An angle-dependent hard-sphere model for atom-diatom chemical

J. Phys. Chem. 1991, 95, 8462-8466

8462

ARTICLES An Angle-Dependent Hard-Sphere Model for Atom-Diatom Chemical Reactions Eric A. Cislason* Department of Chemistry (M/C I I I), University of Illinois at Chicago, Box 4348, Chicago, Illinois 60680

and Muriel Sizun LCAM (Laboratoire No. 281 associi au CNRSJ,University of Paris-South, Biit. 351, 91405, Orsay Cedex, France (Received: April 12, 1991) A simple expression for the reactive cross section QRfor the collision of A + BC, where A, B, and C are hard spheres, is derived. The spheres B and C are initially touching. For the case that A hits B the model assumes that reaction occurs whenever the projection of the collision energy along the A-B axis exceeds the activation energy, which depends upon the A-B-C bond angle. All of the parameters in the final result can be obtained unambiguously from the potential energy surface for the reaction. The one uncertainty in the calculation is the fraction of the BC zero-point vibrational energy that is available for overcoming the activation energy. The model is compared to the extensive trajectory studies of the 0 H2reaction by Alfassi and Baer. At collision energies below 90 kcal/mol very good agreement for the reaction cross sections is obtained if one assumes that 50% of the H2energy is available for overcoming the activation energy. In the energy range 100-1OOO kcal/mol the model result for QRagrees well with the sum of the cross sections for reaction and collision-induced dissociation obtained by Alfassi and Baer. The reactive model derived here is also compared with other simple classical models of chemical reaction.

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I. Introduction In recent years it has become possible to carry out theoretical calculations for chemical reactions such as A + BC AB + C +AC+B (1) using accurate classical’ and/or quantum-mechanical2 methods. However, simpler treatments of this type of reaction can often give considerable insight. An example of a simple model that has proven to be quite useful is the “lineof-centers” model? Recently, more sophisticated models have been developed which treat the two reactants (A and BC) as spheres, but which assume that the activation energy for reaction depends upon the relative orientation of A and BC at the moment of contact of the hard spheres. The classical version of this model is referred to as the “angular dependent line-of-centers” reaction model.es It has been extended to treat the reactant BC as any convex body, such as an ellipsoid, which is physically more rea~onabie.”~ Similar models have been developed from a quantum-mechanical point of +

(1) See, for example: Raff, L. M.; Thompson, D. L. In Theory ofChemicaf Reuclion Dynamics; Vol. 111; Baer, M., Ed.; CRC Press: Boca Raton, 1985; p 121. (2) See, for example: Baer, M. In Theory of Chemical Reuction Dynamics; Bacr, M., Ed.; CRC Press: Boca Raton, 1985; Vol. 1, p 91. (3) Present, R. D. Proc. Natl. Acad. Sei. U.S.A 1955, 41, 415. (4) Smith, 1. W. M. Kinetics and Dynamics of Elementary Gas Reactionr; Butterworths: Boston, 1980; p 8 1. (5) Smith, I. W. M. J . Chem. Educ. 1982, 59, 9. (6) Levine. R. D.; Bernstcin, R. B. Chem. Phys. Lett. 1984, 105, 467. (7) Blais, N. C.; Bernstein, R. B.; Levine, R. D. J . Phys. Chem. 1985,89, 10. (8) Connor, J. N. L.; Whitehead, J. C.; Jakubetz, W. J . Chem. Soc., Faraday Trons. 2 1987,83, 1703. (9) Evans, G. T.; She, R. S. C.; Bernstein, R. B. J. Chem. Phys. 1985,82, 2258. (10) Bernstein, R. B. J . Chem. Phys. 1985, 82, 3656. (11) She, R. S.C.; Evans, G. T.; Bernstcin, R. B. J . Chem. Phys. 1986, 84, 2204. (12) Janssen, M. H. M.; Stolte, S.J . Phys. Chem. 1987, 91, 5480. (13) Evans, G. T.;van Kleef, E.; Stolte, S. J. Chem. Phys. 1990,93,4874. (14) Jellinek, J.; Pollak, E. J . Chem. Phys. 1983, 78, 3014. (15) Pollak, E.; Wyatt, R. E. J . Chem. Phys. 1983, 78, 4464.

0022-3654191 12095-8462302.50/0

these are often referred to as ‘reactive infinite-order sudden approximations” for reactive scattering. Predictions of the models have been compared with the results of quasiclassical trajectory studies in a number of cases with mixed results. One problem with the classical models described above is that the radius of the hard sphere or hard ellipsoid representing BC is somewhat uncertain, and it is usually treated as a variable parameter in the calculations. Recently, we have developed a collision mode1’8-20that treats each of the three atoms A, B, and C as hard spheres, with B and C initially touching. This is a simple extension of the sequential impulse nod el.^'-^^ Our model includes a procedure for determining the “hard-sphere” radii from the potential energy surface. Since real atoms are compressible, these radii decrease slowly with collision energy. Reactive cross sections calculated in the highenergy limitI9 give nearly quantitative agreement with quasiclassical t r a j e ~ t o r i e s . ~In ~ addition, the collision model gives considerable insight into the dynamics of high-energy In two recent we illustrate the use of this method to compute total scattering cross sections from a known potential energy surface. Excellent agreement was obtained with more exact calculations. In this paper we extend our collision model to treat reaction 1. It is assumed that reaction occurs whenever the projection of the collision energy along the atomatom axis (AB or AC) exceeds ~~~

~~

~~

~~

(16) Pollak, E. Chem. Phys. Lett. 1985, 119, 98. (17) Last, I.; Ron, S.;Baer, M. Isr. J . Chem. 1989, 29, 451. (18) Gislason, E. A.; Sizun, M. Chem. Phys. 1989, 133, 237. (19) Gislason, E. A,; Sizun, M. Chem. Phys. Leu. 1989, 158, 102. (20) Gislason, E. A.; Sizun, M. J. Chem. Phys. 1990, 93. 2469. (21) Bates, D. R.; Cook, C. J.; Smith, F. J. P m .Phys. Soc. London 1964, 83, 49. (22) Mahan, B. H.; Ruska, W. E. W.; Winn. J. S.J . Chem. Phys. 1976, 65, 3888. (23) Safron, S.A.; Coppenger, G. W. J . Chem. Phys. 1984, 80, 4907. (24) Alfassi, 2.B.; Baer. M. Chem. Phys. 1981, 63, 275. (25) Sizun, M.; Parlant. G.; Gislason, E. A. Chem. Phys. 1989,133,251. (26) Gislason, E. A.; Polak-Dingels, P.; Rajan, M. S.J . Chem. Phys. 1990. 93, 2476.

0 1991 American Chemical Society

Hard-Sphere Model for Atom-Diatom Chemical Reactions

The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8463 where A hits C, so the total cross section for A hitting the molecule BC is Qta

Figure 1. The instant of collision between atom A and atom B of the diatomic molecule BC. For the case shown here the radii of the three hard spheres are the same. The angle between the line segments AB and BC is denoted ye. (a) Normal picture of the collision. (b) The same collision, but pretending that A is a point mass colliding with spheres of radii ( R A RB)and (RA Rc). The distance between the centers of B and C remains the same as in (a).

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the activation energy, which varies with the A-B-C orientation angle. Simple results are obtained for the reactive cross section, and most of the parameters in the final result are obtained unambiguously from the potential energy surface for the reaction. The one uncertain variable is the fraction of reactant zero-point energy that is available for overcoming the activation barrier. The results are compared with the extensive quasiclassical trajectory (QCT) calculations of Alfassi and Baer,24who used the potential surface of Johnson and Winter.27 The model agrees well with the QCT results over a wide range of collision energy. 11. Theory We consider the collision of a hard sphere A (radius RA)with the hard sphere molecule BC (radii RB, Rc) for the special case that B and C are identical so that RB = Rc. (The more general case RB # Rc is easily treated,20but the results are more complicated.) A typical collision where A hits B is shown in Figure 1, At the moment of contact the angle between the AB and BC axes is denoted yB. The collision is simpler to visualize if one pretends that A is a point mass colliding with two hard spheres, each with radius (RA RB). This equivalent picture is shown in Figure 1 b. Since the distance between the centers of B and C remains the same (2RB),the volumes of the two spheres overlap as shown. The excluded volume occurs because there are certain points on the surface of atom B that cannot be hit by A because A is blocked by C, and vice versa. It is apparent from Figure l b that the largest possible value for ye is given by ymr, where

+

COS Ymax

-RB/(RA + RB)

(2)

Values of greater than ymXcorrespond to collisions where A hits C. For the case A hits C the maximum value of yc is also Ymrx-

The collision of atom A with atom B can be described by an arom-atom impact parameter b and the orientation angle y (Figure 1). In our earlier paperm it was shown that to a very good approximation the joint probability distribution function P(b,y) is given by the product P , ( b ) P 2 ( y ) ,where

Pl(b) = 26/(R,4 + R B ) ~ 0 5 b -< (RA+ RB) (3a) P 2 ( y )= sin y / 2 0 Iy 5 ymax (3b) In this case the total cross section for A-B collisions is given byzo

(4)

Since B and C are identical, the same result is obtained for Qc, (27) Johnson, B. R.; Winter, N. W.J . Chem. Phys. 1977,66, 4116.

= *(RA + RB)(RA+ ~ R B )

(5)

In our earlier paper20 we showed that this formula slightly overestimates the true cross section, since it counts a small number of collisions twice. However, if R A 2 RB = Rc,the error in Qm is 1% or less. The two bond lengths (RA+ RB)and (RA+ ~ R B ) , which appear in eq 5 , can be easily estimated from a known potential energy surface. The model is readily adapted to treat reactive scattering. In general, there is an activation energy E,(?) for the reaction which depends on the molecular bond angle at the moment that A hits B (or C). We assume that reaction occurs whenever the projection of the relative collision energy E along the A-B line of centers is greater than Eact(y), This condition can be written

E[1 - b2/(RA + RBI2]

(6)

Eact(Y)

Combining eqs 3a and 6 gives the angle-dependent reactive cross section for the case A hits B: QR(Y) = *(RA + R B ) ~-[ Eact(y)/El ~

if E 1 (7a)

=0 if E < Ea,(y) (7b) This cross section should be averaged over y by using eq 3b and then multiplied by two to include collisions where A hits C. In what follows we assume that Em(?) increases monotonically from y = 0 to y = ymax,defined in eq 2. Thus, the maximum value of the activation energy is E,(y,). The reaction threshold occurs a t E = Ea,(y=O). In the energy range E,,(y=O) IE I Eact(ymax) the total reactive cross section is given by QR

= *(RA + R B ) ~ L " ~sin T

[ 1 - Eact(y)/El

(8)

where yo is defined by Eact(yo) = E. At higher energies where E 2 Ea,(ymax)the cross section can be written QR = *(RA + R E W A+ 2 R ~ ) [ 1- (Eact)/E(l

- COS ~ m A 1 (9a)

-

The results in eqs 8 and 9 are continuous a t E = Ea,.Jymx). In addition, it is apparent that, in the limit E a,QR in eq 9 equals QtOtdefined in eq 5 . It is instructive to compute Q R for the widely used functioned

Eact(y)= A + B(l - COS 7) (10) Substituting this activation energy function into eqs 8 and 9 gives QR = * ( R A

+ RB)2(E- A)'/2BE '

AIEIA+B(l-cosr,)

+

= *(RA + RB)(RA 2 R ~ ) ( 1- [A B(1 - COS y m , ) / 2 ] / E ) A+B(l-cosy,,,)IE (11)

The functional form of Q R has been seen but our result displays the precise bond lengths that must be known to compute QR. At low collision energy the reactive cross section depends only on ( R A + RB),the distance between the incoming atom and the struck atom at the transition state. It does not depend on the BC bond length, so it is irrelevant whether this bond length has increased or not in the transition state. The model can be tested against quasiclassical trajectory (QCT) calculations carried out on a known potential energy surface. A number of technical points must be considered first, however. The first involves the zero-point vibrational energy of the reactant molecule BC. At collision energies near threshold the reaction occurs in a vibrationally adiabatic manner, and only part of the zero-point energy can be used to overcome the activation barrier.8J4 At higher collision energies, however, we expect the initial AB collision will become vibrationally sudden, so that all of the

Gislason and Sizun

0464 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 zero-point energy would be available to overcome E,,(y). Later in the paper for the system 0 + H2we carry out three calculations assuming that O%, SO%, and 100% of the zerepoint energy of H2 can be used to overcome the activation energy. It is seen that the intermediate case gives the best agreement with the QCT results. Presumably the optimal percentage will vary from system to system. A calculation carried out assuming all the zero-point energy is available should give a rigorous upper bound to the true cross section. The procedure to determine the appropriate bond lengths from the potential energy surface is fairly straightforward. At low collision energy only the bond length (RA RB) is needed. This is taken directly as the A-B distance at the (linear) transition state. Since A and B are assumed to be spheres, the distance (RA+ RB) is not expected to vary with the bond angle y. The additional bond length (RA 2RB) is only needed at higher collision energies, where we expect the collision to be vibrationally sudden. For this reason (RA 2RB) taken to be the sum of (RA + RB),described above, plus one-half of the equilibrium bond length r,(BC). Finally, our work at very high energiesI9has shown that the bond length (RA+ RB) will shrink at these energies because the repulsive potential between A and B is softer than for hard spheres. This potential can be estimated from the potential energy surface as a function of the A-B distance for a fixed B-C distance. Further details of this are given in ref 19, and an example for the 0 + H2 system is given later. This proves to be a very important effect for 0 H2, because the 0-H potential is fairly "soft". At high collision energies two additional complications arise. The first is that collision-induced dissociation (CID) to give A B C becomes possible. In this case, we expect that the model prediction for Q R should give a good estimate of the sum of the reactive cross section plus the CID cross section. If an estimate of the reactive cross section alone in this energy range is desired, the model described in ref 19 can be used. The second complication with the model at high energies is that it assumes that chemical reaction (and/or CID at sufficiently high energies) occurs in every collision where the activation energy is overcome. (This is the fundamental assumption of all simple theories of chemical reaction, including classical transition-state theory.28) Classical trajectory calculation^^^ show, however, that at higher energies the A atom will sometimes rebound before reaction occurs, leaving BC as the stable product. This process is referred to as "recrossing". Since the reaction model described above does not take this into account, it overestimates the reactive cross section. In fact, the difference between the model cross section and the QCT cross section should be a measure of the importance of recrossing in a given system.

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111. Reaction of 0 + H2on the Surface of Johnson and Winter We have chosen this system to compare with our model because Alfassi and BaerU have camed out a set of QCT calculations over a wide range of collision energies. The potential surface they used was that of Johnson and Winter.27 We have computed the potential energy surface and determined the activation energy for a number of orientation angles y. We obtained a good fit to the results with the following quadratic equation E,,,(y) = 12.50

+ 6.46(1 -COS

7)

+ 15.45(1 - COS 7)'

(12)

where the units are in kcal/mol. (This function is measured from the bottom of the H2 potential well.) The cross section Q R for the quadratic activation function E,,(?) = F G( 1 - cos y) K(l - cos y)2 can be computed from eqs 8 and 9 to be

+

+

+

Q R = [ r ( R A + R B ) ~ -COS ( ~ yo)2G/2E][1 (4K/3G)(1 - cos 7011 (13)

for F 5 E I E,, where E,, is the activation energy at the maximum value of y, defined in eq 2. Thus (28) Levine, R. D.; Bernstein, R. B. Molecular Reactions Dynamics and Chemical Reacrioiry; Oxford University Press: New York, 1987; p 182.

=F

E,,,

+ G( 1 - COS 7")

+ K( 1 - COS ~) ,T

(14) (Gardiner and L e ~ i n have e ~ ~ considered a similar problem.) In eq 13 the angle yo is obtained by solving the equation Ea,(yo) = E. The analytical solution is 1

-COS

yo

= (G/2K)([1

For energies greater than E,, QR

+ 4K(E - F ) / G 2 ] ' / 2- l }

(15)

the reactive cross section is

= Qtotl1 - [F + G(1 - COS ymmx)/2

+ K(1 - COS ~ m u ) ~ / 3 l / E I (1 6 4

Qtot *(RA + RB)(RA+ 2RB) (16b) The results in eqs 13 and 16 reduce to those of eq 11 when K = 0. The 0-H bond length, equivalent to (RA RE), has been estimated as a function of relative collision energy E from the potential energy surface in linear configuration via eq 5 of ref 19. The result in angstroms is Ro + RH = 1.118 0 I E 5 5 5 . 2 kcal/mol (17a)

+

= (124.6/E)0.'369

55.2 kcal/mol IE

(17b)

The initial value is the 0-H bond length at the transition ~ t a t e . 2 ~ In all of the calculations we used RH = 0.371 A, half of the equilibrium bond length of H2. Since (Ro RH)shrinks at collision energies above 55.2 kcal/mol, the total cross section QGa, defined in eq 16b. becomes smaller, and at the same time the maximum bond angle ymaxincreases. For the bond length given in eq 17b combined with RH = 0.371 A, the two effects can be summarized as 1 - COS ymax= 1 0.294E0.'369 (18d

+

+

Qtot= ( 4 . 9 8 6 / ~ O . ~1~-~ COS ~ ) (ymax)

(18b)

for E 1 55.2 kcal/mol. The reactive cross section Q R can then be obtained by inserting these results into eq 16a.

IV. Results and Discussion The model reactive cross section Q R has been computed for the reaction 0 + H2 from eqs 13-18 for three assumptions concerning the zerepoint energy (ZPE) of the reactant molecule. In the first calculation it was assumed that none of the ZPE could be used to overcome the activation energy. In that case, E,,(y) is given by eq 12. This gives the minimum value of QR. In the second computation it was assumed that the entire ZPE (6.2 kcal/mol) of H2 was available. The effect was to lower the first term in eq 12 from 12.50 to 6.30 kcal/mol. The rest of the calculation was unchanged. This gives the maximum value of QR. The third calculation assumed that half of the ZPE could be used; in that case, the first term in eq 12 becomes 9.40 kcal/mol. The three curves are shown in Figure 2 for the energy range 0 IE 5 90 kcal/mol. In this energy region CID to give 0 + H + H is not possible. Also shown in Figure 2 as circles are the QCT cross sections obtained by Alfassi and Baer.24 It is seen that the model calculation which assumes that 50% of the ZPE is available to overcome the activation energy agrees very well with the QCT results over the entire energy range shown. The other two curves lie somewhat above (all ZPE available) and somewhat below (no ZPE available) the QCT data. The comparison between Q R and the QCT results indicates that recrossing is not an important phenomenon in this system. Even if we assume that above E = 45 kcal/mol the initial 0-H collision is vibrationally sudden, implying that one should consider the uppermost curve for Q R , the QCT cross sections lie no more than 15% below the model values. There are two likely explanations for this. The first is that the true H-H bond length at the transition state on this potential surface, 0.95 A, is quite extended compared to rc(H2) = 0.74 A. This suggests that the H-H bond is nearly broken before the H atom is hit by the 0 atom. Second, (29) Gardiner, W. C.; Levine, R. D. Chem. Phys. 1987, 111, 1.

Hard-Sphere Model for Atom-Diatom Chemical Reactions

The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8465

7-----7

t

I

41

Relative Energy (kcdmole)

IO

-

Figure 2. Reactive cross sections for the reaction 0 + H2 OH + H as a function of relative collision energy. The three curves are computed from the model as indicated in eqs 13, 16, and 18. The upper curve assumes that all of the H2zero-point energy is available to overcome the activation energy of the reaction, whereas the middle and lower curves assume that 50% and 0%. respectively, of the zero-point energy is available. The circles are the trajectory results of Alfassi and Baer [ref 241. At these energies collision-induced dissociation to give 0 + H + H is not possible.

the 0 atom is much heavier than the target atoms. Both factors make it difficult for the 0 atom to rebound away from the struck H atom faster than the H-H bond is broken. In Figure 3 the three model calculations of QR are shown extended to 1000 kcal/mol, just above the highest energy considered by Alfassi and Baer.24 Above 100 kcal/mol the ZPE of H2 is relatively unimportant, and the three curves are nearly superimposable. It is seen that in each case QR falls by about 20% from this peak value near 100 kcal/mol as E increases. This is completely due to the decrease in the total scattering cross section Qta, expressed in eq 18b. For collision energies less than 55.2 kcal/mol, Qtot = 5.23 A2. The curve Qtot is also shown in Figure 3. The difference between Qtotand QR gives the cross section for collisions where A hits either B or C, but the final product is A + BC. This can only occur when the activation barrier to the reaction is not overcome. It is clear that this cross section is rapidly decreasing at high energy. As discussed earlier, collision-induced dissociation to give 0 H H becomes possible at E = 100 kcal/mol, and by E = 300 kcal/mol it is found to be the major product channeLZ4 However, we expect that at high energies the model cross section QR should be a good approximation to the SUM of the reactive and CID cross sections. Therefore, in Figure 3 we also show this sum as computed by Alfassi and Baer.24 It is seen that all three model curves agree fairly well with the QCT cross sections above 100 kcal/mol, although the curves lie systematically above the data between 250 and 900 kcal/mol. This may be due to the uncertainty in our estimate of the 0-H interaction potential, which gave rise to eq 17b. It is instructive to compare the model developed here with other simple classical models of chemical reaction. The earliest models treated the reactant BC molecule itself as a sphere- (rather than each atom individually as in our model). Although this is a simple picture, there are two points which must be treated carefully. The first is to decide how big the sphere is. If one uses the notation of section 11, a logical choice for the sphere radius is (R, 2RB), the distance from the center of A to the center of BC in the linear configuration. However, our calculations (eq 8) suggest that this choice will overestimate the cross sections near threshold; the better choice is (RA+ RB)at low energies. In fact, the earlier modelses have treated the sphere radius as an adjustable parameter. Our model does not permit this. The second potential problem with the earlier models is that at any point on the sphere there are two orientation angles which must be considered, yB (the A-B-C angle) and yc (the A-C-B angle). At low collision energies only

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+

100 Relative Eoergy (kcabole)

loo0

Figure 3. Reactive cras sections for the collision of 0 + H2as a function of relative collision energy. The three lower curves are computed from the model as indicated in eqs 13, 16, and 18. The upper, middle, and lower curves assume that 1001, 50%, and O S , respectively, of the H2 zero-point energy is available to overcome the activation energy of the reaction. The uppermost curve is the total scattering cross section Qw. For energies less than 55.2 kcal/mol, this is 5.23 A2;at higher energies QtMis given in eq 18. The circles show the sum of the reactive cross section (to give OH + H) and the collision-induced dissociation cross section (to give 0 + H + H) computed by Alfassi and Baer [ref 241 using

classical trajectories. Below 100 kcal/mol only chemical reaction is possible, but above 300 kcal/mol dissociation is the major product channel.

the smaller of the two angles can contribute to reaction, but at higher energies both can. Thus, a decision must be made on how to treat the two angles. Whatever decision is made, it is important that the reactive cross section QR obtained from the model equals the total scattering cross section Qtotat high energies. (This is guaranteed by eqs 5 and 9 in our model.) This problem normally manifests itself in the subtle decision of whether or not to multiply the model cross section by two to take into account the fact that A can collide with both ends of the reactant sphere. This point is discussed further by Gardiner and L e ~ i n e . * ~ The simple classical models have been extended to treat the reactant BC molecule as an ellipsoid of r e v ~ l u t i o n , which ~ ' ~ is a more accurate description of the molecular shape. In principle, these models should allow very accurate calculations of the reactive cross section,provided that the size of the ellipsoid can be estimated accurately from the potential energy surface. To date, however, these size parameters have been treated as adjustable parameters in the theory. In our opinion, the advantage of the model developed in this paper is that it is simpler to implement. A problem with all of the simple models of reaction, including ours, is how to treat the zero-point energy (ZPE)of the reactant molecule. We have seen that the reaction of 0 Hz can be treated well by assuming that 50% of the ZPE is available for overcoming the activation energy of the reaction. However, it is clear from Figure 2 that unless that fraction is known from an independent source, the model predictions are very uncertain near threshold. Our work has shown that the relative importance of the ZPE is smaller for systems with large activation barriers and at higher collision energies. The fact that not all of the classical ZPE is available for overcoming the barrier has been known for a long time.'O It occurs because the system behaves vibrationally adiabatically near thresh01d.I~Classical trajectories at low collision energies can be used to map out the vibrationally adiabatic barrier to reaction as a function of bond angle. Such calculations have been carried out for the reaction of F + Hz,I4and the results have been used in conjunction with the angular dependent line-of-centers model* to estimate QR for this system. The model cross sections

+

(30) Karplus, M.; Porter, R.N.; Sharma, R.D. J . Chcm. Phys. 1965,13,

3259.

J. Phys. Chem. 1991, 95, 8466-8473

a466

agree well with trajectory calculations on the same system near threshold, but at higher energies QRfalls below the exact results. It is interesting that only 8% of the H2ZPE is available to overcome the activation barrier at threshold. We have also examined the F H2reaction using the barrier function denoted EoH(-y) in the paper of Connor et a1.8 Our model calculations actually agree with the trajectory calculations at the highest collision energies considered in ref 8 (30 and 40 kJ/mol), but only if we assume that 100%of the ZPE of H2is available. This result leads to the remarkable conclusion that the reaction of F + H2goes from being vibrationally adiabatic at threshold to vibrationally sudden at 30 kJ/mol. It would be useful to compare the model with trajectory calculations at higher energies to confirm this point.

+

In retrospect, the 0 + H2system was an excellent one to apply the reaction model to. The trajectory data of Alfassi and Baer2' are fit very well at all energies between 10 and 900 kcal/mol by assuming that 50% of the ZPE of H2can be used to overcome the activation barrier. (Above 100 kcal/mol the results are insensitive to the fraction of ZPE available.) The agreement between the model and the trajectories suggests that there is relatively little recrossing in this system. It should be interesting to apply this model to other reactions. Acknowledgment. Support from the NSF-CNRSExchange Program is gratefully acknowledged. Registry NO. H,12385-13-6; 02,7782-44-7.

Solution Photophysics of 1- and 3-Aminofiuorenone: The Role of Inter- and Intramolecular Hydrogen Bonding in Radlationiess Deactivation Richard S. Moog,* Nancy A. Burozski, Monica M. Desai, William R. Good, Celeste D. Silvers, Department of Chemistry, Franklin and Marshall College, Lmcaster, Pennsylvania 17604-3003

Peggy A. Thompson, and John D. Simont Department of Chemistry, University of California at San Diego, La Jolla, California 92093-0341 (Received: April 19, 1991; In Final Form: July 1 , 1991)

Solvent effects on excited-state photophysics of 1-aminofluorend-one (I-AF) and 3-aminofluorend-one (3-AF) are examined. SolventdependentStokes shifts indicate that the lowest energy electronic transition for both species has substantial chargetransfer character. In addition, the influenceof solvent on nonradiative rate constants is investigated. Increased radiationless deactivation of 3-AF in protic solvents relative to that in aprotics is observed and is attributed to intermolecular hydrogen bonding. In contrast, an unusual solvent dependence for k,, of I-AF is observed. The nonradiative rate constant decreases with increasing solvent polarity and is lower in protic solvents than in aprotics. A 1:l hydrogen-bonded complex of 1-AF with 2,2,2-trifluoroethanol (TFE) is observed in heptane solutions, and k,, for this complex is approximately a factor of 4 lower than for 1-AF in pure heptane. These results suggest that the intramolecular hydrogen bond in I-AF provides a more efficient deactivation pathway than an intermolecular hydrogen bond to the solvent.

Introduction excited-state photophysics of these two fluorenones in a variety of protic and aprotic solvents may lend insight into the role of During the past decade, molecules that undergo large changes in charge distribution following photoexcitation, often due to a subsequent intramolecular electron transfer, have been intensively (1) (a) Chapman. C. F.; Fee, R. S.;Maroncelli, M. J. Phys. Chem. 1990, 94,4929. (b) Maroncelli, M.; Fleming, G.R. J. Chem. Phys. 1987,86,6221. studied. Species that exhibit this type of behavior have been used (c) Castner, E. W.; Bagchi, B.; Maroncclli, M.; Webb, S.P.; Ruggiero, A. as molecular probes for dynamic solvation (as shown by an obJ.; Fleming, G. R. Ber. Bunrenges. Phys. Chem. 1987, 92, 363. (d) Castner, served time-dependent Stokes shifP3) and test cases for various E. W.; Maroncelli, M.; Fleming, G. R. J . Chem. Phys. 1987, 86, 1090. continuum and molecular theories of solvent r e l a ~ a t i o n . ~ - ~ ~ (2) Kahlow, M. A.; Kang, T.-J.; Barbara, P. F. J . Chem. Phys. 1988,88, 2372. (b) Nagarajan, V.; Brearley, A. M.; Kang, T.-J.; Barbara, P. F. J . Time-resolved studies of these systems have contributed to an Chem. Phys. 1987,86,3183. increased understanding of solvent effects in solution photophysics; (3) (a) Su,S.-G.;Simon, J. D. J. Phys. Chem. 1987, 91, 2693. (b) Su, several review articles on this topic have recently appeared.'619 S.-G.;Simon, J. D. J . Phys. Chem. 1989.93.753. (c) Simon, J. D.; Su, S.-G. Many of the previous studies rely on the assumption that there Chem. Phys. 1991, 152, 143. (4) Mazurenko, Yu. T. Opt. Spectrosc. (Engl. T r o d . ) 1974, 36, 283. are no specific interactions on a molecular level between the solvent (5) (a) Bagchi, B.; Oxtoby, D. W.; Fleming, G.R. Chem. Phys. 1984,86, and the probe. In some cases, however, these studies have focused 257. (b) Castner, E. W., Jr.; Fleming, G. R.; Bagchi, B.; Maroncelli, M. J . on molecules containing amino groups, since the presence of this Chem. Phys. 1988,89,3519. (c) Castner, E. W.; Fleming, G.R.; Bagchi, B. functional group often enables a large charge separation in the Chem. Phys. Lett. 1988, 143, 270. (d) Maroncelli, M.; Fleming, G.R. J . Chem. Phys. 1988,89, 875. excited state. In hydrogen-bonding solvents, the role of specific (6) (a) van der Zwan, G.;Hynes, J. T. J. Phys. Chem. 1985,89,4181. (b) solventsolute interactions on the observed dynamics (Stokes shift van der Zwan, G.; Hynes, J. T. Physica A (Amsterdam) 1983, 121A, 227. or electron transfer) has not been unambiguously addressed. In (c) van der Zwan, G.; Hynes, J. T. Chem. Phys. Lett. 1983, 101, 367. this paper the photophysical properties of I-aminofluoren-9-one (7) (a) Loring, R. F.; Yan, Y.J.; Mukamel. S.J . Chem. Phys. 1987.87, 5840. (b) Loring, R. F.; Mukamel, S.J . Chem. Phys. 1987,87, 1272. (c) (1-AF) and 3-aminofluoren-9-one (3-AF) in solution are examined. Mukamel, S.;Yan, Y. J. Acc. Chem. Res. 1989, 22, 301. These two molecules were chosen because they have similar (8) (a) Sumi, H.; Marcus, R. A. J . Chem. Phys. 1986, 84, 4859. (b) geometrical and electronic structure, but only I-AF can form an Marcus, R. A.; Sutin, N . Biochim. Biophys. Acra 1985. 811, 275. intramolecular hydrogen bond. This intramolecular interaction (9) Wolynes, P. G. J . Chem. Phys. 1987, 86, 5133. (10) Friedrich, V.; Kivelson, D. J . Chem. Phys. 1987,86,6425. has been observed to have an influence on the extent and effects (1 1) (a) Rips, 1.; Klafter, J.; Jortner, J. J . Chem. Phys. 1988, 88, 3246. of intermolecular interactions.2*22 Thus, a comparison of the (b) Rips, I.; Klafter, J.; Jortner, J. J . Chem. Phys. 1988, 89, 4288. 'Camille and Henry Dreyfus Teacher Scholar 1990-1995.

(12) (a) Wei, D.; Patey, G.N. J. Phys. Chem. 1990,94, 1399. (b) Agmon, N. J . Chem. Phys. 1990, 94, 2959.

0022.365419 1/2095-8466%02.50/0 0 1991 American Chemical Society