


TABLE IlL (Continued).


Hcc( 1,1) -0.0226 CC bond energy in benzene

Hcc(2,0) -0.0061 CC double bond in ethylene

Hcc(3,O) 0.0173 CC single bond in ethane

Hcc( 1,2) 0.0149 CC single bond in isobutane

Hcc(2, 1) 0.0160 CC single bond in cyclohexane

Hc( 1,0) -0.0984 Atomization energy of CH2

HcH( 2 ,O) -0.2878 Atomization energy of CH3HcH( 3 ,0) -0.4507 Atomization energy of methane

HcH(0, 1) -0.2479 CH bond energy in acetylene

HcH(0, 2 ) -0.3221 CH bond energy in benzene

Hc,(!,1) -0.3344 CH bond energy in ethylene

HcH(2,1) -0.4438 CH bond energy in ethane

HcH(0,3) -0.4460 Tertiary-HC bond energy in isobutane

HCH (1,2' -0.4449 CH bond energy in cyclohexaneBHCH (1, 1 )C -0.17325 Centered differenceaC

a-(2,0) -0.09905 Centered differenceac

alcH(0,2) -0.17615 Centered difference5H

aHcH ( 1,1 )8H -0.09795 Centered difference8H

F1,1,1) 0.1264 CC triple bond in acetylene

F12,2,1) 0.0605 Average energy of bonds in (CH3)2C=C(CH3 )

and (CH3)HC=CH(CH3) equal double bond

Ftl,2,1) 0.0120 Atomization energy of HC=CH2

F!1,3,1),FI,3,2) -0.0903 Single bond in H3C--CH

F10,3,1),F(0,3,2) -0.0904 Single bond in H3C-C

F0,2,2) -0.0269 Conjugated double bond in C=CHCH2)

FT0,2,l) 0.0427 Double bond in C=CH2

F0,1,1) 0.0996 Atomization energy of C2H

F11,1,2) 0.0108 Atomization energy of CH2CCHaF(3, 1, 1)3 -0.0950 Centered difference



aF(2,3,2) 0.01345 Centered differenceai


aF(3,4,2) 0.045 15 Centered differenceai

aF(3,4, 1) 0.045 15 Centered differenceai



ber of bonds times the bond strength), leads to, for exam- and Chadi w-bonded molecule ss reconstructions on theple, a tetrahedral structure for methane and linear struc- diamond I 11 j surface are given along with values calcu-ture for ace:ylene. Because lone electron pairs are not ex- lated by Vanderbilt and Louie using local-density-plicitly treated in this empirical potential, however, some functional (LDF) methods." The values given by thestructures are linear that experimentally are bent, such as empirical potentials were calculated using a slab ten lay-methylene.51 Second, although the procedure for defining ers thick with 16 atoms per layer and periodic boundaryNi"j incorporates conjugation in an approximate way, it conditions perpendicular to the surface. The bottom twowill not give the subtle differences in bond orders for con- layers were held rigid and the top eight layers were al-jugated ring systems predicted by even simple molecular- lowed to completely relax under the influence of each oforbital methods. This procedure, however, is able to in- the potentials to the minimum-energy configurationsclude to a first approximation nonlocal effects without starting from the positions reported by Chadi for thethe need for diagonalizing a matrix or going beyond various reconstructions.55 The potentials predict that thenearest-neighbor interactions, hence drastically reducing (undimerized) ir-bonded chain reconstruction is energeti-computational requirements for large systems. Third, the cally preferred, in agreement with the LDF result" andcarbon-carbon single bond energies derived for hydrocar- with a number of experimental studies."7.5 If radical en-bons are smaller than those in diamond. This difference ergetics is not correctly described, the Tersoff-type poten-also occurs in more-elaborate least-squares-fitting pro- tials given in Refs. 28 and 29 incorrectly predict the re-cedures for determining carbon-carbon bond energies, laxed bulk-terminated surface to be the most stable.59

and can be attributed to both the weakening of the bonds Experimental studies have shown that upon exposuredue to hydrogen and steric repulsions.52 Fourth, barriers to atomic hydrogen the ar-bonded chain reconstructionfor rotation about carbon-carbon bonds (especially dou- converts to a hydrogen-terminated surface with the car-ble bonds) and nonbonded interactions such as van der bon atoms reverting to positions that are characteristic ofWaals forces have not been included. With the potential the bulk. 5 ".' 0' 6 To test the potentials, a monolayer of hy-written as a sum over pairs, the former effects can be drogen atoms was placed above the surface carbon atomsmodeled, and methods for incorporating both types of in- on the relaxed bulk-terminated surface and the ar-bondedteractions within this formalism are being developed. Fi- chain reconstruction and the systems were relaxed. Po-nally, the price one pays for using a simple classical ex- tential I predicts energies of -1.00 eV/(surface atom)pression is that properties such as resonance effects53 (i.e., and -0.61 eV/(surface atom) for the hydrogen-covered4n +2 Hickel stability) and Woodward-Hoffman. rules53 bulk-terminated surface and hydrogen-covered ir-bondedmay not necessarily be obeyed. Care should therefore be chain reconstruction, respectively, relative to the cleanused when interpreting reaction mechanisms suggested it-bonded chain reconstruction plus gas-phase H2 mole-by o.npirical potentials, and where possible they should cules. Potential II yields corresponding energies ofbe reexamined using other less-empirical techniques. - 1.13 eV/(surface atom) for adsorption on the bulk-

terminated surface and -0.63 eV/(surface atom) for ad-sorption on the ar-bonded chain reconstruction. Hence

IIL PREDICTED ENERGETICS AND BONDING the potentials correctly predict that hydrogen adsorptionis energetically stable, and that adsorption on a bulk-

To test the transferability of the potential given in the terminated surface is favored over the reconstructed sur-preceding section, the energy and structure of a variety of face. Side views of the equilibrium structures for the twosmall hydrocarbon molecules and the diamond I 11 and hydrogen-ter.aiinated surfaces are shown in Fig. 1.10011 surfaces have been examined. Shown in Table IV The structure and energetics of the dimer reconstruct-are atomization energies given by the empirical potential ed 10011 surface of diamond are also of interest, and de-and corresponding experimental values. The experimen- tailed experimental studies have just recently been com-tal values were derived from heats of formation without pleted. Potential I predicts a stabilization energy ofcorrecting for zero-point energy or other finite tempera- -5.49 eV/dimer relative to the bulk-terminated surface,ture effects. The overall agreement is good, with energies and a surface dimer bond length of 1.38 A. Potential IIbeing reproduced to within 1% or better for 81% of the predicts a slightly less stable reconstruction with an ener-molecules tested. While this level of agreement may be gy of -5.20 eV/dimer relative to the bulk-terminatedsomewhat fortuitous given the approximations made in surface, and a surface dimer bond length of 1.43 A.estimating the experimental values, it does demonstrate These values were obtained using a slab eight layers thickthat the potential can describe chemical bonding for with 32 atoms per layer. Periodic boundaries were useda wide range of hydrocarbon molecules. The worst fits for, the two directions perpendicular to the surface, theare for the molecules CH3CH=C=CH2 and bottom two layers were held rigid, and the remainder ofH2C=C=CH2 for both potentials, and for cyclopro- the system wa. relaxed to the minimum-energy structurepene using potential If. For the former two molecules starting from a surface reconstruction consisting of athe potential function treats the two double bonds as part 2 X I arrangement of surface dimers. The dimer length isof a conjugated system, and so the bond energies are re- stretched from the double bond distance in ethylene forduced. This is an example of where our analytic approxi- both potentials, and the bond energy is correspondinglymation for defining conjugation breaks down. weakened. This is due to the strain induced by the lat-

In Table V the energetics predicted by the potential for tice. The monohydride phase, where a hydrogen atom isthe relaxed bulk-terminated, Pandey ar-bonded chain, 4 bonded to each atom in a dimer pair, was also examined


TABLE IV. Atomization energies for various hydrocarbon molecules. Experimental values were derived from heats of formationusing energies of 7.3768 eV for carbon and 2.375 eV for hydrogen.

Potential I Potential II Experimental valueMolecule (eV) (eV) (eV)

Alkanes methane 17.6 17.6 17.6ethane 29.7 29.7 29.74propane 42.0 42.0 42.0'n-butane 54.3 54.3 54.3'i-butane 54.3 54.3 54.44n-pentane 66.5 66.5 66.6'isopentane 66.5 66.5 66.6'neopentane 66.8 §6.8 66.74

cyclopropane 35.5 35.0 35.8'cyclobutane 48.7 48.5 48.2*cyclopentane 61.4 61.3 61.4'cyclohexane 73.6 73.6 73.6'

Alkenes ethylene 23.6 23.6 23.60propene 36.2 36.2 36.0'1-butene 48.5 48.5 48 .5b

cis-butene 48.8 48.9 48.6'isobutene 48.4 48.4 48 .7 b

(CH 3)2C - C(CH3 )2 73.2 73.3 73.4b

cyclopropene 28.2 27.3 28.8b

cyclobutene 42.4 42.0 42.4'cyclopentene 55.7 55.7 55.6 b

1,4-pentadiene 55.0 55.0 54.8b

CH 2 = CHCH = CH2 41.8 41.9 42.6b

CH 3CH =C= CHz 40.4 40.5 42.1H2C=C=CH2 27.8 27.9 29.6b

Alkynes acetylene 17.1 17.1 17.1 "propyne 29.4 29.4 29.7'1-butyne 41.7 41.7 42.0b

2-butyne 41.7 41.7 42.2b

Aromatics benzene 57.5 57.5 57.5'toulene 69.6 69.6 70.1'1

1,4-dimethylbenzene 81.8 81.8 82.6'ethylbenzene 81.9 81.9 82.5'

ethenylbenzene 76.2 76.2 76.5b

ethynylbenzene 69.8 69.8 69.9'naphthalene 91.4 91.4 91.2b

Radicals CH2 7.8 7.8 7.8cCH3 - 12.7 12.7 12.r

H3C2H2 25.7 25.7 25.5b

H2C2H 18.9 18.9 18.9C2H 12.2 12.2 12.2'

CH2CCH 24.5 24.5 25.8b

n-C3H, 37.9 38.0 37.8'i-C3H, 38.3 38.3 38.01t-C4H, 50.5 50.5 50.5ephenyl 52.7 52.7 52.7

'From heat of formation at 0 K ('ro.:., "7ef. 4 1).'From heat of formation at 30C K [from H. M. Rosenstock, K. Draxl, B. W. Steiner, and J. J. Heron, J. Chem. Ref. Data 6, Suppl. I,

1-774 (1977)].'Calculated from P.ef. 46.'dCalculated from Ref. 45.

'From heat of formation at 300 K (from Ref. 52).(From heat of formation at 0 K [from H. M. Rosenstock, K. Draxl, B. W. Steiner, and J. 1. Heron, J. Chem. Ref. Data 6, Suppl. i, 1-774 (1977)].


TABLE V. Energetics (in eV/surface atom) for various structures of the I 1 I I surface of diamond.

r-bonded n-bondedIdeal I X I Relaxed I X I chain molecule

Potential 1 0.0 -0.24 -1.10 -0.32Potential II 0.0 -0.23 -1.03 -0.25LDF* 0.0 -0.37 -0.68 0.28

'From Ref. 56.

using the potentials. The predicted carbon-carbon sur- tion reactionface dimer bond length is 1.63 A for potential I and 1.60A for potential II, both of which are stretched from the CH 4 +H-"H 3C-H-H-"CH3 +H 2 (20)gas-phase single bond lengths. Potentials I and II predict was examined using the empirical potentials. This reac-changes in energy for hydrogen adsorption of -2.72 tion was chosen because the potential-energy surface haseV/dimer and -2.96 eV/dimer, respectively, relative to been well characterized in the gas phase. Followingthe clean dimer reconstructed surface and gas-phase Hj. Walsh,62 the reaction was examined for a linearHydrogen adsorption is therefore predicted to be energet- configuration along the reaction coordinate assuming C3.ically stable, in agreement with cxperiment. symmetry, with the energy at each point minimized with

The abstraction of hydrogen from hydrogen- respect to the other carbon-hydrogen bond distances andterminated diamond surfaces has been proposed as being angles. The barrier for reaction (20) is 0.52 eV using po-a rate-determining step in the CVD of diamond films, tential I, which agrees with the experimental energyso the potential-energy surface for the hydrogen abstrac- threshold of 0.52 eV,' 3 and which can be compared to the

value of 0.58 eV for the barrier estimated by Walch usingab initio total-energy techniques.62 The barrier along thelinear reaction coordinate for potential I is at a carbon-hydrogen bond distance of 1.30 A and a hydrogen-hydrogen bond distance of 1.02 A. The values calculatedby Walch are 1.47 A for the carbon-hydrogen distanceand 0.93 A for the hydrogen-hydrogen distance. Poten-tial II yields a barrier of 0.72 eV at a carbon-hydrogenbond distance of 1.19 A and a hydrogen-hydrogen bondlength of 1.14 A. This barrier is somewhat higher thanthe experimental value, and so potential I should yield abetter description of hydrogen abstraction.

IV. . AERGETICS FOR STRUCTURESRELATED TO DIAMOND CVD

To begin to quantify energetics for surface structuresassociated with the CVD of diamond films, we have cal-culated the energetics for hydrogen abstraction from thediamond I I fIl and 100 11 surfaces, and the energetics forthe chemisorption of methane, acetylene, and theirrespective radicals at terrace sites on the I 1 II1 surface ofdiamond. More systematic studies of structures and dy-namics of hydrocarbons reacting with diamond surfacesand their relevance to the CVD of diamond films areplanned to be presented at a later time.

Removing a hydrogen atom from a hydrogen-terminated I 11 surface was found to be endoergic by4.13 eV for potential I and 4.15 eV for potential II.These numbers are close to the estimate obtained by sub-tracting the relaxation energy given in Table V for the re-laxed bulk-terminated surfaces from the carbon-hydrogenbond strength of 4.36 eV. Combining these values withthe gas-phase bond strength of 4.75 eV for H 2 yields a net

FIG. 1. Side views of hydrogen-terminated diamond I 1 I exoergicity of =0. 6 eV for abstraction of hydrogen fromsurfaces. The top picture is the n--bonded chain reconstruction the I I I I I surface by a gas-phase hydrogen atom. Be-and the bottom picture is the bulk-terminated surface. cause of the bonding topology of this surface, the energy

42 EMPIRICAL POTENTIAL FOR HYDROCARBONS FOR USE IN ... 9469

TABLE VI. Predicted energetics and intramolecular carbon-carbon bond lengths for various single molecules chemisorbed on ter-race sites on a hydrogen-terminated diamond I II surface.

Potential I Potential IIPotential energy Bond length Potential energy Bond length

Molecule (eV) (A) (eV) (A)

Hydrogen atom -4.1' -4.2aMethyl radical - 3.7a -4.0'Acetyl radical -3.9 1.20 -4.18 1.29Hydrogen molecule - 3.6b - 3.6b

Acetylene - 5.0b 1.33 -4.9b 1.39Ethylene -4.3 b 1.59 -4.3b 1.57'Relative to a hydrogen-terminated surface with one radical site and the gas-phase molecule.bRelative to a hydrogen-terminated surface with two adjacent radical sites and the gas-phase molecule.

required to femove a second adjacent hydrogen atom is tures. For acetylene and ethylene, two hydrogen atomsessentially the same as that required to remove the first, bonded to adjacent surface carbon atoms were removedso radical sites on this surface should be randomly distri- and the molecules were placed with the carbon atoms inbuted. Removing a hydrogen atom from the monohy- each of the molecules approximately over the open sitesdride phase of the dimer reconstructed diamond 10011 (i.e., with the intramolecular carbon-carbon bonds paral-surface was found to be endoergic by 4.44 and 4.49 eV us- lel to the surface). The resulting energies and intrarnolec-ing potentials I and II, respectively, so abstraction by a ular carbon-carbon bond lengths for each of the mole-gas-phase hydrogen atom is exoergic by =0.3 eV. Re- cules are given in Table VI. Energetics for hydrogen ad-moving a second hydrogen atom from the other carbon in sorption is given for comparison. The radicals C2H andthe surface dimer pair, however, is endoergetic by only CH 3 both form carbon-carbon single bonds without large2.9 eV with potential I and 3.1 eV with potential II. This changes in bonding character within the molecules,decrease in endoergicity is consistent with the formation which. le:ds to the similar binding energies. For the ad-of a partial double bond between the two surface carbon sorption of acetylene and ethylene each molecule formsatoms. Hence the dimer reconstructed 10011 surface two single bonds with the surface and the order of thewith radical sites should be composed primarily of a mix- carbon-carbon bond in the molecules is reduced by 1.ture of hydrogen-terminated carbon-carbon dimers with The carbon-carbon bond in the chemisorbed acetylene issingle bonds and nonhydrogen-terminated double-bonded predicted to be equal to the gas-phase double bondcarbon-carbon dimers. length. Hence in contrast to the dimer reconstruction on

To quantity the energetics of adsorption on terrace the 1001 surface the constraint of the lattice does not in-sites on the I 111 surface, chemisorption energies for hibit the formation of a double bond. The difference inboth single molecules and monolayer coverages of ace- energy between the chemisorption of acetylene andtylene and ethylene molecules and ethynyl and methyl ethylene is consistent with their heats of hydrogenation.radicals have been calculated using the empirical poten- Hydrogenation of acetylene to ethylene is exoergic bytials. For the chemisorption of single radicals, a hydro- 1.72 eV relative to H 2 plus acetylene, while the hydro-gen atom on a hydrogen-terminated I111 J surface was genation of ethylene to ethane is exoergic by 1.35 eV.replaced with the molecule (with the carbon-carbon bond Given in Table VII are energies for the chemisorptionin the ethynyl radical perpendicular to the surface), and of a monolayer of each of the molecules discussed abovethe systems were relaxed to the minimum-energy struc- on the I 111 surface. In each case the initial

TABLE VII. Predicted energetics and intramolecular carbon-carbon bond lengths for a monolayer of various molecules chem-isorbed on the diamond 11 surface. The energies are relative to a relaxed clean surface and the gas-phase molecules.

Potential I Potential I1Bond length Bond length

Molecule Potential energy (A) Potential energy (A)

Hydrogen atom -4.2 eV/atom' -4.3 eV/atom"Methyl radical -3.2 eV/molecule' -3.0 eV/molecule"Acetyl radical -4.2 eV/molecule' 1.20 -4.3 eV/molecule 1.29Hydrogen molecule -3.7 eV/moleculeb -3.9 eV/moleculeb

Acetylene -5.2 eV/moleculeb 1.33 -5.2 eV/moleculeb 1.39Ethylene -4.0 ev/moleculeb 1.58 -3.9 eV/moleculeb 1.57'One surface atom per chemisorbed molecule.

bTwo surface atoms per chemisorbed molecule.


configurations were generated by removing the hydrogen parametrized version of Tersoffs empirical-bond-orderatoms terminating the surface bonds and translating the formalism which includes terms that correct for an in-chemisorbed single molecules across the surface. Each herent overbinding of radicals. Nonlocal effects haveconfiguration was then relaxed to the minimum-energy been incorporated via an analytic function that definesstructure. The energies for adsorbed hydrogen, ethynyl conjugation based on the coordination of carbon atomsradicals, and acetylene molecules decrease (i.e., become that neighbor carbon-carbon bonds. Because onlymore strongly bound) from single molecules to mono- nearest-neighbor interactions are incorporated, the func-layers, while because of the interactions between non- tion is very quickly evaluated and can therefore be usedbonded hydrogen atoms the energies for methyl radicals in large-scale molecular-dynamics simulations. Futureand ethylene molecules both increase, refinements to the potential include incorporating bar-

While a specific mode of CVD growth cannot be unam- riers for rotation around carbon-carbon bonds and non-biguously deduced from these energies, they do suggest bonded interactions such as van der Waals forces.some possible trends. First, abstraction of a single hydro- Atomization energies predicted by the function com-gen atom from a 11 1 surface should be favored over pare well to experimentally derived energies for a wideabstraction from a dimer reconstructed 1001 surface. range of hydrocarbon molecules. The function predictsAbstraction of a second hydrogen atom, however, should the ir-bonded chain reconstruction to be the energeticallybe strongly favored on the reconstructed 10011 surface favored structure on a clean diamond I I III surface insince the bonding topology of the surface reconstruction agreement with local-density-functional calculations, andallows formation of double bonds on the surface. Second, correctly predicts hydrogen adsorption on a bulk-displacement of a hydrogen atom from a hydrogen- terminated surface to be favored over the ir-bonded chainterminated diamond I I I I surface by either ethynyl or reconstruction. The structure and energetics of the di-methyl radicals is not strongly energetically favored. mer reconstructed diamond (EIlI surface and itsEthylene and especially acetylene, however, can energeti- monohydride phase were also ex. ed using the empiri-cally displace hydrogen from terrace sites provided that a cal potential.gas-phase hydrogen molecule is formed (rather than two Energetics for a limited number of static surface struc-H atoms). A reaction sequence that might lead to this, tures that are relevant to the CVD of diamond films werehowever, is unclear, although a sequence similar to that also calculated using the empirical potential. Theseproposed by Frenklach and Spear may be possible.' Fi- structures included the energies required to abstract by-nally, the potential suggests that adsorption of a mono- drogen- atoms from diamond II I l and dimer recon-layer of methyl radicals on the I III surface is not an en- structed 10011 surfaces, and energies for methyl andergetically favorable configuration compared to hydrogen ethynyl radicals and acetylene and ethylene molecules ad-adsorption. The predicted energies given here for a sorbed on terrace sites on the I I l I I surface of diamond.monolayer of methyl are uncertain at best since nonbond- More detailed and systematic studies of bonding energiesed steric interactions between hydrogen atoms have not and reactive dynamics using this potential will bebeen included in the potential. A better estimate could presented in the future.probably be made using other potentials such asAllinger's molecular mechanics. 32 ACKNOWLEDGMENTS

B. Dunlap, R. Mowrey, M. Page, and C. T. White ofV. SUMMARY NRL are thanked for helpful discussions and B. J. Gar-

rison, R. Carty, and D. Srivastava of Pennsylvania StateAs a first step toward using molecular-dynamics simu- University are thanked for helpful discussions and for

lation techniques to study the CVD of diamond films, we performing initial tests of the potential. S. J. Harris, G.have developed an empirical potential-energy function Kubiak, M. Pederson, and S. M. Valone are also thankedthat can model intramolecular chemical bonding in a for providing copies of their work prior to publication.wide range of hydrocarbon molecules as well as diamond This work was -supported in part by Office of Navaland graphite lattices. The analytic function is a highly Research through Contract No. N0014-89-WX-24146.

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REPORT DOCUMENTATION PAGE 0 No. 07 -0o 88

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1. AGENCY USE ONLY (Leave bank) 1 2. REPORT DATE 73. REPORT TYPE AND DATES COVERED 10/1/89-11/5/90 Technical Rept. (Interim) 9/30/90

4. TITLE AND SUBTITLE S. FUNDING NUMBERS

Empirical Potential for Hydrocarbons for Use in C: NaOO14-89-WX-24146N00014-90-WX-24103Simulating the Chemical Vapor Deposition of Diamond Films PE: 0601153N

PR: 0601153O6. AUTHOR(S) PR: RR011-03-OK

D.W. Brenner TA: 412n006---02WU: 61-3299-00

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONNaval Research Laboratory REPORT NUMBER

Washington, DC 20375-5000 Technical Report #11

(Code 6119, C.T. White)

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Office of Naval Research AGENCY REPORT NUMBER

800 North Quincy Street Technical Report #11Arlington, VA 22217-5000(Code 1112AI, D.H. Liebenberg)

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12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODEThis document has been approved for public release andsale, distribution of this document is unlimited.

13. ABSTRACT (Maximum 200 words!

An empirical many-body potential-energy expression is developed for hydrocarbons that can modelintramolecular chemical bonding in a variety of small hydrocarbon molecules as well as graphite and diamondlattices. The potential function is based on Tersoff's covalent-bonding formalism with additional terms thatcorrect for an inherent overbinding of radicals and that include nonlocal effects. Atomization energies for a widerange of hydrocarbon molecules predicted by the potential compare well to experimental values. The potentialcorrectly predicts that the s-bonded chain reconstruction is the most stable reconstruction on the diamond {111}surface, and that hydrogen adsorption on a bulk-terminated surface is more stable than the reconstruction.Predicted energetics for the dimer reconstructed diamond {100} surface as well as hydrogen abstraction andchemisorption of small molecules on the diamond {111} surface are also given. The potential function is shortlyranged and quickly evaluated so it should be very useful for large-scale molecular dynamics simulations ofreacting hydrocarbon molecules.

14. SUBJECT TERMS i 15. NUL'.rR OF PAGES

* many-body potentials, hydrocarbons, molecular dynamics Cj 16 PRIC-E CODE

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UNCAS IFI I UNCLASSIFIED UNCLASSIFIEDNSN 7540-01-280-5500 S ,ad :-m 299 (Rev 2.89)

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c isale, distribution of this document is unlimited. 13. abstract (maximum 200lvo'd_) an...

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