Unimolecular Decomposition of Triatomics
The Journal of Physical Chemistry, Vol. 83, No. 8,
1979 933
Exit-Channel Coupling Effects in the Unimolecular Decomposition of Triatomics Don 1.. Bunker, Karin R. Wright, Department of Chemistry, University of California, Irvlne, California 92717
Wllliam L. Hase,* Department of Chemistry, Wayne State University, Detroit, Michigan 48202
and F. A. Houle" Arthur Amos Noyes Laboratory of Chemical Physics, California Institute of Technology, Pasadena, California 9 1125 (Received May 18, 1978) Publication costs assisted by the National Science Foundation
A survey was carried out, by trajectory methods, of the behavior of triatomic unimolecular systems having a potential barrier in the product channel. Principal variables were atomic mass combinations, molecular geometry, degree of coupling of triatomic bending forces to product motion, and angular momentum content of the initial conditions. Distributions were found for conversion of potential energy at the barrier to product translational energy. Broadening and shifting effects on these distributions were observed and are discussed. The observed exit-channel effects are compared with experimental measurements of product energy distributions.
Introductioin The question of the influence of exit-channel coupling on unimolecular behavior has recently become of some importance. It arises in cases where the critical configuration for RRKM theory occurs with interatomic distances such lhat further product separation, or other progress along a reaction coordinate, involves appreciable changes in potential energy. Forces will then continue to act even though the system has left the region of configuration space where statistical behavior is expected. Such things as product energy distributions may be affected thereby. Since attention was directed to this possibility by Marcus1 it has been discussed in connection with two experiments, both concerning reactions initiated by halogen additions to double bonds, one in a molecular beam environment2 and the other studied by arrested relaxation chernilumine~cence.~In addition, a trajectory study has appeared in which molecules of a specialized type (resembling CD3C1and CD,H) have had their behavior ascertained in siome detail.4 Our intended contribution is to survey a wide variety of simpler triatomic models, not necessarily inspired by specific experiments, in order to form an impression of the probable magnitude of these effects and of what variables will have the itnost influence on their relative importance.
Method We proceeded by straightforward adaptation of the minicomputer edition5 of our publicly available generalpurpose A+RC program.6 These programs provide a facility for the user to prescribe his/her own interatomic potential, and this was done in the standard way. The portion of the program that generates starting conditions was replaced entirely, as described blelow. The routine that evaluates final trajectory properties was also modified so as to operate at two different places along the reaction coordinate, rather than one. The potential energy was U = MAB+ MBC+ l/,kH(a - ~ r o + ) ~J - sKL (1) Mij
= Dij(1 - exp[-Pij(rij H = 1 - tanh (UTAEI c )
+
(2) (3)
0022-3654/79/2083-0933$01 .OO/O
= cos-'
[(TAB2
+ rBC2 - rAC2)/2rABrAC]
J = 6 0 " -~ rAB0l2 ~ exp[-y(rAB - rAB0)2] K = 1 - sech [b(rAB - rABO)]
(4) (5)
(6)
L = tanh [b(rAB - rABo)]
(7) The conventional Morse parameters in (2) had the fixed 198.12 kcal; Om = PBC = 2.15 values DAB = 99.115, DB A-l; rmo = rBCo= 0.957 The nominal force constant k was 0.7 mdyn 8, rad-l; except for DBc, masses, and variable geometry, the model is similar to a water molecule at low vibrational am litude. The parameters a and b were fixed at 2 and 3.14 respectively. The value of a regulates the rapidity (not the geometry, which is variable) with which the bending force constant dies out as the bonds are extended. A smooth maximum in potential energy, 22 kcal higher than the value at large TAB, is produced at rm = 2.2 8, by the fixed parameters s = 30 kcal, 6 = 100 kcal A-2, and y = 1 k2. Three things were systematically varied. The molecule was bent (b) or linear (1) according to whether a. = r or a. = 2 ~ / 3 .It was internally (I) or externally (X) coupled according to whether c = -1.8 or -3.0a7 This regulates whether the attenuation of the apparent bending force constant occurs before or after the potential maximum is encountered as rAB extends. Finally, various atomic mass combinations were used. The conventional designations L (light), M (medium), and H (heavy) are adopted; L is 2 amu, M is 16, and H is usually 128 in other studies,8 but has to be 64 in this case for reasonable computer economics. The abbreviation (for example)
1.
LMM - Xb (8) means the calculation in which the particle masses are mA = 2 and mB = mc = 16, and in which the molecule is bent and externally coupled. The mass combinations considered to be of interest were LLL, LMM, MMM, HMM, and HHH. It is always the first atom, A, that is detached. Initial values of coordinates and momentag were generated by a technique called progressive sampling, designed to provide an approximately random sampling of phase space. This procedure is an adaptation of one of the three sampling methods originally devised to study the uni-
0 1979 American Chemical Society
934
The Journal of Physical Chemistry, Vol. 83, No. 8,
1979
molecular decomposition of larger systems, where truly random sampling is a problem of monumental difficulty.l0J1 The success achieved with progressive sampling prompted our use of it in the present work.12 The methodology, which has been described in detail elsewhere,l0J will only be summarized here. Chains of progressively sampled trajectories were generated as follows. The initial condition for the first trajectory in a chain was chosen by orthant sampling.1° After 50 integration cycles a 12-dimensional vector a of the coordinates and momenta was saved for use in generating the initial condition for the next trajectory. (The integration step size, after tuning, varied between 1.25 X and 5 X s; cycles per trajectory were correspondingly between 1000 and 4000.) To initialize the next trajectory a 12-dimensional random vector r was chosen. The (a,r)angle was then bisected to produce a new r. This procedure was then repeated until the (a,r) angle was less than 0.25 rad. The vector r is uniformly scaled to match the target energy, and then used to initialize the next trajectory. Constant energy hypersurfaces for the problem are toroidal in character, with six axial and six radial directions, but distorted in such a way that the vector r very rarely failed to intersect them. Inner and outer intersections of the toroid were chosen randomly. One characteristic of progressive sampling is the presence of a transient at the zero-lifetime edge of the molecular lifetime distribution. To decrease the short-time bias and thereby make lifetime distributions easier to evaluate, a set of initial conditions were rejected if the total energy was either more than 90% or less than 10% kinetic. The total energy was always 140 kcal/mol. With the sampling scheme just described, a substantial amount of this, perhaps 30 kcal/mol in a typical case, will be rotational. (Rotational energy is defined as Erot= l/zw.L, where w and L are the angular velocity and momentum, respectively. Such rotational energization is not dissimilar to what occurs in (for example) molecular beam situations. Since we also wished to study rotationally thermal reactions, we provided the additional option to reject every rotational energy that exceeded 2 kcal/mol, to approximate a thermal distribution. The sampling will accordingly be described in the results as “thermal” or “hot”, according to whether this was or was not done. Relative translational energies of A-BC separation and rotational energies of BC were evaluated by standard formulas of mechanics, extensively published and as present in the standard A+BC program. These were ascertained both a t the barrier position and when the trajectories were terminated at large (4 A) TAB. Lifetimes were also recorded when TAB passed 2.2 A. This calculation was run on a minicomputer (Hewlett-Packard 2108-MX). It was intended to be a first demonstration of the utility of such a system for dynamical calculations, but owing to unusual difficulties encountered with the sampling procedure, two other project^^^^^^^^^ preceded it. Details of the computing environment are given in ref 11, 13, and 14. Like the others, this project involved several months of overnight computing, with results retained internally for tabulation when the batch was terminated.
Results The quantity of immediate interest is the difference between relative translational energies at TAB = 4 A and ?-AB = 2.2 A. A narrow peak of this quantity at 22 kcal would indicate simple and direct conversion of potential to translational kinetic energy after the RRKM critical configuration is passed. Displacement to higher values
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corresponds to a further withdrawal of energy from the internal degrees of freedom of the products. Displacement to lower energy indicates transformation in the reverse direction, from product separational motion to vibration and rotation. These results are shown in Figure 1 (hot sampling) and Figure 2 (thermal sampling). Most but not all of the parameter combinations have been surveyed. The comparison of sampling methods was motivated by speculation that effects to be discussed below, most prominent for HHH,would turn out to arise from approximate rotational adiabaticity when high angular momentum is present. This did not prove to be the case. There are no important differences between Figures 1 and 2 (note that the horizontal scales have been shifted to accomodate a few of the histograms) and our conclusions are independent of rotational temperature throughout. Histograms of the diatomic rotational energy difference between TAB = 2.2 and TAB = 4 A, not illustrated, merely spread about 0 to a half-width of 5 or 10 kcal. Correlation of individual translational and rotational energy differences was looked for and not found. The least important parameter is the molecular geometry. There is some indioation of an average shift of a few kilocalories in the LLL comparisons. Everywhere else there are no statistically significant variations. The nature of the coupling to the bending forces, which we expected to be of more significance, has almost as little importance as the geometry. The largest effect is seen for linear HMM and HHH in Figure 1 (hot sampling). The tendency is for externally coupled linear cases to have
Unimolecular Decomposition of Triatomics
The Journal of Physical Chemistry, Vol. 83,
LLL-IP
HHH-IP
Flgure 3. Lifetime distributions, thermally sampled, corresponding to s with 0 at the left in the upper half 011 Figure 2. Bar width is each case.
distributions shifted 1co higher energies than those for the corresponding internally coupled models. This fact is in accord with expectations based on exit channel coupling effects.15 However, the statistics of the calculations are not sufficiently refined to make a definitive statement about the importance of internal vs. external coupling. Clearly the particle masses are the most important determinants of both the breadth and position of the distributions. Light leaving atoms tend to produce narrower distributions, and heavy ones cause modest but noticeable shifts to lower energies. Both these effects are easily seen as related to the velocity of departure of the A atom, and its interaction with the remaining diatom. Before discussing this, however, we must survey our results for possible indications of intrinsic non-RRKM behavior. Our data are not very good in this respect, since the production of lifetime distributions was not the primary purpose of the calculation. Inspection should be of thermally sampled cases only, and disregarding the leftmost (short-lifetime) histogram bar on account of sampling transients.1° Typical data, corresponding to the upper half of Figure 2, are presented in Figure 3. All that can be said is that their net behavior after transient correction appears to be adequately exponential and with essentially the same decay constants. Distributions of translational energy at the barrier point are broad, featureless, and indistinguishable. We have not resolved any non-RRKM features a t this (low) level of precision. It is uncertain whether previous calculations16 would have led us to expect any, and thus there is no particular reason to assume that we have been studying anything other than the exit channel effects themselves.
Discussion This calculation and that of McDonald and Marcus4 on CD3Cl and CID3H are more nearly complementary than contradictory. McDonald and Marcus began with chemical activation-like startmg conditions whereas we sought to be as thermal as possible, except in the case of overall rotation. The difference of behavior between reactions with diatomic (BC) and polyatomic (CD,) products would surely be expected to be of some importance. Nevertheless there are some similarities. Such effects as they found seem to be associated with heaviness or lightness of the departing atom in somewhat the same way as for us. Excess translational energy, relative to what there would be if there were no barrier, is of course found in both cases. A most striking finding from our calculations is that there is not a direct conversion of potential energy at the exit-channel barrier to product translational energy.
No. 8, 1979 935
Instead we find rather broad distributions for conversion of potential to translational energy. The distributions are the narrowest for the LMM situation, but even here the distributions have a breadth of -10 kcal/mol out of 22. For the models with equivalent masses or those where a heavy atom is detached the width of the distribution may be as large as 20 kcal/mol. Lee and co-workers have made extensive measurements of product translational energy distributions following halogen addition to double bonds.17 The most extensive study is that of Farrar and Lee2 where F atoms are allowed to react with ethylene to yield H atoms and fluoroethylene. In that work they found broad product translational energy distributions with average values greather than those predicted by RRKM theory even with the assumption that potential energy at the exit-channel barrier is directly converted to product translational energy. The additional inclusion of “statistically adiabatic” bends in their calculations still did not bring the calculated RRKM and experimental product translational energies into agreement. Our LMM models most closely resemble the experimental situation of Farrar and Lee.2 For these models we find the exit-channel effects broaden the translational energy distributions. In addition, the averages of the distributions for the two externally coupled LMM models are greater than the potential energy barrier in the exit-channel, 22 kcal/mol. It is possible that these two exit-channel effects, a broadening and a shift to higher translational energies, could explain the results of Farrar and Lee. However, it should be noted that the broadening we find in our LMM models is not nearly as extensive as that seem by Farrar and Lee. We feel that a straightforward extension of the results for these triatomic models to the more complicated F + C2H, H + C2H,F reaction is not possible. Therefore, whether the results obtained by Farrar and Lee are due to intrinsic non-RRKM behavior as they suggest2 or due to exit-channel effects remains an open question. In conclusion, our calculations suggest that it will be difficult to make generalizations about exit-channel effects. We find that they strongly depend upon the mass of the departing atom and depend to a lesser extent on the nature of the coupling in the exit-channel between the vibrational motions and reaction coordinate. These statements only directly apply to our triatomic models and exit-channel effects may be even more complex for molecules with more degrees of freedom.
-
Acknowledgment. The assistance of Charlotte Slater in the early stages of the project was of value to us. We thank Professors R. A. Marcus and E. R. Grant for a critical reading of the manuscript and many valuable comments. Support was provided by the National Science Foundation, to whom we are grateful. References and Notes R. A. Marcus, Faraday Discuss Chem. SOC., 55, 379 (1973). J. M. Farrar and Y. T. Lee, J. Chem. Phys., 65, 1414 (1976). J. F. Durana and J. D. MacDonald, J. Chem. Phys., 64, 2518 (1976). J. D. MacDonaldand R. A. Marcus, J. Chem. Phys., 65, 2180 (1976). S. Chapman, K. R. Wright, D. L. Bunker, A. Gelb, and J. Santamaria, Quantum Chemistry Program Exchange Catalog 316, 1976. S. Chapman, D. L. Bunker, and A. Gelb, Quantum Chemistry Program Exchange Catalog 273, 1975. Internally coupled means the force constant was attenuated before crossing the potential energy maximum. For the externally coupled cases, the force constant is attenuated after the potential energy maximum is crossed. N. C. Blais and D. L. Bunker, J. Chem. Phys., 39, 315 (1963). RektN.5 Cartesian coordinates were used in the trajectoty calculations. In this coordinate representation the center of mass motion is eliminated, and there are only SIXcoupled differential equatlons for
The Journal of Physical Chemistty, Vol. 83, No. 8, 7979
J. D. Kelley and H. H. Harris
a triatomic system: D. L. Bunker, Method. Comput. fhys., 10, 287 (1971). D. L. Bunker and W. L. Hase, J . Chem. fhys., 59, 4621 (1973). E. R. Grant and D. L. Bunker, J . Chem. fhys., 68, 628 (1978). Progressive sampling gives more random initial conditions than does simulation of thermal excitation: D. L. Bunker and S. A. Jayich, Chem. fhys., 13, 129 (1976).
(13) D. L. Bunker, Ber. Bunsenges. fhys. Chem., 81, 155 (1977). (14) J. Santamaria and D. L. Bunker, Chem. fhys., 23, 243 (1977). (15) R. A. Marcus, J. Chem. Phys., 62, 1372 (1975); Ber. Bunsenges. fhys. Chem., 81, 190 (1977). (16) D. L. Bunker, J . Chem. fhys., 40, 1946 (1964). (17) Y. T. Lee, Ber. Bunsengs. Phys. Chem., 78, 135 (1974), and references therein.
Curve-Crossing Trajectories. Vibronic Excitation in N,+-He J. 0 . Kelley” McDonnell Douglas Research Laboratories, St. Louis, Missouri 63 166
and H. H. Harris* Department of Chemistfy, University of Missouri-St.
Louis, St. Louis, Missouri 63 121 (Received September 26, 1978)
Publication costs assisted by the McDonnell Douglas Corporation
Collision-induced vibronic excitation in atom-diatom systems is studied using Monte Carlo-selected classical trajectories on two intersecting diabatic potential surfaces. The calculations are “quasi-classical” in the sense that the initial diatom conditions are selected from a quantized 300 K vibration-rotation distribution. The results are compared with those obtained from a previously developed analytical model. For systems which exhibit a surface crossing which is energetically accessible, the vibrational distribution in the upper electronic state approaches that predicted by the Franck-Condon principle as the collision energy increases, and the rotational distribution approaches that of the initially unexcited diatom. These conclusions are consistent with experimental data.
Introduction At sufficiently high relative velocities (usually lo5 m/s and above), the vibrational state distribution in excited electronic states resulting from collisions between ions and diatomic molecules or diatomic ions and neutral atoms has been observedl to be very near that given by the Franck-Condon (FC) factors connecting the initial, ground electronic state vibrational distribution to the vibrational wave functions of the excited electronic state. As the collision energy is lowered, the vibrational distribution becomes energy dependent and can deviate significantly from that expected from consideration of the FC factors al~ne.l-~ A polarization-distortion model has been suggested5 to explain the deviation from FC behavior in monatomic ion-diatom charge-transfer excitation (CTE). In this model, the electronic transition is assumed to occur vertically from a diatom potential curve which is distorted by the ion. In order to obtain sufficient distortion of the diatom potential curve for agreement with experiment, unrealistically small distances of closest approach are often required;6more importantly, the polarization model is not applicable to collision-induced excitation (CIE) processes involving a neutral atom and a diatomic ion. Such CIE processes, however, also exhibit “non FC” vibrational distributions in the excited state at lower collision energies and approach FC distributions with increasing collision energy.ll4 In previous publication^,^,^ we and our colleagues proposed a model applicable to both CTE and CIE processes to qualitatively explain the energy dependence of the total excitation cross sections and excited-state vibrational distributions. A key feature of this model is that deviations from FC behavior result from direct translation-vibration energy exchange on both the ground- and excited-state potential surfaces. If these surfaces have an avoided 0022-3654/79/2083-0936$01 .OO/O
crossing (or cross in the diabatic formalism actually employed) that is energetically accessible, then for increasingly higher collision energies the approach to FC behavior is predicted. In order to obtain analytical expressions for the excitation cross sections and vibrational distributions, several simplifying approximations were introduced in the previous calculation. The two potential surfaces were assumed to be spherically symmetric about the diatom center of mass, and the angular dependence of the coupling matrix element was neglected. The diatom vibrational wave functions on both surfaces were taken to be harmonic oscillator wave functions. Motion in the relative N2+-He coordinate was treated classically, and straight-line trajectories were employed to generate a time-dependent pmturbation which coupled the electronic states and produced vibrational excitation. While these calculations qualitatively reproduced the collision energy dependence of the excitation cross section and vibrational distribution in the excited diatom, it seems desirable to explore the behavior of this type of model without making many of the approximations required for analytical solution. In the present work, we treat the potential surfaces more realistically (although in an ad hoc fashion), and the dynamics on each surface are computed by solving the complete set of classical equations of motion. The electronic transitions are treated using a “surface hopping”’ approach with the probabilities given by a three-dimensional generalization8 of the Landau-Zener relation at the crossing seam. In these calculations, total energy and angular momentum are conserved, and rotational as well as vibrational excitation can be studied.
Method of Calculation Potential Energy Surfaces. The ad hoc potential energy surfaces used here were constructed as pairwise sums of 0 1979 American
Chemical Society
