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A priori rate constant for the reaction of BH with NO
CHEMICALPHYSICSLETTERS
Volume 168,number 2
27 April 1990
A PRIORI RATE CONSTANT FOR THE REACTION OF BH WITH NO L.F. PHILLIPS Chemistry Department, UniversityofCanterbury, Christchurch, New Zealand Received 3 I May 1989; in final form 13 February 1990
The reaction involves two short-lived collision complexes, BHNOf and HBONt, which rearrange to products BO+NH via the same cyclic transition state. The overall rate constant depends on the rates of complex formation by capture over the centrifugal barrier in the dipole-dipole potential, and the rates of unimolecular decomposition and rearrangement of the complexes. RRKMtype calculations give acceptable values for the rate of rearrangement, but give unreasonably large values for the rate of dissociation because the system is small. Good agreement with experiment is obtained when the reciprocal of the collision duration is used for the complex dissociation rate.
1. Introduction
The reaction of BH with NO is a fairly typical radical-radical reaction, in that it is fast, with a rate constant that has a small, negative temperature coefficient [ I] and is independent of pressure [ 21, despite the presence of a deep attractive well corresponding to the bound species HBON [ 3 1. As in a previous study of the reaction of NH2 with NO [ 4 1, the rate constant has been calculated by using collision theory to obtain the rate of formation of HBONt by capture over the centrifugal barrier in the dipole-dipole potential, and using RRKM-type calculations to obtain the rates of dissociation and rearrangement of the complex. Our initial attempt to model this reaction [ 1] was unsatisfactory in that, although the calculated rate constants were in good agreement with experiment, in order to obtain this agreement it was necessary to make untestable assumptions about the nature of the transition state for dissociation of HBONt back to reactants. The present work avoids such assumptions but encounters a new difficulty, in that the results show that the HBON complex is actually too small for its rate of dissociation at fixed total energy E to be calculated in the usual way, from numbers and densities of states, with the formula ME) = w(E)lhp(E)
,
(1)
where w(E) is the number of energy levels of the transition state above the energy barrier for dissociation and p(E) the density of states of the dissociating species. The complex lifetime obtained by averaging eq. ( 1) over a thermal energy distribution, assuming the transition state for dissociation to lie at the top of the average centrifugal barrier for the association process, is less than the reciprocal of the average collision duration, as obtained from the capture-rate calculation. Such a calculation can be meaningful only if it leads to a dissociation rate smaller, and a lifetime longer, than that corresponding to an elastic collision. For the rate of rearrangement of the complex, the rate constant is smaller and this difficulty does not arise. In practice, for the BH+NO system, it has proven to be acceptable to take the reciprocal of the collision duration as the dissociation rate, and it seems likely that the same approach will be applicable to other small systems. A second aspect in which the present work differs significantly from the previous study is the calculation of capture rates. The present calculations used a new computer program [ 5 ] which obtains capture cross-sections for angle-dependent potentials by treating collisions with different angular factors separately, rather than by averaging over the attractive angular factors as before [ 6 1. The new program provides for the inclusion of London forces and other weak forces, such as dipole-quadrupole forces, in the
0009-2614/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland)
197
Volume 168,number 2
CHEMICALPHYSICSLETTERS
attractive potential and allows replacement of the short-range Morse potential by a hard-sphere potential. Also, in addition to the usual requirement that the energy difference between the extremes of the anisotropic potential be large enough to stop the reactants from rotating relative to one another, the algorithm requires successful collision trajectories to reach a minimum “repulsive wall” radius (usually set at r, for the Morse potential) once the centrifugal barrier has been crossed. There are also options to allow calculation of the mean collision duration and of the mean barrier height and radius at a given temperature, values calculated for individual trajectories being weighted according to the product of collision velocity and capture cross-section. Ion-dipole and dipole-dipole capture rates obtained with the new program typically differ by less than 10% from those obtained with the old program or by other workers (for comparisons see ref. [ 51).
2. The capture rate Table 1 shows data used in the calculation of capture rates for the BH +NO system. The isotropic part of the interaction potential comprised the sum of the London potential, given approximately by [ 71 V=-3~A(YBhuAVB/2(VA+Vg)r6,
(2)
where VAand VBare frequencies of the lowest electronic transitions of species A and B and the other symbols have their usual meanings, and the two dipole-induced dipole potentials V=-~c~a~(3cos~8,+1)/2r~
(3)
with the term in parentheses averaged to the value 2.5. The angular factor in the dipole-dipole potential
Table 1 Data used in capture-rate calculations
Dipolemoment (D) Polarizability ( 10mz4cm’) London frequency v, (cm-‘) HBON Morse parameters: r.=1.364A, D,=132kJ/mol, 198
BH
NO
1.27 3.3 2x 10’
0.16 1.68 5x104
w.=1184cm-’
27 April 1990
v= - p,& ( 2 cos e, cos e, -sin BAsin $ cos $)/r3
(4)
(angles defined as in ref. [ 71) was evaluated for sixty values of each of the three angles in the range 0- 180’ ; the value of the factor in parentheses ranged from + 2 to - 2. The number of values of the angular factor in each of twenty-four sub-ranges between +2 and -2 was then calculated, normalised to a sum of unity, and used as a weighting factor for the crosssection obtained with the angular factor of that subrange. Further details are given in ref. [ 5 1. At small values of r, such that the Morse potential was deeper than the most strongly bound combination of the other potentials, and provided the angular factor in eq. (4) was positive, the Morse potential superseded all the others. The first three columns of table 2 contain temperatures and capture rates calculated using Morse and hard-sphere potentials, the hard-sphere radius being set at r, for the Morse potential. The Morse and hard-sphere capture rates are almost identical at low temperatures, but the hard-sphere rate falls off markedly at high temperatures. This is because the anisotropy of the dipole-dipole potential alone is insuffL!ient to stop the rotation of the reactants at large J, so that for the hard-sphere potential it is only a diminishing population of low-J reactants that contributes to the capture rate when the temperature is increased. The Morse well is needed to quench rotations at high temperatures. Capture rates calculated using the Morse potential are larger than experimental rate constants (available in the temperature range 250-350 K) by about a factor of 2. Mean collisional lifetimes (calculated as twice the time required to travel from the top of the barrier to the repulsive wall) are given for Morse and hardsphere potentials in columns four and five of table 2 and, for the case of the Morse potential, the time spent actually within the Morse well is given in column six. It would appear, from the vaIues in column six, that most of the calculated collision duration is spent outside the Morse well. However, this conclusion is weakened somewhat by the known long-range characteristics of Morse potentials, namely, that they go too rapidly to zero at large distances. A second consequence of the present treatment of the transi-
CHEMICAL PHYSICS LETTERS
Volume 168, number 2
27 April 1990
Table 2 Results of capture-rate calculations .) T
10 20 30 50 75 100 150 200 250 300 350 400 500 600
&Morse)
k(h-s)
t(Morse)
f(h-s)
t (well)
r
V
h
Vl
2.14 2.43 2.26 2.07 1:93 1.84 1.79 1.80 1.85 1.91 1.98 2.05 2.20 2.33
2.14 2.44 2.31 2.18 1.99 1.75 1.30 0.97 0.74 0.58 0.46 0.38 0.27 0.20
32.3 32.5 30.2 26.6 21.1 17.5 12.0 9.11 6.87 5.40 4.48 3.79 2.91 2.37
32.6 32.8 30.3 24.9 16.3 11.3 5.83 3.47 2.38 1.71 1.33 1.08 0.74 0.56
0.208 0.207 0.206 0.202 0.197 0.193 0.187 0.183 0.180 0.177 0.175 0.173 0.168 0.163
14.0 12.2 11.0 9.88 8.92 8.22 7.20 6.52 5.97 5.5% 5.30 5.08 4.78 4.58
0.034 0.065 0.106 0.202 0.332 0.484 0.848 1.27 1.72 2.17 2.62 3.06 3.94 4.17
4.9 6.3 7.5 8.6 9.5 10.7 13.0 15.1 17.0 18.7 20.1 21.1 22. I 24.0
1.9 2.4 2.8 3.3 3.6 4.0 4.9 5.7 6.4 7.0 7.5 7.9 8.5 9.0
‘I Units: T, K, rateconstants k, lo-“’ cm’ molecule-’ SK’;collision duration 1,ps; barrier radius r (between centres of mass), height V, kJ/mol; oscillation frequency v, cm- I.
tion between long-range and Morse potentials now appears, namely, that the region where the transition occurs sets an effective lower limit to the radius of the centrifugal barrier over a significant range of collision energies. As the ‘collision energy is increased from a value near zero, the centrifugal barrier moves to shorter gnd shorter distances until it ‘reaches the region where the long-range l/r” potential is superseded by the steeply falling Morse potential. From this point on, a large increase in collision energy is required to bring about a significant decrease in the radius of the centrifugal barrier. Because of this effect, which is unlikely to be entirely an artifact, longer collisional lifetimes are obtained with the Morse potential than the hard-sphere potential at high temperatures. These results show the importance of understanding the behaviour of bound potentials at long range [ 8 1. The procedure adopted here, of switching discontinuously between the Morse and long-range potentials, is certainly suspect, though probably adequate for the present purpose. The last four columns of table 2 contain values for the mean radius and height of the centrifugal barrier, as obtained with the Morse potential, and values of the dipole oscillation frequency calculated for the BH and NO Qecies separated by the mean barrier radius. The oscillation frequency (s- ’ ) for a molecule with moment of inertia I is given by
A;barrier
v= (1/2lr)Jorll.
(5)
or, in cm-i, F=mhc,
where B is the rotational constant (s-l) -dV/d&cuO
(6)
and (7)
expresses the dependence of the potential energy on the angle 8. The value of a is obtained by differentiating (4) for angles 0 close to zero, with cos Q averaged to zero, which gives, for the dipole-dipole potential a=2pApJr3.
(8)
At large separations, such that the calculated vibmtional frequency is less than twice the corresponding rotational constant, the motion of the dipole is better regarded as either a hindered or free rotation, as described later.
3. The potential surface Energies, geometries and vibrational frequencies of intermediates and transition states in the BHfNO system have been calculated by Harrison and Maclagan [ 31. In this work, their vibrational frequencies are used with a scaling factor of 0.89 [ 91. 199
Volume 168, number 2
There are two bound species, HBON and HBNO, of which only the first is capable of undergoing rearrangement to low-energy products (BO + NH ) with no activation barrier. Kinetically, HBON is much more likely to be formed than HBNO because the HB-ON orientation if favoured and the HB-NO orientation opposed by the dipole-dipole potential; and the possibility of forming HBNO is ignored in the present study. With respect to the BH and NO reactants at Q’K, trans-HBON is calculated to lie at - 133.0 kJ/mol, the rectangular transition state HBON* at - 57.3 kJ/mol, and the products BO+NH at - 193.5 kJ/mol (experimental value: -177 kJ/ mol [lo]). The collision complex with orientation BH-NO is also favoured by the dipole-dipole potential but does not correspond to a chemically bound species. However, this complex is capable of rearranging to BO+NH, via the same HBON* transition state as the HBONt complex, so it cannot be neglected. At r=rC for the Morse potential of HBON, BHNO is bound to the extent of about 2 kJ/mol. Its formation rate is essentially equal to the hard-sphere capture rate, because quenching of rotational energy by the Morse potential occurs only for the HB-ON orientation. At long range, the BH-NO and HB-ON otientations are equally favoured; thus the effective capture rate for forming species that can rearrange to BO+NH is the average of the hard-sphere and Morse capture rates, and the average collision duration, measured from the barrier, is the mean of the hard-sphere and Morse collision durations.
4. Rates of rearrangement and dissociation The reaction scheme and labels for rate constants are shown in fig. 1. The quantities cyoand cul represent the fraction of the HBON) transition state that dissociates to products when formed via complexes HBONt and BHNOi, respectively, a fraction 1-cr going to form the other complex. It is reasonable to assume that rearrangement to form the other complex is unimportant because of the relatively large volume of phase space available to the products BO+NH. (This kind of assumption is liable to be invalidated by dynamics for such a small system, but in the present instance it appears that the results re200
27 April 1990
CHEMICAL PHYSICS LETTERS BH + NO
HB + ON k-l 0
kll k-11
\
/ HBONt
BHNOt
(I-“b
k10
k&2;-yzo
I
-me I*
lj___N
al k21
ao k20 II BO t NH
Fig. 1. Reaction scheme for the BH+NO system. BHNO’ and HBONt are short-lived complexes; BHON* (four-membered ring) is a transition state. The transition states for complex dissociation are not shown.
quire (Yto be close to 1 for both channels because the experimental rate constant is only slightly smaller than the average of the Morse and hard-sphere capture rates. ) A steady-state analysis for the scheme of fig. 1 gives d[products]/dt=
[BH] [NO]
x~2r(ko+Lo)(k,
+L)
-(1-a,)(l-~,)~,,~zol}-‘.
(9)
The rate constant kzo for rearrangement of the HBONt complex via the BHON* transition state was calculated using eq. (1) in the same manner as before [ 41, with results as shown in the second column of table 3. The energy range was too small for results to be obtained for a temperature of 10 K, and for temperatures below 200 K it was necessary to use an energy increment of only 20 cm-r in the BeyerSwinehart routine and to scale the results by the ratio of rate constants for an increment of 100 cm-’ to those for 20 cm-’ as calculated at higher temperatures (scaling factor = 1.03). The rearrangement is seen to be fast, but not unreasonably so. The short lifetime with respect to rearrangement ensures that the rate constant for the overall reaction cannot
Table 3 Rate constants kzo, k- ,o and k, T(K)
.)
k 20
k-m
cc
4.43(11) 4.85(11) 5.18(11) 5.35(11) 5.46(11) 5.67(11) 5.91(11) 6.15(11) 6.39(11) 6.66(11) 6.87( I 1) 7.36(11) 7.85(11)
7.08( 1I) 1.03( 12) 1.79( 12) 2.77( 12) 3.78( 12) 5.65( 12) 8.41(12) 1.08( 13) 1.32( 13) 1.58( 13) 1.84( 13) 2.40( 13) 3.03( 13)
2.65(-10) 2.35(-10) 2.21(-10) 2.05(-10) 1.88(-10) 1.71(-10) 1.43(-10) 1.24(-10) 1.12(-10) 1.03( - 10) 0.97(-10) 0.93(-10) 0.88(-10) 0.85(-10)
10
20 30 50 15 too 150 200 250 300 350 400 500 600
27 April 1990
CHEMICAL PHYSICS LETTERS
Volume 168, number 2
‘) Units: s-l for kzo and k_ 10,cm3 molecule-’ s-l for &,, ers of 10 in parentheses.
pow-
show any pressure dependence at ordinary pressures. Values of the dissociation rate constant k_ 1owere calculated for HBONt by assuming the loose transition state to lie at the top of the average centrifugal barrier. The calculated vibration frequencies given in table 2 were used for the doubly degenerate vibrations of the BH and NO dipoles about their centres of charge, with exceptions as follows: when the vibration frequency was between B and 2B for the particular dipole, regarded as a rotor, the motion was treated as a two-dimensional hindered rotation, i.e. as a free rotation with rotational constant equal to the calculated vibration frequency, and when the calculated vibration frequency was less than the corresponding B value, the motion was treated as a twodimensional free rotation with rotational constant B. The resulting values of k_lo are given in the third column of table 3. The calculated values of k_ 1,,are such that the lifetime with respect to dissociation is much less than the collision duration measured from the rotational barrier, and at temperatures of 150 K and above is less even than the calculated residence time in the Morse well. This is physically unreasonable, and implies that the HBON system is one in which the simple inertia of the fragments is a limiting factor for the dissociation rate. (The rate of migration of energy into the reaction coordinate, i.e. of intramolecular vibrational relaxation (IVR), might also limit the
value of k_ ,,,, thereby malting the lifetime longer than the collision duration, but there is at present no means of estimating the rate of IVR in this system. ) Also, if the dissociation rate were actually as large as the calculated value of k_ lo, the rearrangement with the rate constant kzo could not compete and the overall rate constant would be smaller than the capture rate by a factor of about 15 at 250 K, instead of a factor of 1.3. We are therefore led to test the effect of using the reciprocal of the collision duration in place of the calculated k_ 1o.There is some question as to which is the correct value of the collision duration to use: the collision duration measured from the centrifugal barrier seems likely to be the right one when the process that competes with dissociation is one of collisional stabilization, but the time spent actually within the Morse well would seem to be more appropriate for a complex undergoing rearrangement. However, the calculated Morse-well residence times are uncertain to the extent that the long-range behaviour of the Morse potential is incorrect, and they are also so short as to give an unacceptably low value for the overall rate constant unless some other effect, such as slow IVR, is postulated. Since we have no way of treating such an effect, we use the collision duration measured from the barrier. The collision complex with orientation BH-NO is too weakly bound for the rate of rearrangement to be calculated with eq. ( 1), but it is clear that kzl must be much larger than k,, because of the low energylevel density that results from the absence of a deep potential well for BH-NO. For k_, ,, on the other hand, we can again use the reciprocal of the collision duration, which means that k_ , , must be negligible in comparison with k,,. Making the assumption that both cy, and (Y~are equal to 1 and putting kzol(k,, +k-,,)=f,
(10)
we find d[products]/dt=k,CIBH] =f(W+k,,~PHlWl
[NO] .
(11)
This gives the k,, values shown in the last column of table 3. The value for T= 10 K was obtained by assumingfto have the same value, namely 0.935, at 10 and 20 IL 201
Volume 168, number 2
CHEMICAL PHYSICS LETPERS
I
0 0
I
27April iv90
I
300
Temperature/K
I
600
Fig. 2. Calculated and experimental values of the overall rate constant. (Units: cm’ molecule -r 6-r.) Hollow circles: calculated values. Hollow rectangles: experiment, ref. [ 11. Solid diamond: experiment, ref. [2].
The calculated and experimental values of kte are compared in fig. 2, and the agreement is seen to be very good. The calculations do not reproduce the measured temperature dependence exactly, but are at least qualitatively correct. This agreement stems mainly from the fact that the overall rate is only slightly less than the mean capture rate for the HBON and BH-NO orientations, so that the results are not very sensitive to the actual values of kzoand k_ 1o, provided k_ 1ois not too large relative to k20.For the same reason, the results are not very sensitive to the accuracy of the quantum-chemical calculations: raising the barrier to rearrangement by 10 kJ/mol decreases kzoby about 30%, but reduces kte at 300 R by only 6%. Future studies will involve reactions, such as that of NH with NO*, for which the measured rate is much less than the calculated capture rate.
Acknowledgement I am grateful to Dr. H.H. Nelson and coworkers
202
for sending results prior to publication. This work was supported by the New Zealand Universities Research Committee.
References [ 1 ] J.A. Harrison, R.F. Meads and L.F. Phillips, Chem. Phys. Letters 150 (1988) 289. [2] JR. Rice, N.J. Caldwell and H.H. Nelson, J. Phys. Chem. 93 (1989) 3600. [ 3 ] J.A. Harrison and R.G.A.R. Mactagan, Chem. Phys. Letters 146 (1988) 243. [4] L.F. Phillips, Chem. Phys. Letters 135 (1987) 269. [ 5 ] L.F. Phillips, I. Comput. Chem., submitted for publication. [ 6) L.F. Phillips, J. Chem. Sot. Faraday Trans. JJ 83 (1987) 857. [ 7 ] J.O. Hirschfelder, CF. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1967) p. 27. [S]R.J. LeRoy, J.S. Carley and J.E. Grabenstetter, Faraday Discussions Chem. Sot. 62 (1977) 169. [9]D.J.DeFreesandA.D. McLean, J. C&em. Phys. 82 (1985) 333. [IO] J&P. Huber and G. Henberg, Molecular spectra and molecular structure+Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979).
Volume 168,number 2
27 April 1990
A PRIORI RATE CONSTANT FOR THE REACTION OF BH WITH NO L.F. PHILLIPS Chemistry Department, UniversityofCanterbury, Christchurch, New Zealand Received 3 I May 1989; in final form 13 February 1990
The reaction involves two short-lived collision complexes, BHNOf and HBONt, which rearrange to products BO+NH via the same cyclic transition state. The overall rate constant depends on the rates of complex formation by capture over the centrifugal barrier in the dipole-dipole potential, and the rates of unimolecular decomposition and rearrangement of the complexes. RRKMtype calculations give acceptable values for the rate of rearrangement, but give unreasonably large values for the rate of dissociation because the system is small. Good agreement with experiment is obtained when the reciprocal of the collision duration is used for the complex dissociation rate.
1. Introduction
The reaction of BH with NO is a fairly typical radical-radical reaction, in that it is fast, with a rate constant that has a small, negative temperature coefficient [ I] and is independent of pressure [ 21, despite the presence of a deep attractive well corresponding to the bound species HBON [ 3 1. As in a previous study of the reaction of NH2 with NO [ 4 1, the rate constant has been calculated by using collision theory to obtain the rate of formation of HBONt by capture over the centrifugal barrier in the dipole-dipole potential, and using RRKM-type calculations to obtain the rates of dissociation and rearrangement of the complex. Our initial attempt to model this reaction [ 1] was unsatisfactory in that, although the calculated rate constants were in good agreement with experiment, in order to obtain this agreement it was necessary to make untestable assumptions about the nature of the transition state for dissociation of HBONt back to reactants. The present work avoids such assumptions but encounters a new difficulty, in that the results show that the HBON complex is actually too small for its rate of dissociation at fixed total energy E to be calculated in the usual way, from numbers and densities of states, with the formula ME) = w(E)lhp(E)
,
(1)
where w(E) is the number of energy levels of the transition state above the energy barrier for dissociation and p(E) the density of states of the dissociating species. The complex lifetime obtained by averaging eq. ( 1) over a thermal energy distribution, assuming the transition state for dissociation to lie at the top of the average centrifugal barrier for the association process, is less than the reciprocal of the average collision duration, as obtained from the capture-rate calculation. Such a calculation can be meaningful only if it leads to a dissociation rate smaller, and a lifetime longer, than that corresponding to an elastic collision. For the rate of rearrangement of the complex, the rate constant is smaller and this difficulty does not arise. In practice, for the BH+NO system, it has proven to be acceptable to take the reciprocal of the collision duration as the dissociation rate, and it seems likely that the same approach will be applicable to other small systems. A second aspect in which the present work differs significantly from the previous study is the calculation of capture rates. The present calculations used a new computer program [ 5 ] which obtains capture cross-sections for angle-dependent potentials by treating collisions with different angular factors separately, rather than by averaging over the attractive angular factors as before [ 6 1. The new program provides for the inclusion of London forces and other weak forces, such as dipole-quadrupole forces, in the
0009-2614/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland)
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Volume 168,number 2
CHEMICALPHYSICSLETTERS
attractive potential and allows replacement of the short-range Morse potential by a hard-sphere potential. Also, in addition to the usual requirement that the energy difference between the extremes of the anisotropic potential be large enough to stop the reactants from rotating relative to one another, the algorithm requires successful collision trajectories to reach a minimum “repulsive wall” radius (usually set at r, for the Morse potential) once the centrifugal barrier has been crossed. There are also options to allow calculation of the mean collision duration and of the mean barrier height and radius at a given temperature, values calculated for individual trajectories being weighted according to the product of collision velocity and capture cross-section. Ion-dipole and dipole-dipole capture rates obtained with the new program typically differ by less than 10% from those obtained with the old program or by other workers (for comparisons see ref. [ 51).
2. The capture rate Table 1 shows data used in the calculation of capture rates for the BH +NO system. The isotropic part of the interaction potential comprised the sum of the London potential, given approximately by [ 71 V=-3~A(YBhuAVB/2(VA+Vg)r6,
(2)
where VAand VBare frequencies of the lowest electronic transitions of species A and B and the other symbols have their usual meanings, and the two dipole-induced dipole potentials V=-~c~a~(3cos~8,+1)/2r~
(3)
with the term in parentheses averaged to the value 2.5. The angular factor in the dipole-dipole potential
Table 1 Data used in capture-rate calculations
Dipolemoment (D) Polarizability ( 10mz4cm’) London frequency v, (cm-‘) HBON Morse parameters: r.=1.364A, D,=132kJ/mol, 198
BH
NO
1.27 3.3 2x 10’
0.16 1.68 5x104
w.=1184cm-’
27 April 1990
v= - p,& ( 2 cos e, cos e, -sin BAsin $ cos $)/r3
(4)
(angles defined as in ref. [ 71) was evaluated for sixty values of each of the three angles in the range 0- 180’ ; the value of the factor in parentheses ranged from + 2 to - 2. The number of values of the angular factor in each of twenty-four sub-ranges between +2 and -2 was then calculated, normalised to a sum of unity, and used as a weighting factor for the crosssection obtained with the angular factor of that subrange. Further details are given in ref. [ 5 1. At small values of r, such that the Morse potential was deeper than the most strongly bound combination of the other potentials, and provided the angular factor in eq. (4) was positive, the Morse potential superseded all the others. The first three columns of table 2 contain temperatures and capture rates calculated using Morse and hard-sphere potentials, the hard-sphere radius being set at r, for the Morse potential. The Morse and hard-sphere capture rates are almost identical at low temperatures, but the hard-sphere rate falls off markedly at high temperatures. This is because the anisotropy of the dipole-dipole potential alone is insuffL!ient to stop the rotation of the reactants at large J, so that for the hard-sphere potential it is only a diminishing population of low-J reactants that contributes to the capture rate when the temperature is increased. The Morse well is needed to quench rotations at high temperatures. Capture rates calculated using the Morse potential are larger than experimental rate constants (available in the temperature range 250-350 K) by about a factor of 2. Mean collisional lifetimes (calculated as twice the time required to travel from the top of the barrier to the repulsive wall) are given for Morse and hardsphere potentials in columns four and five of table 2 and, for the case of the Morse potential, the time spent actually within the Morse well is given in column six. It would appear, from the vaIues in column six, that most of the calculated collision duration is spent outside the Morse well. However, this conclusion is weakened somewhat by the known long-range characteristics of Morse potentials, namely, that they go too rapidly to zero at large distances. A second consequence of the present treatment of the transi-
CHEMICAL PHYSICS LETTERS
Volume 168, number 2
27 April 1990
Table 2 Results of capture-rate calculations .) T
10 20 30 50 75 100 150 200 250 300 350 400 500 600
&Morse)
k(h-s)
t(Morse)
f(h-s)
t (well)
r
V
h
Vl
2.14 2.43 2.26 2.07 1:93 1.84 1.79 1.80 1.85 1.91 1.98 2.05 2.20 2.33
2.14 2.44 2.31 2.18 1.99 1.75 1.30 0.97 0.74 0.58 0.46 0.38 0.27 0.20
32.3 32.5 30.2 26.6 21.1 17.5 12.0 9.11 6.87 5.40 4.48 3.79 2.91 2.37
32.6 32.8 30.3 24.9 16.3 11.3 5.83 3.47 2.38 1.71 1.33 1.08 0.74 0.56
0.208 0.207 0.206 0.202 0.197 0.193 0.187 0.183 0.180 0.177 0.175 0.173 0.168 0.163
14.0 12.2 11.0 9.88 8.92 8.22 7.20 6.52 5.97 5.5% 5.30 5.08 4.78 4.58
0.034 0.065 0.106 0.202 0.332 0.484 0.848 1.27 1.72 2.17 2.62 3.06 3.94 4.17
4.9 6.3 7.5 8.6 9.5 10.7 13.0 15.1 17.0 18.7 20.1 21.1 22. I 24.0
1.9 2.4 2.8 3.3 3.6 4.0 4.9 5.7 6.4 7.0 7.5 7.9 8.5 9.0
‘I Units: T, K, rateconstants k, lo-“’ cm’ molecule-’ SK’;collision duration 1,ps; barrier radius r (between centres of mass), height V, kJ/mol; oscillation frequency v, cm- I.
tion between long-range and Morse potentials now appears, namely, that the region where the transition occurs sets an effective lower limit to the radius of the centrifugal barrier over a significant range of collision energies. As the ‘collision energy is increased from a value near zero, the centrifugal barrier moves to shorter gnd shorter distances until it ‘reaches the region where the long-range l/r” potential is superseded by the steeply falling Morse potential. From this point on, a large increase in collision energy is required to bring about a significant decrease in the radius of the centrifugal barrier. Because of this effect, which is unlikely to be entirely an artifact, longer collisional lifetimes are obtained with the Morse potential than the hard-sphere potential at high temperatures. These results show the importance of understanding the behaviour of bound potentials at long range [ 8 1. The procedure adopted here, of switching discontinuously between the Morse and long-range potentials, is certainly suspect, though probably adequate for the present purpose. The last four columns of table 2 contain values for the mean radius and height of the centrifugal barrier, as obtained with the Morse potential, and values of the dipole oscillation frequency calculated for the BH and NO Qecies separated by the mean barrier radius. The oscillation frequency (s- ’ ) for a molecule with moment of inertia I is given by
A;barrier
v= (1/2lr)Jorll.
(5)
or, in cm-i, F=mhc,
where B is the rotational constant (s-l) -dV/d&cuO
(6)
and (7)
expresses the dependence of the potential energy on the angle 8. The value of a is obtained by differentiating (4) for angles 0 close to zero, with cos Q averaged to zero, which gives, for the dipole-dipole potential a=2pApJr3.
(8)
At large separations, such that the calculated vibmtional frequency is less than twice the corresponding rotational constant, the motion of the dipole is better regarded as either a hindered or free rotation, as described later.
3. The potential surface Energies, geometries and vibrational frequencies of intermediates and transition states in the BHfNO system have been calculated by Harrison and Maclagan [ 31. In this work, their vibrational frequencies are used with a scaling factor of 0.89 [ 91. 199
Volume 168, number 2
There are two bound species, HBON and HBNO, of which only the first is capable of undergoing rearrangement to low-energy products (BO + NH ) with no activation barrier. Kinetically, HBON is much more likely to be formed than HBNO because the HB-ON orientation if favoured and the HB-NO orientation opposed by the dipole-dipole potential; and the possibility of forming HBNO is ignored in the present study. With respect to the BH and NO reactants at Q’K, trans-HBON is calculated to lie at - 133.0 kJ/mol, the rectangular transition state HBON* at - 57.3 kJ/mol, and the products BO+NH at - 193.5 kJ/mol (experimental value: -177 kJ/ mol [lo]). The collision complex with orientation BH-NO is also favoured by the dipole-dipole potential but does not correspond to a chemically bound species. However, this complex is capable of rearranging to BO+NH, via the same HBON* transition state as the HBONt complex, so it cannot be neglected. At r=rC for the Morse potential of HBON, BHNO is bound to the extent of about 2 kJ/mol. Its formation rate is essentially equal to the hard-sphere capture rate, because quenching of rotational energy by the Morse potential occurs only for the HB-ON orientation. At long range, the BH-NO and HB-ON otientations are equally favoured; thus the effective capture rate for forming species that can rearrange to BO+NH is the average of the hard-sphere and Morse capture rates, and the average collision duration, measured from the barrier, is the mean of the hard-sphere and Morse collision durations.
4. Rates of rearrangement and dissociation The reaction scheme and labels for rate constants are shown in fig. 1. The quantities cyoand cul represent the fraction of the HBON) transition state that dissociates to products when formed via complexes HBONt and BHNOi, respectively, a fraction 1-cr going to form the other complex. It is reasonable to assume that rearrangement to form the other complex is unimportant because of the relatively large volume of phase space available to the products BO+NH. (This kind of assumption is liable to be invalidated by dynamics for such a small system, but in the present instance it appears that the results re200
27 April 1990
CHEMICAL PHYSICS LETTERS BH + NO
HB + ON k-l 0
kll k-11
\
/ HBONt
BHNOt
(I-“b
k10
k&2;-yzo
I
-me I*
lj___N
al k21
ao k20 II BO t NH
Fig. 1. Reaction scheme for the BH+NO system. BHNO’ and HBONt are short-lived complexes; BHON* (four-membered ring) is a transition state. The transition states for complex dissociation are not shown.
quire (Yto be close to 1 for both channels because the experimental rate constant is only slightly smaller than the average of the Morse and hard-sphere capture rates. ) A steady-state analysis for the scheme of fig. 1 gives d[products]/dt=
[BH] [NO]
x~2r(ko+Lo)(k,
+L)
-(1-a,)(l-~,)~,,~zol}-‘.
(9)
The rate constant kzo for rearrangement of the HBONt complex via the BHON* transition state was calculated using eq. (1) in the same manner as before [ 41, with results as shown in the second column of table 3. The energy range was too small for results to be obtained for a temperature of 10 K, and for temperatures below 200 K it was necessary to use an energy increment of only 20 cm-r in the BeyerSwinehart routine and to scale the results by the ratio of rate constants for an increment of 100 cm-’ to those for 20 cm-’ as calculated at higher temperatures (scaling factor = 1.03). The rearrangement is seen to be fast, but not unreasonably so. The short lifetime with respect to rearrangement ensures that the rate constant for the overall reaction cannot
Table 3 Rate constants kzo, k- ,o and k, T(K)
.)
k 20
k-m
cc
4.43(11) 4.85(11) 5.18(11) 5.35(11) 5.46(11) 5.67(11) 5.91(11) 6.15(11) 6.39(11) 6.66(11) 6.87( I 1) 7.36(11) 7.85(11)
7.08( 1I) 1.03( 12) 1.79( 12) 2.77( 12) 3.78( 12) 5.65( 12) 8.41(12) 1.08( 13) 1.32( 13) 1.58( 13) 1.84( 13) 2.40( 13) 3.03( 13)
2.65(-10) 2.35(-10) 2.21(-10) 2.05(-10) 1.88(-10) 1.71(-10) 1.43(-10) 1.24(-10) 1.12(-10) 1.03( - 10) 0.97(-10) 0.93(-10) 0.88(-10) 0.85(-10)
10
20 30 50 15 too 150 200 250 300 350 400 500 600
27 April 1990
CHEMICAL PHYSICS LETTERS
Volume 168, number 2
‘) Units: s-l for kzo and k_ 10,cm3 molecule-’ s-l for &,, ers of 10 in parentheses.
pow-
show any pressure dependence at ordinary pressures. Values of the dissociation rate constant k_ 1owere calculated for HBONt by assuming the loose transition state to lie at the top of the average centrifugal barrier. The calculated vibration frequencies given in table 2 were used for the doubly degenerate vibrations of the BH and NO dipoles about their centres of charge, with exceptions as follows: when the vibration frequency was between B and 2B for the particular dipole, regarded as a rotor, the motion was treated as a two-dimensional hindered rotation, i.e. as a free rotation with rotational constant equal to the calculated vibration frequency, and when the calculated vibration frequency was less than the corresponding B value, the motion was treated as a twodimensional free rotation with rotational constant B. The resulting values of k_lo are given in the third column of table 3. The calculated values of k_ 1,,are such that the lifetime with respect to dissociation is much less than the collision duration measured from the rotational barrier, and at temperatures of 150 K and above is less even than the calculated residence time in the Morse well. This is physically unreasonable, and implies that the HBON system is one in which the simple inertia of the fragments is a limiting factor for the dissociation rate. (The rate of migration of energy into the reaction coordinate, i.e. of intramolecular vibrational relaxation (IVR), might also limit the
value of k_ ,,,, thereby malting the lifetime longer than the collision duration, but there is at present no means of estimating the rate of IVR in this system. ) Also, if the dissociation rate were actually as large as the calculated value of k_ lo, the rearrangement with the rate constant kzo could not compete and the overall rate constant would be smaller than the capture rate by a factor of about 15 at 250 K, instead of a factor of 1.3. We are therefore led to test the effect of using the reciprocal of the collision duration in place of the calculated k_ 1o.There is some question as to which is the correct value of the collision duration to use: the collision duration measured from the centrifugal barrier seems likely to be the right one when the process that competes with dissociation is one of collisional stabilization, but the time spent actually within the Morse well would seem to be more appropriate for a complex undergoing rearrangement. However, the calculated Morse-well residence times are uncertain to the extent that the long-range behaviour of the Morse potential is incorrect, and they are also so short as to give an unacceptably low value for the overall rate constant unless some other effect, such as slow IVR, is postulated. Since we have no way of treating such an effect, we use the collision duration measured from the barrier. The collision complex with orientation BH-NO is too weakly bound for the rate of rearrangement to be calculated with eq. ( 1), but it is clear that kzl must be much larger than k,, because of the low energylevel density that results from the absence of a deep potential well for BH-NO. For k_, ,, on the other hand, we can again use the reciprocal of the collision duration, which means that k_ , , must be negligible in comparison with k,,. Making the assumption that both cy, and (Y~are equal to 1 and putting kzol(k,, +k-,,)=f,
(10)
we find d[products]/dt=k,CIBH] =f(W+k,,~PHlWl
[NO] .
(11)
This gives the k,, values shown in the last column of table 3. The value for T= 10 K was obtained by assumingfto have the same value, namely 0.935, at 10 and 20 IL 201
Volume 168, number 2
CHEMICAL PHYSICS LETPERS
I
0 0
I
27April iv90
I
300
Temperature/K
I
600
Fig. 2. Calculated and experimental values of the overall rate constant. (Units: cm’ molecule -r 6-r.) Hollow circles: calculated values. Hollow rectangles: experiment, ref. [ 11. Solid diamond: experiment, ref. [2].
The calculated and experimental values of kte are compared in fig. 2, and the agreement is seen to be very good. The calculations do not reproduce the measured temperature dependence exactly, but are at least qualitatively correct. This agreement stems mainly from the fact that the overall rate is only slightly less than the mean capture rate for the HBON and BH-NO orientations, so that the results are not very sensitive to the actual values of kzoand k_ 1o, provided k_ 1ois not too large relative to k20.For the same reason, the results are not very sensitive to the accuracy of the quantum-chemical calculations: raising the barrier to rearrangement by 10 kJ/mol decreases kzoby about 30%, but reduces kte at 300 R by only 6%. Future studies will involve reactions, such as that of NH with NO*, for which the measured rate is much less than the calculated capture rate.
Acknowledgement I am grateful to Dr. H.H. Nelson and coworkers
202
for sending results prior to publication. This work was supported by the New Zealand Universities Research Committee.
References [ 1 ] J.A. Harrison, R.F. Meads and L.F. Phillips, Chem. Phys. Letters 150 (1988) 289. [2] JR. Rice, N.J. Caldwell and H.H. Nelson, J. Phys. Chem. 93 (1989) 3600. [ 3 ] J.A. Harrison and R.G.A.R. Mactagan, Chem. Phys. Letters 146 (1988) 243. [4] L.F. Phillips, Chem. Phys. Letters 135 (1987) 269. [ 5 ] L.F. Phillips, I. Comput. Chem., submitted for publication. [ 6) L.F. Phillips, J. Chem. Sot. Faraday Trans. JJ 83 (1987) 857. [ 7 ] J.O. Hirschfelder, CF. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley, New York, 1967) p. 27. [S]R.J. LeRoy, J.S. Carley and J.E. Grabenstetter, Faraday Discussions Chem. Sot. 62 (1977) 169. [9]D.J.DeFreesandA.D. McLean, J. C&em. Phys. 82 (1985) 333. [IO] J&P. Huber and G. Henberg, Molecular spectra and molecular structure+Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979).