2244
J. Phys. Chem. 1982, 86, 2244-2251
Applications of a Stmple Dynamicai Model to the Reaction Path Hamiltonian: Tunneling Corrections to Rate Constants, Product State Distributions, Line Widths of Local Mode Overtones, and Mode Specificity in Unimoiecuiar Decomposition Charles J. CerJan, Shenghua Shl, and Wllllam H. Mlller' Department of Chemistry, and Meteriais end Molecular Research Divlslon of the Lawrence Eerkeby Laboratory, University of California, &r/reky, Californk 94720 (Received: ~ u i y20, 1981)
A simple but often reasonably accurate dynamical model-a synthesis of the semiclassical perturbation (SCP) approximation of Miller and Smith and the infinite order sudden (10s)approximation-has been shown previously to take an exceptionally simple form when applied to the reaction path Hamiltonian derived by Miller, Handy, and Adams. This paper shows how this combined SCP-10s reaction path model can be used to provide a simple but comprehensive description of a variety of phenomena in the dynamics of polyatomic molecules.
I. Introduction The idea of describing a chemical reaction as motion along some reaction coordinate in configuration space is an old one that has been popular a t the qualitative level, e.g., in providing a language for describing complex organic reactions, and has also been pursued more quantitatively for describing reaction dynamics at a more precise level.' Along these lines one of the present authors and co-workers have recently derived an explicit form of the classical Hamiltonian, the "reaction path Hamiltonian", which characterize a general polyatomic system as motion along a reaction path (the steepest descent path in massweighted Cartesian coordinates) plus harmonic oscillator-like deviations about the reaction path in all the (many) directions orthogonal to it.2 This particular formulation of the reaction path idea was carried out with the view of using ab initio quantum chemistry calculations to determine all the parameters in the Hamiltonian, and one of its important features is that all such information is in principle obtainable from a relatively small number of calculations of the potential energy surface. Applications of this approach to date have been to the role of tunneling in the unimolecular r e a ~ t i o n s ~ - ~ HNC HCN H2CO H2 + CO
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+
H2C=C:
-+
HCECH
The most recent development involving the reaction path Hamiltonian is that by Miller and Shi6y7in showing how a relatively simple, but reasonably accurate, dynamical (1)(a) G. L. Hofacker, 2.Naturforsch. A , 18,607 (1963); (b) S. F. Fischer, G. L. Hofacker, and R. Seiler, J. Chem. Phys., 51,3951(1969); (c) R.A. Marcus, ibid., 45,4493,4500(1966);49,2610 (1968);53,4026 (1976);(d) S. F. Fischer and M. A. Ratner, ibid., 57, 2769 (1972);(e) P. Ruasegger and J. Brickmann, ibid., 62,1086 (1976);60,l (1977);(0 M. V. Basilevsky, Chem. Phys., 24,81(1977);(g) K. Fukui, S. Kato, and H. Fujimoto, J.Am. Chem. SOC.,97,1 (1975). (2)W.H. Miller, N. C. Handy, and J. E. Adams, J. Chem. Phys., 72, 99 (1980). See also W.H. Miller, ACS Symp. Ser., 265 (1981). (3)S. K. Gray, W. H. Miller, Y. Yamaguchi, and H. F. Schaefer, J. Chem. Phys., 73,2733 (1980). (4)S. K. Gray, W. H. Miller, Y. Yamaguchi, and H. F. Schaefer, J. Am. Chem. Soc., 103,1900 (1981). (5)Y.Osamura, H. F. Schaefer, S. K. Gray, and W. H. Miller, J. Am. Chem. SOC.,103, 1904 (1981). (6)W.H. Miller and S.-H. Shi, J. Chem. Phys., 75, 2258 (1981). (7)This model is closely related to that of G. L. Hofacker and R. D. Levine, Chem. Phys. Lett:, 9,617 (1971). 0022-385418212086-2244$01.25/0
model-a synthesis of the semiclassical perturbation (SCP) model of Miller and Smith8 and the infinite order sudden approximation (10s)-can be applied to the reaction path Hamiltonian. This leads to explicit closed-form expressions for S-matrix elements (and thus transition probabilities) describing reaction and energy transfer between the reaction coordinate and the transverse vibrational modes of freedom, and thus allows one to deal in a simple way with aspects of the reaction dynamics that involve the interaction between these degrees of freedom. Some other recent work also dealing with the interaction between motion along the reaction coordinate and the transverse vibrational degrees of freedom is that by Kat0 and Mor~kuma.~ The purpose of this paper is to show how the SCP-10s reaction path model can be applied to a variety of phenomena in polyatomic reaction dynamics and thus provide a d i e d description of them all within the same dynamical model. The features that make this interesting from a practical point of view are that the dynamical model is relatively simple, and thus applicable to complex systems, and that the quantities which characterize these phenomena quantitatively are all obtainable in a relatively straightforward way from ab initio quantum chemistry conditions. Section I1 first summarizes the reaction path Hamiltonian and the SCP-10s approximation to the dynamics. It is also shown here how the model describes tunneling through transition states (i.e., the saddle point region) of potential energy surfaces, and application to the test problem H H2 H2 H shows it to be quantitatively useful. Section I11 considers product state distributions of a reaction, in particular a reduced distribution, e.g., the distribution of fiial states for only one degree of freedom, say, summed over all the final states for the other degrees of freedom. Also shown is the specific form taken by the final translational energy distribution. Line widths associated with excitation of overtones of local modes (usually CH stretches) are considered in section IV, and it is shown how the SCP-10s reaction path model provides a description of this phenomenon and also provides a framework for carrying out quantitative calculations. Finally, section V shows how mode specificity in unimolecular rate constants can, in conjunction with a semi-
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+
(8) W. H. Miller and F. T. Smith, Phys. Rev. A , 17,939 (1978). (9)S. Kato and K. Morokuma, J. Chem. Phys., 73,3900 (1980).
0 1982 American Chemical Society
The Journal of Physical Chemistry, Voi. 86, No. 72, 1982 2245
Dynamic Model of Reaction Path Hamiltonian
classical branching model, be described by the SCP-10s reaction path model, and section VI concludes.
11. The Reaction Path Hamiltonian and SCP-10s Approximation For a nonrotating system of N atoms (i.e., with zero total angular momentum) the reaction path Hamiltonian derived by Miller, Handy, and Adams2 is F-1 H(pd,P,Q) = (1/2p2 f/2%(s)28k2) vo(s) k=l F-1 f/Z[Ps &kpkrBk,kf(s)12 k,k‘=l (2.1) [I + 8kBkp(s)I2
This form is especially useful for semiclassical applications since the action variables (nk)are the classical counterpart of vibrational quantum numbers. The unified semiclassical perturbation (SCP) and infinite order sudden (10s)approximation discussed by Miller and Shi6 gives the S-matrix elements (i.e., transition amplitudes) from initial state n of the transverse vibrational modes to final state n’ as Snf,,(E)= - i z W d q (2r)F-1 where
exp[-iAn.q
+ iA$(q)]
An = n’ - n
?!
(2.5)
(2.6)
k=l
where F = 3N - 6 is the number of degrees of freedom, (s,p,) are the mass-weighted reaction coordinate and its conjugate momentum, V&) the potential energy along the reaction path, (Qk,Pk),k = 1, ...,F - 1 the mass-weighted normal mode coordinates and momenta for vibration normal to the reaction path, with frequencies (Uk(s))that are functions of the reaction coordinate. The coupling elements Bkp’(S)couple vibrational modes k and k’, and BkF(s) couples vibrational mode k to the reaction coordinate (which is designated mode k = Fj. The coupling elements Bkp(s) are a measure of how the curvature of the reaction path couples to mode k; the total curvature of the reaction path, K ( s ) , is related to these elements by
The coupling functions Bkp’(S)are essentially a Coriolis-like coupling involving the twist of the vibrational modes about the reaction path as a function of s. The coupling functions, as well as V&) and (fdk(s)),are obtainable from the ab initio quantum chemistry calculation of the reaction path and the force constant matrix along it. The Hamiltonian for the rotating case, J # 0, has also been worked out2 but is more complicated than for J = 0 because of various kinds of rotation-vibration coupling. For most applications it is useful to transform from the vibrational coordinates and momenta (Qk,Pk)to their action angle variables
with
Va(s)in eq 2.7 is the vibrationally adiabatic potential va(s) = vO(s) +
F-1 k=l
(nk + 72) wk(s)
(2.8)
and (qk) are the integration variables q of eq 2.5. (To ensure a symmetric S matrix, the quantum number (nk] n in eq 2.7-2.8 are actually taken to be the averages of the initial and final quantum numbers, i.e., n 1 / 2 ( n n’).) The phase 4o is the WKB phase shift from the vibrationally adiabatic potential, and the phase A4(q) is the contribution from the transverse vibrational modes and is therefore what causes inelastic transitions between the various modes and the reaction coordinate. In most scattering applications one is interested in the limits s1 -m, s2 + O Dbut , we leave open the possibility of other cases. For example, one may wish to choose s1 = 0 and s2 = +a, so that the initial state n corresponds to the transition state and the final state n’ to products. The transition probability of the n n’ transition is, of course, given by
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+
-
-
-
pn:n(E) = ISn,n(E)12
where here the diagonal element Bk,k(s)is defined by (2.4b)
(2.9)
As discussed before,6 eq 2.5 incorporates the infinite order sudden approximation (which would result if the vibrational phase shifts 6k(s) of eq 2 . 7 ~were set to zero), which is correct in the limit that the transverse vibrational motion is much slower than motion along the reaction coordinate, and also the limit of adiabatic perturbation theory, which is correct when the transverse vibrational motion, is much faster than motion along the reaction coordinate. There is thus a reasonable basis for expecting that the model will be at least semiquantitative in fairly general circumstances. Miller and Shi6have applied this SCP-10s reaction path model to one of the standard benchmarks for inelastic
2248
Cerjan et ai.
The Journal of physlcel Chemistry, Vol. 86,No. 12, 1982
I
I
I
0 15
0 20
0 25
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E, i e V )
Reaction probability for collinear H + H, H, + H on the Porter-Karplus potential energy surface. EQ denotes the exact quantum mechanical values (ref 14), VAZC the results of the vibrationally adiabatic zero curvature approximation, and the points the results of the present SCP-10s reaction path model (eq 2.13). Flgure 2.
2 E& ,
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Figure 1. Transition probability for the 0 1 vibrational excitation of H, by coHislon with He, as a function of total energy. H, is modeled as a Morse osdllator. QM, SCP, and HAR denote the essentially exact
I
quantum mechanical results computed for this collinear system (ref 1I), the present results of the SCP-10s reaction path model, and the exact quantum results if H, is treated as a harmonic oscillator, re-
4
spectively.
scattering, the Secrest-Johnsonlo version of collinear He H2 collisions, and it was seen to give quite reasonable results for a wide range of energies and also for multiple quantum transitions. We have also considered the version of this problem in which the H2 oscillator is described by a Morse rather than harmonic potential," and Figure 1 shows the results given by this model for the 0 1transition probability as a function of energy, compared to the exact quantum mechanical values. The model is seen to do surprisingly well. The SCP-10s reaction path model can also be used to describe tunneling through transition state (i.e., saddle point) regions of a potential energy surface. Consider, for example, the standard test problem, collinear H + H2 H2 H. For the total reaction probabilities
+
-
-
+
PdE) =
c lSn:,(E)I2
10-61
1
/
Io-7
-
0 05
0 10
E,
Same as Figure potential energy surface. Figure 3.
0 15 (eV1
2 except for
0 20
the Truhlar-Kuppermann
the curvature term)and symmetry of the barrier taken into account, then A8(q) = 8, sin q with
An
eq 2.5 and 2.7 give (with F = 2)
PR(E)= (2r)-'S2=dq lexp[i40 + iA4(q)]I2 (2.10)
where
K(S)
= Blp(s) is the reaction path curvature and
0
6(s) = SSds'w(s)/(2[V,(s? 0
where the closure relation eihk7-d = 2 r 6(q - qq An
has been used. The action integrals 4o and A4(q) are complex inside the barrier region, Le., where E < Va(s), so that eq 2.10 becomes
Pk(E)= e-28a(2r)'lL2*dq e-2Ae(q)
(2.11)
where 8, is the vibrationally adiabatic barrier penetration integral Bo = X"ds (2[Va(s)-
(2.12a)
