J . Phys. Chem. 1991, 95, 3477-3480 given to ion-molecule chemistry in the condensed phase. This is in part because experimental methods used have lacked the specificity to distinguish charge-transfer reactions involving the transfer of an electron versus the transfer of a proton. Matrix EPR studies have shown that ion-molecule reactions are possible, but FDMR studies reveal their widespread occurrence and the remarkable diversity in radical-cation reactivity which is manifested by different rates of ion-molecule reactions and different
3477
branching among competing reaction channels.
Acknowledgment. We acknowledge the contributions of our collaborators who were involved in this work: Drs. J. P. Smith, S.M.Lefkowitz, M. F. Desrosiers, L. T. Percy, M. G. Bakker, and X.-Z. Qin. This work was performed under the auspices of the Office of Basic Energy Sciences, Division of Chemical Science, US-DOE, under contract no. W-3 1- 109-ENG-38.
ARTICLES The Impulse Approximation in Photofragmentation: How Accurate an Estimate of Fragment Rotation? C . H.Dugan Physics Department, York University, North York, Ontario M3J 1 P3, Canada (Received: September 17, 1990)
Numerical solutions of the classical equations of motion for a rigid rotor dissociating from an atom are compared with results from a widely used, very simple approximate solution of the problem called the impulse approximation. We first show that the approximation gives a good estimate of the rotation of a diatomic fragment dissociated from an atom when the repulsion is along the breaking bond, for potentials whose steepness values are similar to values that have been used in recent calculations of photodissociation. Then we examine a more general potential for which the resultant repulsion acts beyond the end of the rotor and find that an extension of the impulse approximation compares adequately with the results of trajectory calculations. Finally, we compare the extended approximation with trajectories calculated for a potential of a type widely used, having an exponential repulsion and a truncated Lependre expansion in the angle variable. Factors affecting the impulsiveness of a dissociation are discussed.
A durable and useful way to estimate the amount of fragment excitation following photofragmentation is known as the “impulse approximation”. This approximation will be useful only when short-range forces act between the fragments as they dissociate, so that the dissociation occurs very suddenly. The time scale on which the promptness is judged is the period of a bending vibration. In a time short on this scale, the energy that is available to the fragments after the dissociation is converted to kinetic and internal energies of the fragments, before the fragments have moved away from their initial configurations. It requires no solution to the equations of motion to use this estimate, hence its continued use as a “back of an envelope” estimate of fragment energies. This note is concerned with its application to estimating the rotational and translational energies of the fragments following photodissociation of a three-atom molecule ABC into fragments A and BC, a rigid rotor, although it has been employed with larger molecules, and it has been used to estimate vibration in the fragment. As the approximation is employed for a quick estimate of fragment rotation, one frequently finds an additional condition imposed, that the repulsive force acts between A and B-a nearest-neighbor repulsion. This restriction is not necessary for the dissociation to be impulsive, but for comparison we first study dissociations mediated by potentials of this type, comparing the solutions of the equations of motion (“exact solutions”) to the approximate solutions for a range of initial bond angles and for several values of steepness of the repulsion. We study how the impulsiveness of the dissociation is influenced by the steepness of the repulsion and by the mass combination in the molecule. Following that we look at repulsive potentials for which the resultant force is not a nearest-neighbor interaction, showing that
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if one applies the impulse approximation by using the initial resultant repulsion acting on the rotor, one obtains-if the dissociation is impulsive-a useful estimate of the fragment rotation computed from the equations of motion, for a range of initial bond angles for the parent molecule. The early uses of the approximation were made by experimenters to argue about the plausibility of certain intuitive models as explanations of fragment rotation measured in dissociation of cyanogen halides’-3 and ethyl nitrite.4 Recently, the approximation was used in discussion of dissociation of H O N 0 , 5 C2H2,6 methyl nitrite,’ formaldehyde,* and keteneeg In several cases the predictions of the impulse model have been remarkably close to what is observed-within a few percent in the case of HONO, for instance. Details of the rotational distributions have been discussed by using this approximation in ICNIO and BrCN,” and by use of a modified impulsive model, good results have been obtained for O2from both visible and UV photolysis of ozone and for OH rotation after 157-nm photolysis of H20.’* ( I ) Busch, G.E.; Wilson, K. R. J . Chem. Phys. 1972, 56, 3626. (2) Ling, J.; Wilson, K. R. J . Chem. Phys. 1975, 63, 101. (3) Simons, J. P.; Tasker, P. W. Mol. Phys. 1974, 27, 169. (4) Tuck, A. F. J . Chem. SOC.,Faraday Trans. 2 1977, 689. (5) Dixon, R. N.; Rieley, H. J . Chem. Phys. 1989,90,2308. Vasudev, R.; Zare, R. N.; Dixon, R. Chem. Phys. Lett. 1983, 96, 399. (6) Fletcher, T. R.; Leone, S.R. J . Chem. Phys. 1989, 90, 871. (7) Winniczek, J. W.; et al. J . Chem. Phys. 1989, 90, 949. (8) Butenhoff, T. J.; Carleton, K. L.; Moore, C. B. J . Chem. Phys. 1990, 92. - -, -177. . .. (9) Chen, I. C.; Moore, C. B. J . Phys. Chem. 1990, 94, 272. (10) Dugan, C. H . J . Phys. Chem. 1987, 91, 3929. (11) Paul, A. J.; Fink, W. H.; Jackson, W. M. Chem. Phys. Lett. 1988, 153, 121.
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3478 The Journal of Physical Chemistry, Vol. 95, No. 9, 1991
On the other hand, the approximation was unsuccessful in approximating the rotational energy in fragments from OCS" and H2S.14 The form of the impulse approximation tested is the modified version described in ref 12. Application of the a p proximation in its simplest form, the nearest-neighbor approximation, has been strongly criticized recently,Is and this will be discussed later in the paper. Our intention is to assist experimenters who wish to use this approximation, to understand how estimates from the simple models discussed here relate to accurate trajectory calculations based on the sort of potentials that have been used to compare theory and experiment: by how much do the approximate and the exact solutions differ? Beyond this it is the intention of this note to show that the impulse approximation, properly applied, is a correct sudden limit to solutions of the equations of motion which may considerably simplify comparing various models of the disintegration event by approximating the rotational distribution without computing trajectories. Apparently the errors associated with this obvious extension of the impulse approximation have not been described before. The Impulse Approximation The fragment rotation produced in the impulsive limit is obtained by requiring that angular momentum and energy be conserved and that the fragment kinetic energies are imparted entirely while the fragments are in the configuration from which they begin the dissociation process. Thus, conservation of linear and angular momenta gives enough conditions to determine the energies of translation and rotation. This is the way early formulations of this approximation were obtained.I4 Alternatively, an impulsive limit for Newton's second law for the acceleration of the rotor yields the same result.I6 The expression obtained for the fraction of energy imparted to rotation is given below. The formula is slightly more general than the original formulations in that the factor 1 is the distance along the rotor axis of the force on the rotor, from the center of mass of the rotor. A force of this kind may be understood as the resultant of two forces that are applied at the nuclei of the rotor. For example, a repulsive force along AB and an attractive force along AC, for a molecule having a bond angle greater than 90°, will yield a resultant force that acts on the rotor axis beyond the atom B. We study cases in which the net force is applied beyond the end of the rotor. Experimental data for formaldehyde* and keteneg have been discussed recently in terms of such forces. The expression for the fractional energy conversion is the following.
f = [I
+ I(M + MA)/(MMAlz sin2 $')]-I
(1)
This is the fraction (rotation energy)/(available energy). I is the rotor moment of inertia, M is the mass of the rotor, MAis the mass of atom A, and is the angle between the resultant force on the rotor and the rotor axis. We compare predictions from this equation with the results of trajectory calculations that are described next. A dissociation will be called "impulsive" if the energy imparted to the rotor is well approximated by the above formula. In the next sections we shall consider which conditions, other than the steepness of the potential, cause the dissociation to be more or less impulsive.
+'
Integrating the Classical Equations The equations of motion of the atom and the rigid diatomic rotor are solved for motion in a plane. The angular momentum of the parent molecule is zero. We use two forms of interaction
Dugan potential. In one, atom A interacts with the rotor BC according to this potential U(ABC) = U(AB) exp[-k(AB) R(AB)] + U(AC) exp[-k(AC) R(AC)] (2) where U(AB) is the magnitude of the potential between atom A and the B-atom end of the rotor; k(AB) is the inverse-range parameter for this exponential repulsion (or attraction, if U is negative). The exponential repulsion is a familiar feature of model potentials in photofragmentation, but the potential used here has no special significance. It is useful to the present study because it makes the interaction between atom A and the ends of the rotor more explicit than potentials that are more frequently used, such as the potential studied next in which the repulsion depends exponentially on the distance between atom A and the center of mass of the rotor. Such a potential is this U(ABC) = Uoexp(-kR)(1
+ (a)cos2 $)
(3) where R is the distance between A and the center of mass BC and J, is the angle between the rotor axis and the line joining A to the center of mass of BC. (According to Alexander and Berard" such potentials have been used because of the ease of computing matrix elements. Several similar potentials are given in this reference.) After specifying the bond angle of the triatomic molecule, and selecting the interatomic distances, the masses, and the constants of the potential, the equations are solved step by step until the rotor is widely separated from the atom. Then the kinetic energies of translation of the rotor and the atom and the rotational energy of the rotor are evaluated. The initial potential energy is converted to kinetic energy to the extent of a few parts in lo4 or better if a finer computation mesh is used. In the course of this calculation we evaluate the direction of the force acting on atom A in the initial configuration of the parent molecule ABC. The point where this vector crosses the rotor axis and the angle of intersection with this axis give the quantities needed ( 1 and v ) to evaluate the impulse formula in eq 1. Comparison of Approximate and Exact Results In this section we compare fragment rotation following dissociation as obtained from the equations of motion with results of the impulse approximation, using potential functions like (2) and (3). In order to relate these functions to recent calculations of photofragment rotation, we used values of the inverse-range parameter k from 3.0 A-l, as "soft" potential, to 6.0 A-I, intermediate, to 9.0 A-l, a "hard" potential. The following is a representative sample of values of the inverse-range parameter k that have been used in recent calculations of cyanogen halide dissociation. The potentials that we use resemble but are not the same as those that are used in these studies. molecule
ICN ClCN
BrCN
k , A-' 8.5 3.554 4.12, 5.6
reference 18 19, 20 21
Figure 1 shows the results of exact calculations of the fractional conversion of initial potential energy to rotation versus initial bond angle (the angle ABC) using the potential in eq 2. Values of k used were the ones for the soft potential (3.0 A-I) and the hard potential (9.0 and the results of the impulse approximation are also shown. For these graphs U(AC) = 0 so the repulsive force acts directly between A and B; this is the most elementary model of the interaction that was used in some references given previously.
~~
(12) Levene, A . 8.; Valentini, J. J. J . Chem. Phys. 1987, 87, 2583. (13) Straws, C. E.; McBane, G. C.; Houston, P. L.; Burak, I.; Hepburn, J. W. J . Chem. Phys. 1989, 90, 5364. (14) Weiner, B. R.; Levene, H.; Valentini, J. J.; Baronavski, A. P. J. Chem. Phys. 1989, 90, 1403. (15) Schinke, R . Comments At. Mol. Phys. 1989,23,2387; J. Chem. Phys. 1990, 92, 2397. (16) Fisher, W. H.; Carrington,T.; Filscth, S. V.; Sadowski, C. M.; Dugan, H. Chem. Phys. 1983, 82,443.
(17) Alexander, M. H.; Berard, E. J . Chem. Phys. 1974,60, 3950. (18) Goldfield, E. M.; Houston, P. A.; Ezra, G. S. J. Chem. Phys. 1986, 84, 3120. (19) Waite, B. A.; Dunlap, B. I. J . Chem. Phys. 1986, 84, 1391. (20) Barts, S. A.; Halpern, J. B. J. Phys. Chem. 1989, 93, 7346. (21) Wannenmacher, E. J.; Lin, H.; Fink, W. H.; Paul, A. J.; Jackson, W. H. Presented at the 19th Informal Photochemistry Conference (poster PI-38), Ann Arbor, MI (June 1990).
The Journal of Physical Chemistry, Vol. 95, No. 9, 1991 3479
The Impulse Approximation in Photofragmentation
Error
50
(X)
I r-
I
+50
0
-50 I 100
I Bond Angle (degrees)
180
F i p e 2. Fractional error of the approximation plotted against the bond
angle for a potential given in eq 2 for which U(AC) = +(AB). M A = IO, M B = Mc = 1 (amu). Curves are shown for k = 3.0, 6.0, and 9.0 A-I. The error is smallest for the hardest potential, as we expect. Fractional energy conversion at small angles for the softest potential is about 80%, about 50% for the hardest potential. For each calculation of the exact solution, k(AB) = k(AC). Interatom distances as in Figure I. 180
100 Bond Angle (deqrw8)
Figure 1. Fraction of available energy converted to fragment rotation
Error ( X )
+50
1
as a function of bond angle in the molecule ABC. Nearest-neighbor repulsion acts between atoms A and B. Comparisons are shown for each of two mass distributions, between approximate and exact results. Initially, the atoms are separated by I A. Curves are shown for two values of k, the inverse-range parameter: k = 3.0 and 9.0 A-1. The bond angle is the initial value of the angle ABC. Two mass distributions are represented on the graph. The first resembles the mass distribution in the cyanogen halides: MA= 10, MB = Mc = 1. The second, MA= MB = MC = 1, is presented to illustrate how the masses affect the impulsiveness of the dissociation. The graphs show good agreement between the exact and approximate energy values for the short-range potential and less favorable agreement for the longest range potential ( k = 3.0). However, values of angular moments (rather than rotational energies) derived from these two results would show closer agreement and would still be useful in many situations where choices are to be made, based on data, between different types of theoretical models. As was mentioned earlier, we judge a dissociation to be more or less impulsive according to there being more or less agreement between the exact calculation of energy distribution and the impulse approximation. The set of curves for MA = M B = Mc = 1 lets us compare the impulsiveness of a dissociation as affected by the mass of the retreating atom; intuitively, we expect that for a given rotor BC and a given potential function, a dissociation involving a light atom A is more impulsive that one involving a heavy atom. Comparison of the two sets of curves in Figure 1 supports this expectation. Also, the curves illustrate the obvious fact that a light atom at A imparts less rotation to BC than a heavier one does. To permit impulsiveness to be assessed quantitatively in the next examples, we graph the fractional error of the impulse approximation, that is, the difference between the approximation and the exact solution forf, at a given bond angle, expressed as a fraction of the exact solution. Figure 2 shows the fractional error obtained from calculations like those in Figure 1, but for a potential that involves an attraction between A and C as well as a repulsion between A and B. This potential (see the figure caption) yields a resultant force of A on B-C acting at a point farther from the center of mass than atom B. Naturally a force of this kind gives more rotation than a force acting on B or between B and C. Because the force on BC acts at a different point on the BC axis for each value of the bond angle, and because the location at which the force acts is different-for a given bond
-50 100
180 Bond Angle (degrees)
Figure 3. Fractional error of the approximation as a function of bond angle for the potential of eq 3. The asymmetry parameter CY = 0.8, MA = 10, M B= Mc = 1 (amu). Curves are shown for k = 3.0 and 9.0 A-l. The fractional energy conversion at small angles from this potential is about 8% for the softer potential and less for the harder one. Interatom distances as in Figure I.
a n g l e f o r every k value, we have different approximate solutions,
f vs bond angle, for each k value. The curves in Figure 2 relate to only one mass combination, M A = 10, M B= Mc = 1. As the force on the rotor is applied at larger distances from the center of mass (and here, the initial force is always external to the rotor), the rotation is increased and the dissociation is less impulsive for reasons similar to those given in considering the mass effect in Figure 1, where one sees that if the complex separates more slowly, the dissociation is less impulsive. The sign change in the error graph for k = 3.0, at small angles, comes about because the longer range forces emphasize energy conversion well away from the starting configuration, and for small bond angles, the rotor can rapidly move to a configuration of atom-rotor angles that is less favorable for energy transfer. Thus, the impulse approximation is not always a lower limit to the rotational excitation. Figure 3 shows the result of applying the approximation to the potential of eq 3 for two values of k and for a range of bond angles. This potential yields forces that act at points interior to the ends of the rotor, and so there is smaller conversion to rotation than was obtained in calculations using potential (2). In this figure we graph the error Mexact) -f(approx))/f(exact). This potential is similar to potentials used in analyzing photodissociation, although this particular potential is not one of several potentials recently used to analyze experiments on the cyanogen halides. Results obtained here are simply illustrative of how the approximation will work with a potential of this form. The figure shows
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J. Phys. Chem. 1991, 95, 3480-3486
that the error is larger for the less impulsive potential (k = 3.0) but that the agreement between the exact and approximate results would be adequate for distinguishing between rotational distributions obtained from competing theoretical models, since angular momentum values derived from the approximation and the exact solutions would show fractional errors about half of those obtained for the energies. Again, the curve for k = 3.0 shows the error changes sign for reasons described in the last paragraph.
Conclusions This is a summary of what we have demonstrated in the previous sections. First, we tested the simplest form of the impulse approximation-in which the force between retreating atom A and the rotor is along the line AB-using repulsive potentials in which the interaction is a nearest-neighbor repulsion. Values of the steepness of the repulsion were chosen with reference to some recent calculations, and for these values, the approximation gives reasonable values of fractional energy conversion. Next, we used a potential that is more general in having forces directed other than between nearest neighbors, but still using exponential attractions and repulsions. When used in the way we described, the impulse approximation also works well for calculating the fragment rotation over our range of bond angles in the initial configuration. Such a calculation would enable the experimenter to make easy comparison of dynamical models with, for example, the variety of statistical models that are frequently cited. The present study will be extended to a comparison of approximate and exact computations for other sources of rotation: parent rotation and bending vibration. Finally, we tested the extended
impulse approximation applied to a potential of a widely used type involving short-range repulsion and 'bending forces" and showed that the approximation gives good results for short-range repulsion. When will the impulse approximation fail to give an adequate estimate of fragment rotation? The graphs illustate that it can be in error if the potential is not short range, that is, if k is small. But it has been found that even if the potentials in the dissociating molecule are short range, the elementary form of the approximation fails to yield acceptable results. The nearest-neighbor repulsion model does not correctly describe fragment rotation in CICNi5or in NOC12*even though the classical trajectory results obtained with a short-range potential are in good agreement with experiment. Probably the reason for this failure is that the nearest-neighbor repulsion model of the approximation uses an incorrect direction for the initial force on the fragment. The dissociation can still be impulsive, as described in this note, even though the initial force is not directed along the direction AB. It remains to be determined if the extended impulse approximation provides a good approximation to fragment rotation for ClCN. It is clear from the calculations of error given earlier that the impulse approximation gives good estimates of fragment rotation for some potentials that do not satisfy the requirements given by Schinke and his collaborators's~22for the successful application of the impulse approximation. It is evident that the impulse approximation has more usefulness than these recent criticisms suggest. (22) Schinke, R.; Nonella,M.; Suter, H.U.; Huber, J. R. J . Chem. Phys. 1990, 93, 1098.
Triplet Energy Transfer of the Intramolecular System Havlng Benzophenone and Dibenz[ b ,t]azepine at the Chain Ends: Chain Length Dependence Hideaki Katayama, Shogo Maruyama, Shinzaburo Ito, Yoshinobu Tsujii, Akira Tsuchida, and Masahide Yamamoto* Department of Polymer Chemistry, Faculty of Engineering, Kyoto University, Sakyo- ku, Kyoto 606, Japan (Received: September 28, 1990; In Final Form: January 4, 1991)
Intramolecular triplet-triplet energy transfer in a series of polymethylene chains having a benzophenone (BP) group as an energy donor and a dibenz[bflazepine (DBA) group as an energy acceptor (BP-O(CH,),CO-DBA) has been studied by phosphorescencemeasurement and nanosecond laser photolysis. In a rigid solution and PMMA matrix, the quantum yield of triplet-triplet energy transfer is close to unity for the chain lengths shorter than n = 5 . On the basis of the through-space mechanism of energy transfer, phosphorescencedecay curves were analyzed by Dexter's equation in which the distribution of donor-acceptor distance was calculated by the conformationalenergy analysis. The results of the simulation were in fairly good agreement with the experimentallyobserved decay curves. The rate constant of triplet-triplet energy transfer is strongly dependent on the chain length, Le., about one-tenth decrease per every methylene unit, and the rate is much smaller than that of singlet-singlet energy transfer.
Introduction Triplet-triplet (T-T) energy transfer is a fundamental photophysical process.' Many kinds of photochemical reactions proceed via the triplet state and are often initiated by so-called "triplet sensitizer", which transfers its excited energy to a reactant with a high efficiency.2 T-T energy transfer is forbidden by the ( I ) (a) Turro, N. J. Modern Moleculur Phorochemisrry;Benjamin: Menlo Park, CA, 1978;p 306. (b) De Schyver, F. C.; Boens, N. Ado. Phorochem. 1977,10, 359. (c) Thiery, C. Mol. Phorochem. 1970,2, I . (d) Birks, J. B. Phorophysics oJAromotic Molecules; Wiley: N e w York, 1970;p 518. (e) Wilkinson, E Q.Reu. 1966,20,403.(f) Fox, M.A. J . Phorochem.Phorobiol. 1990, 52,617. (2)(a) Galley,, W.; Stryer, L. Proc. Nurl. Acad. Sci. U.S.A. 1968,60, 108. (b) Wu, 2.;Mornson, H. Phorochem. Phorobiol. 1989.50,525. (c) Eisinger, J.; Shulman, R. G. Science 1968, 161, 1311.
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dipole-dipole mechanism but is spin-allowed by the exchange mechanism? Therefore, the critical radius of T-T energy transfer, Ro, is relatively short compared with that of the singlet-singlet energy transfer and is in the range 1.0-1.5 nm.4 Owing to this factor, the rate constants are highly sensitive to the distance of separation between donor and acceptor molecules. In the case of intramolecular energy transfer, the rate will be markedly affected by the intramolecular donor-acceptor distance that is determined by the molecular structure and conformation of the molecules. Many studies on intramolecular T-T energy transfer systems Lamola et aLs studied T-T energy transfer have been (3)Dexter, D.L.J . Chem. Phys. 1953,21, 836. (4)Ermolaev, V. L. Sou. Phys. Dokl. 1962,6,600.
0 1991 American Chemical Society
