J. Phys. Chem. 1996, 100, 15165-15170
15165
Quantum Calculation of the Recombination Rate Constant of H + CO f HCO Jianxin Qi and Joel M. Bowman* Department of Chemistry and Cherry L. Emerson Center for Scientific Computation, Emory UniVersity, Atlanta, Georgia 30322 ReceiVed: May 6, 1996; In Final Form: June 28, 1996X
The rate constant for the recombination reaction H + CO + M f HCO + M is obtained at four temperatures as a function of the collision frequency using a theory recently proposed by Miller [Miller, W. H. J. Phys. Chem. 1995, 99, 12387]. This theory combines the flux-flux correlation function with the classical strong collision assumption to obtain the recombination rate constant as a function of the collision frequency. The expression for the rate constant is evaluated in a basis of complex L2 eigenfunctions of a complex Hamiltonian, given by the real Hamiltonian for the nonrotating HCO system (J ) 0), plus an absorbing potential in the asymptotic potential. The J-shifting approximation is used to obtain the rate constant for non-zero J. The flux-flux correlation function is evaluated at several dividing surfaces, which define the boundary of the complex, and only a fairly minor dependence on this surface is found at low temperatures, but a more substantial dependence is found at high T. The results are compared with those of the standard Lindemann theory and in the low-pressure region with the low-pressure limit of the Lindemann theory. The calculated results are compared with experiment at room temperature, with Ar as the buffer gas, and can be made to agree well with experiment if the total stabilization collision cross section is about 15 bohr2.
Introduction
kr(T) )
There have been few dynamical formulations or studies of collisional recombination1-8
A + B + M f AB + M
(1)
despite its importance in chemical kinetics.9 The collisional recombination of H + CO (and dissociation of HCO) is of importance in combustion; it has been studied experimentally by several groups10-13 and is the subject of this paper. Recall that the standard model for recombination is
Qcompωkrv Qreact ω + krv 1
(5)
The usual expression for Qcomp is just the sum over the quasibound states of the complex of energy Ei; i.e.,
Qcomp ) ∑exp(-βEi)
β ) (kBT)-1
(6)
i
These states decay back to reactants with a state-specific rate ki, and thus, krv in eq 5 is replaced by ki, giving the final (usual) result
kfw
A + B T AB*
(2)
krv
kS
AB* + M 98 AB
(3)
First, A and B form a metastable collision complex; then, the complex is stabilized by a collision with M. The textbook Lindemann model treats AB* in the steady-state approximation and leads, in the simplest form, to the following expression for the bimolecular recombination rate constant:
kr(T) )
ωkfw(T) krv(T) + ω
(4)
where ω is the collision frequency (given by the product kS[M]). Note that the use of the collision frequency in this equation is a statement of the “strong collision assumption”, i.e., that the complex is stabilized in a single gas-kinetic collision. The next level of sophistication is obtained by noting that the ratio kfw/krv is a quasi-equilibrium constant which can be expressed as the ratio of reactant and complex partition functions, Qreact and Qcomp, respectively. Thus, eq 4 can be rewritten as X
Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)01292-0 CCC: $12.00
kr )
1
∑i exp(-βEi)ωki/(ω + ki)
Qreact
(7)
The ki are the unimolecular decay rates, which in the isolated resonance limit are given by Γi/p, where Γi is the width of the ith quasi-bound state. In the quasi-continuum limit, kr is given by
kr )
ωk(E)
∫dE exp(-βE)F(E)ω + k(E) Qreact 1
(8)
where F(E) is the density of states and k(E) is the microcanonical rate of decay (typically given by RRKM theory). Finally, in the low-pressure limit, defined by ω , ki, we have
kr ) Qreact(T)-1∑exp(-βEi)ω i
) [Qcomp(T)/Qr(T)]ω
(9)
It is worthwhile to notice that eq 9 is an upper bound to eq 7. There have been several quantum dynamical formulations of recombination reactions, based on the standard mechanism and with the strong collision assumption.1-3,8 Some time ago, Bowman3 proposed an extension of Smith’s theory of atomatom recombination14 to molecular recombination. In this © 1996 American Chemical Society
15166 J. Phys. Chem., Vol. 100, No. 37, 1996
Qi and Bowman function will not go to the (classical) constant high-frequency (pressure) limit.1,8 Instead the quantum result reaches a maximum and then goes to zero at infinite frequency (pressure). This is due to the stationary property of the quantum correlation function at t ) 0. The quantum turnover occurs when ω is greater than kBT/h.1 In this paper, we present the results of an application of Miller’s new theory of recombination to
H + CO + M f HCO + M
Figure 1. Sketch of the HCO potential along a path connecting the HCO minimum and the H-CO recombination saddle point. The dashed line is the dividing surface boundary which defines the region of the complex.
approach, Qcomp is given by
Qcomp ) ∫dE exp(-βE)TrQ(E)/h
(10)
where Q(E) is the Smith collision lifetime matrix,15 which is obtained from the full multichannel scattering matrix for the A + B scattering system. This definition of Qcomp suffers from the possibility of negative values because TrQ can be negative at energies where resonances do not form. At resonance energies, TrQ is generally large and positive. This formulation of recombination was used recently by Kendrick and Pack in an application to H + O2.7 They compared Qcomp using eq 10 directly, and with two variants of (10). In one, only the positive part of TrQ was used, and in the other, the positive part of TrQ was fit to Lorentzians. They found 15-24% differences in Qcomp using these three approaches, over the temperature range 100-600 K. Very recently, Miller proposed another quantum approach to obtain the recombination rate constant (still within the strong collision assumption).8 Miller defined the metastable states using a dividing surface, which for a one-dimensional model is shown schematically for HCO in Figure 1. This surface, which usually is at the energy barrier, divides the space into a complex region and a product region. When R < Rd, the molecule is considered to be a metastable species or complex. Within this region, the complex can be stabilized with a classical probability
P ) 1 - exp(-ωτ)
1 ∫dt exp(-ωt)Cf(t) Qreact
Calculations We evaluated the flux-flux correlation function using complex L2 eigenfunctions. The method to obtain these eigenfunctions is briefly reviewed below. The Three-Dimensional Complex L2 Calculation for HCO. The complex Hamiltonian describing continuum states of HCO, H ˆ c, is the usual Hamiltonian operator of the HCO system, plus the absorbing potential in the noninteractive region.26,27 For nonrotating HCO, H ˆ c is given (in atomic units) by
(11)
where τ is the lifetime of the complex and ω is the collision frequency. This is essentially a classical picture; the generalization to quantum theory made by Miller was done by using the quantum flux-flux correlation function C(t),16 evaluated at the dividing surface, with the result
kr )
We used the scaled17 BBH potential surface fit to a Legendre polynomial expansion18 to obtain kr as a function of the collision frequency at four temperatures. The resonances of HCO have been extensively studied,17,19-25 and they can be well characterized as isolated and nonoverlapping, at least at lower energies that contribute significantly to kr up to 2000 K. This sparse spectrum of resonances implies that nonresonant scattering states may play a nonnegligible role in Miller’s theory. To investigate this possibility, we calculated kr using several dividing surfaces. For resonances that are spatially localized, there should be a minor dependence on the dividing surface, provided it is not in the strong interaction region. However, for nonresonant scattering states, there will be a greater dependence on the location of the dividing surface, except of course in the asymptotic region. Therefore, the change of the rate constant with respect to the dividing surface mainly comes from the scattering states. We also can estimate the contribution from nonresonant scattering states by comparing the calculated rate constants with those calculated directly from the Lindemann expression, which only includes the metastable states. The paper is organized as follows. The calculations are described next, followed by the results, discussion, and comparison with experiment. The final section contains a summary and conclusions.
(12)
In the limit where the collision frequency is zero, kr is zero because for a nonreactive collision, the integral of Cf(t) over t must vanish. Also from this equation, it is clear that kr does formally depend on the definition of the dividing surface. However, Miller has argued that for processes dominated by long-lived resonances, kr will show only a weak dependence on the dividing surface, provided that surface encompasses a significant fraction of the resonance amplitudes. In addition, it should be noted that the theory based on a quantum correlation
H ˆc ) -
(
)
∂2 1 1 ∂2 1 + + ˆj 2 + 2µH,CO ∂R2 2µCO ∂r2 µH,COR2 µCOr2 V(R,r,γ) - iW(R) (13) 1
where R is the distance of H to the center of mass of CO, r is the CO internuclear distance, and γ is the angle between r and R. ˆj2 is the square of the CO angular momentum operator, µH,CO is the reduced mass between H and CO, and µCO is the COreduced mass. The potential used is the scaled Legendre fit to the BBH potential, as noted above, and W(R) is a Wood-Saxontype absorbing potential. The eigenfunctions of H ˆ c are denoted as ψcj , and the Schro¨dinger equation is
(
)
Γj c ψ 2 j
H ˆ cψjc ) Ej - i
(14)
where Ej - iΓj/2 are the complex energy eigenvalues. For resonances, these eigenvalues are stable with respect to changes
Recombination Rate Constant of H + CO f HCO
J. Phys. Chem., Vol. 100, No. 37, 1996 15167
in the parameters of the absorbing potential, whereas for nonresonant states Ej and Γj change with these parameters. (Cf(t), however, is stable with respect to changes in the absorbing potential parameters.) The eigenvalue problem of Hc was solved using the truncation-recoupling method.28 We use direct products of 10 numerical CO vibration functions and 1000 2d wave functions in R and γ to construct the 3d basis functions of the real Hamiltonian. The primitive basis in R consists of sine functions which extend to R equal 10 bohr. We discarded those products with energies greater than 0.04 hartree, which resulted in 4338 basis functions for the real 3d Hamiltonian. The basis functions for the 3d complex Hamiltonian were the first 1000 eigenfunctions of the real 3d problem. The details of this procedure are described elsewhere.17,22 Evaluation of kr Miller’s flux-flux correlation function expression for the recombination rate constant given by eq 12 can also be written in the more numerically stable form
“broad” resonances and scattering states. Generally speaking, the probability of finding the system in the complex region is much larger for a resonance than for a scattering state. But there is no criterion for how much larger it should be. Here we use a somewhat arbitrary definition, based on the density
Fj(R) ) ∫0 dr∫0 dγ sin γψjc* (R,r,γ)ψjc(R,r,γ) ∞
π
(19)
to supplement the usual criterion based on the width Γj. For bound states, this equation is nothing but the probability for finding the molecule with HsCO distance equal to R. For the unbound states, it also reflects the distribution of R. Thus, the probability to find a system in the complex region is
Pj,in(Rd) ) ∫0 dR Fj(R) Rd
(20)
and the probability of finding a system outside the complex region is approximately
Pj,out ) ∫R dR Fj(R) L
(21)
d
kr )
1 ∫∞dt(e-ωt - 1)Cf(t) Qreact 0
(15)
where Cf(t) is given by16
Cf(t) ) Tr(Fˆ eiHˆ *ct*cFˆ e-iHˆ ctc)
(16)
where Fˆ is the flux operator and tc ) t - ipβ/2. The detailed derivation of eq 15 is given in ref 8. The trace can be easily evaluated with the eigenfunctions of the H ˆ c as the basis set, giving c 2 〉) × Cf(t) ) -[∑(〈ψcn|Fˆ |ψn′
nn′ -(β/2)(En+En′) i(En-En′)t i(β/4)(Γn-Γn′) -(t/2)(Γn+Γn′)
e
e
e
e
] (17)
Substituting (17) into (15) and integrating with respect to t, we obtain
kr(T) )
∑ n,n′
[
ω Qreact
J Shifting
× c (〈ψn′ |Fˆ |ψnc 〉)2e-(β/2)(En+En′)ei(β/4)(Γn′-Γn)
ω + i(En - En′) +
where Rd is the position of the dividing surface and L is the extent of the sine basis used in the L2 problem, which as noted above is 10 bohr. For the present purpose, we put the dividing surface at the recombination barrier, which is Rd ) 4.6 bohr. The states designated as complexes and included in the summation in eq 7 satisfy Pj,in > APj,out. There is no unique way to assign a value to A; however, we have chosen a “generous” value of 0.3. We calculated Pj,in(R) for many complex eigenfunctions, and those that satisfy the above inequality are given in Table 1. All of the states but one listed are the resonance states previously reported.17 Therefore, for these lower energies, the precise value of A is unimportant, and in addition, a slightly different choice of a dividing surface will not change our selection. With the complex eigenvalues of H ˆ c in hand, the decay rate for every metastable state can be easy calculated by the standard relation ki ) Γi/p and inserted into eq 7.
(
)][
Γn + Γn′ 2
i(En - En′) +
(
)]
Γn + Γn′ 2
(18)
This expression was used in our calculations. The calculated rate constants were found to be quite stable with respect to the variation of the parameters of the absorbing potential, the box length, and the size of the basis. Miller has given alternative expressions for kr(T) based on a commutator identity.8 The alternate expression has been used recently by Mandelshtam et al. in an application of Miller’s theory to H + O2 recombination.29 Miller has shown that eq 18 reduces to the standard Lindemann expression given by eq 7 if only diagonal terms that correspond to isolated resonances are retained in the summations in eq 18. In an actual calculation, a definition of the complex needs to be adopted in order to carry out the summation. For a system like HCO, which has a relatively sparse spectrum of resonances, it is relatively easy to identify the narrow resonances; however, it is hard to clearly distinguish
As noted already, the above calculation was for zero total angular momentum. In order to obtain the full rate constant, a summation over angular momentum states is required. In order to describe non-zero total angular momentum states, the Hamiltonian should be changed to
ˆc + H ˆ rot H ˆt ) H
(22)
where H ˆ rot is the appropriate rotational part of the Hamiltonian. Making the approximation that H ˆ rot can be replaced by rotational energy operator, evaluated at a critical nuclear geometry, then ˆ c and Fˆ , and the full correlation function, H ˆ rot commutes with H which we denote by CfF(t), becomes
CfF(t) ) Tr(Fˆ eiHˆ *ct*cFˆ e-iHˆ ctce-βHˆ ) rot
(23)
which immediately gives
CfF(t) ) Cf(t)∑e-βEn
rot
n
) Cf(t)Q* rot
(24)
where Erot rot is the n is the energy of the rotational states n and Q* rotational partition function of the complex. The rotational
15168 J. Phys. Chem., Vol. 100, No. 37, 1996
Qi and Bowman
TABLE 1: Energies and Widths of the HCO Complex States E, cm-1
Γ, cm-1
E, cm-1
Γ, cm-1
1150.2 1344.2 1424.2 1528.5 1567.5 1880.2 2023.4 2086.3 2170.5 2263.9 2361.5 2435.6 2478.4 2667.4 2872.3 2962.0 2998.6 3134.1 3197.5 3286.5 3328.4 3383.4 3442.7 3693.5 3816.0 3824.5 3848.2 3910.4 4002.7 4153.9 4226.3 4319.6 4371.8 4472.8 4665.0 4684.9 4761.8 4823.3 4884.0 5020.8 5027.4 5140.6 5183.6 5221.8 5333.3 5469.0
1.00e-04 0.00700 0.0500 0.590 0.00433 0.323 0.365 0.0100 2.40 34.5 6.50 15.6 40.1 0.436 2.03 4.05 0.670 140 9.78 25.5 69.1 0.132 33.5 1.55 67.4 14.2 1.15 13.5 5.43 40.4 25.3 30.7 61.0 1.75 7.37 9.44 45.9 21.6 159 134 15.8 31.3 0.227 89.9 58.4 6.71
5539.3 5619.5 5670.9 5738.1 5743.9 5833.8 5986.6 5989.1 5999.1 6076.1 6184.6 6246.4 6259.8 6424.8 6432.0 6502.1 6548.7 6615.9 6646.9 6747.4 6755.4 6761.6 6777.5 6794.8 6819.0 6829.7 6840.8 6886.4 6899.0 6918.2 6947.3 6948.6 6957.4 6964.0 6979.2 6983.5 6989.8 7004.1 7016.7 7026.0 7036.7 7038.2 7048.0 7049.3 7065.6
17.1 16.4 55.8 6.41 52.1 19.5 102 110 50.6 82.7 44.3 2.10 59.4 22.6 12.9 61.5 25.2 143 13.9 168 158 145 128 152 124 167 14.9 150 90.7 173 116 170 140 0.925 48.5 139 140 133 161 108 160 145 145 97.5 59.5
constants are calculated according to the stable structure of HCO, and the rate constants correction can also be done by multiplying kr by Q*rot. (There is an uncertainty in the partition function caused by the choice of structure used to determine the rotation constants. Another obvious choice, the recombination barrier, yields a partition function that is 1.8 times larger than Q*rot.) This approximation is the same as the J-shifting approximation which is used in bimolecular reactive scattering.30 It has also been used by Mandelshtam et al. in their recent calculation of the recombination rate for H + O2.29 Results and Discussion The correlation function for H + CO is given in Figure 2 for T ) 1000 K and for three dividing surfaces, as indicated. The negative part corresponds to recrossing, outgoing flux. As seen, the flux leaves the region of the complex between 5 and 10 fs. Figure 3 shows the calculated kr vs ω over a large range in ω, for 293 and 1000 K. As seen, kr goes up to a maximum and then goes down to zero. This behavior has been observed by Thiele1 in his model quantum study of unimolecular dissociation via collision activation. As pointed out by Thiele, this turnover
Figure 2. Flux-flux correlation functions for H + CO at 1000 K evaluated at three different dividing surfaces.
is due to the invalidity of the classical instantaneous collision assumption, which is valid only when ω , kBT, in this ω region.1 It is not reasonable to stabilize half of a particle; hence, the collisional interval should allow the whole particle to enter the complex region. From Figure 2, the average time for a particle to leave the complex region is approximately 8 fs, which is the time required for Cf(t) to fall from Cf(0) to zero. Therefore, the largest physical ω should be smaller than 0.125 fs-1, and the calculated kr is meaningless for ω larger than this value. Furthermore, regions with ω greater than 0.125 fs-1 are far beyond the pressure range for an ordinary gas. Fall-Off Region. Figures 4 and 5 show kr at 293, 500, 1000, and 2000 K for dividing surfaces at 4.6 bohr, which is the top of the barrier, 4.2 and 5.0 bohr. The approximate results from the Lindemann mechanism, referred to as Lindemann, and its upper limit (Qcomp/Qreact)ω are also shown. Any rate constant larger than (Qcomp/Qreact)ω must have a substantial contribution from scattering states. At 293 and 500 K, the rate constants calculated from different dividing surface are much smaller than (Qcomp/Qreact)ω, and the differences between them are small. These small differences at this temperature indicate that the contribution from scattering states is small. This is reasonable since at lower energies, the
Recombination Rate Constant of H + CO f HCO
J. Phys. Chem., Vol. 100, No. 37, 1996 15169
Figure 5. Same as Figure 4 but for 1000 and 2000 K.
Figure 3. Recombination rate constant (kr) at 293 and 1000 K as a function of collision frequency with the dividing surface located at the top of barrier.
expression can be used in this temperature region. Note also that the Lindemann result and its upper bound do not agree very well, until ω is quite small. The reason for this is the existence of several very long-lived HCO resonances, i.e., several ki with very small values. At the higher temperatures, the rate constants from Miller’s theory exceed (Qcomp/Qreact)ω and the Lindemann result. This is because for higher temperatures, an increasing number of scattering states have significant amplitude in the region of the complex, and they cause the increases in rate constants relative to the Lindemann result. The dependence on the rate constants from different dividing surfaces also becomes large for temperatures larger than 1000 K. Again this is due to the increased importance of scattering states in the complex region. Comparison with Experiment. In order to make comparison of our results with experiment at room temperature, with Ar as the buffer gas,10 we need to convert the collision frequency ω to the pressure, which is the experimental independent variable. We use the simplest kinetic expression, which is adequate for the present purpose, i.e.,
P ω ) σν ) CP RT
Figure 4. Same as Figure 2 in the fall-off region for the temperatures indicated. The rate constants labeled by the dividing surface are the flux-based ones.
scattering states do not have significant amplitude in the complex region (due to the presence of the barrier). Thus, Miller’s
(25)
where σ is the (unknown) average cross section for the stabilization reaction, and where ν is the average relative speed between M and metastable HCO. To determine the constant C we did a least-squares fit of experimental data using a simple power function in P. Then at one value of the pressure we equated the calculated rate constant to the experimental fit to determine the constant C. The rate constant from Miller’s theory, using this calibration at one pressure, is plotted in Figure 6 as a solid line, along with the experimental data. As seen the pressure dependence of the experimental rate constant is well reproduced by the calculations. It is of interest to also use eq 25 to estimate the average cross section for stabilization. Doing so, we obtain a value of
15170 J. Phys. Chem., Vol. 100, No. 37, 1996
Qi and Bowman the buffer gas. The calculated rate constant was normalized to experiment at the lowest experimental pressure, and the subsequent comparison at higher pressures showed good agreement. The average total stabilization implied by this agreement was estimated to be 15 bohr2. Acknowledgment. J.M.B. thanks Bill Miller for helpful conversations and encouragement. Support from the Department of Energy, Office of Basic Energy Sciences, for support (DEFG05-86ER13568) is also gratefully acknowledged. References and Notes
Figure 6. Experimental (filled circles) and flux-based recombination rate constants as a function of pressure for Ar as the buffer gas at 293 K.
approximately 15 bohr2. This value is in fair agreement with a calculation of the total recombination cross section we have obtained in a coupled-channel scattering calculation of energy transfer and dissociation cross sections in the Ar-HCO system.31 The value we obtained31(b) is somewhat lower than 15 bohr2; however, the potential used in the scattering calculations was a model sum-of-pairs type and not likely to give quantitative results. Summary and Conclusions We presented quantum calculations of the recombination rate constant of H + CO + M f HCO + M using a new theory of recombination developed by Miller. This theory requires the calculation of the flux-flux correlation function. This was done using complex L2 eigenfunctions for nonrotating HCO, using a realistic potential energy surface. The J-shifting approximation was used to obtain the full rate constant. The theory formally depends on the dividing surface which defines the boundary of the complex region. In practice, this dependence depends on the contribution that nonresonant scattering states make to the correlation function. At temperatures of 293 and 500 K, the dependence of the rate constants on the dividing surface is small, because few scattering states penetrate the complex region, which is separated from the asymptotic region by a barrier of 1.5 kcal/mol. At temperatures of 1000 and 2000 K, the results show a strong dependence on the choice of dividing surface. This dependence indicates that the contribution from the scattering states in the calculation is significant at higher temperatures. Recombination rate constants were also calculated using the standard Lindemann model. At the lower temperature, these agreed fairly well with the correlation function based rate constants. However, at 1000 and 2000 K, the Lindemann rate constants were signficantly lower than the correlation-based ones. A comparison of the calculated rate constant at 293 K was made with experiment in the low-pressure region with Ar as
(1) Thiele, E. J. Chem. Phys. 1966, 45, 491. (2) Mies, F. H. J. Chem. Phys. 1969, 51, 787, 798. (3) Bowman, J. M. J. Phys. Chem. 1986, 90, 3492. (4) Hase, W. L.; Mondro, S. L.; Duchovic, R. J.; Hirst, D. M. J. Am. Chem. Soc. 1987, 109, 2916. (5) Gallucci, C. R.; Schatz, G. C. J. Phys. Chem. 1982, 86, 2352. (6) Cho, S.-W.; Wagner, A. F.; Gazdy, B.; Bowman, J. M. J. Phys. Chem. 1991, 95, 9897. (7) Kendrick, B.; Pack, R. T. Chem. Phys. Lett. 1995, 235, 291. (8) Miller, W. H. J. Phys. 1995, 99, 12387. (9) See, for example: Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions: Blackwell, Oxford, 1990. (10) Ahumada, J. J.; Michael, J. V.; Osborne, D. T. J. Chem. Phys. 1972, 57, 3736. (11) Wang, H. Y.; Eyre, J. A.; Dorfman, L. M. J. Chem. Phys. 1973, 59, 5199. (12) Hikida, T.; Eyre, J. A.; Dorfman, L. M. J. Chem. Phys. 1971, 54, 3422. (13) Timonen, R. S.; Ratajczak, E.; Gutman, D.; Wagner, A. F. J. Phys. Chem. 1987, 91, 5325. (14) Smith, F. T. Kinetic Processes in Gases and Plasmas; Hochstim, A. R., Ed.; Academic: New York, 1969; Chapter 9. (15) Smith, F. T. Phys. ReV. 1960, 118, 349. (16) Miller, W. H.; Schwartz, S. D.; Tromp, J. W. J. Chem. Phys. 1983, 79, 4889. (17) Wang, D.; Bowman, J. M. Chem. Phys. Lett. 1995, 235, 277. (18) (a) Bowman, J. M.; Bittman, J. S.; Harding, L. B. J. Chem. Phys. 1968, 85, 911. (b) Romanowski, H.; Lee, K. T.; Bowman, J. M.; Harding, L. B. J. Chem. Phys. 1986, 84, 2520. (19) Cho, S.-W.; Wagner, A. F.; Gazdy, B.; Bowman, J. M. 1992, 96, 2799. (20) Gray, S. K. J. Chem. Phys. 1992, 96, 6543. (21) Dixon, R. N. J. Chem. Soc., Faraday Trans. 1992, 88, 2575. (22) Wang, D.; Bowman, J. M. J. Chem. Phys. 1994, 100, 1021. (23) Ryaboy, V.; Moiseyev, N. J. Chem. Phys. 1995, 103, 4061. (24) Grozdanov, T. P.; Mandelshtam, V. A.; Taylor, H. S. J. Chem. Phys. 1995, 103, 7990. (25) Werner, H.-J.; Bauer, C.; Rosmus, P.; Keller, H.-M.; Stumpf, M.; Schinke, R. J. Chem. Phys. 1995, 102, 3593. (26) Neuhauser, D. J.; Baer, M. J. Chem. Phys. 1989, 90, 4351. (27) Seideman, T.; Miller, W. H. J. Chem. Phys. 1992, 96, 4412; 1992, 97, 2499. (28) Bowman, J. M.; Gazdy, B. J. Chem. Phys. 1991, 94, 454. (29) Mandelshtam, V. A.; Taylor, H. S.; Miller, W. H. J. Chem. Phys. 1996, 105, 496. (30) Bowman, J. M. J. Phys. Chem. 1991, 95, 4960. (31) (a) Pan, B.; Bowman, J. M. J. Chem. Phys. 1995, 103, 9661. (b) Pan, B.; Bowman, J. M. Unpublished results.
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