943
PARAMETRIC STUDYOF BIMOLECULAR EXCHANGE REACTIONS
A Parametric Study of Bimolecular Exchange Reactions by Don L. Bunker and Tiang-Shing Chang Department of Chemistry, University of California, Irvine, California
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(Received September 2 5 , 1 9 6 8 )
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For the reaction d BC +AB C, we have constructed a treatment in which the degree of adiabaticity or statisticality of the process may be parametrically varied. Reaction probabilities and product energy distributions have been calculated for K C12, cs CHJ, H C12, K HBr, and H Hz. The form of the theory that best describes these reactions is a semistatistical one, somewhat more closely allied to the statistical treatment of Light than to the adiabatic model of Marcus.
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Introduction: Why a Parametric Treatment? The theory of bimolecular exchange reactions can be divided into sectors, The basis of the division is the degree of coupling between reactant and product states. At one extreme we have the adiabatic-type theory advanced by Marcus1 for the treatment of reaction cross sections, In this it is proposed that certain dynamical degrees of freedom not directly involved in the reaction event shall remain in the same quantum state (or shall conserve action) while the reaction occurs, This theory is the logical successor to the transition-state treatment. It requires a separation of the reaction coordinate from all the others and thus endows the reaction products with the best possible “memory” for the initial condition of the reactants. There is a middle ground, populated by usually rather formal theories of the connection of each initial to many final states12beyond which there occurs the fully statistical approach advocated by Light.* Here the products have no memory whatever for initial states, beyond that imposed by conservation laws. Different possible detailed results of a reaction event are probable in simple proportion to the number of states that correspond to them, The two opposite theories have been tested, the statistical more thoroughly than the adiabatic, with mixed results for each. As is entirely reasonable from a theoretician’s viewpoint, the tests have emphasized the truth or falsity of the individual theories. There have as yet been no comparisons of two treatments of the same reaction or attempts to encompass large numbers of laboratory results. It is also possible to organize an inquiry around reactions rather than theoretical models, Are most reactions adiabatic, statistical, or in between? The material we present here is a possible first step toward a treatment that facilitates answering this question. Such a treatment must contain at least one empirical parameter in terms of which laboratory reactions may be meaningfully classified. I n our formulation there are two such parameters-the adiabaticity or statisticality of the various degrees of freedom and the range
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of reaction coordinate over which this prescription operates. Others are possible, but we believe the present experimental evidence is not extensive enough to justify more than two. We have constructed our calculation around a limited three-dimensional mechanical model of a reaction involving three atoms, this being the least complicated situation for which we believe our treatment makes sense. At its most adiabatic, our procedure exceeds the intent of Marcus’ work, since we calculate product energy distributions for many reactions in which some kind of failure of adiabaticity is clearly inevitable. Conversely, at its most statistical, it stops somewhat short of Light’s method; it should be considered a semistatistical model. (The distinction between statistical and semistatistical will be elaborated in the discussion that follows.) On the whole, however, the range of adaptability of our procedure is not too far out of register with that delimited by the landmarks erected by Marcus and Light.
Notation
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Symbolically, the reaction is A BC 4 A B C. Boldface A or S, with subscripts and superscripts, represent particular parameterizations of the theory. The superscript f measures, on a scale of 0-1, the range of reaction coordinate to which the description applies. The conventions associated with this are illustrated in Figure 1 for reactions without and with a potential barrier (the symmetrical case of the latter is shown, for (1) The key papers are: R. A. Marcus, J. Chem. P h y s . , 45, 2138, 2630, 4493, 4500 (1966), and Discussions Faraday Soc., 44, 7 (1967). For application t o H Hz,see R. A. Marcus. J. Chem. P h y s . , 46, 959 (1967), and M . 9. Child, Discussions Faraday S o c . , 44, 68 (1967). A similar treatment of K H I has been given by M. S. Child, Proc. R o y . Soc. (London), A292, 272 (1966). (2) Examples: 0. A. Coulson and R. D. Levine, J. Chem. P h y s . , 47, 1235 (1967); F. T. Smith, ibid., 3 6 , 248 (1952); D. A. Micha, Arkiv Fysik, 3 0 , 411, 426, 437 (1965); L. Hofacker, Z . Naturforsch., 18a, 607 (1963). (3) J. C. Light, Discusstons Faraday SOC.,44, 14 (1967), and P. Pechukas, J. 0. Light, and C. Rankin, J. Chem. P h y s . , 44, 794 (1966), may be used as key references. Alternate theoretical routes have been provided by B. C. Eu and J. Ross, tbtd., 44, 2467 (1966); J.
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Keck, ”Variational Theory of Reaction Rates,” Fluid Mechanics Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1966; and E. E . Nikitin, Teor. i Eksperim. K h i m . , A k a d . N a u k U k r . S S R , 1, 135 (1965). Volume 7S,Number 4
April 1969
DONL. BUNKERAND TIANG-SHING CHANG
944 f
0 .2 .4 .6 .8 1.0
f = 1.0
.5
0
.5
diatom BC. All quantities with an x dependence are to be evaluated along the reaction path, the shape of which will eventually have to be specified. (2) There is enough energy for the adiabaticity of those triatomic vibrations that are so treated between points 1 and 2. Let b = 1 for adiabatic bending (A and Abf), 0 otherwise; and let s = 1 for adiabatic stretching (A and Asf),0 otherwise. Then
1.0
I-
w I-
O
0
a
DISTANCE ALONG REACTION COORDINATE x Figure 1. Labeling conventions for the reaction coordinate, shown with f = 0.6.
CR
- 6($)+ h ( V R f +)[vR - sVs(x)]
+ [tfi2JR(JR + 1)/IR]
simplicity). The figure has been drawn with f = 0.6. The range described by f is between points 1 and 2. The measurement is unidirectional if there is no barrier and bidirectional if there is. Outside the range the reaction is strictly adiabatic. The possible parameterizations are : Sf,semistatistical; A8f, adiabaticity of the BC vibration 3 ABC stretch 3 AB vibration only; Ab', adiabaticity of the BC rotation -+ ABC bend -+AB rotation only; A, full adiabaticity, for which f is irrelevant. The circled symbols in Figure 1 are used as subscripts to designate different points on the reaction path. Reactants and products, each viewed as an atom plus a diatom, are denoted by R and P, respectively. The beginning and end of the f region, inside which the system is treated as a triatomic molecule ABC, are represented by 1 and 2. Quantities to which these subscripts may be applied include: J , rotational quantum number; v, vibrational quantum number; I , moment of inertia of a diatom; v, vibrational frequency of a diatom; E, translational energy; p , reduced mass of a diatom; 4, potential energy on reaction coordinate, measured from zero indicated in Figure 1; x, the reaction coordinate; u, maximum geometric cross section for the reaction or its reverse.
- bJRhVb(x)
for all x1 5 x 2 xz. Symbols which refer to the triatom ABC are the bend and stretch frequencies Vb and v8 and the moment of inertia IC. J z , x,VZ) (3) At point 2, there exists a set of (L, for which it is possible to satisfy the list of conditions (A) through ( G ) . The angle x is between vectors representing L2,the AB-C orbital angular momentum, and J2fi, the AB rotational angular momentum. We apply a geometric approximation a t this point-that ABC passes through a linear configuration so that the problem of composition of angular momentum is simplified. The rotational kinetic energy of such an object can be found by simple vectorial arguments; it is
T
= (L2/21c)
(A) (B) (C) not too Since €2
=
ER
+
If s If b
+
+ 1)fi2/21z] - Lz[Jz(Jz+ 1)J ' / 2 f i ( ~x~ )~ / ~ c (4) [Jt(Jz
1, as in A and A,f, then v 2 = OR. 1, as in A and Abf, then4 J Z = JR. The AB-C orbital angular momentum L2 is large for a reasonable exit cross section CTP. = =
- 62 + h [ ( V R + #)VR - (VZ + +)VZ] +fi2[{J~(J~
+ ~ ) / I R }-
(Jz(Jz
+ 1)/1z)I
(5)
Theory
we require
We formulate the reaction probability per collision P ( L , Q) as a function of total angular momentum L and initial relative translational energy, with the intention of later summing over distributions of these
Lz 5 (2aPP€z/9)1/2 (6) in which p is the reduced mass of AB and C. This follows from the definition of angular momentum in terms of velocity and impact parameter. Test (C) is extended to the rest of the reaction path in (F), below. (D) Energy is conserved a t point 2. From eq 4
P ( L , ER) =
C
cJ'(JR,
VR)/
~ ( J R V, R )
(1)
J R I VR
J R IV R
where the summed P's are the distributions of reactant internal states. The quantity a has a very complicated dependence on many things, Its value is 0 or 1 ; it will be 1 only if all the following conditions are met. (1) There is enough energy for the reaction to get adiabatically from point R to point 1 (applies only to reactions with a barrier) eR
- 6(x)
+ h ( v R + +)[vR
- v(x)]
+ + f i 2 J ~ (+J ~l ) [ I ~ - l - I(%)-'] for all zR 5 x 5
51.
2 0 (2)
Here I ( % )and v(x) refer to the
The Journal of Physical Chemistry
T - [Jz(Jz
+ 1)fi2/21z]
(7) (E) Angular momentum is conserved a t point 2. This means that L, and J2ii, added vectorially a t angle x, must yield a vector of magnitude L. €2
=
(4) This statement I s a fairly low approximation to the actual adiabaticity requirement, but it is tolerable because the parameterizations of most interest did not turn out to rely heavily on it. For a description of how this should have been done, see R. A. Marcus, J. Chern. Phys., 49, 2617 (1968). Equation 100 of this is the one of interest.
945
PARAMETRIC STUDYOF BIMOLECULAR EXCHANGE REACTIONS
(F) There is not too much orbital angular momentum for the adiabatic motion from point 2 to point P Lz 5 L
+J ~ f i
(8)
a requirement the consequences of which we will discuss later, and
Lz I [2QPPE (XI/ r y z
(9)
for all x2 5 z 5 zp, where e(z) =
ER
+ h(vZ +
-
\
- v(x)]
+ $AzJz(Jz+ 1)rIz-l - I($)-']
(10)
' 0
\
( G ) There is enough energy for this adiabatic process; e(z) is positive over the range of eq 9. If f = 1, (F) and (G) may be omitted. The quantum numbers uz and U P are identical, and so are JZ and J p . The distributions of J p and up, symbolically and without normalization, are
UP, J P ) =
(2Jz
~ ( V R ,JR) J R I OR
+ 1)(2Lz + 1).
La,x
(11)
Computational Tactics This theory, unlike others,',a accepts a limited amount of information about the shape of the potential surface. In the present calculation we supply an 11-entry table of values of dJ(z), v,(z), and vb(x); f is restricted to multiples of 0.1. The reaction path is assumed to have the shape shown in Figure 2. The curved part is a quarter-circle,6 and tabulation is a t intervals of 9' from 0 = 0-90'. All tests involved in establishing the value of CY are made a t these points. Summation over JR, U R and all subsidiary variables is done by random sampling. For fixed ER and L, J R and U R are chosen from distributions corresponding to a prescribed temperature, with attention to nuclear
Figure 3. Reaction path profiles: standard case, I; heavy-atom case, 11; light-atom attractive case I and repulsive case, 111; K HBr, IV; H Ht, V. The energetics are correct but the shapes are arbitrary.
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spin statistical effects if they are important. If tests (1) and (2) are passed, J2 is sampled from a (2J2 1)weighted distribution if b = 0, and if s = 0, uz is randomly chosen.6 The maximum value L,, of L2 consistent with requirement (3C) and with eq 8 is next calculated, and Lz is sampled in the range 0 to L,,, with weight? 2L2 1. The x corresponding to (3D) and (3E) are next calculated and compared. If they match within a small Ax, a = 1; otherwise new uz, Jz,and Lz are selected. For Ax = IS0, 1800 repetitions of this cycle are sufficient, as shown by exponential decline of the distribution of number of trials for JR and U R that have a = 1. Ordinarily many fewer trials suffice.
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The Standard Case: Equal Masses, Attractive Surface All of the reaction path profiles used in this study are shown in Figure 3. I n all nonbarrier cases va(x) was distributed linearly between V R and YP, Y ~ ( x )was 100 cm-l all along the reaction path, u p was 100 A2, and the temperature corresponded to that of the relevant laboratory experiments. All bond lengths were derived from the known properties of diatoms. The standard case involves atoms of equal mass, 40 amu (approximately, K Clz) , with ER = erg, V R = 560 cm-', and v p = 280 cm-l. As is expected for
--+ *-0
m"B
+
L!
j cr; c
'n-8
Figure 2.
Shape conventions for the reaction path.
(5) This is a simplifled form of the artiflcial potential proposed by F. T. Wall and R. N. Porter, J. Chem. Phys., 36, 3256 (1962). (6) It would be easy to make s and b into continuous variables if the amount of experimental data warranted it. For example, the probability of 2 ) s could be made proportional to eXp[-S'(OR - t~2)~l. with 0 5 s' 5 m and s' an appropriate function of s. (7) It is debatable whether ZLZ 1 or uniform weighting is more consistent with our geometrically restricted mechanical model. Fortunately, tests show that the effect on the results is not very important.
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Volume 78,Number 4 April 1069
DON L. BUNKERAND TIANG-SHING CF~ANG
946
reactions of high exothermicity, there is nearly zero reaction probability for A,1 and Ab1. A,' fails because so little energy can be accomodated as product vibration, and Ab1 fails because the combination of Vb (x) and large JRusually requires more energy than there is. The best fit, in terms of the knowns properties of these reactions, is S'. Smoothed distributions of up and J p for four values of L are shown in Figures 4 and 5, along with over-all results obtained by summation over
P (L)
=
?TL/uR~'ER,
(12)
Az
with b11 = 100 and p' the reduced mass of A and BC. The JP distribution corresponds roughly to a temperature in the range 1000-2000°K. Reaction probability was 1.00, 1.00, 0.82, and 0.48 for the descending series of L. The effect of decreasingf is shown in Figures 6 and 7. The JPdistribution is less drastically affected than that for UP, which shifts rapidly to small values when f becomes small. This is a very useful way for this parameter to behave.
18
9
0
,27
36
Figure 6. Distributions of up for the standard case So and SO.3: 1, L = 1.6 X 10-26; 2, L = 1.2 X 10-26; 3, L = 0.8 X 10-26, all for So; 4, L = 1.6 X 10-2G for The ordinate has arbitrary units.
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A Heavy-Atom Case: Cs CHJ The A, B, and C masses were 133, 127, and 15; ZJR = 560 cm-l; vp = 280 cm-l; largest L = 2.6 X lodz6 cgsu. Otherwise, this was similar to the standard case. Tests were made for So, S1,As1, and Ab', with results quite similar to those described above. The point of this is that Entemann and Herschbachg have recently discovered that this reaction produces
0
70
140
210
280
350
JP
Figure 7. Distributions of Jp for the standard case So and Labels are the same as for Figure 6.
S0.3,
products having unusually little CsI internal excitation. Since this did not appear for S1,either an unusual parameterization of the theory is required or (as Entemann and Herschbach suggest) an unusual potential surface is involved. In our calculations the shape of the potential surface would be most important for f E 0.5. 0
IO
20
40
30
50
60
"P
Figure 4. Distributions of up for standard case SI: 1, L = 1.6 X 10-26; 2, L = 1.2 X 10P6; 3, L = 0.8 X 10-26; 4, L = 0.4 X 10-26; 5, summed over L. The ordinate has arbitrary units.
0
70
140
2IO
280
350
JP
Figure 5. Distributions of Jp for the standard case S'. Labels are the same as fora Figure 4. The Journal of Phyaical Chemialry
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A Light-Atom Case: H Clz Masses 1,36, and 36, VR = 560 cm-', V P = 3000 cm-I, ER = 4 X erg, and L = 2 X were used. This is another reaction which does not produce the largest possible up, and for which a "repulsive" potential surface has been proposed.1° We studied it with attractive and repulsive reaction path contours (both are shown in Figure 3 ) . The reaction is definitely subject to some constraints not present in the other cases. S' and So are unsuccessful on both surfaces-the reaction probability, though not zero, is too small (a few per cent) to permit up and JPdistributions to be accumulated. There is no (8) R. Grice and P. B . Empedocles, J . Chenz. P h y s . , 48, 5352 (1968). (9) E . A. Entemann and D. R . Herschbach, Discussions Faraday Soc., 44, 289 (1967).
(10) The key reference is now K . G. Anlauf, P. J. Kuntz, D. H. Maylotte, P . D. Pacey, and J. 0.Polanyi, i b i d . , 44, 183 (1967).
947
PARAMETRIC STUDYOF BIMOLECULAR EXCHANGE REACTIONS single simple reason for this behavior, which involves the interaction of eq 4-8. Simultaneous satisfaction of these when L is small requires special values of V R and J R ; most of the successful points were for VR = 1. There is, however, an intermediate range of f for which the reaction has normal properties. At So.son the attractive surface, the probability of reaction is 0.81. Switching to the repulsive surface shifts the reaction to SO.', with probability 0.25. The up distributions are shown in Figure 8. It is the attractive surface that agrees best with the experiments for up. The JPdistributions are mildly non-Boltzmann, peaked a t J p = 35 (attractive) and J p = 25 (repulsive). The experimentally observed bump at low J p has been attributed to rotational relaxation.
pected), and L = 3 X The intermediate stretching frequency increased linearly from 2200 cm-' a t the barrier to 4400 cm-l a t the ends of the curved part of the reaction path. We found that S1 and A,1 had unit probability and that Abl had none. In S', about half the time UP = VR; otherwise vp - V R = 1. These results are both roughly compatible with the conclusions reached by Karplus14 from trajectory studies.
Discussion The tendency of this far from exhaustive first set of tests is to suggest that the best over-all Parameterization is SI,or perhaps S with f somewhere between 0.5 and 1. Since this is a t one extreme of our parameterization range, we have to inquire into the relation between our fully semistatistical case, S', and Light's fully statistical treatment (for which, in effect, f is also 1). There are three differences, aside from those of model sophistication, Our calculation is not reversible-we neglect the possibility that the intermediate (ABC) will reform A BC, whereas Light assigned this full statistical weight. We impose the constraint expressed by eq 8, thus rejecting states accepted by Light in which final orbital and rotational angular momenta have opposite senses and unusually high values. Finally, we incorporate information about the geometry of (ARC), and other theories do not. Light has commented on the fact that his theory does not produce inverted product vibrational distributions. Our procedure does not suffer from this difficulty. It would be pleasant if the explanation turned out to lie in the first two differences noted above. A revised Light-type theory could then, in principle, be constructed, and it would require knowledge only of reactant and product properties. That this is probably a vain hope can be seen from the following construction, based on our mechanical model of the reaction in the S1 modification. Figure 9 shows the semiclassical u, J2 plane for this reaction. (A subsidiary illustration for the infrequent angularHBr, is also promomentum-limited case, e.g., K vided.) All symbols refer to products; E is total energy. The diagonal dashed line arises from energy conservation
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VP
VP
Figure 8. Distributions of up for the light-atom case; So.6with attractive surface and SO.7 for repulsive surface.
We do not believe that our test tells very much about the character of the reaction path contour. Rather, it contributes another item to the evidence that this reaction is kinematically something of a special case.
K
+ HBr
I n some ways this has been an easy reaction to interpret theoretically. The combination of small exothermicity with a large quantity of conserved angular momentum places severe constraints on the outcome of the reaction; J p must be as large as possible. All other aspects of the reaction are subordinate to this. The high J P feature is found experimentally,'' in Light's theory,3 and in trajectory studies.12 It also appears in the present theory (masses 39, 80, 1; V R = 2650, U P = 230; t~ = 1.3 X lo-", L = 1.9 X The reaction may be either SO or S1 with unit probability. There also exists a probability of 0.25 per collision for the A,' parameterization, but Ab1 is unsuccessful.
H
+ Hz
We tested this reaction only for probability of occurrence, since the product states are not experimentally available. The parameters13 were 1 amu for masses, U R = 15 A 2 , Vb = 950, ER = 3.8 x 10-12 (for which a high probability of reaction would be ex-
+
E 2 vhv
+ (J2fi2/21)
The vertical solid line is our constraint of eq 8, J& The final orbital angular momentum will be
L'
b[2p(E - vhv - J2&2/21)]1'2
(13)
2 L. (14)
(11) C. Malte and D. R. Herschbach, ibid.,
4 4 , 176 (1967); see also J. Chem. Phys., 4 3 , 1140 (1965). (12) D . L. Bunker, unpublished results. (13) Based on the potential surface of R . hT.Porter and M. Karplus, J. Chem. Phys., 4 0 , 1105 (1964). (14) M. Karplus, Discussions Faraday SOC.4 4 , 78 (1967); see also R. A . Marcus, i b i d . , 4 4 , 87 (1967).
A . E. Grosser and R. B. Bernstein,
Volume '73,Number 4 April 1969
DON L. BUNKERAND TIANG-SHING CHANG
948
[.hv-h
J2
Figure 9. Plane of final v, J2 permitted by S' parameterization. The smaller diagram is for the angular-momentum-limited case.
with b the exit impact parameter and p the AB - C reduced mass. The over-all angular momentum cannot be composed unless Jfi 2 L - L'. This leads to vhv
5 E - (J2h2/21)- [ ( L - JF~)~/2pb'](15)
which generates the curved line shown in the figure. It is very clear that low v will dominate the final energy pattern, since the largest number of J states will always be associated with them. No amount of manipulation of the partitioning of angular momentum between J f i and L' will relieve this condition. Only a remodeling which weighs against large amounts of product translational energy would help. Such a feature is present in our treatment because of conditions (3D) and (3E). In making these tests a t point 2 on the reaction coordinate, we effectively limit the product translational energy to
LzZ/2Ic- Lz[Jz(Jz
+ l)]1'2fi
COS
x/1c
(16)
that which is associated with the over-all rotation of (ABC) , In reconstructing Figure 9, then, we might in an approximate way require that vhv
+ (J2fi2/21)2 L2/21*
(17)
in which I* is an ABC final moment of inertia that might be prescribed in a number of ways, e.g., from Figure 2 and its conventions. We would obtain the kind of situation depicted in Figure 10 (one of three possible configurations is shown), The product vibrational energy distribution for this figure would rise at low D, attain a flat maximum, and then decrease rapidly. This corresponds with what is observed, and positively identifies condition (3D) as the key feature in our treatment. A theory of product excitation based on
The Journal of Physical Chemistry
I
ALLOWED REGION
J2 h2 = L21/I*
Figure 10. Figure 9 revised t o include a product translational limitation.
Figure 10 and all of its ramifications, however, is only a little less bulky than the main calculation with which we established S' as the most significant parameterization. There are many other ways in which this study might have been carried out. Adopting f = 1, b = 0, and variablea s might lead to results nearly a8 good as those we have achieved and so might a number of other ways of tampering with Light's prescription8 for the relative probabilities of final states. The A alternatives might better be defined in terms of maximum persistence of quantum numbers consistent with conservation laws, rather than in the absolute form we have used. All of these things are clear by hindsight, but it is by no means certain that we would thereby evade the situation that has developed in our study-the predominance of (ABC) properties over those of AB and C. We therefore reach the following tentative conclusion. To make a satisfactory model of A BC 4 AB C, it will probably be necessary to include structural information about the intermediate state ABC. We have done this in a minimal way in the preceding paragraphs, via the single quantity I*. Our calculation as a whole provides another alternative. Both of these, in crude form, appear to be somewhat successful, and we believe that future exploration along this line will be fruitful.
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Acknowledgments. The work was supported by the Army Research Office, Durham, N. C., under Contract DAHC04-67-C-0054. Most o€ the calculations involved purchase of computer time from the North Node of the Campus Computing Network, UCLA. This arrangement would not have been feasible without the spirited cooperation of Mr. Kenneth Tom and Mrs. Dorothy Parkin, whom we thank.
