150
Martin Quack
The Journal of Physical Chemistry, Vol. 83, No. I , 1979
uantitative Comparison between Detailed (State Selected) Relafive Rate Data and Averaged (Thermal) A&sdwfe Rate Data for Complex Forming Reactionss Martin Quack Institut fur Physikalische Chemie der Universitat, 0 3400 Gottingen, West Germany (Received August 14, 1978)
The application of statistical theories to complex forming reactions involving polyatomic species is discussed briefly. The fundamental assumptions and the corresponding criteria of validity of statistical theories are pointed out. Model dependent assumptions with particular reference to phase space theory are introduced. It is shown that removing one of the assumptions of phase space theory by allowing for an angular dependence of the interaction potential leads to an improved overall agreement with experiment. A parametrization for the angular potential is proposed. General and simplified expressions from this statistical model are summarized for thermal cross sections for complex formation, vibrational relaxation cross sections, and product translational energy distributions in molecular beam collisions involving quasi-bound intermediates. Experimental data for thermal recombination reactions of nine different reaction systems, for vibrational relaxation, and for a molecular beam investigation of the reaction F 1- C,H&l- C,&F + C1 are analyzed on these lines. Common trends for the parameter CY of the angular potential are observed and it is demonstrated how quantitative relationships between detailed relative rate data and thermally averaged absolute rate data can be established in terms of this parameter.
I, Introduction We consider the class of bimolecular reactions u v e z* F=x Y (1) which proceed through quasi-bound intermediates Z*, corresponding to stable molecules or stable free radicals in the electronic ground state. Experimentally there are several sources o f information available. (A) Thermally averaged cross sections for the formation of the complex Z* are accessible via the measurement of the high pressure recombination rate coefficients from either sidel (U+ V (+M) Z (+M), e.g.) and of the rate coefficients for the overall reaction. In favorable cases including isotope exchange reactions2 such averaged rate coefficients are accurately known. (B) In recent years it has in several instances been possible to obtain additional detailed information on the same kind of reaction. There have been experiments in molecular beams, in which the reactants are made to collide a t a relatively well defined collision energy with measurement of the product translational energy in the center of mass system of the separating fragment^.^,^ Alternatively, information about product internal state distributions has been obtained using chemiluminescence5 and laser induced fluorescence6 detection. Although these experiments provide a large amount of valuable information, accurate total cross sections for the direct comparison with thermal rate coefficients for the same reaction systems have not been obtained in general. (C) In still another class of experiments on reactions of type (l),the initial vibrational state of the reactants (e.g., a diatomic molecule U in collision with a reactive atom V) has been selected, observing as “reaction” the vibrational relaxation after translational and rotational motions being thermalized on a much shorter time ~ c a l e . ~ - ~ It is the aim of the present paper to discuss theoretical means to quantitatively compare the data from the three sources, in order to obtain consistency checks and to predict the results of one kind of experiment, if experimental results from another kind of‘ experiment are available. In principle, such a comparison should be possible, since all kinetic quantities for a given reaction
+
system depend upon the same scattering matrix S , e.g. detailed cross sections being
+
-
‘Dedicated t o Professor Dr. Wilhelm Jost on the occasion of his 75th birthday. 0022-3654/79/~08~-O150$01 .OO/O
The observable quantities mentioned above are just suitable sums and averages over these detailed cross sections. The problem is that the number of S-matrix elements is by many orders of magnitude larger than the number of parameters that completely characterize the experimental results, i.e., the system would be heavily underdetermined. If we go back to the molecular Hamiltonian from which the S matrix can be computed, the situation improves somewhat, since additional infomation on potential properties (assuming the validity of the Born-Oppenheimer approximation) can be obtained from molecular spectroscopy and ab initio calculations. Still, the most general potential surface for a polyatomic system depends upon too large a number of parameters to be practical. It is therefore necessary to find some simple parametrizations, which are nevertheless sufficiently realistic at least for a restricted class of reactions, and which use less parameters than can be derived experimentally and therefore provide us with a determined or overdetermined system. We shall. discuss such parametrizations in section 3. Even if the potential were known, the computation of the S matrix would be a major problem. As a matter of fact, for moderately complex reactions such as F + CGH&l- C1 + C&5F (3) which has been studied in a beam experiment4and which we shall use for illustration, the large number of open channels would prevent even the compilation of the S matrix (at a total collision energy in the reactants of less than 10 kJ mol-l, there are more than 1O’O vibrational states accessible in the product C6H&1, and correspondingly more rovibrational channels). I t is then out of question to use any of the fully dynamical methods of calculation for inelastic and reactive processes, as is easily estimated from the considerations summarized, e.g., in ref 10. We shall therefore start with a brief discussion of the use of statistical mechanical calculations in section 2. We shall illustrate in section 4 that with quite simple statistical dynamical models for all three classes of experiments 0 1979 American Chernical Society
Rate Data for Complex-Forming Reactions
The Journal 01' Physical Chemistry, Vol. 83, No. 1, 1979
concerning reactions of type (1)quantitative comparisons are possible and reasonably accurate predictions can be made in favorable cases.
2. Statistical S Matrix for Reactive and Inelastic Collisions 2.1. Fundamental Assumptions. For the class of reactions considered of type (1) involving polyatomic molecules and bound intermediates, there are two features which make the use of statistical approximations necessary and possible. First, with a quasi-bound intermediate Z*, scattering proceeds through a large number of resonance scattering states, namely, AE X p(E,J), for a given total collision energy E and total angular momentum J with the resolution AE for the collision energy in the center of mass system and p(E,J) the density of metastable states of Z* a t E and J. AE is supposed to be the statistical uncertainty in the energy, due to random differences in the translational energies of the molecules in a beam (or in a thermal situation, where AE E k T ) , not a pulse time limited energy uncertainty. Under this condition, we can average over the independent contributions from different energies and obtain for each total angular momentum J an energy averaged transition probability ( IS,(E,J)12)E.If many resonances in AE couple states i and f, the phases will average out and the averaged transition probabilities will also show no inore particular fluctuations. This is the €irst fundamental assumption. The aim of a statistical theory would be to compute this average directly (and as accurately as possible). Whether such an average is statistically meaningful depends upon the following condition:ll hEp(E,J) >> 1 (4) Equation 4 applies if all resonances in the range AE are effectively (not necessarily equally) coupled to channels i and f. This can in general be expected for strongly bound intermediates which are not too large. A more general criterion ensures that even in less favorable cases many resonances in the range A E are coupled to each channel, namely
AEdE,J) >> W(E,J)
(5) with the total number of effectively accessible scattering channels W(l3,J). Equation 5 implies that a meaningful average over many resonances can be made even with specific couplling of each resonance to a particular set of channels. We shall illustrate the importance of the less stiingent condition (4) with two metastable intermediates which correspond to known molecules for which the density of resonance states can be estimated by continuation of the bound spectrum. Figure 1 shows the densities of states p(E,J) a t a collision energy of about 0.5 eV for H+ + H2 and D + CD3 (one may think of an atom exchange reaction or of vibrational relaxation). At moderate resolution in a beam experiment (e.g., m / h c N 100 cm-l) condition (4) is fulfilled in lboth cases for low total angular momenta J . However at hiigh angular momenta, the densities of states decrease more than exponentially with J , leading to a rather well-defined limit beyond which a statistical average on the basis (of the first fundamental assumption is no longer meaningful. It must therefore be remembered that even with strongly bound intermediates this assumption leads to a low angular momentum (and, because of the importance of high angular momenta a t high collision energies, also low collision energy) approximation. Thus it can be understood that statistical theories which are based upon dynamical assumptions connected to complex formation are invalid for collisions with high energies and
151
3 (H?+1 6 c
5 8 5
h
5
m
3 .
U
r Y
0
20
40 60 80 3 (CD4)
100
Figure 1. The dependence of the density of metastable states p(E,J) upon total angular momentum J. The energies €refer to the total e n e r g above the zero point level of Hf -t H, and D CD,, respectively (lg x 5 log,, x ) .
+
IS(E,J,
1
M,TI... ) l L
.... i .... W ( E , J ) G ,
N(E,J)
f
Figure 2. The structure of the statistical scattering matrix (cf. explanations in the text).
high angular momenta, as it has been found in trajec1,ory calculations for the H3+system.12 For reactions for which the first fundamental assumption does not apply, we may introduce statistical simplifications on the basis of a different, second /undamental assumption. If in a collision experiment a large number NI of states of the collision partners (U V) is randomly populated initially (i.e., with equal probability and no phase rc.lationships), and if the experimental resolution allows one to distinguish only groups of states after the collision, e.g., with NF states in one such group or level, then it is idlowed to replace the detailed transition probabilities ISfJyby the quantity ( ISf112)fl, which is averaged over the initial and final states as illustrated in Figure 2 (the overall transition probability will be summed over the final states). A statistically meaningful average is obtained, if thc following condition is satisifed:
+
NI,NF >> 1 (6) One must also nod, forget the condition of random phases for the initial staties. This second fundamental assumpl ion is simply the general assumption in the statistical imechanical description of time-dependent p r o c e ~ s e s , ~ ~ J ~ applied to a collision process. We have given this brief discussion of statistical theories of molecular collisions in order to show that in princiiple
152
The Journal of Physical Chemistry, Vol. 83,
No. 1, 1979
Martin Quack
even with a statistical theory the inclusion of all the mechanical properties of the collision system is desirable. Statistical simplifications arise if either or both of the fundamental assumptions apply. One may easily verify that this very general definition of statistical theories applies not only to the theories usually considered as statistical, such as transition state theory,15 phase space theory,16 RRKM theory,17J8 the statistical adiabatic channel model,lSor Miller‘s unified statistical model,20but also to the use of purely classical trajectories (by virtue of the second assumption). To be sure, classical S-matrix theory,21which includes the superposition principle, is not a statistical theory since condition (6) is not satisfied. 2.2. Model Dependent Assumptions. We shall restrict our attention now to approximate statistical models for reaction 1 with intermediate complex formation. In addition to the fundamental assumptions one assumes now that there is a dynamical criterion which separates strongly coupled channels from weakly coupled channels. This is illustrated in Figure 2. The number of strongly coupled channels is W(E,J),while the number of asymptotically open channels is N(E,J). The simplest assumption concerning the structure of the S matrix is that the strongly coupled channels are on the average equally coupled, whereas the weakly coupled channels are not = l/W(E,J) for the coupled a t all. This leads to (ISfi/z)fl strongly coupled channels and ISf,Iz‘ = afi for the weakly coupled channels.ll Other parametrizations have been discussed as but apparently have not been applied in detail so far. From the given structure of the S matrix one may compute inelastic and reactive cross sections using eq 2.23324 One obtains for example the degeneracy averaged integral cross section for a transition from a combination of the levels a of the reactants (U V) to a combination of levels b of the products (X + Y):
+
g, = (2jua+ 1)(2jva+ 1) is the rotational degeneracy of the initial state, k: = 2yE,/ h2is the wave number of the relative translational motion, and W(E,J,a) and W(E,J,b) are the total numbers of strongly coupled channels leading to the levels a and b, respectively. Equation 7 applies to a reaction on one electronic potential surface and disregards nuclear spin. If there are several identical atoms present in reactants and products, one has to modify eq 7 which leads in the simplest case to a correction factor F(b,a)/ [I?,],which depends upon the species raand r b of the states a and b in a suitable symmetry group.26 If we assume the molecules Z* in eq 1 to always be irreversibly removed during their finite lifetime by collisions with other molecules M, we can define a cross section for complex formation: aaC =
- (ZJ + 1)W(E,J,a) 7 T m
g,h,ZJ=O
(8)
Depending upon the dynamical criterion used to compute W(E,J) and W(E,J,a), the cross sections for complex formation depends in a nontrivial manner upon the initial state a. The various statistical models may be distinguished by the particular dynamical criterion which they use in order to compute the number of strongly coupled channels. A particularly simple assumption has been made in the well-established phase space theory.16 The interaction potential between the reactants is described by a onedimensional attractive potential Vel(q)as a function of the
Figure 3. Coordinate system for the potential energy of a triatomic system.
center of mass distance of the reactants (and similarly the product, usually a van der Waals C / q 6 potential, is used for neutral species). The effective channel potential for channel a is then given by Va(q) = Vel(q) + B(q)*l*(l+ 1) + E a (9) A channel belongs to the strongly coupled channels which lead to complex formation if the maximum value Vam=is smaller than the total available energy E. E,, is the internal energy of the separated reactants in channel a and B(q) is the effective rotational constant for the orbital motion, whose quantum number is 1. Since E,, in eq 9 is independent of q , we may also say that for a given center of mass translational energy Et complex formation is possible up to a certain maximum value lm(Eam) as indicated in Figure 2. Let us review the three physical model assumptions of phase space theory. (1) If a complex is formed, it dissociates with equal probability into all strongly coupled channels (Le., ISfiI2= l/W(E,J), see above). (2) Outside the region of the reaction complex as defined by the centrifugal barriers no transitions between different channels occur, the motions are “adiabatic”.26-28 (3) The interaction potential between the two reaction partners contains only a radial term and no angular dependence upon the relative rotation (this leads to eq 9). Although relatively little is known about the validity of any of these assumptions, we shall now investigate the consequences of maintaining the first two while abandoning the third. This corresponds to replacing the “loose” transition state for complex formation in phase space theory by a more general transition state as determined by the complete potential surface within the framework of adiabatic transition state t h e ~ r y . ~ J ~ ~ ~ ~ 3. A Parametrization for the Angular Interaction Potential for the Collision Partners Figure 3 shows the coordinate system for a diatomic molecule U = 12 colliding with an atom V = 3 and forming a complex Z*. For the interaction potential in the center of mass distance q we use a Morse potential: (10) Vel(q) = Dei1 - exp[P(qe - q)Il2
+
with De = Doo Ez, - EZpand p = (F,/2De)lI2(Doois the bond dissociation energy at 0 K, Ez, the total zero point energy of the molecule Z; EZpthe total zero point energy of the reactants U + V, and F the force constant in the coordinate 4 ) . For chemically Lound systems this choice has some advantages over a van der Waals potential,lg although the matter may deserve further attention. For angles different from the angle qe we have to add to this an angular contribution, which we parametrize as follows: (11) V(q,q)= I/zCVnl(q)[1 - COS jn(q - ~ e ) l I
n is the symmetry of the potential (n = 2 if U is a homonuclear diatomic molecule) and the Vn,(q) are related to the barriers for the hindered rotation about the angle
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 153
Rate Data for Complex-Forming Reactions
4000
reactant channels and complex channels in order to define the channel index a. For detail we refer to ref 19, but we note that two kinds of quantum numbers must be Idistinguished: those refering to high frequency motions (stretching vibrations) which maintain their quantum number in the correlation and those refering to low frequency motions which go over into free rotations in the separated fragments and which are correlated “adiabatically”, Le*,without crossings as a rule for channels of the same good quantum numbers. Of course there are nontrivial dynamical assumptions involved in deciding which channels of the same symmetry are allowed to cross and which are not.lg Equations 10 and 14 go over into the phase space theory formula of eq 9 if’ we take a = (or very large). We see however from Figure 4 that the contributions to the channel energies from the angular potential may be quite large if we take a finite a (e.g., 1 A-l) with typical bending frequencies of about 600 cm-l at qe. It would be interesting to have some guidance from ab initio calculations, which values of a would be appropriate as a rule, and if a representation of the potential according to eq lO-l;! is reasonable. So far the only complete potential surface available for a system with a chemically bound intermediate appears to be for the ion H3+.30It turns out that for moderate values of ( q - qe) the potential is almost isotropic (Vn, = 0), corresponding to large values of a > 5 kl.Thus for this system the phase space theory representation of channel energies should be satisfactory, as has been verified by comparison with experiment12 with the reservations concerning contributions from high angular momenta noted in section 2. We shall analyze in the next section experimental evidence, which suggests that a similar conclusion does not pertain for reactions of neutral species. We feel that ab initio calculations on such systems are urgently needed. The one parameter representation of eq 13 and 14 may also need improvement. A complete potential surface cannot be represented by one or two parameters. However, one parameter is about as much as can be squeezed out of the experimental data at present.
-
0
0
1
2
3
4.(q-qe)
Figure 4. Hindered rotor energy eigenvalues as a function of barrier height computed by the solution of the Mathieu equation (points) and by the correlation interpolation procedure of eq 13 (lines) (CY= a,, see also the text).
p, which depend strongly upon q. In the absence of any knowledge about this dependence, we make an Ansatz which is closely related to a Morse potential: Vn,(qe)12 exp[aJ(qe- 4)I + exp[2aJ(qe-- 4)11/3 vnJ(q) (12) This obviously gives the proper limits for q = qe and q = and in between it provides a smooth exponential interpolation. The eigenvalues for a one-dimensional problem as given by the potentiial in eq 11 and 12 can be computed for all values of q (“clamped q approximation”). The result for a model problem is shown in Figure 4 for two channels (in the example we have taken n = 3 and restricted the sum in eq 11 to one term with j = 1, the two channels shown correspond to a doubly degenerate E component). The open circles and squares come from a solution of the Mathieu equation,29with V3(qe)= 8000 cm-l. A t qe we have slightly anharmonic vibrational energy patterns with quantum numbers v whereas a t q = m we have free rotational energy levels (quantum number j ) . The range in between corresponds almost to a smooth exponential. This is shown by the lines, which are obtained with an interpolation formula for energy levels
-
Ea’(q) = [Ea’(qe) -- Ea-’] exp[a(qe- ~ ) +l Ea-’ (13) Since not too much is known about the exact shape of the potential anyway we use the reasonable approximation provided by eqi 13, which can be justified analytically, and extend it to the full three-dimensional problem including overall rotation, and even to collisions of two polyatomics with the general interpolation formula: Ea(q) = [Ea’(qe) - Ea,] e x p [ a ( q , - q ) ] + Ea, + B(q1.P. ( P + 1) (14) and f’ = 1 + ( J - I ) exp[a(q, - q ) ] (15) B ( q ) is the effective rotational constant (mean of the two small rotationd constants). Equation 1 4 has been used in the adiabatic channel modellg and it needs a suitable correlation between
4. Theoretical Analysis of Experimental D a t a
In this section we shall summarize a theoretical analysis of experimental data from thermal high-pressure recombination-dissociation kinetics, vibrational relaxat ion measurements in systems involving bound intermediates, and molecular beam measurements of product translational energy distributions. We shall focus our attention in all cases on the dependence of the theoretical predictions for these data upon the parameter a , which is the only parameter of the theory which cannot be obtained from thermochemical oir spectroscopic properties of the molecules involved. W e shall mention suitable approximation formulae for the various situations encountered. 4.1. Thermally Averaged Cross Sections for Complex Formation. Experimentally, thermally averaged cross sections are obtained from the high pressure recombination rate constant (for two different collision partners)
or from the unirnolecular high pressure limiting rate constant for the dissociation of Z, since by definition of the high pressure limit detailed balance applies hrec
=
hdiss/KC
(1 7 )
with the equilibrium constant for the dissociation reaction KC.
Martin Quack
Tha Journal of Physical Chemistry, Vol. 83, No. 1, 1979
154
Theoretically, one simply needs to average the cross section for complex formation a:(E) in eq 8 over a thermal distribution of both translational energies and internal states of the collision partners U and V, One obtains
TABLE I: Experimental and Theoretical Results on Thermally Averaged Cross Sections for Complex Formation &( Ty10-16 Cnl2 theor Q / A - ~ m
system
Qhts are the internal (rovibronic) partition functions of the collision partners. We have assumed that the electronic partition function of Z is equal to 1 and that only the nondegenerate electronic ground state contributes to reaction, with trivial changes, if this is not the case. We have used in eq 18 the transition state notation1
C exp(-V,,,,JhT)
Q*
C C (2J + 1)W(E,J,a)= Ch(E - Vim,) J=O
(19)
c1 t NO,
(20)
OH t NO,
1
h ( X )= 0(1) for X < 0 (>O) is the unit step function and we have distinguished here level indices a (eq 8) from channel indices i, noting that degeneracy factors in the sums have been removed by counting all channels individually. W ( E )is the total number of strongly coupled channels and Vim the maximum of the adiabatic channel potential for channel i ( q 2 qe). All partition functions are computed to the same zero of energy. Using eq 10 and 14 one can evaluate eq 18-20 using suitable channel counting algorithms. We note in particular, that most of the vibrational (high frequency) coordinates have a negligible change in frequency during the reaction. Therefore one computes a density of states Ph(E) for these motions separately. only for the remaining low frequency motions one has to use the correlationinterpolation procedure of ref 19, obtaining the number of strongly coupled channels in these coordinates to be W,(E,J). The total number of open channels is given by the convolution:
Although eq 21 makes the calculations feasible even for large polyatomic systems, they remain rather cumbersome for practical applications. It has been shown, however, that a reasonable approximation to Q* can be obtained by looking for the minimum of the following factorized expression as a function of q:31 Q'(4) = &,'(4)Q,'(4)Qv,'(q)QEz'(4)
(22)
with In Q l ( q ) = -V,,(q)/KT (cf. eq lo), Q;(q) = hT/B(q) (cf. eq 14) and In Qvr'(q) = [In &",'((le) - In &,,'(.)I X exp[Q.75a(qe- 4)1 + In Qv,'(m) (23) In
QEZ'(~) =
1
-
ZI[EZ(~J- &plI exp[a(qe - 411 + EZP (24)
I-Iere E Z ( q , )= Ez, -- t R C / 2 , eRC being the quantum in the reaction coordinate, cf. eq 10. The interpolation of In Q,' in eq 23 has been shown to be approximately equivalent to a n interpolation of eigenvalues in eq 14, with QV,'(4J = Q,,Z[1- exp(-€,c/kT)ISe/Q~'(4e) Q,,'(a) =
I + NQ
Qvr"Qv,VS,
(25) (26)
The Q,, are rovibrational partition functions of the molecules U, V, and Z, and the S , and S , are symmetry
+ CH, CH, + CH, H
(a)"
1
(PST)
13 4.5 2.7 1.0 0.32h ( 0 . 3 5 ) 0.8 0.5 3.3' 7.9 3.4 F(1) 8.6 4.7 5.9 (400 K)b 7.6 2.6 2.2j 3.0 0.15' 1.7 15.l(l) 18 0.17m ( O . l ) d 10
300 2100 300
14 5.8 5.9 2.6 1.8
9.3f60.8) 3.1e 3.6g(1,3)
2100
O t 0,
HtOH
= i m W ( E )exp(-E/hT)(dE/hT) a
O + NO
expt
300 2100 300 2100
1
"(E)
C1 + NO
T,K
300 2100 300 2100 300 1100 300 2200
300 1300
10.4n (1)
0
0.5 0.7 0.1 0.2 0.2
1.1 0.2
11 5.6 13 6.6 14 6.9 14
8.8 28
17 28
11
0.1 0.9 0.1 0.9 0.02 0.1 2 x lo-$ 2 x 2X
0.3 4 x
lo-'
IO-'
lo-'
5.0 2 0 7 x 10-4 a The value of 01 which fits the data is given in parentheses. This corresponds to a result near 4 0 0 K.37 Computed from the experimental result for the isotope exchange reaction D 4 OH + OD + H.2 The experimental data can be fitted with Q = 0.1 A - ' only if also the value of the Morse parameter 4 is increased to 3 i i - ' . I 9 e Calculated with 01 = 0.8 A;l. Reference 3.3. Reference 34. Reference 35. Reference 36. Reference 38. RefReference 41. Refererence 39. Reference 40. ence 43. ' Reference 42. 3.0'
correction^.^^,^^ Equations 22-25 are very easy to evaluate and quite good approximations. We note that for the limit of a = a, i.e., the thermally averaged cross section for complex formation according to phase space theory, eq 9, one has a very simple formula ("Gorin wit,hout additional approximations:
Q,*
= C(21 + 1) exp[-Vm,,(l)/kTl 1
(28)
Equation 28 uses eq 9 with Ea, = 0 and the Q, are the electronic partition functions of the collision partners. In Table I we summarize experimental and calculated results obtained with these approximation formulae for CY = 1, a = m, and a = 0 (in several cases a check is possible with the more accurately computed valueslg from eq 19 and 20, with satisfactory agreement). As is seen from the table, the phase space theory cross sections are always too high, however, not more than a factor of 7 , i f we exclude the high temperature data for H CH3 and OH NOz. Nevertheless, there is plenty of evidence that rather small, finite values of a are more appropriate for this whole class of radical recombinations. A value of CY = 1 A-1 fits all the experimental data within about a factor of 2, if we exclude again the high temperature data of "OB and CHI dissociation. In the latter case the experimental number probably does not correspond to the high pressure limit, whereas the former poses still an unresolved problem theoretically or experimentally. The regularities in 01 are for predictive sufficient to use a choice of CY = 1 purposes, although it must be remembered that there is no fundamental reason why all molecules should have the same value o f a (no more than having the same value of the Morse parameter p). We shall see, however, that
+
+
Rate Data for Complex-Forming Reactions
The Journal of Physical Chemistry, Vol. 83,No. 1, 1979
TABLE 11: Partially Thermal Cross Sections fi3r Vibrational Relaxation
155
relaxation data confirm the general trends for N observed in the more abundant thermal recombination data. (U'T(" = 0 +- u = 1))/ 4.3. Product Translational Energy Distributions cm2 Measured in Molecular Beam Experiments (Relative Cross Sections). The measurement of product translatheor (a = tional energy distributions in molecular beam experiments system T,K expt %la is the most detailed kind of experiment to be considered 2.07 1.33" I t NO 2100 in the present paper. We note that even in this kind of 1.87 1700/2100" 1.48e c1 -t NO 1.03e 0.58 experiment in general a single initial state is not perfectly 0 i. NO 2700/2100 0.315~ 0.16b 0 -t. 0, 2100 well selected nor is the final state of the products com8.1 300 10.4g H iOH pletely resolved. Thus we can make use of both our Ar t NOa! 2100 0.001= fundamental assumptions of section 2 and need not 10ely a The theoretical cross sections are obtained with the only on assumptiton 1 in connection with the model devalue of cy from high pressure dissociation-recombination pendent assumptions. Various statistical models have kinetics.. With a value of CY = 1A - ' one would predict already been applied to beam experiments with bound u = 0.28 A2. The numbers refer to the experimental and intermediates, A, particularly detailed study12 of the retheoretical temperatures, respectively. u is only weakly action D+ H2 --* HD H+ has given perfect agreement temperature dependent in this range. These values are with phase space theory16 both for product state dietrigiven for comparison. e Reference 8. Reference 7 . 8 Reference 9. butions and total cross sections after correction for a reduced probabilit,y of complex formation a t high energies similar trend:., for a are also found from the two other and angular mornenta. This is not too surprising consources of experimental information which we shall discuss sidering the large value of cy (isotropic potential) consistent now. with the ab initio potential available for this reaction.30 4.2. Partially Thermal Cross Sections for Vibrational In the somewhat particular case of the molecular beam Relaxation. In several recent experiment^^-^ it has been photodissociation of N02,45application of the statistical possible to measure the vibrational relaxation of diatomic adiabatic channel model gave good agreement with the molecules in collisions with reactive atoms proceeding measured product state distributions, if a value of a,= 1.3 through a chemically bound intermediate (e.g., H + OH(u A-l was chosen, being identical with the N value for thermal = 1) HzO* H t- OH(u = O)9). Translation and rorecombination arid vibrational relaxation e x p e r i m e n i ~ . ~ ~ tation are in these experiments thermalized to a temPhase space thleory and the approximate model of ref perature T , on a time scale which is short compared to the 17 have been applied to an extensive investigation of time scale of .the experiment, by collisions with inert gas reactions of fluorine atoms with unsaturated (sometimes at,oms which on the other hand hardly effect the vibrahalogenated) hydrocarbons (see, for example, ref 4 and ref tional state distribution. Under this condition, one can 1 and 3 for reviews). Sometimes agreement between compare the tlhermally averaged experimental cross section experiment and theory has been found and sometimes not. for vibrational relaxation However, no systematic investigation of the influencle of the interaction potential (particularly of its angular part) of reactant and product molecules has been made, so far. It is our aim, here, to show the possibility of such a with the corresponding theoretical cross section, which is systematic investigation by considering the dependence obtained by averaging aba(E)in eq 7 over a thermal disof product state distributions upon the parameter a , tribution in translational energies and in rotation and similar to our investigation of total cross sections and cross summing over the final rotational states in each final sections for vibrahional relaxation. Product state distrivibrational state u' butions are implied in the detailed cross sections of eq 7 by averaging over the distribution of initial states a and U) = h2g,(87rph TQ,uQ,vQ,u)-l X UT(U' initial translational energies present in the beams andl by summing over the final states b, which lead to a product ( 2 J + l)fexp[-j(j + l ) A / h T ] X J=O 1=0 translational energy E,. A result of such a straightforward W(E,J,u,j)W(E,J,u? calculation is shown in Figure 5 for collisions of C1 with NO being initially in the vibrational ground state and various well-defined rotational states a t a common total energy of about 0 5 eV. The value of a = 0.8 k l is chosen QrU = (,kT/A)is the rotational partition function of the to be the one which best fits the experimental high diatomic molecule U (rotational constant A ) , g, is the pressure recombination rate coefficient (Table I). No degeneracy of the electronic ground state of Z supposed molecular beam experiments of this kind are available for to be the only one contributing appreciably to vibrational this system so far. The reason for showing this example relaxation, and the other terms are as defined before. is the strong dependence of the final translational energy Approximations to eq 30 on the lines of eq 22-26 have been distribution upon the initial rotational state (always at the suggested.44 same total energy!). The nature of this dependence is a Table I1 summarizes experimental and theoretical vifunction of the validity of the model assumption ( 2 ) of brational relaxation data. One notes that for all the systems in Table I1 one has corresponding results on ~ ~ ( 7 ' ) approximate adiebaticity in section 2. Therefore, experimental investigations similar to the calculations in in Table I, and therefore a quantitative comparison of Figure 5 could provide a beautiful test of this assumption. vibrational relaxation data with thermal recombination data is possible. This comparison is given in the last A superficial interpretation of the correlation between final column through the theoretical cross section computed translational energy distributions and the initial internal with the valuo of a obtained for the particular molecule state of NO mighl, lead to the conclusion that "energy is from the recombination data. The agreement is seen to not randomized in the intermediate reaction complex be very satisfactory. This implies also that the vibrational ClNO*". Such a statement must be viewed with due (;
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2
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ln
c C
3
t’ .-2 c
e 0
0
1OO0
3000
ZOO0
4000
Figure 5. Statistical product translational energy distributions calculated for collisions of CI with NO ( v = 0, j ) at a total collision energy of about 0.5 eV, for different initial values of the rotational angular momentum j of the NO molecule (see also the text), as indicated.
caution because of the model assumption (1) (Sf1I2 = 1/ W ( E , J )in section 2. As a second example, we shall treat reaction 3, menF + C6H,Cl+ CEHSFCl” CsHjF + C1 (3) -+
tioned in the Introduction, for which a thorough molecular beam investigation exist^.^ Because of the very large number of channels involved in this reaction, a straightforward evaluation of eq 7 is not possible. We shall summarize briefly an appropriate simplified treatment. The product translational energy distribution can be simulated to be
P(E+,AE,E,a)=
P ( E , , S ) is the probability of finding a product translational energy in the range between E, and E, + AE, P(E,,AE,E,,a) is the same, but given some initial total energy E, = E,, + V,,, and some initial reactant level a. The sum E’ represents counting all product levels b giving a translational energy between E, and E , + LE given the total energy E. The normalization factor N is equal to CJ=0p(2J + 1)W(E,J,a). F(E,,) is the distribution function for the initial translational energies and G, the population of the reactant level a determined by the molecular beam conditions. Since a simulation of the not completely known actual experimental conditions will not be attempted, we put F(Eti)= 6(E,, - Eta) with EtObeing the average translational energy (800 cm-’) of the reactants. Concerning G, we assume that only one level is initially populated with j, = 0, since it has been stated that the rotational excitation of the reactant is very low.* Symmetry corrections can be omitted for this reaction, if we assume that the initial state distribution transforms as the regular representation of the symmetry group for this r e a ~ t i o n . ’Equation ~ 32 can be interpreted as a weighted sum over normalized distributions
The weighting factor is (2J + l)W(E,J,a),which with the assumption j, = 0 becomes (2J f l).constant.h(J,,, - J).
Martin Quack
The value of J,,, is computed from the centrifugal potential in the entrance channel. For the general case with j, f 0, the weighting function is less trivial. It remains to properly evaluate the channel numbers. Only the product channel CGH5F+ C1 will be considered, since the other ones can be safely neglected for the experimental conditions (and for the particular purpose of only computing P(EJ in this channel). As in eq 2 1 we can distinguish “high frequency” constant coordinates and “low frequency“’variable coordinates to obtain W(E,J)by making the correlation interpolation procedure of ref 19 only in the low frequency coordinates (see the Appendix for the specification of the coordinates). The number of states in the high frequency coordinates is evaluated by giving the function Wh(E).Close lying channels in the low frequency coordinates are grouped into representative channels of degeneracy g,. The distribution of internal energies of the products is then given by P(E*,AE,E,J) = Cg,[W-h(EI-- V,, + AE) 1
Wh(E1 - V,,)lh(E
-
Vlma+ VI,
-
E J (34)
This implies the translational energy distribution through E1 + E, = E. A great reduction in the number of channels which have to be computed can be achieved by computing only one centrifugal channel with the best quantum number 1 = (SL j z ) l l 2 in eq 15 for the channel with the rotational quantum numbers G,h) in the product c6H5c1. For the calculation below, channel energies for a small, representative number of values of 1 were computed until approximate convergence was obtained in the product state distributions. The sum over J in eq 32 is transformed into an integral which is evaluated by quadrature since the integrand is smooth. The reaction considered here differs from the reactions considered above by the fact that the intermediate C6HjFCl is not a molecule with known properties. Therefore one must rely on rather crude estimates for the molecular properties and the thermochemistry. The values actually used are given in the Appendix. The product energy distributions also depend on the choice of these parameters. Therefore no unique value of a can be obtained from a comparison with the experiment. We present the calculations rather as model calculations for illustrating the general influence of LY on product state distributions. It may be useful to give some idea of the major energy parameters. The initial translational energy is 800 cm the maximum possible final translational energy is 11900 cm-l, the bond dissociation energy for C6H5C1-F is 15 150 cm-’, and for C6H5F-C1it is 5000 cm-’. The total available energy is thus more than three times the dissociation energy. Furthermore, the total zero point energy is even higher, namely about 20 000 cm-’. The average excitation in one vibrational mode is therefore very low. This makes classical approximations useless; the harmonic approximation to the density of states is however quite good. Figure 6 shows the results of the model calculation for values of a = 0,1, and 2 A-’ and the experimental resultn4 It is seen that decreasing the value of N leads to a shift of the maximum of P(EJ to higher energies and to a broadening of the distribution. The value of a = 2 A-’ leads to a maximum of the distribution close to the exprimental distribution. However, the half-width of the theoretical distribution is by almost 30% lower than the experimental half-width. On the other hand, a value of N = 1 A-’ leads to the experimental width in the distribution, but the maximum is shifted by almost a factor of 2 . In considering these discrepancies one should note that
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The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
Rate Data for Complex-Forming Reactions 1.0
0.8
0.2 1000 Et
0
-
/ern-'
2000
Flgure 6. Product translational energy distribution for the reaction F C6H5CI C6ti5F CI. The maximum possible translational energy would be 11 900 crn-‘. The numbers indicate the value of the parameter LY (in k‘) used in the calculations and Exp. is the experimental result of ref 4. The product state distribution from phase space theory and for a = 2 A-’ are practically coinciding as indicated.
+
+
the energy rainge shown in Figure 6 is only a small fraction of the total translational energy distribution, which decreases monotonously at higher energies. The shift of the and the difference in width for cy maximum for cy = 1 correspond to less than 10% of the total energy =2 available for translation a priori. The position of the maximum is the more significant parameter, since introducing an initial rotational state distribution for C6H5C1 with jarnu = 50 (instead of j , BI = 0 in Figure 6) and using a symrnetric function F(E,J of finite width leads to a considerable broadening of the calculated distribution with little shift of the maximum. Disregarding the uncertainties in the other molecular parameters (cf. Appendix) one would thereffore conlclude that the experiments are consistent with a value of cy = 1 to 2 A-l, corresponding more closely to the latter. This is completely in line with our general findings for a obtained from the thermal cross sections for complex formation and from vibrational relaxation cross sections. (There appears to be no measurement of the thermal rate coefficient for the reaction C1+ C 6 H ~ F (+M) +. C6H5FC1which could be compared to the value of LY obtained from the product translational energy distributions.) It should be mentioned that another reason for a broad P(EJ may be a lowered number of vibrational modes being excited in C6H5C1(“incomplete energy randomization”) as has been suggested in ref 4. Furthermore one may have rovibrational relaxation in the long range part of the potential, which may or may not lead to a broader P(E,). It is useful to mention a t the end a generalization from these calculations. Low values of a always lead to a broadening and shift to high energies of P(EJ as compared to the phase space theory result ( a = a), Similarly, low values of cy lead to smaller thermal cross sections than the phase space theory result (loose transition state) in eq 27. One can therefore conclude quite generally from a product translational energy distribution which is shifted to much higher energies than expected from phase space theory, that the rate coefficient for the reverse recombination reaction in the high pressure limit is much lower than computed with eq 27. Quantitative statements are possible with the aid of the parameter a. It would be interesting to have molecular beam investigations for systems in which the properties of the intermediate are well known and for which measurements of the high pressure recombination rate coefficients exist. 5. ConcT‘usion
The phase space theoryL6of complex-forming reactions and the corresponding loose transition state for radical recombinations have found wide application in the past
157
with reasonable although not always completely satisfactory success. We have shown that a definite improvement can be obtained by taking into account the angular part of the interaction potential between the collision partners (Le., abandoning the model dependent assumption (3) of phase space theory, cf. section 2). A suitable Morse-potential-like parametrization of this potential is proposed and it is seen to provide very siniple approximate expressions for channel energies and the corresponding partition functions. The theoretical model thus obtainedl9 allows one to systematically investigate the influence of potential parameters (the parameter a in particular) on both detailed relative rate data and thermally averaged absolute rate data. It also allows for a quantitative comparison of data from different sources. A theoretical analysis of ten different reaction systems reveals very similar trends in the properties of the effective parameter cy for the angular potential for thermal complex formation cross sections, for vibrational relaxation cross sections, and for product state distributions in molecular beam experiments. A most intriguing question which remains is if and when the model dependent assumptions (1) and (2) of phase space theory have to be abandoned in complex forming reactions, for which the fundamental assumptions of statistical theories ((1)and (2)), put forward in the present paper, are appropriate. Another open question concerns the suitability of our parametrization for the angular potential. Ab initio calculations for the appropriate parts of the potential surfaces of reactive systems involving bound intermediates do not appear to exist so far and are urgently needed
Acknowledgment. I am greatly indebted to J. Troe for his interest and many stimulating discussions. This work has also profited from discussions with R. Buss, Y. T. Lee, W. H. Miller, B. Schumann, and K. Shobatake. In the early stages, programs on the eigenvalue problem for one-dimensional molecular motions in general poteni,ials provided by A. I3auder and R. Meyer were most useful. Financial suppoipt by the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
Appendix (1) Molecular Model for t h e Intermediate C6H5~W‘CI. Thirty “high frequency vibrations” are taken to be the vibrations of the C6H5Clproduct,4Ror C6H,F product, for the former in cm 5 X 3050,2 X 1595, 1626,1493,1444, 2 X 1020,2 X 1156,1295,1062,955,831,979,894,753,1225, 806,403, 615, 683, 517, 498, 242, 368; t R C = 200 cm-l, 6 = 1.7 A-l. The “low frequency motions” correspond to two bending vibrations (200 and 240 cm-l) and the rotational quantum number K of C6H6FC1and to the rotational quantum numbers j and k of C6H5C1and orbital quantum number 1. These motions are correlated according to the method of ref 19. (2) Molecular Structure and Rotational Constants. The benzene ring wai4 taken to be a hexagon with four equal C-C bonds of 1.4%A, and five equal C-H bonds of 1.08 A. The sixth carbon I S tetrahedral (irregular),with C-C bonds of 1.52 A, r(C-Cl) = 1.90 A, r(C-F) = 1.45 A, L(CFC1) = llOo,the plane CFCl being perpendicular to the ring plane. The geometry of I he reaction cwrdinate is supposed t o be removal of F or C1 with all angles being kept constant. This may be a poor assumption, but no details are known and it is certainly better than a “quasi-diatomic” treatment considering the center of mass distances only. Rotational constants for C61-15C1Fare as follows: A = 0.094 cm-l, B = 0.0443 cm-’, c = 0.0439 cm-l; for C6HbF:49 A = 0.18892
’:
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The Journal of Physical Chemistry, Vol, 83,No. 1, 1979
cm-l, B = 0.08575 cm-l, C = 0.05897 cm-'; for C6H5C14'A = 0.189229 cm-l, B = 0.052596 cm-l, C = 0.041151 cm-l. (3) Energy Parameters. &"(C6H,C1 F) = 15150 cm-l; Do"(C6H5F-t C1) = 5000 cm-l; initial translational energy E',i = 800 cm-'; total initial energy with respect to C6H5F + C1, E = 11900 cm-l.
+
References and Notes (1) (2) (3) (4)
(5) (6) (7) \ , (8) . . (9) (10) (1 1) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)
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Martin Quack (23) R. D. Levine, "Quantum Mechanics of Molecular Rate Processes", Clarendon Press, Oxford, 1969. (24) M. L. Goldberger and K. M. Watson, "Collision Theory", Krieger, New York, 1975. (25) M. Quack, Mol. Phys., 34, 477 (1977). (26) J F. Hornig and J. 0. Hirschfelder, Phys. Rev., 103, 908 (1956); M. A. Eliason and J. 0. Hirschfelder, J . Chem. Phys., 30, 1426 (1959). (27) L. Hofacker, Z . Naturforsch. A , 18, 607 (1963). (28) R. A. Marcus, J . Chem. Phys., 46, 959 (1967). (29) J. E. Wollrab, "Rotational Spectra and Molecular Structure", Academic Press, New York, 1967. (30) I.G. Csizmadia, J. C. Polanyi, J. C. Roach, and W. H. Wong, Can. J. Chem., 47, 4097 (1969). I am most indebted to Professor Polanyi for sending me a complete list of the computed points of the potential surface of H.', (31) M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem., 81, 329 (1977). (32) W. N. Olmstead, M. Lev-On, D. M. Golden, and J. I.Brauman, J. Am. Chem. Soc., 99, 992 (1977); E. Gorin, Acta Physicochem. URSS, 9, 691 (1938). (33) H. Hippler, Dissertation, ETH Lausanne, 1974. (34) J. Troe, Ber. Bunsenges. Phys. Chem., 73, 906 (1969). (35) H. Hippler and J. Troe, Ber. Bunsenges. Phys. Chem., 75, 27 (1971). (36) H. E. van den Bergh and J. Troe, J . Chem. Phys., 64, 736 (1976). (37) M. L. Dutton, D. L. Bunker, and H. H. Harris, J . Phys. Chem., 76, 2614 (1972). (38) C. Anastasi and I. W. M. Smith, J . Chem. Soc., Faraday Trans. 2, 72, 1459 (1976). (39) K. Glnzer and J. Troe, Ber. Bunsenges. Phys. Chem., 78, 71 (1974). (40) J. T. Cheng and C. T. Yeh, J . Phys. Chem., 81, 687, 1982, 2304 ( 1977). (41) R. Hartig, J. Troe, and H. Gg. Wagner, Symp. (Int.)Combust., [Proc.], 73th, 147 (1969). (42) K. Glnzer, M. Quack, and J. Troe, Chem. Phys. Lett., 39, 304 (1976); K. Glanzer, M. Quack, and J. Troe, Symp. (Int.) Combust., [Proc.], 76th, 949 (1977), see also ref 1 for a review. (43) H. E. van den Bergh, A. B.Callear, and R. J. Norstrom, Chem. fhys. Lett., 4, 101 (1969), see also ref 1 for a review. (44) M. Quack and J. Troe, Ber. Bunsenges. Phys. Chem., 81, 160 (1977). (45) G. E. Busch and K. R. Wilson, J . Chem. Phys., 56, 3626, 3638 (1972). (46) M. Quack and J. Troe, Ber. Bunsenges. fhys. Chem.,79, 469 (1975). (47) T. Beyer and D. F. Swinehart, Commun. ACM, 16, 379 (1973). (48) E. W. Schmid, J. Brandmuller,and G. Nonnenmacher, Z.Elektrochem., 64, 726 (1960). (49) Landolt-B&nstein, "Zahlenwerte und Funktionen", Springer, Heidelberg, 1974.
