214
J. Phys. Chem. 1984,88, 214-221
Contour maps of 1$12, where $ is the resonance wave function for the collinear reaction, show that the resonance wave functions have a very simple structure and allow for unequivocal assignment of quantum numbers (0,3)for F H2 and F H D and (0,4) for F + D2. The assignment is consistent with a previous adiabatic treatment of FH2 in hyperspherical coordinates. The structure of 1$12 is also consistent with interpretations based on adiabatic barriers in natural collision coordinates and on classical resonant periodic orbits. The contour maps directly illustrate the dynamical localization of the resonance wave function to a compact region
+
+
of the potential energy surface in the strong-interaction region between the potential energy barrier in the entrance channel and an effective dynamical barrier in the exit channel.
Acknowledgment. Helpful collaboration with Bruce C. Garrett in related studies of F + H2 and isotopic analogues is gratefully acknowledged. This work was supported in part by the National Science Foundation through Grant No. CHE80-25232. Registry No. Hydrogen, 1333-74-0;deuterium, 7782-39-0;hydrogen deuteride, 13983-20-5; atomic fluorine, 14762-94-8.
A Quasiclasslcal Trajectory Study of Colllsional Excitation in H
+ CO
Lynn C. Geiger and George C. Schatz* Department of Chemistry, Northwestern University, Evanston, Illinois 60201 (Received: July 13, 1983) The results of a quasiclassical trajectory calculation of cross sections for collisional excitation in H (D) + CO at 1-4-eV translational energy are presented and used to interpret recent laser photolysis measurements. A realistic potential energy surface was used in these calculations based on Dunning’s recent ab initio studies. Overall agreement of the calculated results with experiment is generally good, and this enables us to assess in detail what features of the potential energy surface the measured results are sensitive to. For the rotationally summed vibrational distributions, we find that the high u tail of these distributions is almost exclusively due to collisions which form a COH complex. The average COH lifetime is found to be about 3 OH vibrational periods, and the cross sections for complex formation are found to be very sensitive to the H + CO COH barrier. Based on comparisons with experiment we revise this barrier from Dunning’s 1.72-eV value down to 1.52 eV. Many other features of the measured results, such as the average vibrational energy transfer, are found to be primarily sensitive to impulsive collisions of H with either the C or 0 atom. Although the HCO portion of the potential surface was sampled with significant probability, none of the HCO complexes formed had a lifetime of more than one inner turning point. The rotational distributions were found to be composed of a high j component (for which ( j ) increases with increasing u ) arising from impulsive collisions of H with the C atom, and a lower j component (for which ( j ) is constant or decreases with increasing v) arising mostly from collisions with the 0 atom.
-
I. Introduction have developed a new and promising Recently, two technique for studying collisional excitation and chemical reactions in small molecules using fast hydrogen and deuterium atoms. In this technique, excimer laser photodissociation of H2S, D2S, HCl, and other species is used to produce translationally monochromatic H or D atoms with center-of-mass energies between 0.95 and 4.0 eV, and the collisions of these atoms with small molecules are monitored on a microsecond timescale by infrared chemiluminescence. Among the first applications of this technique have been studies of translational to vibrational energy transfer in H CO and D + C 0 3 and translational to rotational energy transfer in H The H (D) CO system is especially interesting from the point of view of these experiments because metastable intermediates corresponding to the formyl radical isomers H C O and COH are energetically accessible. This means that the vibrational and/or rotational distributions might contain information that pertains to regions of the potential surface corresponding to these species. The kinetics of H C O H C O is, of course, familiar from its importance in comb~stion,~ and HCO is known to be stable (by 0.81 eV6) relative to H + CO, with only a small (-0.11 eV7>*)barrier to formation. COH, on the other hand, has never been observed experimentally, but accurate ab initio calculationss indicate that a metastable minimum corresponding to COH does exist at 1.04 eV above H CO, with a barrier of 0.68 eV to dissociation. Adding the last two numbers together yields an estimated barrier to formation of COH from H + CO of 1.72 eV which is nicely within the range of energies spanned by the photolysis experiments. In this paper, we present the results of a detailed quasiclassical trajectory study of H (D) + C O collisional excitation using a
+
+
+
-
+
+
*Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/84/2088-0214$01.50/0
realistic potential energy surface. These results will be used to interpret CO vibrational distributions measured by Wight and Leone3 and rotational distributions reported by Wood, Flynn, and west or^.^ Evidence will be given which suggests that certain features of the vibrational and rotational distributions are sensitive to the presence of metastable COH (COD), and that the barrier to formation of this metastable can, at least crudely, be inferred from the measured results. To summarize the rest of this paper, the section 11, we describe our potential energy surface as obtained by fitting a combination of experimental and ab initio data, and we briefly summarize the trajectory calculations. Section I11 presents the resulting state resolved cross sections, including a detailed analysis of how the region of the potential surface sampled influences the final state distributions. This section also contains an analysis of the experimental data. A summary of our conclusions is presented in Section IV. 11. Theory A . The HCO Potential Energy Surface. Although small portions of the ground (zA’) electronic potential energy surface (1) Magnotta, F.; Nesbitt D. J.; Leone, S. R. Chem. Phys. Lett. 1981,83, 21. (2) Quick, Jr., C. R.; Weston, Jr., R. E.; Flynn, G. W. Chem. Phys. Lett. 1981, 83, 15. (3) Wight, C. A.; Leone, S.R. J. Chem. Phys. 1983, 78,4875. (4) Wood, C. F.; Flynn, G. W.; Weston, Jr., R. E. J. Chem. Phys. 1982, 77, 4776. (5) Ahumada, J. J.; Michael, J. V.; Osborne, D. T. J . Chem. Phys. 1972, 57, 3736, and references therein. (6) Warneck, P. Z . Naturforsch. A 1974, 29, 350. (7) This is the highest of a range of estimates made in ref 8 based on the 0.087-eV activation energy measured by Wang, H. Y . ;Eyre, J. A,; Dorfman, L. M. J . Chem. Phys. 1973, 59, 5199. (8) Dunning, T. H. J . Chem. Phys. 1980, 73, 2304.
0 1984 American Chemical Society
Collisional Excitation in H
+ CO
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 215
TABLE I: LEPS Parametersa
TABLE 11: Parameters for VHCOand V C O H ~
diatomic
De
Re
ae
Sat0 parameter
CO
0.412498 0.127570 0.169880
2.131 80 2.11631 1.834 17
1.21690 1.061 51 1.21393
0.75 0.75 0.75
CH OH a
In atomic units.
of H C O have been studied many times by ab initio and other methods (see ref 8 and references therein), the only previous global surface that we are aware of is that of Carter, Mills, and M ~ r r e l l . ~ This surface was obtained by using the Sorbie-Murre11 methodlo to fit the H C O and C O H equlibrium properties and all the two-body fragment potentials. Accurate barriers to dissociation of HCO and COH, and for the isomerization of HCO into COH, were not available a t the time this fit was developed, and the resulting surface has at least one important defect: the barrier to formation of H C O from H C O is much too high (0.56 eV9 rather than -0.1 1 eV7). In addition, the isomerization barrier is absent on this surface. In view of the recent large-scale configuration interaction calculation by Dunnings for H C O and COH near equilibrium and near the dissociation barriers, and for the H C O * C O H isomerization barrier, we chose to develop a new global potential surface for HCO. This surface is also based on the Sorbie-Murre11 (SM) method, although a somewhat different fitting strategy was used so that we could get all barrier heights and equlibrium energies as well as most geometries and harmonic frequencies within acceptable accuracy. This strategy involves separately fitting the regions near the H C O and C O H minima by using SorbieMurrell-like potentials, and fitting the region near the isomerization barrier with a quadratic function of the coordinates. The barriers to formation of H C O and COH are then fit by adjusting the nonlinear parameters in the HCO and C O H S M functions, and then the three separate pieces of the potential function are blended together with switching functions. In another variation on the standard strategy we have replaced the sum of diatomic pair potentials which is used to describe two-body forces by a LEPS potential for HCO. This means that the "two-body term" actually includes three-body interactions as well, but this form is no more difficult to fit than the sum of pairs, and it provides a more realistic zeroth-order description of the three-atom interaction for geometries near the isomerization saddle point. With this introduction, the global potential energy surface V is given by
j = HCO 2.00000 3.5 0.70 C,j -0.012216 C,j 0.011962 C3j -0.12733 C,j -0.0064779
Ylj Yzj ^/3j
C,j a
where V,,, is the above-mentioned LEPS potential, the parameters for which are given in Table I, and the remaining terms in (1) are the regionally fit three-body terms VHco, V,,,, and VcoH corresponding to the HCO, isomerization, and COH parts of the potential. These regional potentials are multiplied by three switching functions T I ,T,, and T3designed to blend them together at appropriate boundaries. V Hand~ VCoH ~ have the form
5 = ( l - tanh tanh
(l/zYIJslJ))(l
y2T3J'3J)(c1J
'6JS2J2
+ cTJS3J2)
- tanh
('/Y2JS2J))(' -
+ '2JSIJ + C~J'ZJ+ '4JS3J + y2(csJS1J2 + +
'SJ'IP~J+ c 9 ~ S 1 ~ s 3 ~ + '10~~2jS3~) (2)
where j stands for H C O or COH and S,/ (i = 1, 2, 3) is the displacement of the ith internuclear distance RI from its equilibrium distance for species j (Le., S,] = R,- RIJq). The internuclear distances R, are labeled according to the convention that i = 1 denotes the CH distance, i = 2 denotes CO, and i = 3 OH. The coefficients CIJ- CloJin ( 2 ) were obtained by requiring that (9) Carter, S.;Mills, I. M.; Murrell, J. N. J . Chem. SOC.,Farday Trans. 2 1979, I , 148. (10) Sorbie, K.S.; Murrell, J. N. Mol. Phys. 1975, 29, 1387. 1976, 31, 905. Varandas, A. J. C.; Murrell, J. W. Faraday Discuss. Chem. Soc. 1977 62, 92.
0.65 2.6 2.12 0.037658 0.012356 -0.15206 0.068642 0.10772
C6j C,j C,j C,j C,,j
R 2:j R;Bj Ra ;j
j=HCO
j=COH
0.12130 0.18236 0.082001 -0.14262 -0.21165 2.116 3.8817
-0.18526 0.37219 -0.13260 -0,031380 -0.022439 3.601 36 1.848
2.242
2.434
In atomic units,
TABLE 111: Parameters in V.,'
Rise R.*iSO
.Po, deg
+
v = VLEPS+ T IVHCO+ T z K m + T ~ ~ C O H (1)
0.29048
j = COH
V+iso
1.999 2.432 19.79 0.11442
Fzoo F,,,,, Foe, F,,,
F,,, F~,,
a
0.1624 0.3405 -3.816 X lo-' 0.03942 -2.282X 2 . 8 4 0 ~10-3
In atomic units unless noted otherwise.
each regional potential ( VHc0or VcoH) exactly fit the energy, geometry, and harmonic frequencies associated with the regional minimum (HCO or COH). For both H C O and COH, the geometries and frequencies were taken from Dunning's calculation.8 The equilibrium energy (relative to H CO) for C O H was also taken from ref 8, but for HCO, the energy was taken from an estimate based on experimental heats of The motivation here is that other potentials generated with methods comparable to Dunning's usually give geometries and frequencies which are more accurate than energy defects and barriers." Thus where possible we use experimental energy defects and barriers. This means that the energetics associated with the H C O part of the surface will be accurate, but the energetics associated with COH, where experimental information is not available, will probably not. Some estimate of the magnitude of the error can be obtained by noting that Dunning's esimate of the H CO H C O energy defect is 0.68 eV, off by 0.13 eV. An analogous situation applies to the barriers for dissociation of HCO and COH. These were determined by adjusting the three nonlinear parameters yIJ,T ~and ~ , y3Jin ( 2 ) until the energy and geometry of the barrier for species j were fit as well as possible. In this case the assumed barrier locations were taken from Dunning's calculation as was the C O H barrier height, but the H C O barrier height was adapted from experiment,' and is 0.22 eV below Dunning's estimate. Note that the potential in eq 2 is not sufficiently flexible to enable us to fit all the barrier properties exactly given that most of the parameters therein were fit by using the equilibrium properties. Where inaccuracies occurred, we emphasized the fit to barrier height rather than position. Table I1 summarizes the resulting parameters in VHcoand VcoH. Because the isomerization saddle point influences a spatially well-localized portion of the potential associated with quite high energies, the regional potential V,,, was developed by using a quadratic function:
+
+
V,,, = VIS,+ F,,,(R - R'SO)2 + Fozo(R2 - R?)2 Foo,(O
-
+
- 81'o)2+ Fllo(R - RIS")(R2- RPO) + Flo,(R -
+
R''O)(0 - ~9'~") FOl1(R2 - R?)(O - Po) (3) where the variable R is the H atom to C O midpoint distance and 0 is the angle between R and R, such that 0 = 0 corresponds to linear COH. The parameters in (3) were taken directly from Dunning's calculation, and are summarized in Table 111. The switching functions T l , T2,and T3were determined by trial and error with the constraints that (1) they turn on and off quickly enough that the regional potentials are not changed appreciably close to minima or saddle points, and ( 2 ) they must not vary so (!1) Harding, L. B.; Wagner, A. F.; Bowman, J. M.; Schatz, G. C.; Christoffel, K. J. Phys. Chem. 1982, 86, 4312, and references therein.
216
Geiger and Schatz
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
TABLE 1V: Comparison of Potential Surface Properties for
Minima and Saddle Pointsa I
I
I
parameter
I
-51 -5
I
I
I
1 3 5 X Figure 1. Potential energy contours of the fitted surface as a function of the H atom location for a fixed CO distance of 2.432 bohr. Contours are spaced in 0.02-au energy increments starting at -0.44 au. The po-
-3
tential of separated H
-1
+ CO is -0.412486
au at equilibrium.
rapidly that the potential has significant ripples. The functions used (with all parameters in atomic units) are
T , = yz(l - tanh (9.0(R1 - R3 - 0.14))) T2 = exp[-((R1
+ R3 - 4.652)2/2.9)] exp[-((R,
- R3 -
0.358)2/0.09)]
T3 = yz(l
+ tanh (7.O(R, - R3 - 0.61)))
(4)
Note that Tz is chosen to be unity at the isomerization saddle point (where R1 R3 = 4.652, R, - R3 = 0.358),and the other functions are chosen to turn on and off at geometries -0.2 bohr on either side of this saddle point. B. Properties of the Potential Energy Surface. Figure 1 shows potential energy contours of the fitted surface as a function of the H atom location relative to a fixed C O distance equal to that for the isomerization saddle point. The HCO and COH wells are clearly evident, and the potential surface near each well is smooth. The isomerization saddle point occurs with the H atom closer to the 0 atom than to C, and near this saddle point the potential has a few ripples, mostly at high energies relative to H + CO. Although there are no cusps on our potential, Figure 1 clearly shows that in a few spots the transitions between regional potentials are fairly abrupt. These were not found to interfere with trajectory integrations, however. Since we found that making the transitions less abrupt lead to poorer fits elsewhere, further optimization was not attempted. Table IV compares several properties of each saddle point and minimum on our surface with those from Dunning's calculation. The fit to Dunning's results (or experiment where appropriate) is essentially exact for the two minima and reasonable for all the saddle points. All the barrier heights are essentially exact, and most C H or OH saddle point distances are within 0.2 bohr of the desired values. The poorest fit is for the HCO saddle point bending angle which is off by 21O. Since this saddle point plays only a minor role in the trajectory results, we have chosen not to attempt further improvement. One other defect in the surface is that a steepest decents path from the isomerization saddle point does not go directly to HCO. (The one going to COH is direct.) Instead, it goes first to shallow (0.07 eV) minimum (at RCH= 2.65, Rco = 2.27, and ROH= 2.1 l ) , over a second barrier (RcH = 2.76, Rco = 2.26, and ROH = 2.46), and then to separated H CO rather HCO. This w a r s because of the strongly repulsive "wall" seen in Figure 1 for perpendicular H + C O orientations. This feature is due to the LEPS part of eq 1 and could not be removed without substantially altering our surface in the C O H and H C O wells. It should be noted that, although the steepest descents path does not allow for direct isomerization, it is still possible for trajectories to isomerize due to the presence of centrifugal barriers.
+
+
present surface HCO Minimum V relative to H + CO, eV -0.81 2.116 RCH 2.242 RCO HCO angle, deg 126 harmonic frequencies, cm-' 2920, 2147, 1148
(-0.8 1)' 2.116 2.242 126 2921,2147, 1150
HCO Saddle Point 0.1 1 RCH 3.346 RCO 2.182 HCO angle, deg 139 harmonic frequencies, cm-' 2043, 222, 8991'
(0.ll)d 3.263 2.172 118 2060,422, 1043i
V relative to H
+ CO, eV
V relative to H
+ CO, eV
COH Minimum
ROH RCO
COH angle, deg harmonic frequencies, cm-'
1.04 1.848 2.434 114 3620, 1305, 1043
COH Saddle Point 1.72 2.530 ROH 2.3 26 RCO COH angle, dee 125 harmonic frequencies, cm-' 1620, 388, 2051i V relative to H
+ CO. eV
V relative to H
+ CO, eV
Isomerization Saddle Point 2.64 2.83 2.30 2.18
RCH
RCO ROH
Dunningb
1.04 1.848 2.434 114 3620,e 1305, 1043 1.72 2.328 2.258 120
2.30 2.505 2.432 2.147
a In atomic units unless noted
otherwise. Reference 8. Reference 7. e These are a revised set of frequencies not reported in Dunning's original paper but based on the force constants reported therein.
'Reference 6 .
C. Details of Trajectory Calculations. The standard threedimensional quasiclassical trajectory method12 was used to calculate cross sections for collisions of H with C O at relative translational energies Eo of 0.95, 1.84, 2.30, 2.51, 3.19, and 4.0 eV, and for D C O at E = 2.2 and 2.4 eV. Most of our results refer to batches of 1000 trajectories run with an initial vibrational state u, = 0, and rotational state j , = 7, although some batches of 5000 trajectories were also used (as will be noted). j , = 7 refers to the most probable rotational state of CO at 300 K. Several runs using j , other than 7 were considered, but no significant variation of the transition probabilities with initial j , over the range j , = 0-20 was noted. The maximum impact parameter b,,, was determined separately at each energy. The value of b,,, needed to converge the u 2 1 cross section was used, and this typically had a value of 4.6 bohr. Final state binning was done by rounding off the vibrational action (as defined in ref 12b) and rounding down the rotational action to the nearest integer multiple of h . The only nonstandard part of the trajectory work concerns two procedures used to tag trajectories associated with hitting certain regions of the C O molecule. In the first procedure, trajectories were analyzed to determine the smallest CH and OH distances during each collision. Those with the C H distance smaller than OH were regarded as collisions in which H hits the C while those with O H smaller than C H were regarded as H hitting 0. The cross sections calculated with each batch are labeled QSH and respectively, with vj denoting the final vibration/rotation quantum
+
QF,
(12) (a) Porter R. N. Annu. Reu. Phys. Chem. 1974, 25, 317, and refer-
ences therein. (b) Porter, R. N.; Raff, L. M.; Miller, W. H. J . Chem. Phys.
1975,63,2214. Note that the right-hand side of eq 49 in this reference should be divided by 27r.
Collisional Excitation in H
+ CO
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984 217
TABLE V: Rotationally Summed Integal Cross Sections and Average Rotational Quantum Numbers for H
Vu)
Qu
V
(j0,H)
QO“ ,
QJCOH)
E , = 2.30 eV (b,,,
1 2 3 4 5
14.6 f 0.8 5.1 f 0.4 0.75 f 0.16 0.51 f 0.12 0.11 f 0.04
14.9 16.4 17.8 15.0 11.2
6.4 f 0.5 2.4 f 0.3 0.62 f 0.15 0.48 i 0.12 0.11 k 0.04
1 2 3 4 5 6
15.2 f 0.8 5.7 f 0.5 1.7 i 0.2 0.56 f 0.13 0.25 f 0.08 0.042i 0.024
15.5 17.9 17.6 19.2 13.3 16.3
6.2 f 0.6 2.4 f 0.3 1.3 i 0.2 0.44 i 0.11 0.24 f 0.08 0 . 0 4 2 i 0.024
13.2 13.9 14.6 13.9 11.2
(~JCOH))
+ CO
QCY
(jC,H,
8.2 f 0.7 2.7 i 0.33 0.14 f 0.06 0.033 f 0.033
16.2 18.6 32.2 30.9
Q ~ ( H C O ) V~(HCO))
= 4.59 bohr)
0.47 i 0.10 0.22 f 0.06 0.052 f 0.030 0.053 i 0.024 0.076 f 0.029
19.6 18.1 11.7 13.0 9.7
0
20.8 19.3 32.2 30.9
0
E o = 2.51 eV (b,,,
TABLE VI: H
12.6 15.6 13.2 13.6 13.3 16.3
= 4.67 bohr) 0.37 f 0.09 23.7 0.37 i 0.09 22.3 0.35 f 0.09 17.3 0.13 f 0.05 10.9 0.10 f 0.04 12.5 0.042 i 0.024 16.3
5.5 f 0.5 2.6 f 0.3 0.1 f 0.06 0.033 f 0.033
9.1 f 0.7 3.1 f 0.4 0.40 f 0.11 0.13 f 0.07
6.7 f 0.6 3.0 f 0.3 0.40 f 0.11 0.13 f 0.07
0 0
21.1 20.4 31.7 38.5
0 0
+ CO Relative Vibrational DistributionsP = Q u / X u = , ‘ Q , experimental resultsa
trajectory results
E , = 2.3 eV 0.69 i 0.04 0.24 i 0.02 0.036 i 0.008 0.024 f 0.006 0.005 f 0.002
V
1 2 3 4 5 6
0
column a
17.4 19.7 31.7 38.5
Reference 3.
1
E , = 2.51 eVb
fluxes
probabilitics
0.64 f 0.01 0.25 f 0.01 0.069 f 0.002 0.028 f 0.003 0.012 f 0.002 0.003 f 0.001 2
0.75 f 0.15 0.14 f 0.01 0.07 ?: 0.01 0.008 +- 0.008 0.015 i 0.008
0.74 f 0.15 0.15 f 0.01 0.08 f 0.01 0.01 f 0.01 0.02 f 0.01 0.01 i 0.01 4
0.007
i
0.007
3
These results were obtained by using 5000 trajectories.
numbers. The analo ous rotationally summed cross sections are denoted Qf” and Q,8H. In the second analysis procedure, we labeled those collisions having the minimum C H distance smaller than the C H saddle point distance in H C O as “forming” HCO (with cross section Q”,(HCO)), and those having the minimum OH distance smaller than the OH saddle point distance in COH as “forming” C O H (cross section Q,(COH)). We use the term “forming” here only to mean that the region of the potential surface associated with a certain species was sampled. The question of the lifetimes associated with these species will be addressed separately. In a handful of trajectories we found that both the C H and OH minimum distances were smaller than their respective saddle point values. To avoid double counting, these were binned as forming HCO or C O H depending on which minimum distance, CH or OH, was smaller. Most of these trajectories were associated with perpendicular H CO collisions rather than isomerization from COH to HCO or visa versa. In fact it should be noted that except for certain C O H forming collisions to be discussed below, most of the trajectories exhibited just a single inner turning point corresponding to a direct impulsive collision.
+
111. Results and Discussion
A . Vibrational Distributions. The resulting rotationally summed integral cross sections for H CO(v,=O,j,=7) at translational energies Eo = 2.30 and 2.51 eV are presented in Table V. Included in the table are all the regionally analyzed cross sections introduced in the previous section. Also included in this table are the average rotational quantum numbers ci,) associated with each vibrational state v, and with each analysis procedure. The results in Table V are best discussed by first comparing the relative cross sections therein for v = 1-6 with the results measured by Wight and Leone3 at E = 2.3 eV. This comparison is given in Table VI, with our results at 2.3 and 2.51 eV given in columns 1 and 2, and the Wight and leone results in columns 3 and 4. Column 4 contains the relative probabilities reported in ref 3 while in column 3 we have converted the experimental relative probabilities to relative fluxes by multiplying by the final velocity (for j = 7) and normalizing. This was done since it is the fluxes which should be compared with cross sections. It should be noted here, however, that the comparison in Table VI is not as precise as it may seem because the experiments are not really single-collision experiments. A more accurate com-
+
parison between theory and experiment would require us to average our cross sections over the velocity distributions associated with each of the successive collisions sampled in the experiments. To estimate the influence of this we have calculated the total average energy transfer per collision (including rotationally inelastic scattering) by calculating all the inelastic cross sections at 2.51 eV and by separately calculating the elastic cross section Qelastic using a quantum scattering method with the H C O spherically averaged potential. We find that Qelastlc= 129 bohrZ and that the average energy transfer per collision is 0.076 eV. Since this is only a small fraction of the incident translational energy, it appears that the effect of averaging over successive collisions will be small as long as the number of collisions is small (say, less than 5). Note that the effect of this averaging on our results would be to decrease the probabilities associated with the highest vibrational states. Since, as we will discuss, our single-collision results for the highest vibrational states tend to be too low, this averaging would be expected to worsen the agreement with experiment. Comparing our results with the experimental ones in column 3 of Table VI indicates generally good agreement, especially with our results at Eo = 2.51 eV. In fact, the v = 1, 3, 5, and 6 results are within the respective uncertainties of the calculations and measurements, and the experimental result for v = 4 is so uncertain that the comparison with our result is not too meaningful. The comparison for u = 3 and 5 is apparently better with our Eo = 2.51 eV results than our Eo = 2.3 eV results, which suggests that some feature on our potential energy surface should be shifted by around 0.2 eV from its current value. We will have more to say about this later, but for now we note that this same shift is also seen in the results for D + C O which are presented in Table VII. Here the experimental measurements refer to Eo = 2.2 eV, so we have calculated trajectory results at 2.2 and 2.4 eV. The agreement of our results at 2.4 eV with experiment is roughly comparable in quality to that for H C O at 2.51 eV, and seems better than for Eo = 2.2 eV. Note also that the H CO trajectory probabilities at 2.51 eV are roughly the same as those for D + C O at 2.4 eV. This agrees with an observation made by Wight and Leone,3 and we also find, in agreement with them, that the magnitude of the vibrationally inelastic cross sections for H C O are substantially larger (factors of 2-3) than for D CO. To further test the agreement between our trajectory results and experiment, we have used our trajectory results to directly
+
+
+
+
+
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
218
TABLE VII: D t CO Relative Vibrational Distributions trajectory results V E , = 2.2 eV E , = 2.4 eV 1 0.80 f 0.03 0.72 f 0.05 2 0.18 f 0.01 0.23 f 0.02 3 0.015 f 0.004 0.04 i 0.01 4 0.002 f 0.001 0.002 f 0.001 5 0 0.004 f 0.003 6 0 0 a
Geiger and Schatz experimental resultsa fluxes probabilities 0.80 f 0.1 9 0.12 f 0.01 0.04 f 0.02 0.016 f 0.016 0.0074 f 0.0074 0.0065 f 0.0065
0.79
i: 0.19
0.13 f 0.01 0.05 i 0.02 0.02 2 0.02 0.01 f 0.01 0.01 f 0.01
Reference 3.
ar
2
0 ’900
1
1
,
2000
2100
y;j .’. ‘
2200
2300
Wavenumbers (crn-l) Figure 2. Comparison of measured (v 2 2) infrared spectrum (circles)
with the fit of Wight and Leone (solid line) and with the simulated spectrum (dashed line) associated with our calculated cross sections Qu,. These cross sections were converted to number densities before calculating the spectrum, and the result was normalized to agree with the measured results near the peak in the spectrum. 10 I
n\\ 1
\
/
4-
t
\ /
\\
Figure 4. Cross section as a function of final vibrational state u for H + CO at Eo = 2.51 eV. Included in the plot is the integral cross section Q, (labeled “total”),along with the regionally analyzed cross sections QF” and QJHCO) (“HCO”). (“hit C”),QeH (“hit 0”),Q,(COH) (“COH”), 10
I
4-
\
1900
2000
2100 2200 Wavenurnbers(crn-’l
2300
Figure 3. Comparison of the measured spectrum with the simulated spectrum associated with our rotationally thermalized cross sections (solid line). For reference, the nascent spectrum from Figure 2 (dashed line) has been reproduced. simulate the C O chemiluminescefice spectrum at 2.51 eV. Since the measured results used a cold gas filter, only u 3 2 were considered. The spectrum was simulated with the same intensity expression and resolution function as used by Wight and Leone. The results are presented in Figures 2 and 3. Figure 2 presents the measured spectrum, Wight and Leone’s fit to that spectrum, and the spectrum expected from our calculated nascent vibration/rotation distributions. Wight and Leone’s results refer to thermalized rotational distributions, so in Figure 3, we present a “rotationally thermalized” version of our spectrum along with our nascent spectrum and the measured results. Evidently the agreement of our thermalized spectrum with experiment is as good as Wight and Leone’s fit. The nascent spectrum is not in good agreement because of contributions from highly rotationally excited species. Since the overall agreement of our vibrational distributions with those of Wight and Leone looks good, it is relevant to ascertain what features of the potential energy surface are important in determining these distributions. For this purpose, we study the regionally analyzed cross sections of Table V. Figure 4 presents these cross sections for H C O at 2.51 eV, plotted as a function of u. Comparing first those collisions where H hits C vs. those
+
Figure 5. Cross section vs. u as in Figure 4 but for D + CO at 2.4 eV. where H hits 0, we find that for u = 1,2 Q:” is slight1 higher than while for u = 3-6, Q,””is much higher than Note especially that for u = 5,6 all of the cross section is determined by collisions where H hits 0. Considering now the C O H and HCO formation cross sections, we find in Figure 4 that the cross section for forming HCO is quite large, in fact most of QFH for u k 1 is determined by trajectories which cross the relatively low HCO barrier. The cross section for forming COH is much smaller and is rather slowly dependent on u, but for the highest u’s, a substantial part of QeHis determined by trajectories in which C O H is formed. The results for D C O
+
Collisional Excitation in H
+ CO
The Journal of Physical Chemistry, Vol. 88. No. 2, 1984 219
TABLE VIII: Lifetimes of the HCO (DCO) and COH (COD) Species Formed in H (D) + CO Collisions
TABLE IX: Average Vibrational Energy Transfer' H t CO as a Function of E ,
ITP
species
(E,, eV)
lifetime, s
av no.a
HCO (2.51) COH (2.51) DCO (2.40) DCO (2.40)
1.1 x 10-14 3.0 x 10-14
1 3 1
1.6 x 1 0 4 4 6.1 x 10-14
0.005 0.014 0.017
0.95 1.84 2.30
3
a ITP (inner turning point) average number is defined by using the number of sign changes in the time derivative (from negative to positive) of the H (D) t o closest heavy atom internuclear distance.
2.5 1 3.19 4.00
0.019 0.025 0.032
a CAE)has been calculated by using the rotationally summed integral cross sections Q, (such as in Table V) and the j = 0 vibrational energies E , in the following formula where vmax is the maximum vibrational state allowed at a given E,:
t
,max m = z (E, U=O
E,)QU,YQ, ,=0
I
6
4 Qv
2
n
U
0
2
4
6
8
V
Figure 6. Cross section QD vs. u for H + CO at Eo = 1.84, 2.3, 2.51, and 4.0 eV.
at 2.4 eV, presented in Figure 5, are analogous in that the highest vibrational states observed are due to collisions of D with the 0 atom, usually with formation of COD. We next consider the average lifetimes of the HCO's and COH's which are formed in H CO collisions at 2.51 eV. These lifetimes are defined by the time interval between the first and last crossing of the appropriate saddle point CH or OH distance. Table VI11 summarizes the H CO lifetimes as well as their D CO analogues. Those for HCO and DCO are quite short, corresponding to only one inner turning point of the HC or DC internuclear distance, and indicating a simple impulsive encounter. Those for COH and COD are, on the other hand, somewhat longer, with an average of three OH or OD inner turning points. Some of the COH's lived much longer than this average (many tens of turning points), providing at least some support for the idea that at least some fraction of the COHs formed are behaving statistically. The slow variation of QJCOH) with u in Table V is also reminiscent of this statement. We are now in a position to explain the 0.2-eV shift between theory and experiment noted in Tables VI and VII. Table VI shows that the biggest relative change in the trajectory results in going from 2.3 to 2.51 eV is in the probabilities for u 2 3. Table V shows that most of that change is due to changes in Q,"". Qv(COH) exhibits a corresponding, but smaller change. Since the magnitude of QJCOH) (and to a lesser extent Q,"") depends directly on the height of the H + CO COH barrier, and since the latter barrier is taken from an ab initio calculation which is known to be susceptible to errors in barriers of a few tenths of an electronvolt, it seems likely that the 1.72-eV barrier is the origin of the problem. The 0.2-eV difference between 1.5 1 and 1.3 eV is then the empirically determined shift in the barrier needed to fit our results to experiment. This shift is identical in magnitude with the probable error estimated in the barrier by Dunning13 based on the corresponding error in the HCO barrier. The dependence of our trajectory cross sections on the initial translational energy Eo is displayed in Figure 6 where we plot Qu vs. u a t Eo = 1.8, 2.3, 2.51, and 4.0 eV. The expected increase in vibrational inelasticity with increasing Eo is observed, but it
+
+
+
-
(1 3) Dunning, T. H., private communication.
Figure 7. Cross section Q, vs. j at u = 1 for H + CO at E = 2.51 eV. This is the result of a batch of 5000 trajectories.
should be noted that the cross section at 4.0 eV shows an unusually slow falloff at high u. A regional collision analysis shows this to be due to COH formation. The only experimental information concerning the translational energy dependence of the collisional excitation cross sections comes from measurements of the total emission intensity per H atom by Wight and Leone.3 This is roughly proportional to the average energy transferred into vibration per collision. Table IX presents our calculated ( A E ) / E o values between 0.95 and 4.0 eV. The table indicates an almost linear dependence of ( A E ) / E , on Eo. Unfortunately, no direct comparison with experiment is possible because of the lack of an absolute calibration on the experimental results. The results of Wight and Leone are not inconsistent with a linear functionality although the experimental error bars are too large to rule out other possibilities as well. B. Rotational Distributions. Figure 7 presents a typical final state rotational distribution for H CO, this one for u = 1 at 2.51 eV. The figure indicates that Q , peaks at j = 7, which is identical with the assumed initial rotational quantum state. The distribution is far from Boltzmann-like, however. This is most clearly shown in Figure 8 where Qu,/(2j 1) is plotted on a semilog scale vs. j ( j + 1). The figure shows a rather curved distribution, which roughly fits a 300 K Boltzmann distribution at small j (the straight line labeled 300 K), but has a long tail at higher j . The straight line labeled 3480 K shows a very approximate fit to these high j points using a Boltzmann distribution. The only experimental results with which the present rotational distributions may be compared are due to Wood, Flynn, and Weston! They measured the fluorescence spectrum of collisionally excited CO between 2160 and 2280 cm-', and interpreted peaks in the spectrum at 2180 and 2230 cm-I as due to the predominance
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+
220
The Journal of Physical Chemistry, Vol. 88, No. 2, 1984
lo 0
6
Geiger and Schatz
6 2000
1000 jij
+
3000
11
Figure 8. Semilog plot of QU,/(2j+ 1) vs. j(j + 1) using the same data as in Figure 7.
I
2 /\"=I-6
V
4
6
Figure 10. Average rotational quantum number vs. u for H + CO with E = 2.51 eV. The averages included are (i,) (labeled "total"), (jZH) ("hit C"), ( ~ 2("hit ~ )On), Ci,(COH)) ("COH"), and (j,(HCO)) ("HCO").
.
0
5
Wavenumbers (~6') Figure 9. Simulated spectrum associated with our u = 1-6 and u = 1 vibration-rotational distributions. The instrumental resolution has been assumed to be 30 cm-' in calculating this spectrum. (The same as in ref 4.) To remove the noise in our spectrum which would result from use of the calculated u = 1 cross sections in Figure 8, we have used the T = 300 and 3480 K lines to define smoothed cross sections for use in the intensity calculation. The calculated u = 2-6 cross sections were used directly without smoothing.
of rotational states j = 19 for v = 2 and j = 24 for v = 1, respectively. Figure 9 shows the spectrum to be expected from our nascent trajectory results including all vibrational states. This spectrum was calculated by using a resolution function similar to that in ref 4, and for reference to Figures 7 and 8, we also plot the spectrum from v = 1 only. Both spectra in Figure 9 are similar, and show dominant peaks at 2105 and 2170 cm-' due mostly to the P and R branches of the 300 K part of the v = 1 rotational distribution in Figures 7 and 8. There is also a peak at 2250 cm-' due to the high-temperature (3480 K) part of the v = 1 rotational distribution. Evidently, two or our peaks (Le., 2170 and 2250 cm-') are a t close to the same frequencies as the two peaks seen in ref 4,suggesting reasonable agreement between theory and experiment. The ratio of peak intensities is not the same, however, as in ref 4,and our interpretation of the lower peak (i.e., that is due to v = 1 rather than v = 2) is different. Since the experimental results probably involve some rotational relaxation, it is hard to say at this point whether the differences between our results and experiment are significant. Let us now study the rotational distributions using the regional analysis methods. In Figure 10 we plot the average rotational quantum number from Table V vs. v for H CO at 2.51 eV, along (j,with the corresponding regional averages &"), ",):I( (COH)), and (j,(HCO)). The analogous results for D + CO at 2.4 eV are presented in Figure 11. Examination of these figures indicates that ci,) is not strongly dependent on v. At low v, is roughly the arithmetic average of 0::") and (f"), with 0, ) always larger than 0:"). This makes sense physically since
+
Pi)
0
1
2
3
V
Figure 11. Average rotational quantum number vs. u as in Figure 10 but for D + CO at 2.4 eV.
collisions with the lighter end of the CO molecule will have a larger orbital angular momentum associated with them, and much of this orbital angular momentum would be transferred to rotation in an impulsive collision with a light atom such as H. As v increases, we find that 0:")increases while 0:") either decreases or is constant. Generally for an impulsive collision one would expect that 0')should increase with increasing v since the higher impulse needed to excite higher vibrational states will (for nonlinear collisions) partially be converted into additional rotational angular momentum. This happens for collisions with the C atom, but not for collisions with the 0 atom. However, we have already noted that the high v collisions with the 0 atom are often associated with formation of a COH complex. Under these circumstances we would expect an average j which decreases with increasing u because of the decreasing available rotational phase space. Examination of (j,(COH)) and (j,(HCO)) further confirms these conclusions. Going back to Figure 8, we now see that the high j tail in the distribution is mostly due to impulsive collisions with the C atom. Collisions associated with C O H formation for v = 1 also produce highly rotationally excited CO
J. Phys. Chem. 1984,88, 221-232 as Figure 10 indicates, but Table V shows that the contribution of such collisions to the overall cross section for u = 1 is small. Only the rotational distributions associated with higher u such as 5 or 6 will be sensitive to C O H formation. IV. Conclusion In this paper we have used the quasiclassical trajectory method with a realistic potential energy surface to study collisional excitation in H C O and D + CO. Comparison of our results with experiment has been very good for the most part, and this has enabled us to assess in detail just what features of the potential energy surface the measured results are sensitive to. Probably the most important conclusion of this analysis is that the high u portion of the vibrational distribution is due to formation of a COH complex which lives for a few vibrational periods. The barrier for forming this complex directly determines how large are the high u cross sections, and thus one can use the data to infer (at least crudely) the barrier height. In the present case we found
+
221
that the 1.72-eV barrier predicted by Dunning is about 0.2 eV too high. Other features of the measured vibrational and rotational distributions were found to be less sensitive to the presence of barriers or wells. For example, the average vibrational energy transfer ( AE)is largely controlled by u = 1 and 2 excitation, which primarily comes from impulsive collisions of H with C or 0. The rotational distributions were found to be decomposable into low and high j components, with much of the low j component due to collisions of H with 0, and much of the high j component from collisions of H with C.
Acknowledgment. Helpful discussions with T. H. Dunning, S. R. Leone, C. A. Wight, and R. E. Weston are gratefully acknowledged. This research was supported by NSF Grant CHE-8 115109. Registry No. Atomic hydrogen, 12385-13-6; carbon monoxide, 63008-0; COH, 71080-92-7.
A Theoretical Study of Deuterium Isotope Effects in the Reactions H,
+ CH, and H 4-
CH4 George C. Schatz,+Albert F. Wagner,* and Thomas H. Dunning, Jr. Theoretical Chemistry Group, Chemistry Division, Argonne National Laboratory, Argonne, Illinois 60439 (Received: February 3, 1983; In Final Form: June 13, 1983)
This paper presents ab initio potential surface parameters and transition state theory (TST) rate constants for the reaction Hz CH3 H + CH4, its reverse, and all the deuterium isotopic counterparts associated with it and its reverse. The potential surface parameters are derived from accurate POLCI calculations and include vibrational frequencies, moments of inertia, and other quantities for CH3, H2, CHI, and the H-H-CH, saddle point. TST rate constants are calculated from standard expressions and the Wigner tunneling correction. For H2 CH3 and H2 + CD,, agreement of the rate constant with experiment is good over a broad temperature range, suggesting that the calculated 10.7 kcal/mol barrier is accurate to within about 0.5 kcal/mol. Agreement with experiment for H + CH, using the calculated 13.5 kcal/mol reverse reaction barrier is poorer; a 12.5 kcal/mol barrier is found to provide a more reasonable estimate of the true barrier. Primary isotope effects for the deuterated analogues of Hz + CH, are found to be correct in magnitude at high temperature, but with a weaker temperature dependence than experiment. The calculated secondary isotope effects also appear to be weaker functions of temperature than experiment, although a large uncertainty in the experimental results precludes a quantitative assessment of errors. Our analysis of isotope effects in the H + CH4 reaction is restricted to examining the branching ratios between H and D atom abstraction in the reaction of H with the mixed species CH,D, CH2D2,and CHD,. A combination of reaction path multiplicity, favorable zero point energy shifts, and a greater likelihood of tunneling causes H atom abstraction to predominate over D atom abstraction in H + CH3D and H + CH2DZ,but for H + CHD,, we find that the H atom and D atom abstraction rate constants cross near 700 K, with H atom abstraction dominating at low temperatures and D atom at high.
+
-+
+
-
I. Introduction The reaction H2 CH3 H + CH4and its reverse have long played an important role in the theoretical and experimental development of chemical kinetics. Early studies of the potential surface for this system were done in the 1930’s,’ and by the mid-l95O’s, a detailed comparison between theory and experiment for H2 + CH, and seven isotopic counterparts had already been completed.2 Reflecting general advances in experimental methods, the rate constants for H 2 + CH3 and H CH4 have been remeasured several times in the past few years. Recent reviews have summarized the current status of the measured results for both H2 + CH33and H + CH4.4 Generally speaking, there is good agreement between the rate constant measured by different groups within the past 20 years, although it has been noted that the ratio of forward to reverse rate constant is somewhat at variance with the equilibrium constant a t low temperature^.^
+
+
Consultant. Permanent address: Department of Chemistry, Northwestern University, Evanston, IL 60201. Alfred P. Sloan Research Fellow and Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/84/2088-0221$01.50/0
In this paper we present a detailed theoretical analysis of H2 H CHI, its reverse, and their deuterated isotopic counterparts. This analysis is based on a potential surface which was recently determined by accurate configuration interaction technique^.^ An earlier analysis of the rate constant for just the H + CH4 reaction using this surface6 suggested that the barrier for H CH4 was accurate to within 1 kcal/mol. In this paper we present new ab initio results for the CH, force field which provide the information necessary to evaluate the H, CH, rate constant and isotope effects. Since the barrier height tends to
+ CH3
-+
+
+
+
(1) Gorin, E.; Kauzmann, W.; Walter, J.; Eyring, H. J. Chem. Phys. 1939, 7, 633. (2) Polanyi, J. C. J. Chem. Phys. 1955, 23, 1505. (3) Kerr, J. A.; Parsonage, M. J. “Evaluated Kinetic Data on Gas Phase Hydrogen Transfer Reactions of Methyl Radicals”; Butterworths: London,
1916.
(4) Shaw, R. J . Phys. Chem. ReJ Data 1978, 7, 1179. (5) Walch, S. P. J . Chem. Phys. 1980, 72, 4932. (6) Schatz, G.C.; Walch, S. P.; Wagner, A. F. J. Chem. Phys. 1980, 73, 4536.
0 1984 American Chemical Society
