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Chemical Physics Letters 413 (2005) 151–156

Theoretical rate constants for the reaction BF2 + NF = BF3 + Nof importance in boron nitride chemistry

Simone S. Ramalho a, Alessandra F.A. Vilela a,Patricia R.P. Barreto a,b,*, Ricardo Gargano a

a Instituto de Fısica, Universidade Brasılia, CP04455, Brasılia, DF, CEP 70919-970, Brazilb Laboratorio Associado de Plasma – LAP, Instituto Nacional de Pesquisas Espaciais – INPE/MCT, CP515,

Sao Jose dos Campos, SP, CEP 12247-970, Brazil

Received 28 June 2005; in final form 11 July 2005Available online 11 August 2005

Abstract

A kinetic mechanism was developed for the growth of boron nitride films, including up to 35 species and 1012 gas-phase reac-tions. For 117 elementary reactions, rate constants are available from published experimental/theoretical data and a code was writ-ten to estimate the others using transition state theory. As an example of our approach, we report here results for one of thesereactions, BF2 + NF = BF3 + N of presumable importance for the mechanism, using geometries and electronic energies (includingtheir first and second derivatives), calculated ab initio along the minimum energy path. The reaction proceeds in two steps and rateconstants were calculated for the temperature range 200–4000 K.� 2005 Elsevier B.V. All rights reserved.

1. Introduction

In the past few years there has been an enormousinterest in the growth of boron nitride thin films, be-cause of their great applicability in various technologyfields [1–6]. Boron nitride has different allotropes similarto carbon, the most common of them are the hexagonal(hBN), similar to graphite, and cubic (cBN) phases, sim-ilar to diamond. Boron nitride films can be grown or bychemical vapor deposition (CVD) or by physical vapordeposition (PVD), but normally these films have lowcBN/hBN ratio and high internal stress that make thesefilms useless. Previous knowledge of the chemical pro-cesses which are involved in such growth and controlthe synthesis of either hexagonal or cubic boron nitridephases from the vapor phase is highly recommended to

0009-2614/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2005.07.041

* Corresponding author. Fax: +55 12 3945 6710.E-mail address: patricia@plasma.inpe.br (P.R.P. Barreto).

optimize the boron nitride growth conditions and im-prove the quality of these films.

A mechanism related to surface and kinetic of reac-tions involved in the chemical route of boron nitridegrowth was proposed and developed by one of us[7–9]. This mechanism has a high number of reactionsto be studied: in this Letter, we present some theoreticalresults obtained from ab initio calculations performedfor the BF2 + NF = BF3 + N reaction, in the tempera-ture range of 200–4000 K using our own code [10,11],ab initio calculations as electronic energies, frequenciesand geometries of reactants, products and saddle pointwere performed and the reaction rates and minimumenergy path (MEP) were calculated applying conven-tional transition state theory (TST) plus two differentcorrections for the tunneling effects (Wigner�s [12,13]and Eckart�s [14–17]).

We outline how the Letter is organized. In Section 2,we present the main characteristics of our program. Ourresults and discussion are presented in Section 3.

152 S.S. Ramalho et al. / Chemical Physics Letters 413 (2005) 151–156

2. Calculation of rate constants

The rate coefficients of a bimolecular reaction,A + BC = AB + C, can be written as [10–12,18]

kTST ¼ kBTh

QX z

QAQBC

exp � V Gza

RT

� �; ð1Þ

where QA, QBC and QX z are the partition functions ofthe particles A, BC and transition state (TS), respec-tively, kB is the Boltzmann�s constant, h is the Planck�sconstant, T is the temperature, R is the gas universalconstant and V Gz

a is the potential barrier

V Gza ¼ V z þ eZPE; ð2Þ

where V� is the classical potential energy of the saddlepoint measured from the overall zero of reactants� en-ergy and eZPE is the harmonic zero-point energies (ZPE).

Introducing the transmission coefficient, jW/E(T),Eq. (1) becomes

kTST ¼ jW=EðT ÞkTSTðT Þ; ð3Þwhere jW/E(T) is used to account for the tunneling effectalong the reaction coordinate. The Wigner correction(jW(T)) for tunneling assumes a parabolic potentialfor the nuclear motion near the transition state andtherefore can not be considered as an accurate correc-tion. The Wigner transmission coefficient is given by[12,13]

jWðT Þ ¼ 1þ 1

24

�hxz

kBT

��������2

; ð4Þ

where the imaginary frequency at the saddle point is de-noted by x�.

The other tunneling correction used here, is based onthe one-dimensional Eckart potential barrier ðV Gz

a ðsÞÞ

V Gza ¼ ay

1þ yþ by

ð1þ yÞ2þ c; ð5Þ

where

y ¼ eaðs�s0Þ; ð6Þ

a ¼ DH 00 ¼ V Gz

a ðs ¼ þ1Þ � V Gza ðs ¼ �1Þ; ð7Þ

b ¼ ð2V Gza � aÞ þ 2ðV Gz

a ðV Gza � aÞÞ1=2; ð8Þ

c ¼ eGintðs ¼ �1Þ; ð9Þ

s0 ¼ � 1

aln

aþ bb� a

� �; ð10Þ

a2 ¼ � lðxzÞ2B2V zðV z � AÞ

; ð11Þ

where eGintðs ¼ �1Þ is the sum of the zero-point energyof the reactants and l is their reduced mass.

Solving the Schrodinger equation for the Eckart po-tential barrier yields the transmission probability[14,16,17]

CðEÞ ¼ 1� cosh½2pða� bÞ� þ cosh½2pc�cosh½2pðaþ bÞ� þ cosh½2pc� ; ð12Þ

where

a ¼ 1

2

EC

� �1=2

; ð13Þ

b ¼ 1

2

E � aC

� �1=2

; ð14Þ

c ¼ 1

2

b� CC

� �1=2

; ð15Þ

C ¼ ðhxzÞ2B16DV zðDV z � AÞ

. ð16Þ

The parameters a and b are as defined above.The tunneling corrections, jE(T), is obtained as the

ratio between the quantum mechanical and the classicalrate constant calculated by integrating the respectivetransmission probabilities over the entire energies range

jEðT Þ ¼ expðDV zc=RT Þ

RT

Z 1

0

expð�E=RT ÞCðEÞ dE. ð17Þ

For higher accuracy variational transition state the-ory should be used and also semiclassical methods forthe tunneling effect [12], but the required additionalinformation about the potential energy surface werenot obtained in the present Letter.

The characteristics of the MEP and the rate con-stants, with the Wigner and Eckart tunneling correc-tions, were determined using our own code, which isdescribed in the literature [10,11]. The rate constantswere then written in the Arrhenius form, as

kðT Þ ¼ AT n exp � Ea

RT

� �. ð18Þ

3. Results and discussion

To apply transition state theory, we must know thegeometries, frequencies, and the potential energy forreactants, BF2 and NF, reactant complex, RC, saddlepoint, TS and products, BF3 and N. These propertiesare obtained from accurate electronic structure calcula-tion performed using the GAUSSIAN 98 program [19]. Wedetermined the equilibrium geometry for the reactant,RC, TS and products, as well as the harmonic frequen-cies at the Møller–Plesset (MP2) level with the6-311G(d) basis. The frequencies give the ZPE, used toobtain the barrier in Eq. (2). Fig. 1 shows a schematic

Fig. 1. Atomic numbering of the reaction system for reactionBF2 + NF.

S.S. Ramalho et al. / Chemical Physics Letters 413 (2005) 151–156 153

representation of the species involved here, and givesalso the notation used in Table 1.

Table 1 shows the optimized geometry for reactants,RC, TS and products, and compares also these valueswith experimental and theoretical reference data. Theinteratomic distance and bond angle absolute errors forthe reactants and products are smaller than 0.0136 Aand 9.42�, respectively (when compared with the experi-mental data for BF2) and 0.0059 A and 0.32�, respec-tively (when compared with the theoretical data).

The frequencies determined for the species consideredhere are given in Table 2, and they are also comparedwith the experimental and theoretical reference data.The frequencies are scaled by a factor of 0.9684 to takeinto account of known deficiencies at this level. Table 2shows also the scaled ZPE.

Table 1Geometrical parameters of reactants, reactant complex, saddle point and prod

Parameter BF2 NF B

RBF 1.314 1.(1.30 [20])e (1(1.3195[22])t (1

AFBF 121.42 12(112 [20])e (1(121.10 [22])t (1

RNF0 1.310(1.3173 [25], 1.317 [26])

RBF0

AFBF0

ABF0N

e, Experimental data; t, theoretical data.

Total electronic energies for all species involved inthis reaction were calculated using the optimized geom-etries at MP2/6-311G(d) and different calculation levels.Table 3 compares the calculated total energies, heat ofreaction and barrier using different basis sets. The heatof reaction and barrier are corrected by the scaled zeropoint energy. As there are no experimental results forthis reaction we will use the energies calculated atQCISD(t)/6-311++G(d,p) level, because the heat ofreaction at this level is closer to the heat of reaction cal-culated on the basis of the heat of formation of reactantsand products.

The reaction path proceeds in two steps, the first oneinvolving a fast preequilibrium between reactants andthe prereactive complex (or reactants complex, RC),and the second one yielding the corresponding products,represented by equations

BF2 þ NF �k1

k�1

BF2 � � �NF ðR1Þ

BF2 � � �NF*k2BF3 þ N ðR2Þ

In Fig. 2, the energies of the four critical points of thepotential surface, relative to reactants, have been repre-sented: the isolated reactants, the pre-reactive complex,the transition state and the isolated products. The en-ergy profile shows that the energy of the transition stateis lower than the energy of the reactants. A negativeapparent energy barrier (Vapp = VTS �

PVreac) was

found for this reaction, being 47.65 kcal mol�1 lowerthan the isolated reactants. The existence of a reactantcomplex, about 78.48 kcal mol�1 more stable than theisolated reactants, justifies the sign of Vapp and leadsto a complex mechanism, presenting a reversible firststep with no energy barrier and a non-reversible secondstep with a barrier of 49.15 kcal mol�1.

If k1 and k�1 are the forward and backward rate con-stants for the first step, Eq. (R1), and k2 corresponds tothe second step, Eq. (R2), then a steady-state analysisleads to a rate coefficient for the overall reaction k,which can be written as

ucts at MP2/6-311G(d) (interatomic distance in A and bond angle in �)

F3 TS RC

316 1.3349 1.3138.307 [20], 1.324 [21])e.3135[23], 1.3218 [22], 1.3235[24])t0.0 119.277 120.98220 [20])e20 [22,24])t

1.9361 1.3564

1.4129 2.2852116.031 150.34558.643 38.659

Table 2Harmonic vibrational frequencies (cm�1) and zero-point energy (kcal mol�1) for reactants and products at the MP2/6-311G(d) level

BF2 NF BF3 TS

m1 530.4723 1196.0536 484.6746 260.2038(500.0 [20], 523.7 [27])e (1115.0 [20], 1141.4 [26,25])e (480.7 [20], 480 [28,23])e(552.9 [22], 523.6 [27])t (1138.5 [25])t (484 [23], 507.9 [22])t

m2 1150.2955 484.6746 348.2314(1080.0 [20], 1151.4 [27])e (480.7 [20], 480 [28,23])e(1239.2 [22], 1151.4 [27])t (484 [23], 507.9 [22])t

m3 1417.3907 707.6813 433.3688(1213.0 [20], 1394.0 [27])e (696.7 [20], 691 [28,23])e(507.6 [22], 1393.5 [27])t (697 [23], 737.7 [22])t

m4 882.5994 464.3690(888 [20], 888 [28])e(942.8 [22])t

m5 1463.8987 634.7324(1463.3 [20], 1449 [28])e(1575.1 [22])t

m6 1463.8987 823.3052(1463.3 [20], 1449 [28])e(1575.1 [22])t

m7 1212.7543m8 1378.2451m9 675.5375i

ZPE 4.289 1.656 7.597 7.690

e, Experimental data, t, theoretical data.

Table 3Total energy (hartree) and experimental heat of reaction (kcal mol�1) for the reactants, products, TS and RC, potential barrier and reaction enthalpy(kcal mol�1) including ZPE correction

MP2/6-311G(d) QCISD(t)/6-311G(d) QCISD(t)/6-311++G(d,p) (e)DHf

BF2 �224.1435362 �224.1591327 �224.1702706 �120.20 [29]NF �154.1344995 �154.1600308 �154.1681854 59.56 [30]BF3 �323.9705849 �323.9882954 �324.0021502 �271.08 [29]N �54.4750512 �54.4914202 �54.4931927 112.53 [31]TS �378.3501432 �378.4001576 �378.4171659 –RC �378.4068765 �378.4464181 �378.4635272 –

Vapp �43.50 �49.08 �47.65Vf 37.35 30.78 30.84DH �103.71 �99.20 �96.99 �97.91

e, Experimental data.

154 S.S. Ramalho et al. / Chemical Physics Letters 413 (2005) 151–156

k ¼ k1k2k�1 þ k2

. ð19Þ

According to Singleton and Cvetanovic [32], Galanoet al. [33] and Alvarez-Idaboy [34], the entropy change ismuch larger for the reverse reaction than in the forma-tion of products. Thus, k�1 is expected to be consider-ably larger than k2. Based on this assumption, Eq. (19)can be rewritten as

k ¼ k1k2k�1

. ð20Þ

Applying basic statistical thermodynamic principles, theequilibrium constant for the first step is given by:

Keq1 ¼k1k�1

¼ QRC

QR

exp �ðV RC � V RÞRT

� �ð21Þ

and from TST, k2 is given by Eq. (1), rewritten below as:

k2 ¼kBTh

QTS

QRC

exp �ðV TS � V RCÞRT

� �. ð22Þ

Combining Eqs. (21) and (22) with Eq. (20) the rate con-stant for the overall reaction is obtained by the follow-ing expression:

k ¼ Keq1k2 ¼kBTh

QTS

QR

exp �ðV TS � V RÞRT

� �. ð23Þ

The tunneling correction factor of Wigner, Eq. (4), orEckart, Eq. (17), can be applied to Eq. (23).

The VMEP and V Ga along the MEP as a function of the

reaction coordinates for these reactions are plotted inFig. 3, considering the region between RC and products.One can see from these figures that the V G

a curves are

-100

-80

-60

-40

-20

0

Products

TS

RC

Reactants

Ene

rgy

[kca

l mol

-1]

Reaction coordinate

MP2/6-311G(d) QCISD(T)/6-311G(d) QCISD(T)/6-311++G(d,p)

Fig. 2. Energy profile of the BF2 + NF, at different level ofcalculation.

0 10 20 30 40 5010

20

30

40

50

60

70

80

90

Log

(k)

[cm

3 mol

-1 s

-1]

10000/T [K-1]

kTST

(T)

κW(T)k

TST(T)

κE(T)k

TST(T)

MP2/6-311G(d) QCISD(T)/6-311G(d) QCISD(T)/6-311++G(d,p)

Fig. 4. Arrhenius plot of reaction rate against the reciprocal temper-ature in the range of 200–4000 K for reaction BF2 + NF.

S.S. Ramalho et al. / Chemical Physics Letters 413 (2005) 151–156 155

similar to the VMEP curves. From �1 to �7.5l1/2a0, thepotential curves change only slowly, and they increasegradually after �7.5l1/2a0 and decrease quickly afterpassing the transition state.

The reaction rates are shown in Fig. 4: they were eval-uated using a computational code developed by our re-search group [10], considering conventional TST andalso applying the transmission coefficients of Wignerand Eckart. Since the apparent barrier is negative, thereaction rate decreases as the temperature increases.

The skew angle value for this reaction (56.43�), whichis considered as an intermediate value, indicates that thetunneling effects are not critical here. Furthermore, thisreaction has only heavy atoms and we focus on the rateconstants at high temperatures (1000–3000 K). We haveextended the calculation for the temperature range of

-10 -5 0 5 10-110

-100

-90

-80

-70

-60

-50

-40

-30

Ene

rgy

[kca

l mol

-1]

Reaction coordinate [µ1/2 a

0]

VMEP

Va

G

MP2/6-311G(d) QCISD(T)/6-311G(d) QCISD(T)/6-311++G(d,p)

Fig. 3. Classical potential energy curve (VMEP) and vibrationaladiabatic potential energy curve ðV G

a Þ, between RC and products, asfunction of reaction coordinate for reaction BF2 + NF.

200–4000 K. The Eckart tunneling correction is huge,of order of 1013 at room temperature and 104 at1000 K, so it is clearly over-estimated here.

4. Conclusions

In this Letter, an ab initio study was performed onthe system BF2 + NF. According to the reaction pro-file, this type of reaction occurs in a two-steps mecha-nism. The first step is the reversible formation of aprereactive complex, which occurs without activationenergy, with energy of 78.48 kcal mol�1 lower than thatof the reactants. Consideration of the prereactive com-plex is essential for the determination of the correct en-ergy barrier of the second step of the reaction, that isirreversible.

Comparing the heat of reaction obtained by theab initio calculation and the experimental heat of forma-tion of reactants and products, the best calculation levelanalyzed here is QCISD(t)/6-311++G(d,p). The temper-ature dependence of the overall rate constant incm3 mol�1 s�1 can be described by the expressionk = 1750.8T2.4955 exp(49558/RT). This information willbe of value for the quantitative assessment of the roleof this reaction in the proposed mechanism of the boronnitride film growth.

Acknowledgments

The authors are grateful to the Centro Nacional deComputacao Cientıfica de Alto Desempenho de Campi-nas – SP (CENAPAD-SP) for the provision of computa-tional facilities and also Fundacao de Emprendimentos

156 S.S. Ramalho et al. / Chemical Physics Letters 413 (2005) 151–156

Cientıficos e Tecnologicos (FINATEC) for the financialsupport. S.S. Ramalho and A.F.A. Vilela acknowledgethe Conselho Nacional de Desenvolvimento Cientıficoe Tecnologico (CNPq) for the fellowship which providedher the opportunity to perform this work.

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