2352
J. Phys. Chem. 1982, 86,2352-2358
Energy Transfer, Stabilization, and Dissociation in Collisions of Helium with Highly Excited HO, Charles R. Gallucci and George C. Schatz'+ Department of Chemistry, Northwestern Universify, Evanston, Illinois 6020 1 (Received: November 30, 198 I; In Final Form: February IO, 1982)
This paper considers application of the classical trajectory method to the determination of cross sections and rate constants for energy transfer, stabilization, and dissociation in collisions of He with highly excited H02. The resulting rate constants are used to evaluate several models of the termolecular recombination kinetics in He + H + 02,such as the weak and strong collision models of stabilization. An ab initio potential energy surface is used for the H02 molecule, and the He + HOP intermolecular potential is approximated as a sum of pairs. Cross sections and rate constants are evaluated for two types of initial HOP*ensembles, one which is microcanonical for the HOPvibrations but cold for rotation, and one which is microcanonical for both vibration and rotation. For both the rotationally cold and hot ensembles, the stabilization cross sections were found to decrease rapidly with internal energy Eintand increase rapidly with translational energy Eo above the dissociation threshold, with the c r w sections always smaller for cold HOP*than for hot. This difference between cold and hot is interpreted in terms of the greater efficiency of R T energy transfer for rotationally hot HOP*. For both H + O2 and OH + 0 dissociation, the cross sections increase rapidly with increasing Eint,with the rotationally cold cross sections generally much larger than rotationally hot. The stabilizing influence of centrifugal barriers was found to be important in determining the smaller rotationally hot cross sections. An analysis of stabilization and H + O2 dissociation rate constants indjcates that dissociation is faster than stabilization for cold HOz*, but neither rate constant is more than a few percent of the gas kinetic rate constant. For hot HOP*, stabilizationis much faster than dissociation, and the stabilization rate constant is 14-1870 of gas kinetic. Roughly speaking, hot H02*is pretty well approximated by the strong collision model (with an appropriate efficiency factor), while cold HOP*is better approximated by using the weak collision approximation. These results emphasize the importance of using accurate stabilization rate constant information in the kinetic modeling of termolecular recombination rates, and the potential importance of collision-induced dissociation in the recombination mechanism.
-
I. Introduction One of the most important reactions in the combustion of hydrogen and hydrogen-containing fuels is the recombination of H with O2 in the presence of a third body M to form HOP:
H
+ O2 + M -% H 0 2 + M
(1)
Although the rate constant for this reaction has been measured many times and for several different third bodies,l a theoretical interpretation of it in terms of the competing elementary steps has not yet been successfully developed. Such an analysis would be useful, however, as this reaction is the prototype for many hydrogen atom recombination reactions (H+ CzH2,H CzH4,H + CH,), most of which are poorly understood theoretically at p r e ~ e n t . ~ In - ~ addition, the relatively small number of electrons involved for M = He makes this system a viable candidate for the determination of an accurate potential energy surface. Indeed, Melius and Blint (MB)5 and Dunning et aL6 have already determined ab initio surfaces for the HOP fragment, and Miller' has used the MB surface to study the bimolecular reaction
+
H
+0 , L O H +0
(2)
which competes with reaction 1, obtaining excellent agreement with measured rate constants. One previous study of the kinetics of reaction 1 was attempted by Blint,* who assumed that it could be described by the commonly used two-step mechanism Alfred P. Sloan Research Fellow and Camille and Henry Dreyfus Teacher-Scholar. 0022-365418212086-2352$01.25/0
H
+ O2
k3
H02*
k-3
2
(3)
+
H02* + M HOP M (4) where H02* is a metastable HOP molecule with energy above that needed to dissociate. Blint used classical trajectories and the MB surface to evaluate the equilibrium constant for reaction 3 and combined this with either a strong or weak collision model for the stabilization step 4 (for M = Ar) to evaluate the termolecular rate constant kl = k3k4/k-,. His results indicated that the temperature dependence of the reaction rate constant depends mainly on the assumed collisional stabilization rate constant, with only a weak dependence on the lifetime distribution of H02*. No quantitative evaluation of k4 was attempted, however, and because of this Blint was unable to come to a firm conclusion concerning the validity of the mechanism of reactions 3 and 4. The lifetime distribution of HOz* has also recently been studied by Brown and Miller,g who found that the amount of H02* rotational excitation plays an important role in HO, unimolecular decay. In this paper we present a classical trajectory study of collisional stabilization and dissociation of H02* by He (1) G. Dixon-Lewis and D. J. Williams, Compr. Chem. Kinet., 17, 1 (1977). (2) D. G. Keil, K. P. Lynch, J. A. Lowfer, and J. V. Michael, Int. J . Chem. Kinet., 8, 825 (1976). (3) W. A. Payne and L. J. Stief, J . Chem. Phys., 64, 1150 (1976). (4) J. H. Lee, J. V. Michael, W. A. Payne, and L. J. Stief, J . Chem. Phys., 68, 1817 (1978). ( 5 ) C. F. Melius and R. J. Blint, Chem. Phys. Lett., 64, 183 (1979). (6) T. H. Dunning, S. P. Walch, and M. M. Goodgame, J. Chem. Phys., 74, 3482 (1981). (7) J. A. Miller, J. Chem. Phys., 74, 5120 (1981). (8) R. J. Blint, J . Chem. Phys., 73, 765 (1980). (9) N. J. Brown and J. A. Miller, J . Phys. Chem., 86, 772 (1982).
0 1982 American Chemical Society
The Journal of Physical Chemistry. Vol. 86, No. 13, 1982 2353
Colllsions of Helium with Highly Excited Hop
atoms. The results of this study will be used to assess the usefulness of the various strong and weak collision models of reaction 4 and to study the contribution of the often neglected dissociation reaction H02*
+ H e k b - H + O2 + He
(5)
to the termolecular rate constant. In addition, we study the competition between reaction 5 and the more endoergic dissociation process H02* + H e 2 OH
+ 0 +He
w -_
-_
(6)
In all of these processes, we examine the dependence of the cross sections and rate constants on the initial phase space distribution used for H02* and on the He atom translational energy. For the two dissociation reactions 5 and 6, the distribution of vibration-rotation states in the diatomic molecule products is studied. This is actually the first theoretical trajectory study of collisional dissociation processes in a metastable triatomic or larger polyatomic molecule,10and in many respects our goal in this initial work is to obtain qualitative rather than quantitative information concerning the possible rate processes. We will not attempt to evaluate kl in this paper (although this could easily be done by using Blint’s mechanism) for, as will be evident from the results, any kinetic model based on reactions 3 and 4 alone may be inadequate. In addition, the MB surface (which is used to describe the H 0 2 fragment dynamics) needs some refinement as the 0.094-eV barrier in the H + O2 channel is probably too higha6 In the next section (section 11)we discuss details of the potential surface and of the trajectory method used, while in section I11 we present and discuss cross sections, energy-transfer moments, and final state distributions for reactions 4-6. An analysis of the collisional stabilization and dissociation rate constants is presented in section IV, along with comparison with the predictions of various models. The conclusions of the paper are summarized in section V. 11. Theory Potential Energy Surface. Labeling the four-atom HeHOO system as ABCD, and defining Rij as the i-to-j distance (i, j = A, B, C, D), we can represent the potential V as v = VHO,(RBC,RBD&D)+ VHeHOz(RAB,RAC,RAD) where VHoz is a slightly modified version of the MB ab initio potential for H 0 2 and VHeHOz is an assumed intermolecular potential. The MB potential surface was modified slightly to remove a cusp in it which occurs at C , geometries and which caused difficulties in the trajectory integrations. The cusp arises because of a discontinuous switch between the two internal angles O> and O< (see ref 5 ) which is made at that geometry as the H atom rotates about OF The cusp was removed by replacing cos O> everywhere in VHol by 1 /J(1 + tanh yA) cos t9> + (1- tanh yA) cos e,] where A = RBC - R B D and y = 10ao-’. This function smoothly changes from cos O> to cos O< as the H atom rotates about 02,and its use in VHoPcauses only small changes in the potential away from the CPvconfiguration. A summary of the energetics of the modified MB surface is indicated in Figure 1. Note that relative to separated (10)One previous study of collisional energy transfer is D. L. Bunker and S. A. Jayich, Chem. Phys., 13,129 (1976).
j
ha2
Figure 1. Schematic of Meiius-Blint6 Hoppotential surface (as slightly modified; see text). Measuring energies relative to the H i0, minimum, there is a 0.094-eV barrier located at R , = 3.725a0, R , = 2.341ao,Run = 5.250a0and a 1 . 8 6 4 well at R, = 1.826a0,Roo = 2.578s0,Row = 3.514a0. The equilibrium 0-0and 0-H distances are 2.316a0 and 1.846a0,respectively. The OH 0 channel is 0.6 eV endoerglc, wlth no barrier to dissociation. Also shown are the zero-point energies for each of the stationary points.
+
+
H 02,the H 0 2 is stable by 1.86 eV (2.26 eV experimentall+”) and that the OH + 0 channel is 0.60 eV more endoergic than H 02.In addition, the MB surface has a 0.094-eV barrier in the H + O2entrance channel, and this defines the classical threshold energy for dissociation (relative to the energy of separated H + 02). This threshold energy is accidentally close to the 0.1-eV quantum threshold energy for dissociation to H O2 (u = 0), although it is well below the 0.21-eV vibrationally adiabatic potential at the saddle point. The intermolecular potential VHeHoz is approximated by a sum of pair potentials VH~O(RAC) + VHeo(RAD) + VHeH(Rm). Following Suzukawa et al.,12 we take VHeoto be the He-Ne interaction potential and VHeH to be the He-He ~ 0 t e n t i a l . l Although ~ the use of pair potentials to model the intermolecular interactions in this case is probably a significant approximation, recent ab initio calculations have shown14that such potentials can provide a correct zeroth-order description and that, for qualitative purposes, this approximation is often adequate. Trajectory Calculations. Three-dimensional quasiclassical trajectories were integrated for the He H02* system using a standard fifth-order predictor-corrector algorithm with a time step of 1.2 X s. Initial conditions were randomly sampled, with the H02* initial phase space distribution taken to be microcanonical. This is actually an assumption which is consistent with the common usage of RRKM theory to determine the unimolecular rate constant k3, but it may or may not mimic the distribution of H02*’s actually produced by chemical activation. Although we plan future calculations in which the initial H02* either is prepared directly by H O2 collisions or is indirectly inferred through solutions of a master equation for collisional excitation/deexcitation in H02*, the present work will consider only microcanonical ensembles. In defining a microcanonical ensemble for H02*, we consider two limiting treatments of rotational motions. In the first (which we term “rotationally cold”), the microcanonical distribution is chosen with the constraint that the molecular rotational angular momentum
+
+
+
+
(11)S. N.Foner and R. L. Hudson, J.Chem. Phys., 36,2681(1962). (12)H. H. Suzukawa, M. Wolfsberg, and D. L. Thompson, J. Chem. Phys., 68,455 (1978). (13)C. H. Chen, P. E. Siska, and Y. T. Lee, J. Chem. Phys., 59,601 (1973);A. L.J. Burgmans, J. M. Farrar, and Y. T.Lee, ibid., 64,1345 (1976). (14)M.J. Redmon, R. J. Bartlett, B. C. Garrett, G. D. Purvis, P. M. Saatzer, G. C. Schatz, and I. Shavitt in “Potential Energy Surfaces and Dynamics Calculations”,D. G. Truhlar, Ed., Plenum Press, New York, 1981,p 771.
2354
The Journal of Physical Chemistty, Vol. 86, No. 13, 1982
is zero. In the second (“rotationally hot”) the initial internal energy E,, is distributed randomly between rotation and the three vibrational modes. The actual distribution of rotational energies produced in chemical activation experiments is probably somewhere between these two limits. Without detailed kinetic modeling, however, it is difficult to define it precisely. If vibration-rotation coupling is not important, then in general we expect that at low pressures the H02* rotational distribution should approximate the distribution produced by H O2 collisions (presumably relatively cold), while at high pressures it should approach a Boltzmann distribution at the ambient temperature. In defining the microcanonicalphase space distribution for either hot or cold H02* we used an algorithm related to that of Bunker, Hase, and Chapman.15 In this, the molecular coordinates and momenta are first randomly sampled by using the harmonic normal mode-rigid rotor (symmetrictop) Hamiltonian to define approximate states at a given total energy. After the Cartesian coordinates and momenta are evaluated, the momenta are then scaled so that the full total energy (including anharmonic and Coriolis terms) equals to the chosen value. If the evaluated potential energy is greater than the chosen total energy at the resulting molecular geometry, then the point is rejected and new sets of initial conditions are calculated. In order to be sure that the initial phase space was well sampled, we sampled a broad range of zero-order states, leading to rejection of about half of the initial points according to the above criteria. In considering H02*molecules a t energies above the H + O2dissociation threshold, we had to choose a somewhat arbitrary cutoff in the allowed H02* internuclear distances in order to avoid sampling molecules which had already dissociated. This was done by rejecting all molecules which before the collision had both 0-H distances greater than 6.6~0or had an 0-0 distance greater than 5 . 7 ~ 1These ~ distances were found to be reasonable “points of no return” on the path to dissociation for essentially all trajectories of interest in this study. As a second precaution, all initially prepared H 0 2 molecules were “aged” for 5.4 X s before collision. This time amounts to about six OH stretch vibrational periods and allows for any molecules initially placed near or beyond the barrier to dissociate (according to the above cutoffs) if their velocities are appropriately directed. The end of each collision was determined by first requiring that the He atom be at least 6.6~0from each of the other three atoms and then assigning dissociation to H + O2when both of the 0-H distances became greater than 6.6~0,dissociation to OH + 0 when the 0-0 distance became greater than 5 . 7 and ~ ~ nondissociation ~ if the trajectory could be further integrated to a He to center of mass of H 0 2 distance of loao without dissociating. The nondissociating HOis were assigned as “stable” if the final internal energy was less than 0.094 eV and “metastable” (i.e., H02*)otherwise. Although it is possible that those HOis considered metastable might still dissociate at a later time, a significant effort was made to adjust the aging and end-point tests so that this was of low probability. To test this, several runs were made in which the end-point HeH 0 2 distance was increased to 12uo,with no change in the end-point binning assignments. Likewise, no change in cross sections was observed when the aging time was in-
+
(15) S. Chapman and D. L. Bunker, J. Chem. Phys., 62, 2890 (1975); W. L. Hase in “Potential Energy Surfaces and Dynamics Calculations”, D. G. Truhlar, Ed., Plenum Press, New York, 1981, p 1, and references therein.
Gallucci and Schatz
creased by 25%. Although some H02*’s do dissociate if the aging time is drastically increased (e.g., to greater than 1 ps), the calculated cross sections were not found to be sensitive to this. This suggests that the H02*’swith finite but long lifetimes have energy-transfer characteristics similar to those with infinite lifetimes. We also tested for collision-induced isomerization of HOAOB to OAOBH and found this to be of negligible probability for stable H02. Metastable H 0 2 was found to isomerize even in the absence of collisions, so in all results presented no distinction between the two metastable isomers is made. Although it is possible to dissociate HOAOB either to OAH or to OBH, we found the probability of the second process to be small ( 0.094 eV gives the cross section for stabilization and Q(H02*) for Eht < 0.094 eV gives that for collisional destabilization. This also means that the elastic cross section is included in Q(H0,) for Eint< 0.094 eV and in Q(H02*)for Eint> 0.094 eV. Of course, the elastic and total cross sections cannot be determined by classical mechanics without assignment of a cutoff impact parameter. Since Blint assumed a hard-sphere cutoff of 13.6~1,for Ar + H 0 2and the potential that we used gives 7 . 2 for ~ ~the sum of ‘12r,(02) r,(OHe), it is likely that 6.1u0 is somewhat small for this cutoff. This means that the cross sections Q(HO,*) for Eht > 0.094 eV and Q(H0,) for Eint< 0.094 eV are probably too small. We include them in the reported results only to study trends. No measurable quantities of relevance to recombination kinetics depend on them.
+
111. Results Figure 2 presents the cross sections for stabilization and dissociation of rotationally cold H02*by He as a function of the initial H02* internal energy Ehtfor Eo = 1 eV. The range of internal energies considered runs from about 0.3 eV below the dissociation threshold (where the symbol H02 rather than H02*should be used in discussing the initial conditions1’) to 0.7 eV above. The qualitative trends in the results are as follows: (1)Q(H02)and Q(H02*)have about the same value at the threshold for dissociation. Q(H02*)drops rapidly for Eint> 0.094 eV to about 5a02and then shows a slower decline which extends several tenths of an eV. The rise in Q(HO,*) below threshold is fairly gradual. The destabilization cross section is 10% of Q(H02)at 0.25 eV below threshold. (2) Q(H+O,), the cross section for dissociation, shows a monotonic increase over the energy range indicated. Note that for Eo = 1 eV, the energetic threshold for dissociation is at Eint= -0.906 eV. In actual practice, Q(16) R. N. Porter, L. M. Raff, and W. H. Miller, J. Chen. Phys., 63, 2214 (1975). (17) For notational simplicity, we generically use the symbol H02* to denote the initial HOz molecule even though HOBis appropriate below threshold. The two symbols are correctly differentiated, however, wherever the distinction is important, particularly in defining final conditions and cross sections.
The Journal of Physical Chemistty, Vol. 86, No. 13, 1982 2355
Collisions of Helium wkh Highly ExcRed HO,
TABLE I : Average Change in Translational Energy ( A E , ) and Standard Deviation u for Nondissociating He + HO Collisions Eht, eV
- 0.193 - 0.030 0.052 0.133 0.133 0.133 0.133 0.161 0.188 0.242 0.514 0.569 0.786
E,, eV
( A E ; ) ,eV
Rotationally Cold HO, 1.0 - 0.057 1.0 - 0.057 -0.056 -0.055 -0.036 - 0.021 - 0.010 - 0.055 - 0.054 - 0.050 - 0.044 - 0.037 -0.052
1.0 1.0 0.5 0.25 0.11 1.0 1.0 1.0 1.0 1.0 1.0
1ow&u , OH+O
+
Flgure 2. Stabilization and dissociation cross sections for He H0,' (cold) vs. internal energy E , (relative to H 0,) for a translational energy of le V. The curve labeled HO, (where open circles label the calculated points) denotes the cross section for formation of stable HO, while that labeled HO,' (filled circles) is the cross section for formation of metastable HOP, that labeled H O2 (lower half-filled circles) is the cross section for dissociation to H O,, and that labeled OH 0 (upper half-filled circles) is the cross section for dissociation to OH 0. The dashed line at E, = 0.094 denotes the classical dissociation threshold. Representative 1u error bars are shown for a few points.
+
+ +
+
+
1oc
Rotationally 1.0 1.0 1.0 0.5 0.25 0.11 1.0 1.0 1.0
Hot HO, - 0.030 - 0.029 - 0.023 0.010 0.029 0.026 - 0.020 - 0.006 - 0.002
N O
50
Figwe 3. Stabilization and dissociation cross sections as in Figure 2 but for He HO,' (hot).
+
(H+02) does not approach lao2until Eht = 0. This provides an indication that transfer of translational energy into internal energy is fairly inefficient in He HOz collisions. This also means that for Eht < 0.094, it is much more probable to collisionally excite H 0 2 to H02*than it is to dissociate it, even though both processes have the same energy gap. (3) Q(OH+O) is similar to Q(H+02),but shifted upward in energy by -0.5 eV. This shift corresponds closely with the difference in threshold energies (0.094 eV for H + O2 vs. 0.60 eV for OH + 0). Figure 3 presents cross sections analogous to those in Figure 2 but for rotationally hot H02* (as defined in section 11). Over the internal energy range Eht = -0.03 to +0.8 eV the average initial H02* rotational angular momentum in the hot ensembles was found to vary smoothly from 24h to 34h. All of the cross sections in Figure 3 have an energy dependence which is analogous to that in Figure 2, but there are some quantitative differences. Q(H+O,) is smaller for hot H02* than for cold H02*, and Q(H0,) is larger above Eht = 0.094 eV. Both of these differences are
+
0.122 0.121 0.129 0.131 0.073 0.042 0.023 0.131 0.130 0.133 0.138 0.153 0.164 0.171 0.177 0.179 0.128 0.112 0.092 0.189 0.202 0.191
in accord with expectations based on two criteria. First, HOz* is stabilized by rotation through the presence of centrifugal barriers. Second, the most efficient mechanism for collision-induced energy exchange is T e R. For rotationally cold H02*, the only possible direction of energy flow between translation and rotation is T R. This adds to the internal energy, causing destabilization. For rotaT are possible tionally hot H02*, both T R and R with similar probability, and the R T process will tend to cause stabilization, thereby increasing Q(H0,). Table I presents the average change in translational energy ( A E o ) for both rotationally cold and hot H02*. Also included is the standard deviation u = [ ( A E o 2 )( AE0)2]1/2 associated with the translational distribution. A t the relatively high translational energy (1eV) considered in Figures 1 and 2, we find that both ( A E o )and u show little variation with Eht,with ( AEo) always negative (corresponding to net transfer of energy from translation to H 0 2 internal motions). (ao) for the cold H02* has values ranging from -0.057 to -0.037 eV over the range of Eht considered, while u varies from 0.121 to 0.164 eV. The fact that u > I(AEo)Iindicates that the width of the translational distribution is much larger than its shift from zero. This also means that, at any H 0 2 initial internal energy, there is approximately an equal probability for an internal energy increase or decrease. At Eint= 0.094 eV, this would cause Q(H0,) and Q(HOz*)to be approximately the same, as any increase in Eintwould cause H 0 2 to become metastable, and any decrease would lead to stable HOP. This explains the crossing of Q(H02)and Q(HO,*) near Eint= 0.094 eV in Figures 2 and 3. We should also note that the (AE,)and u values for hot HOz* are both larger than for cold HOP That ( aEo)is larger is in accord with our previous statements about T R energy transfer. For hot HOz*Table I indicates that both ( AEo) and u do increase systematically with increasing Eint.Presumably this reflects an increase in the T R exchange with increasing energy in rotation. The dependence of Q(H02),Q(H02*),and Q(H+02)on initial translational energy Eo for a given Eht is plotted in Figure 4 for both cold and hot HOz* at Eint= 0.133 eV.
- --
I
2 a
-0.030 0.052 0.133 0.133 0.133 0.133 0.242 0.514 0.786
a , eV
-
-
2958
The Journal of Physical Chemistry, Vol. 80, No. 13, 1982
Gallucci and Schatz
TABLE 11: Average O,(uj)and O H ( u ‘ j ’ )Vibrational and Rotational Actions and Total Internal EnergieP
Ekt, eV
Q(H+o,Ib
(j)
( U)
( E u j ) ,eV
Q(OHtO)C
(U’)
(j’)
( E , , f j f )eV ,
- 0.3 - 0.3 - 0.3 -0.1
6.5 4.3 4.8 4.7
0.184 0.169 0.159 0.238
NS - 0.4 - 0.1
NS 9.6 7.5
NS 0.278 0.307
0.133 0.161 0.188 0.242 0.514 0.569 0.786
1.6 f 3.0 ?r 3.2 f 6.4 f 22.1 ? 29.9 f 43.1 i:
0.4 0.6 0.6 0.8 2.2 2.5 3.0
- 0.2 - 0.0 - 0.1 - 0.1 0.4 0.6 1.0
Rotationally Cold HO,*, E , = 1 eV 23.5 0.246 0 16.8 0.155 0 15.3 0.127 0 16.1 0.137 0.06 f 0.04 15.5 0.241 2.1 k 0.7 17.0 0.28 1 3.0 k 0.8 17.3 0.361 9.8 ? 1.6
0.242 0.514 0.786
1.6 ? 0.6 16.4 f 1.9 32.3 f 2.6
-0.1 0.1 0.7
Rotationally 14.4 21.9 24.8
Hot HO,*, E , = 1 eV 0.115 NSd 0.234 0.1 ?r 0.1 0.357 3.2 f 0.8
E,j is the 0,internal energy (including zero-point energy) for vibrational action u and rotational action j . Primed variables denote the analogous quantities for OH. All actions are in units of h . Dissociation cross section (in a O z for ) H + 0, product, with l o error bars indicated. Dissociation cross section (in a o 2 )for OH + 0 product. Not statistically significant.
value of (AE) switches from positive to negative with increasing E,,, an apparent manifestation of the dominance of R T at low E,,. Table I1 summarizes information about the vibrationrotation distributions in the dissociated diatomics O2and OH. This includes the average vibrational and rotational actions ( u ) and ( j ) and for O2and (u’), ( j ’ ) for OH, as well as the average vibration-rotation energies (including zero-point energy) (E,j) and (E,,Y). The vibrational and rotational actions are the classical analogues of the vibrational and rotational quantum numbers, and often there is good correspondence between averages of the corresponding variables. However, one exception in this respect occurs when the average classical vibrational action is negative, as the smallest possible average quantum number is zero. Table I1 indicates that negative ( u ) and ( u ’) values do occur for some values of Ehv In these cases, we would expect (on the basis of results of other calculations18) that the corresponding quantum ( u ) and ( u ’ ) would be very close to zero and that the quantum dissociation cross section would be smaller than the classical one (i.e., the process is semiclassically forbidden). Table I1 indicates that, except at high Eht,both O2 and OH are vibrationally cold and rotationally hot. For 02, we find that ( u ) is close to zero up to Eht = 0.514 eV, while ( j ) has an average value of about 16h (for cold H02*) except at low Eo where it is somewhat higher. Because of the small rotational constant for O2 (1.74 X lo4 eV), the rotational energy associated with j = 16h is only 0.047 eV, but this still means that the O2rotational temperature is -550 K. The hot H02* results indicate less O2vibrational excitation and more rotational excitation than for cold H02*. For OH, the mean vibrational action ( u ’ ) is very close to its lower limit (-0.5h), while ( j ’ ) has an average value (for cold HOP)of about 5h, corresponding to a rotational energy of 0.069 eV (800 K). The rotationally hot H02* results show about the same ( u ’ ) values but somewhat higher ( j ’ ) values than the cold H 0 2 results. The physical interpretation of many of the results summarized in Table I1 is quite straightforward. First, for Eht close to threshold for either O2 or OH dissociation, most of the intemal energy needs to be concentrated in the bond which is breaking in order for dissociation to occur. This means that the dissociated diatomic will be vibrationally cold, as is observed. A t the same time, the minimum energy path for dissociation to both O2 and OH involves a nonlinear H 0 2 geometry. Thus, any repulsive energy release as the system dissociates will at least partially be
-
Figure 4. StaMllzation and dissociation cross sectkns for He 4- HO,’ vs. translational energy E, for E, = 0.133 eV. The left panel shows cold results whlle the right panel shows hot results. Other notation Is analogous to that in Flgwe 2. Note that the energy threshold to stabllizatlon at this value of E , Is 0.039 eV.
This value of Eht is only slightly above the classical threshold energy, and at this energy we find that the cold and hot values of Q(H+02) show a somewhat different dependence on Eo. The cold Q(H+02) is f i i t e near Eo = 0 and shows a slow rise with increasing Eo, while the hot Q(H+02) is zero below Eo = 0.5 eV and rises rapidly at higher E* The behavior of the cold result is in accord with the notion that dissociation of H02* requires no energy gap, only a collision-induced change in vibrational phase which puts the molecule in a dissociative region of phase space. Adding translational energy might enhance dissociation, but this effect quickly saturates for cold H02*. For hot H02*, evidently dissociation is not efficient until rotational barriers have been overcome, and this requires more translational energy. The cold and hot stabilization cross sections in Figure 4 also show a somewhat different dependence on Eo,with the cold result increasing with increasing Eo and the hot result roughly constant, except for a sudden rise near threshold. These results are also in accord with our earlier T exchange. For the cold result, arguments about R R T is not possible, so stabilization must proceed by V T, and this is effective only at high Eo. For the hot results, stabilization is brought about by R T transfer, and this remains quite efficient even down to relatively low translational energy. Table I presents (So) and u as a function of Eo for Eht = 0.133 eV. As expected, we find that the cold values of (AE,)and u both increase with increasing Eo. The hot
--
-
-
(18)J. W. Duff and D. G. Truhlar, Chem. Phys., 9, 243 (1975).
Collisions of Helium with Highly Excited H02
converted into diatomic rotational excitation. Rotational excitation can also arise from torques exerted by the colliding He atom although the previously noted inefficiency of energy transfer from translational to internal motions makes this mechanism less important and causes the O2and OH internal state distributions to be similar to those obtained from unimolecular decay of O2and OH. For rotationally hot HOz*, there is more energy in H02* rotation, and less in vibration for a given total energy. Near threshold for dissociation, these get converted into more rotational excitation and less vibrational excitation in the diatomic dissociation products, as observed. For cold HOz*, the average diatomic vibrational excitation should increase with increasing ICint,but rotation should not be strongly influenced. For hot HOz*, the amount of both vibrational and rotational excitation should increase with increasing ICint.Table I1 indicates that these expectations are (with minor exceptions) born out.
IV. Analysis of Three-Body Recombination Kinetics In this section we study two aspects of our results as they relate to three-body recombination kinetics. First, the cross sections and rate constants for collisional stabilization from this calculation are compared with various strong and weak collision models. Second, the competition between stabilization and dissociation processes in recombination mechanisms is evaluated. In order to model the recombination kinetics of He + H + O2accurately, it is necessary to know the distribution of internal energies in the complex. An accurate evaluation of this requires detailed modeling of the collisional excitation/ deexcitation kinetics, as well as determining the nascent complex energy distribution through simulation of H + O2 collisions. This is beyond the scope of our analysis,lgalthough the present results do provide some of the input necessary to do such an evaluation. An alternative to this is to use a statistical expression such as a Boltzmann distribution above some threshold energy The logical threshold to use here is just the classical threshold energy (Eht= 0.094 eV), although we should note that the threshold obtained by applying quantum vibrationally adiabatic theory is significantly higher (0.094 + 0.12 eV from Figure 1). Unless we impose a zero-point energy constraint on those trajectories which dissociate (recall the discussion concerning the negative ( u ) and (u') values in the previous section), it seems inconsistent to use this quantum threshold. Once a distribution P(Eint) of H02* internal energies is determined, we can then define rate constants for stabilization k, and dissociation k, via
en,.
where the labels C,H refer to rotationally cold or hot H02*, respectively, and the right-hand integral refers to the average over translational energies Eo of the product of cross section times relative velocity. The cross section Q4 is just Q(H02) for Eint> 0.094 eV, and Q5is Q(H+O2). To evaluate the above double integral, we make two assumptions. First, we assume that the cross section Q(Eo,EhJ(suppressing the superscripts and subscripts) can be expressed as Q(Eo,Eint) = AIEo - g(Eint)]Xe-aEint (19) A paper where some aspects of this analysis has been considered is M. Quack and J. Troe, Spec. Period. Rep. (Chem.Soc.),5,175 (1977).
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 2357
TABLE 111: Parameters Used in Rate Constant Evaluation cold or process hotC 4O 4 5b 5
C H
c
H
A , au 29.2 30.5 1.58 2.07 X lo4
Refers to stabilization (eq 4 ) . C = cold; H = hot. (eq 5). a
X
a , au
0.46 0 0 3.2
111.7 168.1 - 256.1 - 203.1
Refers to dissociation
TABLE IV: Model Calculations o f Collisional Stabilization and Dissociation Rate Constants and Efficiencies
T, K 300 500 1000
T, K 300 500 1000
k d C ,c m 3 / (molecule s )
p,C
h a H , cm3/ (molecule s )
P~~
Stabilization 3.6 (-12) 0.008 7.7 (-11) 5.7 (-12) 0.010 9.3 (-11) 1.0 (-11) 0.013 1.1(-10)
k S C ,c m 3 / (molecule s )
osc
ksH, c m 3 / (molecule s )
Dissociation 0.018 4.1 (-16) 7.8 (-12) 1 . 3 (-11) 0.023 3.2 (-15) 5.7 (-11) 0.072 8.0 (-14)
0.18 0.17 0.14
PiH 9.4 ( - 7 ) 5.7 (-6) 1.0 (-4)
where A, x , and CY are parameters. ET is the translational threshold energy for a given process. For stabilization, E;f = Eint- Frit (i.e., E;f is the minimum energy needed to lower the HOz* internal energy to below dissociation threshold) while for dissociation, E;f = 0. Our second assumption is that the density of internal states of H02* is fairly constant over the relatively small range of Eint which contributes significantly to the rate constant. This allows us to write
and the resulting expressions for kf;:
(2 + LukT)-l
are
+ (2 + akT)-2
(kT)T(x + l)A ( x (1 + akT)-'
for reaction 4
+ 1) X for reaction 5
where r is a y function. The relative efficiency p of stabilization or dissociation is defined as the ratio of k?: to the gas kinetic rate constant. The gas kinetic rate constant is approximatelyequal to ab,,,2(8kT/~p)~/~ where b,, is taken to be the cutoff impact parameter 6.1ao. The parameters A, x , and a used in this evaluation are given in Table 111. x was chosen to be zero for the 4,H and 5,C combinations, as the cross sections in Figure 4 are nearly constant close to threshold. The other parameters were chosen so as to fit the Eo dependence in Figure 4 and the Ehtdependence in Figures 2 and 3 between Eht = 0.13 and 0.24 eV. The resulting stabilization and dissociation rate constants and efficiencies are given in Table IV. All of the rates increase with increasing T, primarily because of the T1I2 dependence of the gas kinetic collision rate. The decreasing stabilization cross section with increasing Eint in Figures 2 and 3 tends to make the rate constants smaller
2358
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982
at higher T , but this effect is of secondary importance in the present results. Table IV indicates that both the cold stabilization and dissociation efficiencies are small (few percent) while the hot stabilization efficiency is much larger, and the hot dissociation efficiency much smaller. In addition, we find that dissociation dominates over stabilization in the cold results, while the opposite is true in the hot results. The relatively slow variation of /34c and P4H with T suggests that the often used hard collision model, where k4 is equated to a constant efficiency factor times the gas kinetic rate constant, is approximately correct. The cold value of p4cis much smaller than is normally assumed (p = 0.25 is a common value2), but the hot value of 0.14-0.18 in Table IV is pretty close. The dominance of dissociation over stabilization for the cold results has important consequences for recombination rate theory, as normally collision-induced dissociation is entirely neglected. If one includes eq 5 with the mechanism of eq 3 and 4, then, in the high-pressure limit, the apparent bimolecular rate constant changes from k3 to k3k4/(k4 + k5). Thus, at T = 1000 K, the apparent bimolecular rate constant would be reduced by a factor of 7 because of dissociation effects. This provides one explanation for the very small transmission coefficients which are sometimes needed in recombination rate constant evaluations to rationalize the high-pressure rate constant values.2 This explanation is not sufficient however if the H02* is rotationally hot, as then dissociation is unimportant compared to stabilization. Another class of stabilization models is based on the weak collision approximation, wherein a master equation which describes transitions between various states of the complex is solved subject to an assumed transition probability function.20 A typical form for the transition probability function is given by the exponential gap formula f‘(Eht,J’i,t) = (y/2)e-IEint-E’intl/r where Eintand E:nt are the initial and final internal energies, and y is a parameter. Although other expressions for the transition probability are occasionally used (Blint used a one-sided exponential gap, in which P was assumed to vanish for E!,,, > Eint),the above formula has been shown to approximate the results of trajectory calculations reas0nab1y.l~However, Table I does indicate one problem with the above formula: it assumes ( hEint)= -( hEo)= 0 whereas in reality the value is small but not zero. If we ignore this error, then the parameter y can be related to the standard deviation of the translational distribution via y = v‘2a. Table I indicates that, for the cold results, a varies roughly linearly with Eo and is independent of Eht, so that y N 0.2Eo. Thus, for a typical average relative translational energy E, N 1/2kT,we find y N O.lkT. For the hot results, a is almost independent of Eo even at relatively low translational energies (because of the dominance of low energy gap R T processes). Thus, in this case, y = 0.15 eV independent of temperature. The value of YH is in the range of y values which Blint found to give a temperature-dependent stabilization factor which
-
(20) A. J. Stace and J. N. Murrell, J.Chem. Phys., 68,3028(1978);T. Mulloney and G. C. Schatz, Chem. Phys., 45, 213 (1980).
Gallucci and Schatz
matches the apparent experimental value for HOz reasonably. The value of yc is much smaller than yHexcept a t unobtainably high temperatures. Such a small value supports the use of a weak collision model to describe excitation and stabilization. The hot value is more akin to what might be appropriate for a strong collision model to be valid.
V. Conclusion In this paper we have studied collisional excitation and dissociation in H 0 2 and have used the results to study models of kinetic processes of relevance to termolecular recombination. The resulting stabilization and dissociation rate constants were found to depend strongly on the assumed distribution of complex rotational energies, with dissociation rate constants larger than stabilization for rotationally cold H02* and much smaller for rotationally hot HOz*. These effects were found to be correlated with the relative ease of R T transfer in hot H02*, together with the influence of rotational centrifugal barriers on H02* lifetimes. In addition, the collisional stabilization efficiency was found to be similar to that assumed in the strong collision model for hot H02*,but not cold HOz*. When the dissociation rate constant is higher than that for stabilization, the high-pressure rate constant for formation of H 0 2 is depressed relative to its value when dissociation is neglected. This was found to be important for cold H02* but not hot. We also studied collisional dissociation of HOz* to give OH + 0, and found it to be very similar in energy dependence to the H + O2 dissociation process although shifted upward in energy by the difference in respective energetic thresholds. While this study has provided much insight into the dynamical basis for several poorly understood approximations of termolecular rate theory, there are still several problems to be solved before a quantitative treatment of HOz kinetics will be possible. Among these is the development of a detailed master equation model for the collisional excitation/deexcitation/dissociation processes which includes a description of the H02*rotational energy distribution. Without such rotational information, it is not possible to assess which of the two limiting cases considered here (cold or hot) best describes the HOz kinetics. Another problem which needs to be addressed is the development of more accurate potential energy surfaces, both for H 0 2 and for rare gas + H 0 2 systems. The presence of a barrier in the H O2 dissociation channel appears to be significant in determining both collisionless and collisional dissociation thresholds and presumably has a large effect on the temperature dependence of the termolecular rate constant. Finally, an important but often ignored problem in modeling the termolecular kinetics is the accurate treatment of quantum effects. This includes not only the rather obvious influence of tunneling but also the proper definition of dissociation thresholds (as discussed in section IV), and the problem of maintaining microscopic reversibility when incorporating collision-inducedprocesses in the chemical activation and dissociation steps of the termolecular recombination mechanism.
-
+
Acknowledgment. This research was supported by NSF Grants CHE-7820336 and CHE-8115109. G.C.S. acknowledges helpful discussions concerning this work with Dr. A. F. Wagner and Professor J. M. Bowman.
