Bond-order potentials with split-charge equilibration: Application to C-, H-,and O-containing systemsM. Todd Knippenberg, Paul T. Mikulski, Kathleen E. Ryan, Steven J. Stuart, Guangtu Gao et al. Citation: J. Chem. Phys. 136, 164701 (2012); doi: 10.1063/1.4704800 View online: http://dx.doi.org/10.1063/1.4704800 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v136/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 136, 164701 (2012)

Bond-order potentials with split-charge equilibration:Application to C-, H-, and O-containing systems

M. Todd Knippenberg,1,2 Paul T. Mikulski,1 Kathleen E. Ryan,1 Steven J. Stuart,3

Guangtu Gao,4 and Judith A. Harrison1

1Departments of Physics & Chemistry, United States Naval Academy, Annapolis, Maryland 21402, USA2Department of Chemistry, High Point University, High Point, North Carolina 27262, USA3Department of Chemistry, Clemson University, Clemson, South Carolina 29634, USA4US Department of Agriculture, ARS National Center for Cool and Cold Water Aquaculture, Kearneysville,West Virginia 25430, USA

(Received 5 February 2012; accepted 4 April 2012; published online 26 April 2012)

A method for extending charge transfer to bond-order potentials, known as the bond-orderpotential/split-charge equilibration (BOP/SQE) method [P. T. Mikulski, M. T. Knippenberg, andJ. A. Harrison, J. Chem. Phys. 131, 241105 (2009)], is integrated into a new bond-order potential forinteractions between oxygen, carbon, and hydrogen. This reactive potential utilizes the formalism ofthe adaptive intermolecular reactive empirical bond-order potential [S. J. Stuart, A. B. Tutein, andJ. A. Harrison, J. Chem. Phys. 112, 6472 (2000)] with additional terms for oxygen and charge interac-tions. This implementation of the reactive potential is able to model chemical reactions where partialcharges change in gas- and condensed-phase systems containing oxygen, carbon, and hydrogen. TheBOP/SQE method prevents the unrestricted growth of charges, often observed in charge equilibrationmethods, without adding significant computational time, because it makes use of a quantity whichis calculated as part of the underlying covalent portion of the potential, namely, the bond order. Theimplementation of this method with the qAIREBO potential is designed to provide a tool that canbe used to model dynamics in a wide range of systems without significant computational cost. Todemonstrate the usefulness and flexibility of this potential, heats of formation for isolated molecules,radial distribution functions of liquids, and energies of oxygenated diamond surfaces are calculated.© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4704800]

I. INTRODUCTION

Due to the technological importance of Si- and C-containing materials, several empirical potential energyfunctions, including a class of potentials known as thebond-order potentials (BOP), have been developed forthese materials.1–13 These potentials have been particu-larly successful modeling a wide range of processes. Forhydrocarbons, the reactive empirical bond-order (REBO)potential has been used to model mechanical properties inunfilled nanotubes,14 properties of clusters,15 the tribologyof diamond,16–19 and diamond-like carbon,20–23 as well asstresses at grain boundaries.24, 25 This hydrocarbon REBOpotential analytically describes energies and forces of varioussolid structures of carbon and hydrocarbon molecules as afunction of local coordination, types of neighboring atoms,and the degree of conjugation. The many-body nature of thepotential allows the bond energy of each atom to depend onits local environment. Therefore, REBO potentials allow forchemical reactions and the accompanying changes in bondhybridization. Intermolecular interactions and torsional inter-actions were added to the second-generation REBO potential9

to create the adaptive intermolecular REBO (AIREBO) po-tential for hydrocarbons.10 The AIREBO potential has beenused to model structural properties of single-walled carbonnanotubes,26 elastic properties of graphene nanoribbons,27

sputtering of hydrocarbons,28–30 the tribology of self-assembled monolayers,31–33 and pyrolysis of hexadecane.34

To realistically model sliding-induced chemical changesof carbon-based materials in the presence of water usingmolecular dynamics (MD), the potential energy functionmust include charges that have the ability to change duringthe simulation. Rappé and Goddard35 introduced the chargeequilibration (QEq) approach to calculate the charge distri-bution around a molecule. Matrix methods or an extendedLagrangian approach36 can be used to obtain the chargeson each atomic site. In the extended Lagrangian approach,charge is a dynamic variable that evolves in a manner thatmirrors the evolution of atomic positions in MD; thus, thisapproach seems ideally suited to examine chemical reactions.Most applications of this approach involve a one-to-onecorrespondence between the update of atomic charges andatomic positions. However, this need not be the case.37

The Lagrangian approach requires the definition of acharge-neutral entity, such as a molecule or the simulationcell. Yu et al.38 recognized the problem with defining acharge-neutral simulation cell when simulating Si/SiO2 in-terfaces and modified the self-energy terms of the chargeoptimized many-body potential to circumvent it, while Ricket al.36 defined individual water molecules as charge-neutralentities. Nistor et al.39 addressed this problem by introduc-ing the concept of split charges that allow charge to flow onlybetween covalently bonded neighbors. An additional draw-back to QEq methods when simulating large molecules isthat they yield atomic charges and molecular dipole moments

0021-9606/2012/136(16)/164701/11/$30.00 © 2012 American Institute of Physics136, 164701-1

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164701-2 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

that are too large.40 The traditional solution to this problemis to constrain sub-entities of large molecules to be chargeneutral.41 However, where chemical reactions occur in simu-lations, it is not possible to adhere to a pre-determined fixedset of molecule-based, charge-neutral entities.

Recently, we developed a method to integrate chargeequilibration with bond-order potentials that does not re-quire the assignment of charge neutrality to molecules.42

The BOP/split-charge equilibration (SQE) method is a bond-centered approach, where the charge of atom i is the sum ofall charges transferred to it across each of its bonds, whichhas been referred to as SQE.39 In the BOP/SQE method, eachbond is partly decoupled from its bond network and settlesinto its own equilibrium by connecting the bond order ofeach bond to an amount of shared charge in each bond. Splitcharges cannot grow in size beyond a shared-charge limit.This method was shown to reproduce the correct qualitativebehavior of the dipole moments of long-chain alcohols as afunction of chain length.42

To study the dynamics of the interactions of oxygen-containing materials with hydrocarbon surfaces during slid-ing, we have developed a fluctuating charge, MD code thatutilizes the BOP/SQE method and a reactive potential en-ergy function capable of modeling chemical reactions in C-,O-, and H-containing systems. This new potential utilizes theformalism of the AIREBO potential for the covalent bond-ing terms and will be referred to as the qAIREBO potential.The qAIREBO contains additional covalent, REBO param-eters for O-containing materials and Lennard-Jones (LJ) pa-rameters for oxygen. Furthermore, additional terms have beenadded to describe all possible charge-charge interactions forC-, O-, and H-containing systems. The qAIREBO potentialand MD code discussed here have been developed in the samespirit as the REBO family of potentials. That is, they are de-signed to model dynamics in a broad range of C-, O-, andH-containing materials with a reasonable level of accuracy,while keeping the analytic form as simple as possible but stillable to model chemical reactions.

While there is at least one other potential that is capa-ble of modeling both electrostatics and chemical reactionsin C-, O-, and H-containing systems, simulations that utilizethe qAIREBO potential are approximately an order of magni-tude faster than simulations that utilize the potential known asReaxFF.43, 44 The limited number of potential parameters andthe way charges are implemented in the BOP/SQE methodallow simulations to be significantly faster than with othervariable-charge codes. In fact, due to the method of chargeimplementation, simulations that utilize qAIREBO to studyhydrocarbons are only slightly slower than simulations thatmake use of the AIREBO potential. In what follows, the func-tional form of the qAIREBO potential and the details of theparameterization for the new covalent REBO interactions, LJparameters for O, as well as charge parameterizations for C,O, and H are presented. In addition, the implementation ofthis potential within the context of the BOP/SQE method isdiscussed. To demonstrate the flexibility of this potential, thefluctuating-charge MD code and the qAIREBO potential wereused to calculate heats of formation C-, O-, and H-containingmolecules, radial distribution functions for water and ethanol,

as well as to calculate energies of diamond surfaces with O-containing adsorbates.

II. METHOD

The qAIREBO potential between two atoms, i and j canbe compactly written as the sum of the covalent (REBO)bonding energy between the atoms, and contributions fromtorsional rotations about the bond, dispersion interactions, andelectrostatic energy,

EqAIREBO({ri}, {qij }) = V REBO({ri}) + V tors({ri})+V LJ({ri}) + V QEq({qij }). (1)

The term V REBO utilizes the form of the second generationREBO potential developed for the covalent interactions of hy-drocarbons by Brenner9 and subsequently applied to othersystems.6, 12, 45 Rotation about single bonds is modeled bya torsional potential V tors , intermolecular interactions by aLennard-Jones potential V LJ , and V QEq is the fluctuating-charge interaction between atoms.

To determine the number of neighbors of atom i, andthus the hybridization of that atom, BOPs require the cal-culation of a bond order bij. The BOP/SQE method42 uti-lizes the bond-order approach along with the QEq model ofRappé and Goddard,35 to determine how charge is sharedacross each bond, +q on one side and −q on the other. In theoriginal QEq model, charges were atom-centered entities. TheBOP/SQE method adopted here is one of many bond-centeredapproaches39, 46–52 to QEq. In the present context, the bond-centered approach to QEq in the context of a reactive BOP,allows for a natural treatment of the processes of bond break-ing and bond forming. What follows here is focused on howthe BOP/SQE approach is implemented in qAIREBO.

Within the context of the BOP/SQE method, the chargeon atom i, Qi, is defined as the sum of all charges transferredacross each of its bonds, qij ,

Qi =∑

j

qij =∑

j

f ij qmaxij . (2)

Here, the charge delivered to atom i across bond ij is ex-pressed as a fractional amount (a number between −1 and +1with f ij = −f ji) of the maximum amount of charge for pos-sible transfer, qmax

ij . The maximum amount of charge for pos-sible transfer is directly set by the BOP, while QEq establishesthe fractional amount of transfer. It is from this formulationin terms of f ij and qmax

ij that bond forming and breaking ishandled. As long as a bond is recognized to exist, the chargedynamics will evolve yielding a fractional transfer between−1 and +1. However, weak bonds that are just beginning toform, or on the verge of being completely broken, never ex-hibit a large charge transfer because the scaling factor qmax

ij isvery small.

The final bond order used to describe a covalent bondbetween atoms i and j is an average of elementary bondorders bij, which characterizes the bond’s strength in thecontext of atom i in its local chemical environment, and bji,which characterizes the bond’s strength in the local chemicalenvironment of atom j; there are also correction terms that

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164701-3 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

are applied to generate a final bond order.9 The connectionbetween the BOP and QEq is implemented by taking themaximum allowed charge transfer across bond ij, in units ofthe fundamental charge, to be equal to the minimum of thetwo elementary bond orders,

qmaxij = min(bij , bji). (3)

Taking the minimum is a conservative choice that allowsappreciable charge transfer only in cases where the bondis assessed to be a strong one from the perspective of thechemical environments on both sides of the bond. At theappearance of a nascent bond, the fractional amount oftransfer f ij is initialized to zero and evolves through chargeequilibration as the bond strengthens and qmax

ij grows in stepwith the minimum, elementary bond order. As a bond breaks,the charge dynamics associated with the continued evolutionof f ij continues unimpeded by the weakening of the bond;however, the actual charge transferred across the bond issmoothly turned off as qmax

ij approaches zero. Once a bond isfully broken, meaning qAIREBO no longer assigns a bondorder to the pair, the ij pair is removed from the charge-bondnetwork to which charge equilibration is applied. If an ij pairfrom such a broken bond were to reform a bond, the pairwould begin anew with an fij of zero.

At each time step, the total atomic charge Qi on everyatom is calculated before the next iteration of the QEq dynam-ics. The electrostatic potential energy of the system is calcu-lated as

V QEq =∑

k

(χkQk + 1

2J 0

kkQ2k

)+

∑i �=j

J shiftedij (rij )QiQj ,

(4)where

J shiftedij (rij ) =

⎧⎪⎨⎪⎩

Jij (rij )

[1 −

(rij

Rcut

)2]2

, rij < Rcut

0, otherwise.

Here, Jij(rij) is the unit-charge Coulomb interaction evaluatedthrough the use of Slater orbitals,35, 36, 47 while the term insquare brackets introduces a spherical cutoff that turns off theelectrostatic interaction at Rcut. This form for the sphericalcutoff gives no discontinuities in slope at the cutoff for boththe electrostatic potential and force and is one of the formsavailable in the CHARMM molecular dynamics program.53

The cutoff is chosen to be Rcut = 9 Å. The LJ cutoff distanceis also set to 9 Å, so that only a single pair list is required.As a result, qAIREBO run times, though larger, are nonethe-less comparable to AIREBO run times, a critical feature in thecontext of qAIREBO’s intended applications.

The electrostatic potential can be described, as in Eq. (4),as a function of atomic charges, rather than charges trans-ferred across bonds. The bond-centered approach to QEq isused as a convenient method for smoothly evolving the atomiccharges in the context of a dynamic system, where bonds canbreak and form.

With qmax set by the BOP, all that remains is to establishhow the fractional charge transfer f ij is to be evolved at eachsimulation step. An extended Lagrangian approach is adoptedfor the evolution of f ij that parallels the MD evolution of

TABLE I. Parameters for charge equilibration.

Atom type J0AA(r = 0) χ i

C 8.6530 5.3061H 17.3350 5.0780O 14.1631 8.3767

particle trajectories.36 A maximal-damping approach is takenwhereby the fractional charge transferred across bond ij, f ij

is updated solely by means of a charge acceleration calculatedas

d2f ij

dt2= −k

⎛⎝

⎡⎣χi + J 0

iiQi +∑k �=i

JikQk

⎤⎦

−⎡⎣χj + J 0

jjQj +∑k �=j

JjkQk

⎤⎦ + ε

|f ij |(1 − |f ij |)

⎞⎠ .

(5)

The quantities in square brackets can be considered as evolv-ing electronegativities for each of the atoms of the ij pair withthe parameters χ i and χ j being the neutral-atom electronega-tivities associated with each atom type. The term added to theevolving electronegativity is a penalty term that inhibits largefractional charge transfers and strictly constrains |f ij | < 1.This constraint prevents any charge transfer from exceedingthe maximum set by qmax and acts a restoring force that op-poses charge transfer in either direction across a bond. Theessential properties of this penalty term are that it goes tozero as f ij goes to zero, and it diverges to infinity as |f ij |approaches 1. While other functions obey these limits,42 theform utilized here for the penalty term is a simple one thatdoes not affect QEq appreciably when charges are small yetalso becomes significant long before |f ij | gets close to one.The practical result of the penalty term is that the fractionalcharges never get close to one even in the case of liquid wa-ter where polarity differences between atoms are apprecia-ble. A value of εij = 0.982 in conjunction with the standardQEq atomic parameters (electronegativity and hardness, seeTable I) results in a dipole moment of 1.976 D for water,54

using rOH = 0.9687 Å and rHH = 1.520 Å.The proportionality constant k in the above expression

for the charge acceleration provides some flexibility overhow rapidly the QEq dynamics are evolved. When k is large,one QEq step can be taken for each MD step. Alternatively,a smaller scale factor could be adopted along with multipleQEq steps for each MD step.37 The above approach suggestsa QEq process that unfolds in time; however, there is notime scale here that is physically motivated aside from thegeneral idea that QEq should take place much faster thanthe evolution of atomic motions. Many QEq steps for eachMD step (enough steps for charges to stabilize suitably)would be a default approach if simulation run time were not aconcern. The value of 0.008236 eV−1 fs−2 for k used herein,coupled with the maximal damping approach, is suitable for

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164701-4 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

a one-to-one correspondence between MD position updatesand charge equilibration steps.

The parameters used in the charge equilibration schemeare presented in Table I. Each atom’s electronegativity, χ i,and hardness, J 0

ii , are taken from work by Müser et al.,39

which optimized the original atomic parameters of Rappéand Goddard to fit a variety of molecules containing only C,H, O, and Si. The electronegativity and hardness are used tocalculate the self-energy term for atom i,

Vself = χiQi + 1

2J 0

iiQ2i . (6)

These contributions summed over all atoms are seen abovein the expression for the system electrostatic potential energy(Eq. (4)).

The inclusion of charges in the qAIREBO potential re-quires some modification of LJ interactions involving chargedhydrogen atoms. Most three-site atomic models of water usedfor MD simulations involve an LJ interaction between theoxygen atoms only.55–58 However, there are many systems,and in particular the variety of hydrocarbon systems accu-rately described by the original AIREBO potential, which dorequire the description of LJ interactions involving hydrogen.

A practical way to bridge these extremes is to smoothlyturn off the LJ interaction as a hydrogen atom becomes morepositively charged. In models of liquid water where chargesfluctuate, the varying charge of the hydrogen atoms withinwater molecules stays typically above +0.3e. This is taken asan upper limit for hydrogen atoms beyond which the LJ in-teraction is completely turned off. As a lower limit, hydrogenatoms with no charge or negative charge have LJ interactionsfulled turned on.

A polynomial switching function which turns off the in-teraction according to the charge of the positively charged hy-drogen atom relative to the upper limit x = qH/(0.3e) is used,

f (x) = 1 − 3x2 + 2x3. (7)

In the case of hydrogen-hydrogen interaction, the larger of thetwo hydrogen atom charges is used to assess the LJ interactionbetween the pair.

The remaining terms in the qAIREBO potential, V REBO

and V tors (Eq. (1)) are short-ranged terms and can be looselythought of as the covalent-bonding portion of the potential.The V REBO term has the form originally used by Brenneret al.9, 59 to model hydrocarbons and subsequently applied toother covalent systems.6, 10, 12 The qAIREBO model utilizesa REBO-type potential for all covalent bonding interactions.Because the REBO potential developed by Brenner9, 59 wasparameterized for C–C, C–H, and H–H interactions only, ad-ditional parameters are required for C–O, O–H, and O–O in-teractions. It should be noted that a REBO potential for C–O,O–H, and O–O interactions, known as OREBO, was devel-oped by Sinnott and coworkers.12 Some parameters from thatwork are adopted herein without change; however, shortcom-ings identified with the OREBO potential when dealing withoxygen-containing molecules necessitated some changes tothe OREBO formalism.

Detailed discussions of the second-generation REBO po-tential, V REBO , have been given in Refs. 59, 9, and 10. Forcompleteness, a brief description, along with tabulated valuesfor all parameters, is given here. REBO potentials have thegeneral form

V REBO =∑

i

∑j (>i)

[V R(rij ) − bijV

A(rij )], (8)

where V R and V A are repulsive and attractive pair potentials,respectively. The bond-order term, bij, is a many-body termthat modulates the attractive part of the potential based onatom coordination, bond angle, and conjugation effects. Re-pulsive and attractive interactions between atoms i and j aredescribed by

V Rij = wij (rij )

[1 + Qij

rij

]Aij e

−αij rij , (9)

V Aij = wij (rij )

3∑n=1

B(n)ij e−β

(n)ij rij . (10)

The wij function switches the REBO potential off when thedistance between atoms grows too large for covalent interac-tions to be present. Values for all parameters in V R and V A

are given in Table II.The bond-order term, bij, in Eq. (8) determines the

strength of a bond in its local bonding environment, withlarger bij values indicating stronger bonds. It is roughly equiv-alent to the usual chemical concept of bond order, and issimply a means of modifying the strength of a bond due tochanges in local environment. It is given by

bij = 1

2

[bσπ

ij + bσπji

] + πrcij + πdh

ij , (11)

and the terms bσπij and bσπ

ji represent the covalent-bonding in-teractions between atoms and are given by

bσπij =

⎡⎣1 +

∑k(�=i,j )

wik(rik)gi(cos θjik)eλjik

+Pij (NHi ,NC

i , NOi )

⎤⎦

− 12

. (12)

The πrcij term is needed to describe radical structures, such

as the vacancy formation energy in diamond, and to accountfor non-local conjugation affects, such as those found in ben-zene. It depends on whether a bond between atoms i and jas radical character and is part of a conjugated system.9 Thefunction is three-dimensional spline that depends on the to-tal number of neighbors bonded to atoms i and j, as well asthe local conjugation. The complete functional form, as wellas the values of the spine at the knot points, are given inRefs. 9 and 10. The πdh

ij depends on the dihedral angle forcarbon-carbon double bonds and is also described in Refs. 9and 10. The function wik(rik) ensures that interactions includenearest neighbors only.

The function gi modulates the contribution that eachnearest neighbor makes to the empirical bond order,

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164701-5 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

TABLE II. Parameters for the qAIREBO potential. Covalent-bonding parameters between carbon and hydro-gen are from the original REBO potential8, 9 with minor modifications. Covalent-bonding parameters involvingoxygen are modeled using the covalent-bonding terms in the OREBO potential12 with slight modifications.

Parameter CC CH HH CO OH OO

Qij (Å) 0.313460 0.340776 0.370 9.132 0.1235 0.4065αij (Å−1) 4.746539 4.1025498 3.536 3.554 1.628 1.173Aij (eV) 10953.544 149.94098 31.6731 81.0576 717.1495 685.255

B(1)ij (eV) 12388.792 32.355187 28.2297 268.043 884.504 1105.0

B(2)ij (eV) 17.567065 0.0 0.0 0.0 0.0 0.0

B(3)ij (eV) 30.714932 0.0 0.0 0.0 0.0 0.0

β(1)ij (Å−1) 4.7204523 1.434458 1.708 2.344 1.704 1.325

β(2)ij (Å−1) 1.4332133 1.0 1.0 1.0 1.0 1.0

β(3)ij (Å−1) 1.3826913 1.0 1.0 1.0 1.0 1.0

ρij (Å) 0.0 1.09 0.7415887 1 0.96 1εij (eV) 0.00284

√εCC + εHH 0.00150

√εCC + εOO

√εOO + εHH 0.00674

σ ij (Å) 3.40 12 (σCC + σHH) 2.65 1

2 (σCC + σOO) 2.30 3.176εiCCj (eV) 0.307855 0.1789 0.125 . . . . . . . . .

depending on the bond angle θ jik between vectors rji andrki . The gC function for carbon is a piecewise functioncomposed of three, sixth-order polynomial splines in cos(θ ).Values for spline coefficients are given in Refs. 9 and 10. Theparameter λjik is a correction term designed to improve thepotential energy surface for hydrogen atom abstraction fromhydrocarbons.9, 59 To obtain accurate atomization energiesfor small molecules, a correction term Pij is applied to thebond-order function. In this work, this correction is a functionof the number of hydrogen NH

ij , carbon NCij , and oxygen

NOij neighbors of atom i. In contrast, a two-dimensional

cubic-spline Pij (NHi , (NC

i + NOi )) is utilized in the OREBO

potential.12 The adoption of the three-dimensional splinefunction provides additional flexibility when fitting energiesof a broad range of C-, H-, and O-containing molecules.

The angular dependence of different atomic hybridiza-tion states is controlled partially by gi(cos θ ijk) (Eq. (12)),which was originally parameterized as a quintic spline to ac-count for energy differences of bonds at different angles be-tween carbon and hydrogen atoms only.9, 10, 59 The inclusionof oxygen requires a new angular function, GO, because thehybridization of oxygen differs from that of carbon. Whencarbon or oxygen is the central atom in a bond, the angular-dependent energy function, gi, is given by

gi =⎧⎨⎩

GC = g(1)C + S ′(tN (Nij ))

[g

(2)C − g

(1)C

]Carbon central atom,

GO = a0 + a1[a2 − cos(θ ∗ π/180◦)]2 Oxygen central atom.(13)

The OREBO potential originally fits the angular depen-dence for oxygen-containing bonds GO using the bond energyand angle in O3, which has a minimum energy value at 118.2◦.This angular function is adequate for molecules that have an-gles centered around oxygen if either of the nearest neighborsis carbon or oxygen but results in inaccurate bond angles ifthe neighbors are hydrogen due to the unique bonding natureof water. To achieve energetically favorable bond angles forall bonding cases when oxygen is a central atom, separate pa-rameters are needed for GO depending on its nearest neigh-bors. If the central atom is oxygen, then the nearest neighborsare identified and hydrogen is treated differently than carbonor oxygen. As long as there is one O–H bond present in anangle, then the angular dependence will follow that of water,leading to a smaller bond angle than if oxygen is bonded tocarbon or oxygen. A plot of the two parameterizations of the

angular function when the central atom is oxygen (GO) is pro-vided in Figure 1, with the parameters for each function givenin Table III.

The tricubic-spline coefficients, and the species used tofit each coefficient for the Pij corrections in Eq. (12) are givenin Table IV. These coefficients are added to the bij term to im-prove the energy associated with individual bond types. Thecoefficients were chosen by least-squares minimization of er-rors for individual Pij values between experimental and simu-lated enthalpies of formation at 298 K.

The original AIREBO potential10 was a combinationof the covalent interactions from the second-generationREBO potential, torsional interactions from dihedral angles,and long-range interactions described by a Lennard-Jonespotential. The inclusion of the torsional potential is neededto accurately describe the potential energy of molecules

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164701-6 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 20 40 60 80 100 120 140 160 180

GO

(θ)

θ(degrees)

No Neighboring HydrogenNeighboring Hydrogen

FIG. 1. Angular contribution to the bond order in Eq. (12) for angles aroundoxygen. Two distinct functions are required to capture the different behaviorof hydrogen with oxygen or carbon neighbors.

that rotate about single bonds, such as ethane. Because theAIREBO potential was parameterized for hydrocarbons only,a single torsional potential was used for every torsionalinteraction, where the value of ω represents the dihedralangle about a carbon-carbon bond,

V torsij (ω) = ε

[256

405cos10(ω/2) − 1

10

]. (14)

The functional form in Eq. (14) was selected because it isable to model the rotational behavior around a carbon-carbonsingle bonds while not being explicitly parameterized for in-dividual hydrocarbon molecules. This is important for a reac-tive potential in which molecular configurations can changeduring the course of a simulation. This function exhibits tripleminima and the values of ε are unchanged from the AIREBOpotential.10 The value of εHCCH was determined by fitting theexperimental barrier of 2.9 kcal/mol for rotation about theC–C single bond and ethane. The values of εHCCC and εCCCC

were fit to the torsional barriers in propane (3.4 kcal/mol) andthe 0.90 kcal/mol energy difference in the anti and gaucheconformations of butane, respectively.

Torsional interaction centered around a carbon-oxygenbond, such as the torsional angle composed of hydrogen-oxygen-carbon-carbon in ethanol, also has three local min-ima; thus, the same torsional potential as ethane is used.In contrast, if the torsional angle is centered on an oxygen-oxygen bond, as in the case of peroxides, a separate torsionalpotential is needed to describe the rotation. A function thatcontains three local minima cannot adequately describe thebarriers to rotation due to the hybridization of oxygen. Forexample, the torsional potentials for hydrogen peroxide andmethyl peroxide both contain two minima.60 For peroxides,

TABLE III. Parameters for angular function GO in Eq. (13).

Angle type a0 a1 a2

Carbon-oxygen-oxygen − 0.014 0.07 − 0.478Carbon-carbon-oxygenOxygen-oxygen-hydrogen − 0.025 0.07 − 0.16

TABLE IV. Interpolation points for the tricubic spline Pij

(NH

i , NCi , NO

i

)in Eq. (12). All values not listed, and all derivatives, are zero at integral valuesof NH

i , NCi , and NO

i .

NHij NC

ij NOij Pij Fitting species

PCC

2 0 0 − 0.00932849 Ethene3 0 0 − 0.0268434 Ethane1 2 0 − 0.124 Isobutane1 1 0 0.001119 cis-2-butene2 1 0 − 0.130 c-C6H12

2 0 1 0.14532 Ethanol0 0 1 − 0.142415 Ketene (ethenone)1 0 1 0.42380015 Ethenol

PCH

1 0 0 0.209336 Methylene (CH2)2 0 0 − 0.06449615 CH3

3 0 0 − 0.345725 Methane0 1 0 − 0.050 Acetylene1 1 0 − 0.211803 Ethene2 1 0 − 0.341821 Ethane0 3 0 − 0.1019 Isobutane1 2 0 − 0.121803 c-C6H12

0 2 0 − 0.141927 cis-2-Butene0 3 1 0.054102 1,1-dimethyl ethanol

(2-methyl-2-propanol)1 1 1 0.2832 Ethanol2 0 1 0.286315 Methyl peroxide

PCO

0 0 1 − 0.213764 Carbon dioxide0 2 0 − 0.2 2-butanone1 1 0 − 0.241803 Acetaldehyde2 0 0 − 0.257291 Formaldehyde0 1 1 0.25803 Acetic acid1 2 0 − 0.276279 Cyclobutanol2 1 0 0.0 2 methyl, 1 propanol

POH

1 0 0 − 0.0184116 Water0 1 0 − 0.0322632 Methanol1 1 0 0.3 –OH diamond(111)

(1 × 1)POO

0 0 0 − 0.0471844 Oxygen diatomic1 0 0 0.0000862469 Hydrogen peroxide0 0 1 − 0.00905 Ozone0 1 0 0.00950 Methyl peroxide

POC

0 1 0 0.192575 Dimethyl ether1 0 0 0.599668 1,1-dimethyl ethanol

(2-methyl-2-propanol)

the torsional potential is modeled by the following equation:

V torsij (ω) = Acos6

(ω

2

)− Bcos6

(ω

2+ φ

2

)

− Bcos6

(ω

2− φ

2

)+ 2Bsin6

(φ

2

). (15)

A plot of the potentials for torsional angles centeredabout carbon-carbon and oxygen-oxygen bonds is shown inFig. 2, which highlights the need for different equations.

The parameters A, B, and φ were optimized using aseries of peroxide molecules: hydrogen peroxide, methyl

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164701-7 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 90 180 270 360

V (

eV)

θ (degrees)

Ethane TorsionalHOOH Torsional

FIG. 2. Comparison of the torsional potentials for hydrogen peroxide andethane. The hydrocarbon torsional potential has a triple well with equivalentmaxima and minima. The peroxide potential has a double well with a singlemaximum.

hydroperoxide, and dimethyl peroxide. The parameters werefit using high-level quantum mechanical calculations.60 Thevalues for each parameter and torsional interaction type aregiven in Table V.

It should be noted that while the torsional potentialpresented in Table V is a double-well function, with min-ima of approximately 112◦ for HOOH and 118◦ for COOHand COOC, high-level quantum mechanical calculations pre-dict the existence of three minima for dimethyl peroxide(a COOC dihedral angle) with the absolute minimum occur-ring at 180◦.60, 61 In the case of dimethyl peroxide, the tor-sional potential has zero slope in the region from 100◦ to 180◦

when calculated using differing quantum mechanical meth-ods, with only the highest level calculations showing the tripleminima. This behavior is not well characterized by the cur-rent torsional potential; therefore, this potential does not re-produce the correct minimum geometry when comparing toquantum mechanical calculations. Experimental findings in-dicate an equilibrium structure not at 180◦ (Ref. 62) but ratherat 120◦ ± 10◦. Despite the differences between experimentaland theoretical findings, the torsional pontential for COOCdihedral angles will maintain the double-well shape as thoseof HOOH and HOOC/COOH peroxides.

Intermolecular interactions in the qAIREBO potential areimplemented via a unique adaptive algorithm and a LJ poten-tial in exactly the same manner as the AIREBO potential.10

The Lennard-Jones 12-6 potential depends on atomic param-eters σ and ε (Table II) and takes the following form:

V LJij = 4εij

[(σij

rij

)12

−(

σij

rij

)6]

. (16)

TABLE V. Parameters for peroxide torsional potential in Eq. (15).

Torsion type A B φ

HOOH 0.24471 0.09396 1.7085COOH 0.28702 0.05438 1.0107COOC 5.6211 2.6062 0.11547

TABLE VI. Lennard-Jones potential switching function for any pair ofatoms containing oxygen in the qAIREBO potential.

Switch min max

bOO 0.80 1.11bCO 0.90 1.20bOH 0.90 1.20

At certain distances, the strength of the adaptiveLennard-Jones potential depends in part on the bond orderthat the atoms would have if they were in bonding range.10

The parameters that determine this switching functionfor any pair of atoms containing oxygen are provided inTable VI. The switching function parameters for any com-bination of carbon and hydrogen atom pairs are unchangedfrom the AIREBO potential.

III. RESULTS AND DISCUSSION

A. Isolated molecule properties

Recently, Mikulski et al. demonstrated that any bond-order potential could be extended to include charge trans-fer between atoms through a modification of the SQEformalism.42 Without significant optimization of any chargeparameters, it was shown that the BOP/SQE method did a rea-sonable job predicting dipole moments of linear chain alco-hols and did not exhibit the unrealistic growth of charges thatis often associated with QEq methods.46, 63 To highlight onebenefit of treating charges with the bond-centered (BOP/SQE)approach, dipole moments were calculated for a set of 20, lin-ear alcohol molecules using density functional theory (DFT)and using the qAIREBO potential and the BOP/SQE method.The geometry of each alcohol molecule was constrained tomaintain linearity in the longer alcohol chains. These dipolemoments, as well as the dipole moments from our previouswork,42 are shown in Figure 3.

From analysis of these data, it is clear that the dipolemoments calculated using the qAIREBO and the BOP/SQE

1.4

1.6

1.8

2.0

2.2

2.4

2 6 10 14 18

Dip

ole

Mom

ent (

Deb

ye)

# Carbon in Linear Alcohol

Gaussian / B3LYP

Previous BOP/SQE

New BOP/SQE

FIG. 3. The dipole moments of 20 different linear alcohol molecules arecompared using DFT (B3LYP 6-31G*), the BOP/SQE method with AIREBOand one charge parameter,42 and the fully parameterized qAIREBO potentialwith the BOP/SQE method.

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164701-8 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

TABLE VII. Comparison of molecular heats of formation between the qAIREBO potential and experiment at298 K. RMSD is the root-mean-square deviation between the two sets of heat of formation values. ZPE is thezero point energy for each molecule.

ZPE HqAIREBO

f Hexpt

f Diff

Molecule (kcal mol−1) (kcal mol−1)a (kcal mol−1)b (kcal mol−1)

CH2 10.407 92.39 91.095 − 1.375CH3 18.715 34.821 33.453 − 1.368Methane 28.379 − 16.827 − 17.89 1.062Ethane 47.209 − 20.399 − 20.08 − 0.319n-propane 82.492 − 9.130 − 25.02 15.29Butane 83.341 − 33.679 − 30.06 − 3.619Pentane 101.368 − 38.976 − 35.09 − 3.886Isobutane 83.027 − 42.604 − 32.07 − 9.995Methanol 32.293 − 47.827 − 49.0 1.173Ethanol 50.399 − 57.194 − 56.12 − 1.0741-butanol 86.495 − 70.742 − 66.0 − 4.7421-pentanol 101.353 − 75.746 − 70.66 − 5.0862-methyl-1-propanol 86.229 − 77.926 − 67.83 − 10.0961,1 dimethyl ethanol (2-methyl-2-propanol) 85.446 − 107.303 − 74.72 − 32.584Cyclobutanol 72.663 − 50.929 − 34.60 − 4.800Cyclopentanol 91.395 − 90.301 − 58.08 − 32.261Tetrahydrofuran 73.669 − 6.8011 − 44.02 37.220Cyclohexane 107.360 − 42.493 − 29.78 − 12.713Isopropyl alcohol 68.010 − 96.715 − 65.18 − 31.5351,3-propanediol 72.114 − 100.402 − 97.60 − 2.803Acetylene 16.714 53.361 54.19 − 0.829Ethene (ethylene) 32.144 5.01 12.52 − 7.513cis--2-butene 68.188 − 7.411 − 1.83 9.241Pentene 86.401 − 7.615 − 5.0 − 2.615Etheneol 35.600 − 29.357 − 30.59 1.233Formaldehyde 16.840 − 28.482 − 27.701 − 0.781Formic acid 21.294 − 114.515 − 90.54 − 23.97Acetic acid 38.946 − 109.177 − 103.32 − 5.800Dimethyl ether 50.391 − 36.910 − 44.0 7.090Ethenone (ketene) 19.922 − 18.301 − 11.4 − 6.907Methyl ketene (1-propen-1-one) 38.466 − 21.472 − 20.55 − 0.9222-methyoxypropane 86.020 − 100.50 − 60.24 − 0.8302-butanone 70.922 − 32.147 − 57.03 24.88CO2 7.273 − 94.200 − 94.05 − 0.150Water 13.280 − 56.430 − 57.79 1.360Hydrogen peroxide 16.480 − 30.493 − 32.531 2.047Ozone 4.634 32.491 30.099 − 1.608Methyl peroxide 34.443 − 41.62 − 31.31 − 2.0071-methylethyl hydroperoxide 70.161 − 68.918 − 47.11 − 21.81RMSD . . . . . . . . . 13.95

aCalculated using Hcarbonf = −170.589 (kcal mol−1), H

H2f = −101.948 (kcal mol−1), and H

O2f = 119.727 (kcal mol−1).

bFrom Ref. 64.

method are very close to those calculated using first-principles. There is a slight oscillation in the dipole momentscalculated using DFT, while the same behavior is not as evi-dent using the qAIREBO potential. In addition, the completeparameterization of the qAIREBO potential has significantlyimproved the dipole moment values compared to our earlierpublication.42

The enthalpies of formation at 298 K for a variety ofC-, O- and H-containing compounds calculated with theqAIREBO potential are shown in Table VII. For compari-son, the experimental heats of formation are also shown. Withthe exception of cyclic compounds, the heats of formation

agree with the experimental values to within ∼10 kcal mol−1.The root-mean-square deviation between the experimentalheat of formation and the value calculated with qAIREBO is13.95 kcal mol−1. This level of accuracy is acceptable for apotential that is designed to have a minimal number of pa-rameters, so that it is computationally efficient, while still be-ing applicable to a wide-range of C-, O- and H-containingmaterials. It should also be noted that heats of formation ob-tained with the qAIREBO potential will differ somewhat fromthose calculated with the second-generation REBO (Ref. 9) orthe AIREBO potential due to the reparameterization of the Pij

terms.

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164701-9 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

2 3 4 5 6 7 8 9 10

g OO

(r)

r (Å)

X−rayqAIREBO

FIG. 4. The oxygen-oxygen radial distribution for water calculated usingthe qAIREBO potential and compared to the experimental radial distributionfrom Soper.54

B. Liquid properties

While the qAIREBO potential was not explicitly de-veloped to model the structure of liquids, an examinationof the structure of several liquids provides an interestingtest of the transferability of the potential. Because water isubiquitous, it is important that the qAIREBO potential do areasonable job reproducing the properties of liquid water. Theoxygen-oxygen, oxygen-hydrogen, and hydrogen-hydrogenradial distribution functions of water at 298 K are shown inFigures 4–6. For comparison, the experimental radial distri-bution functions are also shown.54 It is clear from analysisof these data that the qAIREBO potential does a reasonablejob reproducing the positions of the first and second peaksin the radial distribution functions. It should also be noted,however, that the qAIREBO potential produces a somewhatmore structured water sample as evidenced by the structurein the radial distribution functions at large distances, whichis absent in the experimental data. The O–H bond length(0.981 Å), bond angle (100.26◦), and dipole moment(1.88 D) of an isolated water molecule calculated using this

0

0.5

1

1.5

2

2.5

2 3 4 5 6 7 8 9 10

g OH

(r)

r (Å)

X−rayqAIREBO

FIG. 5. The oxygen-hydrogen radial distribution for water calculated usingthe qAIREBO potential and compared to the experimental radial distributionfrom Soper.54

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2 3 4 5 6 7 8 9 10

g HH

(r)

r (Å)

X−rayqAIREBO

FIG. 6. The hydrogen-hydrogen radial distribution for water calculated usingthe qAIREBO potential and compared to the experimental radial distributionfrom Soper.54

new potential also compare well with the experimentallydetermined values of 0.9575 Å, 101.51◦, and 1.855 D.54 Incomparison, the OREBO potential12 is unable to reproducethe bond angle or dipole moment of water because its oxygenangular function was optimized using ozone and terms tomodel charge are not included.

A comparison of the oxygen-oxygen radial distributionfunction for ethanol calculated using the qAIREBO and theoptimized potentials for liquid simulations (OPLS) (Ref. 65)potentials is shown in Figure 7. The qAIREBO potential doesa reasonable job of reproducing both peaks in the radial dis-tribution function obtained using the OPLS potential althoughthe first peak is shifted to somewhat larger separations.

C. Oxygenated diamond surfaces

Under ambient conditions, the presence of hydrogen,oxygen, and water in the atmosphere means that a varietyof surface terminations can occur on diamond. The predictedadsorption energies of these chemisorbed species on dia-mond relative to the surface with no adsorbates are shown in

0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

5

2 2.5 3 3.5 4 4.5 5 5.5 6

g OO

(r)

r (Å)

ExperimentqAIREBO

FIG. 7. The oxygen-oxygen radial distribution for ethanol calculated usingthe qAIREBO potential and compared to the simulated OPLS results fromSaiz et al.65

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164701-10 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

TABLE VIII. Comparison of values calculated with qAIREBO to DFT results for C(111) and C(100) surfaces.Energies and bond lengths are in eV and Å, respectively. DFT values are in parenthesis. Energies for H-, O-, andOH- terminated surfaces are relative to the clean surface with no adsorbates. Eclean is the energy per carbon atom.

Property C(111) 1 × 1 C(111) 2 × 1 C(100) 2 × 1

Eclean −7.30 −7.35 −7.24EH (100%) −4.2705 (−4.89)a −2.818 (−4.44)a −4.326 (−4.30)b

dH-C 1.088 (1.11)b 1.125 (1.10)b 1.082EOH (100%) −5.886 (−4.34)a −4.3774 (−3.99)a −5.119 (−4.14)b

dH-O 0.922 (0.995)a 0.91 (0.997)a 0.901dO-C 1.339 (1.324,c 1.326b) 1.300 (1.407,a 1.45b) 1.347EO (100% bridge) . . . −3.150 (−5.23)a . . .EO (100% on top) −3.049 (−4.24)a −2.649 (−4.85)a −5.618 (−5.97)b

dO=C 1.272 (1.326,a 1.32b) 1.276 (1.195,a 1.45b) 1.273 (1.19)b

EO (50% on top) −3.606 (−5.3)c −3.6905 (−5.74)b . . .EO (33% on top) . . . −4.858 . . .EO (25% on top) −3.741 (−5.57)c −5.342 . . .

aDFT calculations from Loh et al. (Ref. 66).bDFT calculations from Petrini and Larsson (Ref. 67).cDFT calculations from Derry et al. (Ref. 68).

Table VIII for the C(111) and C(100) diamond faces. For theC(111) crystallographic phase, the (1 × 1) configuration and(2 × 1) Pandey chain configuration69 are both examined. The(2 × 1) configuration of the C(100) crystallographic face isstudied.70 Each surface is minimized using the qAIREBO po-tential and the difference in energy between two surfaces canbe described as an adsorption energy,

Eads = Esurf ace − Eclean − nEads

nsurf

, (17)

where Eclean represents the energy of the pristine diamond sur-face in the absence of any chemisorbed species, Esurface is theenergy of the diamond surface with nsurf adsorbates (H, O, orOH) chemisorbed to the surface, each of which has n isolatedadsorbates species with an energy Eads.

Because qAIREBO retains the terms related to cova-lent bonding in diamond and the angular function of thesecond-generation REBO potential for carbon, it inheritssome of the strengths and weaknesses of the underlying po-tential REBO formalism. For example, quantum mechanicalcalculations,67, 69 the REBO potential,9 and the qAIREBO po-tential all predict that in the absence of any adsorbates, theC(111) (2 × 1) Pandey chain,71 which exhibits π -bonding be-tween two carbon atoms on the surface, is more energeticallyfavorable than the unreconstructed (1 × 1) surface. In the caseof the qAIREBO, it is more favorable by 0.61 eV per surfacecarbon atom. DFT calculations of Petrini and Larsson67 andVanderbilt and Louie69 predict that the (2 × 1) reconstructedsurface is more favorable by 0.5 eV and 0.68 eV per surfaceatom, respectively.

For a full monolayer coverage of hydrogen on each ofthe diamond surfaces, the qAIREBO potential predicts theC(100) (2 × 1) surface to be slightly more energetically fa-vorable than both the C(111) (2 × 1) and C(111) (1 × 1) sur-faces. In contrast, DFT calculations predict that the C(111)(1 × 1) surface is the most energetically favorable. With theexception of the H-terminated C(100) (2 × 1) surface, ad-sorption energies calculated with the qAIREBO potential for

hydrogen-terminated diamond surfaces agree reasonably wellwith quantum calculations66, 67 despite the incorrect energeticordering. The REBO potential (Ref. 9) is also unable to pre-dict the correct energetic ordering of the hydrogen-terminatedsurfaces; however, it does correctly predict that the C(111)(1 × 1) surface is the most energetically favorable.

For the chemisorption of oxygen-containing molecules,a full monolayer of hydroxyl groups was attached to each ofthe surface on-top sites and the surface adsorption energieswere calculated. The qAIREBO potential correctly predictsthat the attachment of a full monolayer of hydroxyl is mostfavorable on the C(111) (1 × 1) surface and it predicts thecorrect energetic ordering of the three surfaces with hydroxyladsorbates in the on-top sites.66, 67 The potential also predictsthe correct energetic ordering when oxygen is absorbed onthe on-top site and that this is most favorable on the C(100)(2 × 1) surface.

Surfaces with submonolayer oxygen coverages were con-structed by randomly placing oxygen adsorbates in a surfaceon-top sites. For both C(111) surfaces, the potential predictsthat oxygen termination becomes more energetically favor-able on a given diamond surface as the number of oxygenadsorbates decreases. Reducing the number of oxygen adsor-bates reduces the repulsion between oxygen atoms and thusthe energy in agreement with DFT calculations.68

On the C(100) (2 × 1) surface, adsorption of oxygen toa bridge site and to an on-top site were both considered. Inagreement with DFT calculations, the potential predicts thatthe adsorption to the bridge sites is more energetically favor-able compared to the on-top sites.66 However, as was the casewith hydrogen termination, the energy values do not agreewell with DFT calculations.

IV. SUMMARY

In this work, a new bond-order potential for oxygen,carbon, and hydrogen has been integrated into a molecu-lar dynamics code that makes use of the BOP/SQE method.

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164701-11 Knippenberg et al. J. Chem. Phys. 136, 164701 (2012)

This method allows charge to be integrated naturally into thedynamics via the use of the bond order. This is an advanta-geous method for including charge because it prevents the un-realistic growth of charges, it is computationally fast, and theuse of the bond as the charge-neutral entity allows chemicalreactions to be modeled. This new qAIREBO potential wasshown to do a reasonable job of predicting the heats of for-mation of isolated molecules, radial distribution functions forwater and ethanol, and the energies of oxygenated diamondsurfaces. Due to the computational efficiency of the BOP/SQEmethod, simulations that utilize the qAIREBO are onlyslightly slower than simulations that utilize the AIREBO po-tential and an order of magnitude faster than other potentialscapable of modeling chemical reactions in similar systems.

ACKNOWLEDGMENTS

J.A.H., P.T.M., and K.E.R. acknowledge support fromthe National Science Foundation IAA 1129629. M.T.K. alsoacknowledges partial support from the Office of Naval Re-search (ONR) Grant No. N0001411WX21417. G.A.O. andJ.A.H. acknowledge partial support from AFOSR Grant No.F1ATA00130G001. The authors would also like to thank J.David Schall for helpful discussions.

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