J. Phys. Chem. 1995,99, 6540-6547
6540
Rate Constant Determinations for HBO Calculations
+ F Channels from ab Initio Reaction Path
Maribel R. Soto Laboratory for Computational Physics and Fluid Dynamics, Code 6410, Naval Research Laboratory, Washington, DC 20375-5344 Received: August 24, 1994; In Final Form: December 21, 1994@
Understanding the detailed mechanism of boron combustion in the presence of fluorine has important consequences to the development of energetic materials. From a recently proposed reaction mechanism,' one of the dominant reactions for which very little kinetic or mechanistic information is available is HBO F FBO H. Ab initio multiconfigurational methods have been used to study two reactive pathways stemming from the reactants HBO F. These are HBO F HF BO and HBO F H(F)BO H FBO. The optimized structures and harmonic vibrational frequencies of the stationary points are reported, and other features of the potential energy surface of HBOF are discussed. Furthermore, the rate constants were calculated by using transition-state theory and variational transition-state theory. For reaction 2, AHR = -20.7 kcal/mol, EA = 2.7 kcal/mol, and k*(T) = (5.08 x 10-'6)T'.77 exp(-1666/T) cm3 molecule-! s-I. Reaction 3a is two and a half times more exothermic than reaction 2 and proceeds without a barrier. For this reaction, AHR = -56.5 kcaYmol and k3=(T) = (4.22 x 10-'4)T'.98 exp(353/T) cm3 molecule-' s-l. Reaction 3b is endothermic with AHR = 9.9 kcal/mol, E A = 13.0 kcal/mol, and k3b(T) = (7.00 x 10'3)T0,0'exp(7817/T) s-I. These calculations predict that reaction 1 proceeds through the formation of an H(F)BO complex which dissociates to form FBO H. At low temperatures, the complex formation dominates the HBO F reaction, but at temperatures exceeding 1500 K it competes with the abstraction pathway. These findings have important implications to the development of the reaction mechanism because they predict the formation of a complex which was not previously included in the model, and they provide kinetic parameters that are not available experimentally.
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I. Introduction
HBO
The combustion of boron in the presence of fluorine is potentially a highly energetic process. Understanding the detailed mechanism of this process is essential in the development of new energetic materials. To this end, R. C. Brown and C. E. Kolb at Aerodyne Research, Inc., jointly with R. A. Yetter, F. L. Dryer and H. R. Rabitz at Princeton University' are developing a realistic gas-phase kinetic model for the homogeneous chemistry of B/O/H/C/F combustion systems. In their preliminary work they have compiled a reaction mechanism that consists of more than 100 elementary reactions of 25 different species. Furthermore, using gradient sensitivity analysis, they have identified reactions leading to the production of FBO as critical to the mechanism. However, their model remains incomplete because there is not enough kinetic information available on these reactions. One of the reactions which was identified as critical to the mechanism is HBO +F-FBO
+H
(1)
From experimental* and theoretical3 heats of formation the exothermicity of this reaction is estimated to be 50 kcdmol. However, information such as the barrier, rate of reaction, or mechanism, which is valuable to the model, is not available. In order to address the lack of kinetic data for reaction 1 and in an effort to increase the present understanding of fluorineenriched boron. combustion, ab initio multiconfigurational methods have been used to study reactions of HBO F. There are three reactive pathways stemming from HBO F. These are
+
@
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Abstract published in Advance ACS Abstracts, April 1, 1995.
+
+ F-HF + BO
+ F - H(F)BO - FBO + H HBO + F - HBOF - BOF + H
HBO
(2) (3a,b) (4)
In reaction 2 the fluorine abstracts a hydrogen from HBO to form HF BO. From heats of this reaction has an estimated exothermicity of 21.1 kcai/mol. In reaction 3, the fluorine adds to the boron in HBO to form a complex, which falls apart to give FBO H. As mentioned previously, the overall reaction has an estimated exothermicity of 50.0 kcaY mol. Finally, the fluorine can add to the oxygen in HBO to form HBOF. This complex can fall apart to give BOF H or it can go back to reactants. Previous work4 has shown that the BOF isomer is a high-energy local minimum, and that it is highly unstable to rearrangement to FBO. The large barrier required to form BOF makes it unlikely that this reaction occurs to an appreciable extent. Since the primary motivation of this work was to understand reaction 1 and processes that could compete with it, only reactions 2, 3a, and 3b have been addressed. The approach used in this study was to map out stationary points of the potential energy surface (PES) of HBOF which corresponded to the reactants, products, and transition states of reactions 2, 3a, and 3b. The PES was calculated by using the complete-active-space self-consistent field (CASSCF) method, and it was further refined by doing single-point energy calculations with the multireference configuration interaction (MRCI) method. Furthermore, reaction path (RP) calculations were done to characterize regions of the PES in the vicinity of the saddle points. This information was especially valuable in obtaining
+
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This article not subject to U.S. Copyright. Published 1995 by the American Chemical Society
+
Rate Constant Determinations of HBO
+ F Channels
rate constants by using canonical variational transition-state theory (CVT).
More explicitly, for reaction 2 the 313 space consists of the BH a, BH a*,and the F radical orbitals on the reactants’ side, which become the HF a, HF a*,and B radical on the products’ side. For reaction 3a the 313 space is different for HBO F. In this case the 313 space includes the BO n,BO E*, and F radical on the reactants’ side, which become the BF a, BF a*,and 0 radical on the products’ side. Similarly, for H(F)BO the 313 reference space is different for reactions 3a and 3b. In reaction 3a the 313 space of H(F)BO consists of the BFa, BFa*, and the 0 radical, but in reaction 3b it consists of the BHa, BHu*, and 0 radical. The final 313 MRCI wavefunction had 1.2 million configurations. The larger 515 MRCI wavefunction had 9.1 million configurations. The basis sets used were the correlation consistent polarized valence double-l; (cc-pVDZ), the augmented cc-pVDZ (aug.ccpVDZ), and the correlation consistent polarized valence triple-5 (cc-pVTZ), all of which were developed by Dunning and his c o - w ~ r k e r s . ~The ~ ’ ~ cc-pVDZ basis consists of a (9s5pld) primitive set contracted to [3s2pld] for the first-row atoms and a (4s) primitive contracted to [2s] and augmented with a (lp) polarization function for hydrogen. The aug.cc-pVDZ is the cc-pVDZ basis set augmented by (lslpld). The cc-pVTZ basis consists of a (10s5p2dlf) primitive set contracted to [4s3p2dlfl for first-row atoms and a (5s) contracted to [3s] and augmented by a (2pld) polarization set for hydrogen. Reaction path (RP) calculations were done for reactions 2, 3a, and 3b. That is, paths of steepest descent or minimum energy paths (MEPs) from the saddle points toward the reactants and toward the products were obtained along the 11110 CASSCF PES. The MEPs were calculated by using the corrected local quadratic approximation (CLQA).” At several points along the MEPs the energies, harmonic vibrational frequencies of the modes orthogonal to the path and zero-point energies were calculated. Furthermore, single-point energy calculations at the 313 MRCI level were done for points along the MEPs. Rate constant calculations were done by using transition-state theoryI2and canonical variational transition-state theory (CVT).I3 In transition-state theory, the transition state is assumed to be at the saddle point on the RP and the rate constant is calculated by the following expression:
+
11. Theoretical Method The first step in doing a study of this kind is to find a wavefunction that describes the reaction properly. In multiconfigurational methods such as CASSCF, the wavefunction is tailored by the basis set and the complete active space (CAS). Calculations were done by using different basis sets in order to assess their effect on the relative energies. However, for CASSCF calculations, finding a good wavefunction also means finding the CAS that best describes the reaction in question. In general, the active space is chosen to include the orbitals and electrons that are directly involved in the bond breaking and bond forming. For example, in reaction 2 a BH bond is broken and a HF bond is formed, and the radical electron goes from the fluorine to the boron. The smallest active space that could be used to describe this reaction consists of a threeelectron-in-three-orbital (313) space of the radical and the aforementioned bonds. That means that the 313 CASSCF wavefunction includes all single, double, triple, etc. excitations that are possible by exciting these three electrons among the three orbitals. That is, it consists of a full configuration interaction (CI) in the restricted orbital space of three electrons in three orbitals. In many cases, this small CAS can describe a reaction p r ~ p e r l y .However, ~ there are cases in which other electrons and orbitals play an important role in the reaction, even though they are not a part of the direct bond breaking and bond Finding the “best” multiconfigurational wavefunction entails exploring the effect of including other electrons and orbitals in the active space. For this purpose, five different CASSCF wavefunctions were used to obtain the optimized structures of the reactants and products of reaction 2. These were: 313, 515, 717, 919, and 11110. The 313 consisted of the BH a and a* and the F radical on the reactants’ side. These orbitals evolved into the HF a and u* and the B radical on the products’ side. For the larger wavefunctions the 313 active space was increased systematically as follows: adding the BO E and x* for the 515, adding the 0 2p lone pair and empty B 2p orbital for the 717, adding the BO a and a* for the 919, and adding the 0 2s lone pair for the 11110. For the other stationary points on the HBOF PES that are discussed in this paper, such as the saddle point for reaction 2 and the products and saddle points for reactions 3a and 3b, the optimized structures, harmonic vibrational frequencies, and zeropoint energies were obtained at the 11/10 CASSCF level. The calculations for HBO F, HF BO, and FBO H were done as supermolecules, where the molecular fragments were placed 10 A apart. In many cases it is desirable to obtain energy differences at a high level of theory such as CI or MRCI. However, the cost of optimizing geometries at these levels can be prohibitive. In this event, it is helpful to do calculations at a higher level of theory using the geometries optimized at a lower level of theory.8 In this study, single-point energy calculations were done by using MRCI. These calculations include all doublet configurations that result from single and double substitutions from the configurations in a defined CASSCF reference space. These calculations exclude excitations out of the 1s core orbitals of oxygen, boron, and fluorine and into the corresponding virtual orbitals. The MRCI calculations done here used a 313 and 515 CASSCF reference space and are referred to as 313 and 515 MRCI. In order to obtain energy differences that were consistent using the 313 reference space, it was necessary to calculate two different energies for HBO F and H(F)BO.
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J. Phys. Chem., Vol. 99, No. 17, 1995 6541
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k(T) = (kBT/h)(Q*(T)/Qr(T))eXP(-EA/kBT)
(5)
In this equation, kB is the Boltzmann constant, T is temperature, h is Planck’s constant, Q* is the partition function of the transition state, Qr is the partition functions of the reactants, and EAis the barrier or activation energy. In CVT, the position of the transition state along the MEP is optimized at each temperature so as to minimize the calculated rate constant.
where
k G T ~ ~= , s(~,T/~)(Q‘~(T,~)/Q,(T)) > exp(- v,,,(~Y~,T) (7) Here QcT(T,s)is the partition function for the bound degrees of freedom orthogonal to the reaction path at s, and VMEPis the value of the potential along the RP at s. All calculations were done by using MESAI4 on several supercomputer centers. These were the C90 and Cray YMP at CEWES in Vicksburg, MS; the Cray YMP at the Arctic Region Supercomputer Center (ARSC); and the Cray YMP-EL at the Naval Research Laboratory (NRL).
6542 J. Phys. Chem., Val. 99, No. 17, 1995
Sot0
TABLE 1: Heats of Reaction (kcaYmol) for the Abstraction Reaction Obtained by Using Different Wavefunctions CC-DVDZ aug.cc-pVDZ CCDVTZ
*
~~
~~
HF 313 515 717 919 11/10 313 MRCI" 515 MRCP a
TABLE 2: Optimized Structural Parameters and Harmonic Vibrational Frequencies for the Reactants and Products of the Abstraction Reaction
1.5 -4.6 -5.1 -6.0
-4.8 -10.6 -11.2 -11.7 -11.9 -12.0 -20.5 -21.6
-.10.0 RBO RBH VI
v2 v3(2)
ZPe
Geometry optimized at the 11/10 MCSCF level.
RHF VHF
All of the results displayed in the tables and discussed in the article are in the following units; MR'S in kcdmol, bond lengths in A, bond angles in degrees, frequencies in cm-I, zeropoint energies in kcdmol, energies in hartrees, and temperatures in K.
111. Abstraction As mentioned previously, reaction 2 has an estimated exothermicity of 21.1 kcdmol. This estimate was obtained from the heats of formation2q3of the products and reactants; however, these are not all experimental values. The value for HBO was obtained theoretically by Nguyen, Vanquickbome, Sana, and Leroy3 at the MP4SDTQ/6-311++G (3df,2p) level with optimized geometry from the MP2/6-3 1G (d,p) level. They predict the accuracy of their values to be within f 2 kcdmol. Hence the estimated exothermicity of the abstraction has at the very least an uncertainty of f 2 kcdmol. Table 1 shows the heats of reaction obtained for the same reaction by using different wavefunctions. In this table, HF represents the Hartree-Fock wavefunction and 3/3, 5/5, etc. represent the different CASSCF wavefunctions. The geometry was optimized at each of these levels except for the 3/3 and 5/5 MRCI calculations, which used the geometry optimized at the 11/10 CASSCF level. The results in this table display two significant characteristics: First, as the size of the active space increases, the predicted exothermicity increases regardless of the basis set. Second, the exothermicities obtained with the aug.cc-pVDZ basis are approximately 6 kcdmol larger than those from the cc-pVDZ basis. Only one calculation was done using the cc-pVTZ basis and it gives similar results as the same wavefunction using the aug.cc-pVDZ basis. In this case, the larger basis set did not improve the description of the system. Moreover, it apepars that the diffuse functions in the aug.ccpVDZ basis set are necessary to describe the highly electronegative F atom properly.I0 Therefore, this basis set was used for the remaining calculations in this study. At the CASSCF level the 11/10 wavefunction using the aug.cc-pVDZ basis predicts the highest exothermicity of 12.0 kcal/mol; nonetheless, this result does not agree closely with the experimental estimate. Single-point calculations were done by using 313 and 515 MRCI wavefunctions with the optimized geometry obtained at the 11/10 level. These calculations predict exothermicities of 20.5 and 21.6 kcdmol, respectively. Including the correction due to the zero-point energy yields AHR's of -20.7 and -21.8 kcal/mol. These values are within 0.4 and 0.7 kcal/mol of the experimental estimate. These deviations are less than the uncertainty of f 2 kcal/mol in the estimated exothermicity of -21.1 kcal/mol obtained from heats of formation. Table 2 shows the structures, harmonic vibrational frequencies, and zero-point energies obtained for HBO, HF, and BO at the 11/10 CASSCF level. The BH and BO bond lengths obtained for HBO are within 0.02 8, of the experimental
ZPe RBO VBO
ZPe a
11110
exptl
HBO 1.217 1.191 2762.2 1791.7 762.7 8.70
1.201" 1.167 2811.1 1825.6 754.4
HF 0.9 18 4122.1 5.90
0.917b 4138.3
BO 1.218 1862.4 2.66
1.205 1895.7
Reference 15. Reference 2.
TABLE 3: Optimized Structural Parameters and Harmonic Vibrational Frequencies for the Saddle Point of the Abstraction Reaction 11/10 RBO RBH RHF VI
VZ(2)
1.214 1.331 1.235 1924.4 7 10.0
11/10 v3
v4(2) v5
ZPe
455.4 123.2 -2598.3i 5.79
measurement^.'^ Other theoretical calculation^^.'^.-'^ for HBO show smaller deviations from experiment for the structural parameters. For example, DeFrees, Binkley, and McleanI7 obtained values that were within 0.006 8, for the BO bond length and 0.002 8, for the BH bond length using CI with single and double substitutions (CISD) and a 6-311G** basis set. The longer bond lengths predicted here are due to the tendency of the CASSCF wavefunction to overestimate bond distance^.'^ This tendency also affects the harmonic vibrational frequencies and leads to these values being underestimated. For example, the harmonic vibrational frequency for the BH stretch is lower than the experimental value by 49 cm-I. The agreement with experiment for the BO stretch and the HBO degenerate bend is much closer, Le., with 34 and 8 cm-I, respectively. The comparison of structural parameters for HF and BO to experimental measurements* shows closer agreement than for HBO. The calculated bond len ths for these molecules are within 0.001 8, for HF and 0.01 for BO of the experimental values. The calculated frequencies are within 16 cm-I of the experimental values for the HF stretch and within 33 cm-' for the BO stretch. Overall, the agreement of the structural parameters and the frequencies obtained at the 11/10 CASSCF level with experimental values is very close. Therefore, the 11/10 CASSCF wavefunction with the aug.cc-pVDZ basis set was used to characterize other stationary points on the PES of HBOF. In order to make predictions about the rate of the abstraction reaction it was necessary to characterize the saddle point. Table 3 shows the optimized structural parameters, zero-point energy, and harmonic vibrational frequencies of this stationary point on the 11/10 CASSCF FW. The imaginary frequency due to the reaction coordinate is also shown but was not used to calculate the zero-point energy. For this reaction the structure at the saddle point is linear and the BH bond length is stretched by 0.14 8, or 12% of its value at the reactants. The BO bond length is shorter than it was at the reactants by 0.003 A. Presumably it lengthens by 0.004 8, throughout the remainder
1
+ F Channels
Rate Constant Determinations of HBO RP for HBO
+ F ---z
HF
J. Phys. Chem., Vol. 99, No. 17, 1995 6543
+ BO
Frequencies dong the CASSCF RP of HBO + F --HF + BO
\ 4
MRCI
-10
-0.6 -0.4
Figure 1. RP of HBO
-0.2
0
0.2
arc length
(5)
0.4
0.6
+ F - HF + BO.
of the reaction path to reach 1.218 8, at the products. At this level of theory, the barrier (including the zero-point energy correction) is predicted to be 17.7 kcal/mol. A single-point energy calculation using a 3/3 MRCI wavefunction and the optimized structure from the 11/10 CASSCF level predicts the barrier to be 2.7 kcdmol. This value is about 85% smaller than the CASSCF barrier. The CASSCF wavefunction significantly overestimates the size of the barrier. Unfortunately, a single-point energy calculation at the 5/5 MRCI level was computationally prohibitive and could not be obtained for comparison. However, since the A.HR obtained at this level was slightly more exothermic than the one at the 3/3 MRCI level, it is likely that the barrier at the 5/5 MRCI level would be slightly smaller than 2.7 kcdmol. RP calculations at 11/10 CASSCF level were done by using the CLQA method at a step size of 0.05 amul/*bohr. On the RP the saddle point is located at 0.0 bohr for the arc length parameter s. Negative values of s refer asymptotically to the reactants, and positive values of s refer asymptotically to the products. The MEP was calculated 10 steps toward the reactants, HBO -I- F, and 10 steps toward the products, HF -IBO. At each step on the MEP, the 11/10 CASSCF energy and the harmonic vibrational frequencies for the 3N - 6 degrees of freedom (for a linear transition state) transverse to the RP were calculated. The resulting RP without including the zero-point energy correction is shown in Figure 1. Also shown on this figure is the curve resulting from 3/3 MRCI calculations of the points on the 11/10 CASSCF path. For the purposes of determining the energetics of chemical reactions, the ground-state vibrationally adiabatic energy curve, which is the result of the RP plus the zero-point energy, is more significant than the curves shown in Figure 1. However, this curve could not be obtained because of problems describing the lowest lying frequencies in the region surrounding the transition state. That is, Figure 2 shows the behavior of the harmonic vibrational frequencies in this region. The lowest lying frequencies are transitional modes that evolved from motions that were rotations at the reactants. As can be seen in the figure, the transitional modes become imaginary from s = -0.05 toward reactants and from s = 0.40 toward the products. This is a consequence of the particular choice of coordinates for solving the vibrational problem in directions transverse to the MEP. That is, these motions are best described by a nonEuclidean geometry.20 However, the RP, as well as the harmonic vibrational frequencies, is calculated by using Car-
-1000
-0.8 -06 -04 -0.2 0 0 2 arc length (s)
04
06
Figure 2. Behavior of the 3N - 6 harmonic vibrational frequencies transverse to the Rp of the abstraction reaction. This figure is only for a small area in the transition-state region. tesian coordinates. At the stationary points the choice of coordinate system does not pose a problem for the calculation of the harmonic vibrational frequencies; however, for other points on the RP the Cartesian coordinates do not provide a good description. This dependency on the choice of coordinates, which has been discussed by Natanson, Garrett, and Truhlar,21 is exacerbated for transitional modes that evolved from rotations. Therefore, without a good description of all of the harmonic vibrational frequencies along the RP, it was impossible to calculate a meaningful ground-state vibrationally adiabatic curve. Nonetheless, a reasonable estimate of the barrier was obtained by using the MRCI result at the saddle point. As can be seen in Figure 1, the maximum on the MRCI curve occurs at s = -0.05. At this point, the energy is 5.8 kcdmol. At s = 0.0 the MRCI energy is only 0.2 kcaymol lower at 5.6 kcal/mol. The latter energy corrected for the zero-point energy has been discussed before and it is 2.7 kcal/mol. If the MRCI energy at s = -0.05 could be corrected for the zero-point energy, it would be a little higher than 2.7 kcdmol, possibly as little as 0.2 kcal/ mol higher. Recall that in the previous discussion of the barrier at the 5/5 MRCI level it was expected to be a little lower than 2.7 kcal/mol. In fact, the combination of both effects, Le., lack of inclusion of some electron correlation and the position of the maximum on the RP, may result in some cancellation of error. However, without further evidence I can only suggest that 2.7 kcal/mol is a reasonable estimate for the barrier of the abstraction reaction. The temperature-dependent rate constant was calculated by using transition-state theory and 2.7 kcal/mol for the barrier. The rate constant was not calculated variationally because of the difficulties in defining the ground-state vibrationally adiabatic energy curve. Furthermore, anharmonicity and tunneling effects were not taken into account. These effects can influence the rate constant ~ubstantially,'~ but the error due to the neglect of these effects may be smaller than the error induced by the imprecise determination of the barrier height.I6 The resulting Arrhenius curve is shown in Figure 3 . The rate constant can be expressed in the conventional three-parameter form, k(T) = A F exp(C/T), as k2(T) = (5.08 x 10-'6)T'.77 exp(-l666/T) cm3 molecule-' s-l.
Sot0
6544 J. Phys. Chem., Vol. 99, No. 17, 1995 Rate Conrtrntr of HE0 + F Channels -9
-11
,
,
I
l
l
/
w
\
--
H + OBF
*1’
1
H\
\
-m -12--
j
FJ”O
I-
Figure 4. Energy diagram for the reactants, products, and transition
\
states of reactions 1 and 2. \,
-l
TABLE 5: MRCI Energies of the Stationary Points in hartrees
.9:-
-:GI +Complex
stationary point
- 41 -
0
0.5
1
1.5
2
2.5
3
3.5
+
1ooon
Figure 3. Calculation of the temperature-dependentrate constant for channels of HBO
+ F.
11/10
exptl
exutl
1.356 1.213 1.347 117.8 120.7 2544.7
VI
REO REF
1.219 1.304 2056.5
VI
v2 v3 v4 V5
v6 zPe FBO 1.207“ 1.283 2078.9
~2
~3(2) zpe
1389.7 1065.2 968.1 880.6 411.5 10.39 966.2 506.6 5.77
+
313 MRCI
515 MRCI
-199.9709“ -199.9759’ -200.0036 -200.0648‘ -200.06 86d -200.04 17 -199.9620 -199.9878 -200.0382
-199.9800 -200.0145 -200.0729 -200.0483
a The 313 space contains the following orbitals: BH u, BH u*, and F radical. The 313 space contains the following orbitals: BO il, BO n*, and F radical. The 313 space contains the following orbitals: BH u, BH u*, and 0 radical. dThe 313 space contains the following orbitals: BF u, BF u*, and 0 radical.
H(F)BO
REO RBH RBF LHBO LFBO
HF BO H(F)BO FBO H TS for reaction 2 TS for reaction 3a TS for reaction 3b
TABLE 4: Optimized Structural Parameters and Harmonic Vibrational Frequencies for H(F)BO and FBO 11/10
+F
HBO
TABLE 6: Ea’s and A&% in kcaVmol (Including Zero-Point Energy Corrections) Obtained at the MRCI Level by Using Optimized Structures from the 11/10 CASSCF Level 502b
reaction 1 2 3a 3b
‘
Reference 22. Reference 23.
IV. Complex Formation The optimized structures, harmonic vibrational frequencies and zero-point energies for the complex, H(F)BO, and the products, FBO H, were also obtained by using an 11/10 CASSCF wavefunction. These results are shown in Table 4. The complex is trigonal planar and FBO is linear. A comparison with experimental measurements can only be made for FBO. For this molecule the BO and BF bond lengths are within 0.01 and 0.02 A of the experimental values.22 The harmonic vibrational frequencies are within 22 cm-’ for the BO stretch and 5 cm-’ for the degenerate FBO bend.23 The calculated structure and frequencies are in close agreement with the experimental values. The energies of the minima and saddle points were obtained at the 3/3, and wherever possible at the 5/5 MRCI level. These results are displayed in Table 5 , and Table 6 displays the EA’S and MR’S of reactions 1-3. Recall that in the 3/3 reference space in order to obtain heats of reaction in which the reference space of the reactants and products contained the same electrons and orbitals, different 3/3 MRCI calculations had to be done for HBO F and H(F)BO, depending on which reaction was being considered. The two different energies obtained for HBO F and H(F)BO are reported in Table 5 . The MR’S reported in Table 6 are the ones obtained by using the energies in which the reference space of the reactants and products for each reaction is consistent. Therefore, M R ( ~ f) M ~ ( 3 b in ) Table 6 because of the change in reference space
+
+
+
+
313 EA
313 AHR
515 AHR
2.7 -7.4 13.0
-47.4 -20.7 -56.5 9.9
-45.8 -21.8 -56.5 10.8
for H(F)BO. Figure 4 is an energy diagram (including zeropoint energy corrections) obtained from single-point 3/3 MRCI calculations using the geometries optimized at the 11/10 CASSCF level. For consistency, the energy differences shown in this diagram are the ones obtained by using the same 3/3 reference space for all of the stationary points. For the diagram, the 3/3 space considered is the one which includes the BH (I, BH o*,and F radical orbitals at HBO F and their correspondifng orbitals at the other stationary points. Here, m ~ ( 1 = ) MR(3a) f M ~ ( 3 b ) . This diagram shows that the H(F)BO complex is the lowest-lying species of the ones studied here. In fact it is almost three times lower in energy than the products from the abstraction reaction. From this figure the reaction is expected to have an exothermicity of 57.2 kcdmol. However, this estimate is misleading because the electrons directly involved in the reaction were not included in the reference space of the MRCI wavefunction. By use of the 3\31 reference space, which includes the BO n,BO n*,and F radical at the reactants and the BF (I, BF o*,and 0 radical at the products, the heat of reaction is predicted to be -56.5 kcdmol. Although, this result is a little smaller it still shows that the complex formation reaction is considerably more exothermic than the abstraction reaction. The exothermicity predicted at the 5/5 MRCI level is also -56.5 kcdmol. Typically, as the exothermicity of a reaction increases, the barrier decreases. Therefore, it is
+
Rate Constant Determinations of HBO
+ F Channels
J. Phys. Chem., Vol. 99, No. 17, 1995 6545
TABLE 7: Optimized Structural Parameters and Harmonic Vibrational Frequencies for the Transition States of Reactions 3a and 3b 11/10
RP for HBO
t F
---> H(F)BO
11/10
TS of 3a RBO RBH RBF LHBO LFBO VI
1.244 1.192 1.848 160.5 92.1 2739.4
v2 v3 v4 v5 v6
zPe
1571.2 778.2 138.4 306.4 - 1085.1i 8.78
TS of 3b RBO RBH RBF LHBO LFBO VI
1.243 1.603 1.319 95.2 159.3 1877.8
v2 V3 v4
v5 v6
ZPe
984.0 686.9 602.9 557.9 -911.2i 6.74 -25
expected that the complex formation reaction should have a smaller barrier and proceed faster than the hydrogen abstraction reaction. The saddle points of reaction 3 were characterized at the same CASSCF level as the other stationary points. The optimized structural parameters, harmonic vibrational frequencies, and zero-point energies are shown in Table 7. Once again the imaginary frequencies are shown but were not used to calculate the zero-point energies. At the saddle point of the complex formation reaction, or reaction 3a, the activated complex is planar and nonlinear, with the F perpendicular to the HBO fragment. The BO and BH bond lengths are almost the same as they are in the reactants, and the BF distance is very long at 1.8 A. This is a very “early” transition state. The barrier obtained at the 11/10 CASSCF level (including zero-point energy correction) is 14.8 kcallmol. At the 3/3 MRCI level, the barrier is reduced to -7.4 kcaVmo1. That is, the complex formation essentially has no barrier. For reactions like this one that proceed without a barrier, the rate constant must be calculated by using the variational version of transition-state theory.24 In this method the transition state is not necessarily located at the saddle point but at a point along the RP where the reaction flux is a minimum. In fact, the location of the transition state changes with temperature. At low temperatures the location of the transition state is dominated by the Boltzmann factor, but at high temperatures it is dominated by the partition functions of the points on the RP. In order to calculate the rate constant by using CVT, information about the RP must be available. Specifically, the area of the RP in the immediate vicinity surrounding the saddle point must be known. For this reaction the RP was calculated the same as before, Le., at the 11/10 CASSCF lvel using the CLQA method with a step size of 0.05 amu1/*bohr. The MEP was calculated 15 steps toward the reactants, HBO F, and 15 steps toward the products, H(F)BO. At each step on the MEP, the energy and the harmonic vibrational frequencies for the 3N - 7 degrees of freedom (for a nonlinear transition state) transverse to the RP were calculated. In this case, the harmonic vibrational frequencies along the RP were well-described, hence the ground-state vibrationally adiabatic energy curve was obtained. T h i s curve is shown in Figure 5. Also shown in this figure is the energy curve, including the zero-point energy correction, resulting from 3/3 MRCI calculations at each step on the MEP. The energy maximum or the barrier on the 11/10 CASSCF energy curve is 14.9 kcaVmol above the reactants and it occurs at s = -0.05. That is, with the inclusion of zero-point energy the location of the barrier moves slightly closer to the reactants.
+
/
/
I
!
-0.8-0.6 -0.4 -0.2 0
:
!
!
0.2 0.4 0.6 0.8
arc length (I)
+ F - H(F)BO.
Figure 5. Ground-state vibrationally adiabatic energy curves on the 11/10 CASSCF and 3/3 MRCI PES of HBO
On the 3/3 MRCI energy curve the maximum is -0.4 kcaVmol and it is at s = -0.75. In fact, the energy curve obtained at the 3/3 MRCI is very different from the one obtained at the 11/10 CASSCF level. However, the 3/3 MRCI energy curve is a more realistic description of a reaction without a barrier. The rate constant for reaction 3a was calculated by using CVT and the information from the 3/3 MRCI energy curve. No attempt was made to correct for anharmonic effects. By use of the same three-parameter form discussed in the previous section, the temperature-dependent rate constant can be expressed as k3,(T) = (4.22 x lO-I4) exp(353/T) cm3 molecule-’ s-I. The resulting Arrhenius curve is shown in Figure 3, where it can be easily compared to the curve from reaction 2. As can be seen in this figure, this reaction is much faster than the abstraction reaction at low temperatures by 2 orders of magnitude. At temperatures exceeding 1500 K the two reactions proceed at about the same rate. This information has important implications for the development of the gas-phase model of fluorine-enriched boron combustion for several reasons besides providing kinetic parameters. First of all, the model included only two product channels for HBO F, which were direct FBO formation, Le., reaction 1, and the abstraction reaction. From this study, it is very unlikely that reaction 1 is the mechanism by which FBO formation occurs, since complex formation is very facile. When FBO is formed from HBO F, it probably goes through reactions 3a and 3b. Second, the model did not include any reactions of H(F)BO. From these calculations it is evident that the formation of H(F)BO is fast and dominates the reaction of HBO F at temperatures below 1500 K. At higher temperatures it competes with the abstraction reaction. For combustion purposes it is very important to understand chemical reactions in the temperature range 1800-3000 K.’ Since reaction 3a competes with reaction 2 in this temperature regime, it must be included in the model. Furthermore, the fact that the complex is formed easily implies that other reactions with H(F)BO must be considered and their temperature dependence in the hightemperature regime must be studied. Finally, the possibility that there is a pathway from HF BO to H(F)BO cannot be ruled out at this time. Recent experimental work by S. Anderson and his group at Stony Brook Universityz5have found that the elimination of an HFBO complex is the dominant surface reaction when HF reacts with boron clusters. Also they found that complex formation is more likely than the formation of
+
+
+
+
Sot0
6546 J. Phys. Chem., Vol. 99,No. 17, 1995 RP for H(F)BO ---> FBO
arc length
+
Rate Constant Calculstlon for H(F)EO ---> FBO + H
H
0
(0)
-
Figure 6. Ground-state vibrationally adiabatic energy curves on the 11/10 CASSCF and 3/3 MRCI PES of H(F)BO H FJ30.
+
FBO. These findings are consistent with the premise that there
+
could be a facile pathway from HF BO to H(F)BO. More importantly, their findings coincide with these calculations that the complex will be formed before FBO can be formed.
0.5
1
1.5
2
2.5
3
3.5
1000/T
Figure 7. Calculation of the temperature-dependent rate constant for reaction 3b using CVT. resulting Arrhenius curve is shown in Figure 7. The temperature-dependent rate constant in the three-parameter form is k36 = (7.00 x lOI3) To.o’ exp(-7817/T) SKI.
VI. Summary and Conclusions
V. FBO Formation The optimized structure, harmonic vibrational frequencies, and zero-point energy for the saddle point of reaction 3b are given in Table 7. At this point the molecule is planar and nonlinear. This structure is very similar to the structure at the saddle point of reaction 3a, except, in this case, the H is perpendicular to the FBO fragment. The BH bond distance is very long at 1.603 A. The BO and BF bond distances are very close to what they are at the products. In the process of going from H(F)BO to FBO H, the BO and BF bond distances have to shrink by 0.137 and 0.043 A, respectively. At the saddle point they have already shrunk by 0.113 and 0.028 A, which is 82% and 65% of the total change, respectively. This reaction is endothermic; the predicted endothermicity at the 3/3 MRCI level is 9.9 kcal/mol. With the larger 5/5 MRCI wavefunction, it is 10.8 kcaYmo1. RP calculations were done in the same way as previously described. In this case, single-point 3/3 MRCI calculations were done for only 10 steps in the reactants’ direction and 15 steps in the products’ direction. Since the reaction is endothermic it was more important to characterize the products’ side. Figure 6 shows the energy curves at both levels. The energy maximum or the barrier on the 11/10 CASSCF energy curve is 8.6 kcal/ mol above the reactants and it occurs at s = -0.05. On the 3/3 MRCI energy curve the maximum is 14.5 kcaYmol and it is at s = 0.40. This change in the transition state toward the products is consistent with Hammond’s postulate that the transition state for an endothermic reaction will be closer to the products on the RP. Without the RP, one could assume that the maximum of the 3/3 MRCI curve would be at the saddle point of the 11/10 CASSCF curve. At that point a 3/3 MRCI calculation predicts a barrier of 13.0 kcal/mol. This value is 10% lower than the value from the 3/3 MRCI energy curve and this error would have been introduced into the rate constant calculation if the RP had not been evaluated. Rate constant calculations were done by using CVT and the 3/3 MRCI energy curve. Tunneling and anharmonic effects were not included in the calculation of the rate constant. The
+
Features of the PES of HBOF have been characterized at the 11/10 CASSCF level. Improved energy predictions have been obtained at the 3/3 MRCI level. The structures and harmonic vibrational frequencies for several stationary points have been discussed. Furthermore, transition-state theory and CVT have been used to obtain rate constants for reactions 2, 3a, and 3b. There are several important conclusions that result from this work. First of all, the most likely mechanism for reaction 1 is through complex formation and subsequent FBO elimination, i.e., reactions 3a and 3b. The first step is highly exothermic and the second step is endothermic. Second, complex formation is very facile and dominates the reaction of HBO F at low temperatures. Further proof that complex formation is facile is the experimental work of Anderson25 and his group. They found that complex formation dominates the reaction of boron oxide clusters with HF. In fact a pathway from BO HF cannot be ruled out at this time. Third, at temperatures exceeding 1500 K, complex formation competes with the abstraction reaction. The rate constants and barriers obtained here should be incorporated in the combustion model of B/O/H/C/F systems along with the recommendation that the model include reactions of the H(F)BO complex.
+
+
Acknowledgment. It is my pleasure to acknowledge fruitful discussions with Dr. Michael Page of North Dakota State University. This work was supported by the Office of Naval Research (ONR) through the Mechanics Division. Also, the computational work was done with grants for supercomputing time through the Department of Defense’s High Performance Computing Modernization Plan (HPC-MP). References and Notes (1) Brown, R. C.; Kolb, C. E.; Yetter, R. A.; Dryer, F. L.; Rabitz, H. Gas Phase Kinetics Modeling and Sensitivity Analysis of BIHIOICIF Combustion Systems; Report, ARI-RR-964; 1993; Vol. 93, Abstract NO. 328, p 536. (2) JANAF Thermochemical Tables, 2nd ed.; National Bureau of Standards: Washington, DC, 1971.
Rate Constant Determinations of HBO
+ F Channels
(3) Nguyen, M. T.; Vanquickborne, L. G.; Sana, M.; Leory, G. J . Phys. Chem. 1993, 97, 5224. (4) Nguyen, M. T.; Groarke, P. J.; Ha, T. Mol. Phys. 1992, 75, 1105. (5) Soto, M. R.; Page, M. J . Phys. Chem. 1990,94, 3242. (6) Soto, M. R.; McKee, M. L.; Page, M. Chem. Phys. Lett. 1991,187, 335. (7) Page, Soto, M. R. J . Chem. Phys. 1993, 99, 7709. (8) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Znitio Molecular Orbital Theory; John Wiley & Sons: New York, 1986; p 95. (9) Dunning, T. H., Jr. J . Chem. Phys. 1989, 90, 1007. (10) Kendall. R. A,: Dunnine. T. H.. Jr.: Harrison. R. J. J . Chem. Phvs. 1992.96, 6796. (11) Paw, - M.; Doubledav, C.: McIver, J . W., Jr. J . Chem. Phys. 1990, 93, 5634. (12) Smith, I. W. M. Kinetics and Dynamics of Elementary Gas Reactions; Butteworths: London, 1980. (13) Truhlar, D.; Isaacson, A,; Garrett, B. C. In Theory of Chemical Reaction Dvnamics: Baer. M.. Ed.: Chemical Rubber: Boca Raton, FL. 1985; Vol. iV, p 65. (14) Saxe, P.; Martin, R.; Page, M.; Lenesfield, B. H. MESA ((Molecular Electronic Structure Applicatiok).
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J. Phys. Chem., Vol. 99,No. 17, 1995 6547 (15) Kawashima, Y.; Endo, Y.; Hirota, E. J. Mol. Spectrosc. 1989,133, 116. (16) Page, M. J . Phys. Chem. 1989, 93, 3639. (17) DeFrees, D. J.; Binkley, J. S.; McLean, A. D. J . Chem. Phys. 1984, 80, 3720. (18) Tynell, J. J . Phys. Chem. 1979, 83, 2906. (19) Roos, B. In Advances in Chemical Physics; Lawley, K. P., Ed.; John Wiley & Sons: Chichester, 1987; Vol. 69, p 399. (20) Kraka, E.; Dunning, T. H., Jr. In Advances in Molecular Electronic Structure Theory; Dunning, T. H., Ed.; JAI: Greenwich, 1990; pp 129173. (21) Natanson, G. A.; Garrett, B. C.; Truong, T. N.; Joseph, T.; Truhlar, D. G. J . Chem. Phys. 1991, 94, 7875. (22) Kawashima, Y.; Kawaguchi, K.; Endo, Y.; Hirota, E. J. Chem. Phys. 1987, 87, 2006. (23) Snelson, A. High Temp. Sci. 1972, 4, 318. (24) Hase, W. L.; Wardlaw, D. M. In Bimolecular Collisions;Baggott, J. E., Ashfold, M. N., Eds.; Burlington House: London, 1989; p 171. (25) Anderson, S. Private communication. JP942261Z
