J. Phys. Chem. 1986, 90, 3063-3072
3063
FEATURE ARTICLE Selectivity of Elementary Molecular Processes Associated with Energy Transfer and Chemical Reaction Stuart A. Rice Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (Received: December 26, 1985)
This paper contains a review of experimental and theoretical work concerned with the selectivity of collision-induced vibrational relaxation and proposes a new method for achieving selectivity of chemical reaction. The paper is based on the Debye Award Lecture and emphasizes work in the author's laboratory.
I. Introduction The fascination and the usefulness of chemistry depend on the selectivity of the reactions used to transform one molecule into another. The selectivity of a reaction can, of course, depend on the exploitation of a favorable equilibrium between species, on a favorable difference between the rates of competing processes, or on a combination of equilibrium and kinetic preferences. For those cases in which the time scale of the chemical transformation is very much longer than the time between molecular collisions-very often the situation when liquid solutions are employed-the use of empirical rules and model calculations to design a selective reaction is well-advanced, as attested to by the many successes of synthetic organic chemistry. Of course, the apparent simplicity implied by these rules and calculations arises because the time scale for reaction is such that only the averaged effects of the elementary dynamical processes induced by collisions, or by the excitation mechanism, need be accounted for. In contrast, photoinduced reactions of isolated molecules can occur on a time scale comparable with those of the elementary dynamical processes induced by the excitation. Clearly, it is reasonable to expect the selectivity of reaction to be different with and without averaging over the dynamics following excitation. It is worth noting that even in solution some energy-transfer processes m u r rapidly enough that full averaging over elementary dynamical processes does not take place; one such case arises in the photoinduced isomerization of stilbene.' From the point of view of fundamental theory, one method for improving the selectivity of reactions is to control, to the extent consistent with the principles of quantum mechanics, the selectivity of elementary dynamical processes. Yet we know surprisingly little about the selectivity of the elementary dynamical processes which follow, say, photoexcitation or a single collision between molecules. For example, we do not know, except in general terms, how such selectivity depends on the magnitude of anharmonic coupling and/or rotation-induced coupling of the vibrational motions of a molecule. Indeed, many of the extant analyses of elementary dynamical processes in a molecule consider them to be so complex that a statistical treatment is warranted. It is not surprising that such analyses draw attention away from examining the possibility of controlling the selectivity of a process. There are many ways in which a molecule can be made reactive and many ways in which energy can be transferred within one molecule and between colliding molecules. In this paper I focus attention on what must be considered only a first step in learning about the selectivity of those elementary dynamical processes that (1) G. R. Fleming, private communication.
0022-3654/86/2090-3063$01 .50/0
influence the-selectivity of a chemical reaction. Specifically, I shall consider the characteristics of vibrational energy redistribution arising from atom-plyatomic molecule collisions and from the dissociation of an atom-polyatom van der Waals molecule. Sections I1 and I11 provide a brief summary of experimental data which support the conclusion that collision-induced vibrational energy relaxation and vibrational energy redistribution associated with van der Waals molecule dissociation are, at the elementary process level, quite selective with respect to state-to-state transitions. Section IV contaitls a comparison of the qualitative features of the data available for several molecules, and section V sketches a number of theoretical approaches intended to explain the observed selectivity of energy transfer, Given the nature of this symposium I trust I will not offend anyone by relying primarily on work done in my laboratory for the material of sections 11-V. Once it is accepted that elementary dynamical processes can be selective even when the number of final states accessible from some prepared state is very large, it is possible to image conditions which take advantage of that selectivity. That is, it is possible to devise various schemes for active control of energy transfer or chemical reactivity. Whether or not such schemes will work depends on the accuracy of our description of the molecular dynamics and the robustness of any particular scheme with respect to variation in molecular parameters. One suggestion for controlling chemical reactivity is sketched in section VI. 11. Collision-Induced Vibrational Energy Redistribution:
Some Data Collision-induced vibrational relaxation has been studied for many years? However, until recently the experimental techniques used were based on the measurement of some bulk property of the medium, e.&, the dispersion of the velocity of s o ~ n d . Such ~ measurements yield, typically, only an averaged relaxation time which provides no direct ififormation about state-testate transition probabilities. It is not surprising, then, that the early analyses of vibrational energy redistribution in atom-plyatom collisions focus attention on the dependence of the relaxation cross section on the relative kinetic energy and the density of vibrational states and say little about expected state-to-state energy-ttansfer propensities in a field of many final states. The introduction of tunable laser sources has permitted direct study of collision-induced vibrational energy redistribution. A (2) See, for example, the old revkw T. L. Cottrell and J. C. McCoubrey, Molecular Energy Transfer in Gases, Butterworths, London, 1961, and the newer review W.Miller, Ed., Dynamics of Molecular Collisions, Plenum, New York, 1976. (3) See, for example, K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves, Academic, New York, 1959.
0 1986 American Chemical Society
3064 The Journal of Physical Chemistry, Vol, 90, No. 14, 1986
seminal paper on this subject, by Parmenter and Tang,4 mapped the relaxation pathways for collisions between 'B2u(61e2g)benzene and a variety of atoms and molecules. Very slightly later Chernoff and Rice5 determined the relaxation pathways for collisions between Ar and a large number of vibronic states of 'B2 aniline. The latter measurements have been extended to cover many different collision partners for the 'B2 aniline by Rice and co-workers6 and by Pineault, Cracker, Hedstrom, and Struve.' Other systems that have been studied include 'B3" pyrazine + Ar,8 'Bl difluorodiazirine Ar,9 lBju naphthalene + Ar,Io 'A, glyoxal + several species," 'A2 s-tetrazine several species,12 and lA, difluorobenzene several species.') The results of these studies lead to four broad conclusions: (i) The cross section for atom-polyatom collision-induced intramolecular vibrational energy transfer is large, typically an appreciable fraction of the geometric cross section. (ii) Only a subset of the energetically allowed relaxation pathways are utilized. (iii) The selectivity of the collision-induced energy transfer is dominated by intramolecular processes. (iv) Although transitions with Au = f l are usually more important than those with IAul > 1, the relative importance of single and multiple quantum number changes in transitions varies considerably from system to system. For present purposes we shall not consider other detailed results of the experimental studies cited. One data set illustrating conclusion ii is displayed in Figure S1, and a data set supporting conclusion iii is displayed in Figure S2 (see supplementary material paragraph). All of the studies referred to above involve collisions with a thermal distribution of relative kinetic energy corresponding to T == 300 K. Rice and co-workers14devised a method of studying collision-induced vibrational relaxation with a thermal distribution of relative kinetic energy corresponding to a very low temperature, as low as T = 1 K. The general theory of collisions between neutral particles predicts that at very low relative kinetic energy the cross section is inversely proportional to the initial relative momentum.15 The energy range in which this contribution to the inelastic cross section will dominate its behavior will vary from system to system. Leaving aside this general dependence on relative momentum, conventional semiclassical theories of collision-induced vibrational relaxation, such as that due to Schwartz, Slawsky, and Herzfeld,16 predict that the vibrational relaxation cross section decreases as the collision energy decreases and that it becomes vanishingly small for collision energies near 1 cm-I. Rice and co-workers have studied very low energy collision-induced relaxation in systems involving 12(3n0,+)r14 C6H5NH2('B2),' C6H5CH3('B2),18 C6H5F('B2),18C6H6('B2,,),'* and C2H2O2('AU);I9
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(4) C. S. Parmenter and K. Y. Tang, Chem. Phys., 27, 127 (1978). (5) D. A. Chernoff and S. A. Rice, J . Chem. Phys., 70, 2521 (1979). (6) M. Vandersall, D. A. Chernoff, and S. A. Rice, J . Chem. Phys., 74, 4888 (1981); S. M. Cameron, M. Vandersall, and S. A. Rice, J . Chem. Phys., 75, 1046 (1981). (7) R. L. Pineault, R. Cracker, J. F. Hedstrom, and W. Struve, J . Chem. Phys., 80, 5545 (1984). (8) D. B. McDonald and S. A. Rice, J . Chem. Phys., 74, 4907 (1981). (9) M. Vandersall and S. A. Rice, J . Chem. Phys., 79, 4845 (1983). (10) D. B. Moss, S. H. Kable, and A. E. Knight, J. Chem. Phys., 79, 2869 (1983). (11) H. ten Brink, Ph.D. Thesis, Universiteit van Amsterdam, 1979. (12) M. W. Leeuw, Ph.D. Thesis, Universiteit van Amsterdam, 1981. (13) D. L. Catleff, K. W. Holtzclaw, D. Krajnovich, D. B. Moss, C. S. Parmenter, W. D. Lawrence, and A. E. Knight, J . Phys. Chem., 89, 1577 (1985); D. J. Muller, W. D. Lawrence, and A. E. Knight, J . Phys. Chem., 87,4952 (1983); W. D. Lawrence and A. E. Knight, J . Chem. Phys., 77, 570 (1982). (14) J. Tusa, M. Sulkes, andS. A. Rice, J . Chem. Phys., 72, 5733 (1980). (15) E. P. Wigner, Phys. Rev., 73, 1002 (1948); R. G. Newton, Scaftering Theory of Waves and Particles, McGraw-Hill, New York, 1966; R. B. Bernstein, Ed., Atom-Molecule Collision Theory. A Guide for the Experimentalisf, Plenum, New York, 1979. (16) R. N. Schwartz, Z . I. Slawsky, and K. F. Herzfeld, J. Chem. Phys., 20, 1591 (1952). (17) J. Tusa, M. Sulkes, S. A. Rice, and C. Jouvet, J . Chem. Phys., 76, 3513 (1982). (18) C. Jouvet, M. Sulkes, and S. A. Rice, Chem. Phys. Lett.. 84, 241 (198 1 ) .
Rice Knight and co-workers have studied systems involving C6H6('B2J20 and CloHlo('B3,).10 The results of these studies lead to three broad conclusions: (v) Very low energy atom-polyatom collisions do lead to efficient vibrational relaxation. (vi) The energy ranges in which the collisions are efficient are different for vibrations with different point group symmetries. (vii) There is a similarity between the relaxation pathways accompanying predissociation of a van der Waals molecule and the corresponding very low energy collision (see section 111). Data sets illustrating conclusions vi and vii, and by implication also v, are displayed in Figures S3 and S4, respectively. All of the studies cited above involve collisions between atoms and excited molecules, so much of the information available concerns depopulation pathways. By the principle of microscopic reversibility, dynamical effects which influence a deexcitation cross section must also influence the corresponding excitation cross section. Gentry, Giese, and c ~ - w o r k e r s have ~ ~ - used ~ ~ a crossed molecular beams method to study state-to-state energy transfer in collisions of neutral molecules. This very important work both complements and extends the previously referred to studies and leads to the conclusion that collision-induced excitation of the rotationally cold ground vibrational state of a polyatomic molecule is very selective. For example, collision-induced vibrational excitation of p-difluorobenzene by He atoms occurs predominantly in mode 30 even at kinetic energies which correspond to many quanta of excitation of other modes of vibration.24 It is further implied, in contrast with conclusion iii of section 11, that level mixing is not the origin of the large cross section for excitation of mode 30. 111. Vibrational Energy Redistribution Associated with van der Waals Molecule Dissociation: Some Data Interest in the investigation of van der Waals complex relaxation dynamics was stimulated by the report, in 1976, by Smalley, Levy, and Wharton, of the fluorescence excitation spectrum of the HeI, molecule.25Since that time workers in that laboratory have studied the van der Waals complexes of iodine with many specie^.^^-^* The predissociation lifetime of the He12 molecule is found to be dependent on the iodine vibrational level initially excited and ranges from 38 to 221 ps,28 with the variation of the predissociation lifetime with the iodine vibrational quantum number strongly nonlinear. The predissociation lifetimes for the and deuterium32complexes are found to be somewhat shorter. As for the product I2vibrational-state distribution accompanying the photodissociation, for He,12 the Au12 = -n channel is dominant, with a branching ratio of 10.95 for u12 = 20.4 Such relaxation "selection rules" are also observed to dominate the I2 vibrational-state distributions for the other complexes, though in general the minority dissociation channels are more significant than in (19) C. Jouvet, M. Sulkes, and S. A. Rice, J. Chem. Phys., 78, 3935 (1983); M. Sulkes,C. Jouvet, and S. A. Rice, Chem. Phys. Lett., 9 3 , l (1982). (20) A. E. Knight, private communication. (21) G. Hall, K. Liu, M. J. McAuliffe, C. F. Giese, and W. R. Gentry, J . Chem. Phys., 78, 5260 (1983). (22) G. Hall, K. Liu, M. J. McAuliffe, C. F. Giese, and W. R. Gentry, J . Chem. Phvs.. 81. 5571 (1984). (23) K. Lh,'G. Hall, hi. J. McAuliffe, C. F. Giese, and W. R. Gentry, J . Chem. Phys., 80, 3494 (1984). (24) G. Hall, C. F. Giese, and W. R. Gentry, J . Chem. Phys., 83, 5343 (1985). (25) R. E. Smalley, D. H. Levy, and L. Wharton, J. Chem. Phys., 64,3266 i ~1976) ~
~ , ~ , .
(26) W. Sharfin, K. E. Johnson, L. Wharton, and D. L. Levy, J . Chem. Phys., 71, 1292 (1979). (27) R. E. Smalley, L. Wharton, and D. H. Levy, Chem. Phys. Left.,51, 392 (1977). (28) K. E. Johnson, L. Wharton, and D. H. Levy, J . Chem. Phys., 69,2719 ( 1 978). (29) K. E. Kenny, K. E. Johnson, W. Sharfin, and D. H. Levy, J . Chem. Phys., 72, 1109 (1980). (30) G. Kubick, P. S. H. Fitch, L. Wharton, and D. H . Levy, J . Chem. Phys., 68, 4477 (1978). (31) K. E. Johnson, W. Sharfin, and D. H. Levy, J . Chem. Phys., 74, 163 (1981). (32) J. E. Kenny, T. D. Russell, and D. H. Levy, J . Chem. Phys.. 73, 3607 (1980). (33) K. E. Johnson and D. H . Levy, J . Chem. Phys., 74, 1506 (1981). ~
Feature Article the case of the helium c o m p l e x e ~ . ~ ~ * ~ ' The incorporation of polyatomic molecules in van der Waals complexes greatly enlarges the set of possible vibrational redistribution pathways accompanying photodissociation since, in general, there will be many vibrational levels lower in energy than, or within a span of kBTof, the initially prepared level of an excited complex. One of the first systematic studies of a van der Waals complex involving a chemically bound polyatomic molecule is that of the Ar-tetrazine complex by Brumbaugh, Kenny, and Levy." These workers recorded the dispersed fluorescence spectra resulting from excitation of six single vibronic levels of the complex, ranging in energy from the zero-point vibrational level to the 6a2level (==1400 cm-' above the origin of the excited electronic state.) In all cases the dominant feature in the dispersed spectrum is fluorescence from the vibronic level of the complex initially prepared by the laser. The remaining molecules undergo vibrational predissociation (fluorescence from bare tetrazine in a different vibrational state) or vibrational relaxation (fluorescence from Ar-tetrazine in a different vibrational state). In the latter case, the residual tetrazine vibrational energy has presumably been redistributed into the vibrational coordinates associated with the van der Waals bond without causing dissociation. From the branching ratios and the lifetimes of the tetrazine single vibronic levels involved, the rates of vibrational relaxation/predisscciationare found to be -lo8 s-', independent of the complex vibrational level initially excited. The Ar-tetrazine system has also been studied by Rettschnick and co-worker~.~~ Detailed studies of glyoxal van der Waals c o m p l e x e ~ ~have ~~~' been carried out by Halberstadt and Soep. The most extensive set of data is for the dissociation pathways of H,-glyoxal. The distribution of vibrational energy in the glyoxal fragment is found to be highly selective. These workers invoke a "spectator mode" model to interpret the relaxation dynamics of combination levels. According to this model the dissociation channels active for a combination level will be a superposition of the dissociation channels active for the constituent fundamental vibrations. For example, H2-glyoxal excited to the 8l level dissociates to yield vibrationless bare glyoxal with a branching ratio of unity, while complexes excited to 5' dissociate to yield primarily 7'. The spectator mode model predicts that the dominant dissociation pathways for the 5'8' level of the H2-glyoxal complex will populate the and 8'7' levels. This is the behavior observed. Halberstadt also reports the product vibrational-state distributions resulting from the dissociation of glyoxal complexed with deuterium, argon, neon, and helium.37 These data are, however, extremely limited in scope. A number of investigations of the dissociation dynamics of ethylene complexes have been camed These studies have utilized infrared lasers to prepare the complexes in an excited vibrational level of the ground electronic state and have focused (34) D. V. Brumbaugh, J. E. Kenny, and D. H. Levy, J. Chem. Phys., 78, 3415 (1983). (35) J. J. F. Ramachers, H. K. van Dijk, J. Laugelaar, and R. P. H. Rettschnick, Furaduy Discuss. Chem. Soc., 75, 183 (1983). (36) N. Halberstadt and B. Soep, Chem. Phys. Letr., 87, 109 (1982). (37) N. Halberstadt, Ph.D. Thesis, Universit€ de Paris-Sud, 1982. (38) M. Casassa, D. S. Bow, and K. F. Janda, J . Chem. Phys., 74,5044 (1981). (39) M. Casassa, D. S. Bomse, J. L. Beauchamp, and K. C. Janda, J . Chem. Phys., 72, 6805 (1980). (40) M. A. Hoffbauer, K. Liu, C. F. Giese, and W. R. Gentry, J . Chem. Phys., 78, 5567 (1983). (41) D. S . Bomse, J. B. Cross, and J. J. Valentini, J. Chem. Phys., 78,7175 11983). (42) M. Casassa, C. Western, F. G. Calli, D. E. Brinza, and K. Janda, J . Chem. Phys., 79, 3227 (1983). (43) M. A. Hoffbauer, K. Liu, C. F. Bilse, and W. R. Gentry, J. Phys. Chem., 87, 2096 (1983). (44) M. A. Hoffbauer, C. F. Giese, and W. R. Gentry, J. Chem. Phys., 79, 192 (1983). (45) M. A. Hoffbauer, C. F. Giese, and W. R. Gentry, J. Phys. Chem., 88, 181 (1984). (46) J. Geraedts, S. Setiadi, S. Stolte, and J. Reuss, Chem. Phys. Lett., 78, 277 (1981). (47) J. Geraedts, S.Stolte, and J. Reuss, Z . Phys. A , 304, 167 (1982). \ -
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3065 on determining the absorption line width, dissociation product angular distributions, and time-of-flight distributions. One of the most striking findings is that the photodissociation action spectrum line widths are insensitve to the excess energy of excitation (i.e., excess with respect to the van der Waals bond energy). Although direct evidence for the following conclusion is lacking, the available data strongly imply that the photodissociation line width is determined by vibrational relaxation within the van der Waals molecule via coupling of a vibrational level of the polyatom to those van der Waals molecule modes which are internal rotations (or libration^).^^ There is also a strong inference that selective state-to-state couplings, rather than scrambling of the energy, dominate the dynamics of the relaxation process. An extensive study of vibrational energy redistribution associated with dissociation of van der Waals molecules of He and lBzu benzene and Ar and lBZubenzene has been reported by Stephenson and Rice.@ Some of their data are displayed in Figure s5. The results of these studies, and another mentioned below, lead to two broad conclusions: (i) Vibrational energy redistribution associated with van der Waals molecule dissociation is very selective. (ii) Although transitions with Au = f l are often important, and they do dominate the energy transfer in the He12 system, there are many instances in atom-polyatom systems in which they do not dominate the branching into final states.
IV. Some Comparisons of Data Even though the data base is very limited, it is worthwhile making comparisons between the patterns of vibrational energy redistribution observed for different van der Waals molecule dissociations, as well as with the patterns of the corresponding collision-induced vibrational energy redistributions. The discussion that follows will focus attention on the behavior of those van der Waals molecules involving chemically bound molecules with vibrational coordinates similar to those of benzene, along with the vibrational relaxation induced in these molecules by collisions with rate gas atoms. As noted, Levy and co-workers have studied the excited-state dynamics of the Ar,-tetrazine complex.34 When this complex is excited to the 6a' level (702 cm-' of vibrational energy), the dominant relaxation process is vibrational predissociation to form 16a' tetrazine, with a net loss of 447 cm-' of vibrational energy.% Vibrational relaxation to the 16a2level of the complex also occurs, with about half the branching ratio of the predissociation process. In this case 192 cm-' of vibrational energy is redistributed into the van der Waals vibrational coordinates. These relaxation pathways resemble those observed for the Hel-benzene complex excited to the 6l level (521 cm-' of vibrational for which case the dominant relaxation channel involves population of the 162 level, with relaxation to the 16l level being of secondary importance.s1 When He2-benzene is excited to the 6' level, the contribution of the 16l channel increases relative to that of the 162channel. Given that the He,-complex relaxes primarily to the 16* level, it is reasonable that the 16l population increases when two helium atoms predissociate and/or provide additional degrees of freedom to absorb vibrational energy. We note that the 162level is triply degenerate, while the 16' level is only doubly degenerate, so that the dynamical discrimination between these levels is not as great as that indicated in the branching ratios. The fact that about 80-90% of the relaxation from the 6l level of (48) A. Mitchell, M. J. McAuliffe, C. F. Giese, and W. R. Gentry, J . Chem. Phys., 83,4271 (1985). (49) T. A. Stephenson and S.A. Rice, J . Chem. Phys., 81,1083 (1984). (50) s-Tetrazine photodissociates. In the text "relaxation process" is used in the sense of excluding photon emission and those nonradiative processes that are characteristic of the chemically bound molecule. The discussion considers only those vibrational relaxation or vibrational predissociation processes that pertain to the van der Waals molecule itself. (51) The observed emission was assigned, by Stephenson and Rice, to 11' and 16'. In light of the small energy gap between 11' and 6' (-4 cm-l) it is likely that only 16' is involved in the relaxation process. The experimental spectral resolution does not permit a definitive assignment in this case. (52) Parmenter and Tang consider the 1 1' channel to be populated via near-resonant energy transfer from 6'; see footnote 51.
Rice
3066 The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 He,-benzene and the 6aI level of Ar,-tetrazine populates the same set of levels is suggestive of an inherent similarity in the relaxation dynamics of these complexes. The reversal of the importance of the 16, (16aI) and 16* (16a2) relaxation channels likely reflects the competition between vibrational relaxation and predissociation in the tetrazine complex. Note that because the Arl-tetrazine binding energy is a t least 254 predissociation cannot populate the 16aZ level. N o information regarding the Helbenzene binding energy is available. If it is less than the 45-cm-l separation of the 16, and 6, levels, then vibrational predissociation alone may dominate the He-benzene relaxation dynamics. The similarities between the He,-benzene and Arl-tetrazine complex relaxation dynamics are less obvious when the 6l16, (6a' 16a2) levels are considered. The tetrazine complex relaxes primarily via vibrational predissociation to populate the 6b' level (loss of 856 cm-' of vibrational energy), while vibrational relaxation to the 6a116a' level of the complex (-255 cm-' of vibrational energy) occurs with an efficiency approximately 30% that of the predissociation. In contrast, 80%of the He, complexes lose 238 cm-' of vibrational energy in the benzene coordinates to populate the 6I16l level. Excitation of the He,-benzene complex appears to populate the 6l level. Note again that the 6a116a1level in tetrazine cannot be populated by vibrational predissociation, while the difference between the benzene 61162and 6l16' levels (238 cm-I) almost certainly exceeds the binding energy of the helium complexes. The only level common to the studies of the Ar-benzene and Ar-tetrazine complexes is 6, (6a'). In the former case there is no evidence for any vibrational relaxation/predissociation while the tetrazine complex does undergo these processes. Note, however, that the binding energy of the Ar-benzene complex is likely at least twice that of the Ar-tetrazine complex. Based on the comparison between the benzene and tetrazine complexes it seems, therefore, that energy mismatches can play an important role in determining the vibrational energy redistribution in van der Waals complexes. Bernstein, Law, and Schauer have reported a study of Heaniline c ~ m p l e x e s . ~They ~ find that when the 6al level of the complex is excited, the 16a1,00, and I' (NH, inversion mode) levels are populated. Note that, unlike the relaxation of 6l He-benzene, 16a2is not populated. Of course, the important aniline inversion mode has no analogue in the benzene vibrational spectrum. Bernstein and co-workers have also excited Heaniline complexes to the lobZlevel. Their dispersed fluorescence spectra indicate that the 16a1, lob1, 16a1,and Oo levels are populated. The corresponding Hebenzene relaxation is strikingly different: the 1O2 He,-benzene complex relaxes to the 6I1O1 level with a branching ratio of over 0.80. Parmenter and Tang have studied the vibrational energy transfer that occurs when helium atoms collide, at room temperature, with benzene excited to the 6' level.4 These workers find that the dominant energy-transfer process is the endothermic population population of the 6l16' level (branching ratio ~ 0 . 7 5 ) . Energy transfer to the 16l and/or 11' levels occurs with a branching ratio of = 0.20.52 Population of the 16,, 4l, and Oo levels accounts for only about 5% of the relaxation, at most. V. Theoretical Analyses: Some Calculations and Some Concepts
Consider, now, how the broad conclusions listed in sections I1 and I11 can be understood in terms of the properties of the intraand intermolecular potential energy surfaces and the dynamics thereon. It will be seen that only a few of these conclusions can be satisfactorily interpreted by using available theoretical analyses; support for the other conclusions is limited to qualitative arguments about expectations of the consequences of certain formal theoretical representations of the dynamical processes. A . Low-Energy Atom-Diatom Collision-Induced Vibrational Energv Redistribution. Consider, first, the case of very low energy (53) E. R. Bernstein, K. Law, and M. Schauer, J . Chem. Phys., 80, 207 (1984).
atom-diatom collisions, specifically collisions between He and Iz(3110u+), The purpose of the analysis is to determine whether or not the interpretation of the experimental data proposed by Tusa, Sulkes, and RiceI4 is consistent with the available information concerning the potential energy surface of the system. These investigators proposed that the cross section for vibrational relaxation is of the order of magnitude of the geometric cross section for a small energy range from zero to a few cm-' relative kinetic energy. It is then shown that for He-I, collisions Au = -1 transitions are dominant and that it requires n successive collisions to generate Au = -n transitions (see Figure S6). In contrast, for Ar-Iz collisions half the population corresponding to ik, = -2 is generated by a twoquantum transition and the other half from successive singlequantum transitions. Since a successful fit of the kinetic theory-hard sphere collision/relaxation cross section model to their data requires that only very low energy collisions are effective, Sulkes, Tusa, and Rice suggest that orbiting resonances contribute to the relaxation process observed. They also note that the particular collision-induced downward transitions monitored in the relaxation of 12(3~0u+) resemble those observed in the predissociation of excited HeI, and NeI,, both of which produce 12(3~0u+) with Au = -1, whereas in the predissociation of ArI, Au ranges up to Due to the complexity of quantum-mechanical inelastic scattering theory, only very simple systems can be analyzed without the use of severe approximations. The He-I, system, in the limit of very low collision energy, is a favorable case for detailed calculations. This system has been studied by Cerjan and Rice,s4 who represented the potential energy surface as a sum of atomatom Morse functions. The details of their close-coupling calculations can be found in ref 54. The variation of the calculated cross section with increasing translational energy is displayed, for the processes (24, j 25,0), (24, j 25, 2), (0, j 1, 0), and (0, j 1, 2), in Figure S7,A, B, C, and D, respectively. For the (24, j ' 25, j ) cross sections only zero total angular momentum was included in the total cross section sum, thereas J = 0, 1, 2 are included in the (0,j'- 1,j)results. These calculations show that the qualitative interpretation of the enhanced low-energy vibrational relaxation cross section proposed by Rice and coworkers is correct: the cross section is of the order of the geometric cross section near zero energy and becomes vanishingly small for energies greater than a few cm-I. Calculations of the rotational relaxation cross section, for the same potential surface, have also been carried These calculations, though, are not exact since a rotational infinite-order sudden approximation was used. That is, it was assumed that the basis set expansion for the entire wave function is restricted to one vibrational manifold. This approximation is suggested by the success of a similar analysis of the vibrational predissociation of He-Iz. With this restricted basis set expansion, total cross sections (summed over all contributing total angular momentum J) were obtained for several different sets of parameters for the atomatom Morse functions. Figure S8 presents the (25,O 25,j) processes for one parameter set. For comparison, the uncorrected data of Tusa, Sulkes, and Rices6 are also included. (It should be noted that these data are for the n = 28 state rather than the n = 25 state.) Overall, it is clear that the calculations provide qualitative support for the interpretation of the experimental observations. The Cerjan-Rice calculations confirm, for a potential surface thought to be a reasonable approximation for the He12(3~0,+) system, both that extremely low energy collisions can lead to very efficient vibrational relaxation and that the relaxation cross section drops to a very small value at relative kinetic energies greater than, say, 5 cm-I. However, close-coupling calculations are not easy to interpret in terms of concepts such as orbiting resonances. To check that part of the Sulkes-Tusa-Rice analysis of the experimental data, Gray and Rices7 have developed a Feshbach reso-3.28329
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(54) C. Cerjan and S. A. Rice, J . Chem. Phys., 78, 4952 (1983). (55) V. Sethuraman, C. Cerjan, and S.A. Rice, J . Phys. Chem., 87,2087 (1983). (56) J. Tusa, M. Sulkes, and S.A. Rice, Proc. Natl. Acad. Sci. U.S.A., 77, 2367 (1980).
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The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3067
nance theory of collision-induced relaxation which is similar in spirit to the theory of predissociation of van der Waals molecules by Beswick and J ~ r t n e r . ~They " show that the existence and decay of resonances corresponding to certain van der Waals molecule metastable states at low collision energy can, indeed, lead to substantial increases in the vibrational relaxation cross section. The idea that van der Waals metastable states can be responsible for certain scattering resonances is certainly not new. Indeed, an established theoretical technique for characterizing van der Waals metastable states is to carry out scattering calculations; see, for example, the work of Ashton, Child, and HutsonSgon the rotational predissociation of ArHCl and the work of Beswick and Jortne$* and most recently Delgado-Barrio et a1.60 on the vibrational predissociation of model T-shaped He12. The Gray-Rice theory differs from those cited in two ways. First, Gray and Rice are concerned with the converse problem of predicting the scattering behavior given the van der Waals molecule metastable-state information. Second, they treat general resonances and not just the closed product channel Feshbach resonances discussed in the above references. In a significant early contribution to the subject of the role of van der Waals complexes in collision-induced relaxation, Ewing61 demonstrated that the thermally averaged cross section can increase as temperature decreases if van der Waals complexes form. Ewing, in fact, estimated the van der Waals complex decay rates in a fashion similar to that of Beswick and J ~ r t n e r . ~Ewing's " approach, however, assumed the van der Waals complexes to be real chemical entities and included their formation (with the aid of a third body) and decay in a classical chemical kinetics fashion. This assumption is not necessary. Before it can be decided whether or not scattering resonances make an important contribution to very low energy collision-ind u d vibrational relaxation, it must be shown that the potential energy surface used supports said resonances. Gray and Rice have studied the states supported by four potential energy surfaces purported to be reasonable representations for the He 12(311 +) system; the results for J = 0 are shown in Figure S9. CleaAy, these surfaces do support resonances, but the positions are somewhat different for each surface. Gray and Rice have used their approximate theory of very low energy collision-induced vibrational relaxation to describe the inelastic processes on the potential surfaces whose resonances are displayed in Figure S9. Their results lead to several general conclusions: (i) The combination of very low collision energy and high initial vibrational excitation can lead to a large cross section for collision-induced vibrational relaxation. (ii) If the potential energy surface supports a low collision energy resonance, the rate of vibrational relaxation can be enhanced considerably above that characteristic of (i) above (see Figure S10). (iii) Small changes in the potential energy surface can lead to very different distributions of resonances and associated energies and widths. Because of the enhancement of the contribution to the cross section as that resonance is moved to lower collision energy, small differences in the distribution of resonances lead to very different weightings of the dynamical processes that determine the relaxation cross section. Recent experiments62 and related approximate scattering calculation^^^ for u' = 0 1, 2, 3 transitions in He 12(X'Z,) collisions have shown no clear evidence of resonance contributions to the cross section. However, Gray and Rice have shown that the importance of resonances shows up much more clearly when the diatomic target is more highly excited; for example, uu-l+ is 1/100 of 0,,z5..24. By the principle of microscopic reversibility,
+
-
+
(57) S. Gray and S.A. Rice, J. Chem. Phys., 83, 2818 (1985). (58) J. A. Beswick and J. Jortner, J. Chem. Phys., 69, 512 (1978). (59) C. J. Ashton, M. S.Child, and J. M. Hutson, J . Chem. Phys., 78, 4205 (1983). (60) G. Delgado-Barrio, P. Villareal, P. Mareca, and J. A. Beswick, Int. J. Quantum Chem., 27, 173 (1985). (61) G. Ewing, Chem. Phys., 29, 253 (1978). (62) G. Hall, K. Liu, M. J. McAuIiffe, C. F. Giese, and W. R. Gentry, J . Chem. Phys., 81, 5577 (1984). (63) D. W. Schwenke and D. G. Truhlar, J . Chem. Phys., 81, 5586 (1984).
a similar difference must characterize the excitation cross sections. Moreover, although the HeI2(IZ,) and He12(3n, +) potential surfaces are believed to be similar, the sensitivity of the dynamics to the nature of the potential surface makes it difficult to draw any inferences about the role of resonances on the lZ, surface given information about the 37r4+ surface. B. Atom-Polyatom Collision-Induced Vibrational Energy Redistribution. As demonstrated in the preceding section, the theory of very low energy atom-diatom collision-induced vibrational relaxation, and also rotational relaxation, reproduces the principal features of the available data. However, the sensitivity of the theoretical predictions to small changes in the potential energy surface of the system leads to considerable uncertainty oncerning their accuracy. Given the crudity of the available data, it is clear that further progress will depend on the development of refined techniques to study very low energy collisions and on improvement of the accuracy of the atom-diatom potential energy surface of interest. The theory of atom-polyatom collision-induced vibrational relaxation is very much less developed than that for atom-diatom collisions, in part because of the complications introduced by intramolecular coupling of vibrations and of vibrations and rotations and by the possible dependence of intramolecular mode mixing on the intermolecular force field. For although there exists an elegant formal theory of molecule-molecule scattering, knowledge of the relevant potential energy surfaces is meager and under many conditions of interest there are a vast number of open channels representing different distributions of translational, rotational, and vibrational energy in the final state of the system. Given these complications, it is not surprising that a fully satisfactory reduction of the scattering theory formalism to a useful algorithm has not yet been accomplished. Several of the analyses of collision-induced vibrational energy transfer,64for conditions such that the deBroglie wavelengths of the collision partners are small relative to the range of the potential, depend on the model introduced by Schwartz, Slawsky, and Herzfeld,I6 later modified by Tanczos and S t r e t t ~ n . Despite ~~ the gross oversimplifications inherent in this model, it yields some useful insights concerning the relative magnitudes of the cross sections for different relaxation pathways. The model proposed by McDonald and Rice,66which describes the influence of molecular symmetry on the energy-transfer process by projecting normal mode motions along a collision trajetory, assuming that only atom-atom interactions are important and sampling the configuration space of possible collision trajectories, provides some qualitative guidance to interpretation of the selectivity of the collision-induced vibrational relaxation. However, the crudities of the several analyses of this category of model t h e o r i e ~ ~ ~ , ~ ' seriously undermine confidence in the quality of the predictions. I will not discuss these theories further. Rolfe and Rice6" have reported a classical trajectory study of energy transfer in Ar-CF2N2 collisions. Although classical trajectory studies of collision-induced energy transfer have been carried out for a number of systems, the emphasis in the calculations is usually on the total vibrational and/or rotational energy cross sections;69in contrast, Rolfe and Rice focus attention on the selectivity of energy transfer in the collision. The conceptual design of the calculation is simple: Given the normal mode structure of the molecule (IB1 CF2 N2) and a multidimensional potential energy surface, and assuming classical mechanics is an adequate representation of the real molecular mechanics, Rolfe and Rice (64) C. S. Parmenter, J. Phys. Chem., 86, 1735 (1982). (65) F. I. Tanczos, J. Chem. Phys., 25,439 (1956); J. L.Stretton, Trans. Faraday Soc., 61, 1053 (1965). (66) D. B. McDonald and S.A. Rice, J . Chem. Phys., 74, 4918 (1981). (67) See also A. Miklauc and S.F. Fischer, Chem. Phys. Lett., 44, 209 (1976); J . Chem. Phys., 69, 281 (1978); A. Miklauc, Mol. Phys., 39, 855 (1980); J . Chem. Phys., 72, 3805 (1980). (68) T. J. Rolfe and S.A. Rice, J . Chem. Phys., 79, 4863 (1983). (69) See G. C. Schatz in Topics in Current Physcs, Vol. 25, J. M. Bourran, Ed., Springer-Verlag, West Berlin, 1983; W. R. Gentry in Atom-Molecule
Collision Theory. A Guide for Experimentalists, R. B. Bernstein, Ed., Academic, New York, 1979, Chapter 12.
3068 The Journal of Physical Chemistry, Vol. 90, No. 14, 1986
Rice
Cerjan, Lipkin, and Rice have implemented the DWBA-corattempt to track the translational energy to internal energy and relation function representation of atom-polyatomic molecule the normal mode to normal mode vibrational energy transfers. scattering for the case of very low energy collisions. Following They fiid (as have othed9) that classical mechanics is inadequate M i ~ h a , and ~ * transforming the collision-induced transition rate to a degree sufficient to influence the interpretation of normal to a time correlation function representation involving the scatmode to normal mode energy transfer; e.g., the zero-point energy tering operators for the given atom-atom interaction, they show of the molecule redistributes over the normal modes and also that, by expanding the timedependent terms in the rate expression couples to rotation. Perhaps better put, the normal mode motions to first order in the normal mode displacements of the molecule (and frequencies) which are so useful in the analysis of the and by retaining only the downward transitions, several general spectroscopy of a polyatomic molecule are not the proper coordinates for the description of atom-polyatom collisions. S ~ h a t z ~ ~ features of the collision-induced vibrational relaxation may be inferred. First, the distorting effect of the potential, and possible has shown how to define a better set of coordinates for the demultipolar couplings, are contained in time-independent terms scription of atom-polyatom collisions. Unfortunately, except for which also contain the molecular geometry. The magnitudes of very small (say triatomic) molecules, the calculation of these these factors will significantly affect the possible scattering coordinates is extremely difficult, and they have been little used. outcomes, controlling the amplitudes of the different potential Despite the difficulty mentioned, classical trajectory calculations contributions to the relaxation rate. Second, if each of the sucof collision-induced normal mode to normal mode vibrational cessively higher order processes (with respect to changes in normal energy transfer do lead to interesting conclusions, the principal mode quantum numbers) decreases in magnitude, corresponding ones being the following:6s (i) The collision-induced normal mode to an expected decrease in coupling as the separation of the states to normal mode energy transfer is very selective,just as is observed. increases, then the vibrational transitions to nearest-neighbor levels (ii) Despite thermal averaging, some idiosyncratic features of the will generally be favored. For the weak-interaction potentials potential energy surface are reflected in the functional dependence of the average energy transfer on impact parameter. (iii) The assumed to pertain to the very low collision energy process, this assumption is probably valid, as manifested by the observed branching pattern characteristic of collision-induced normal mode propensity rule for one vibrational quantum transfers. Third, if to normal mode energy transfer is likely rather sensitive to details special symmetry restrictions constrain the internal motion of the of the potential energy surface and not just to its overall properties. system, then it is to be expected that certain vibrational energy Clary and c o - w ~ r k e r have s ~ ~ developed an approximate-but transfer processes, which might otherwise be dominant, are not apparently rather accurate-theory of atom-polyatom collisions. allowed or greatly suppressed. The results published to date concern collision-induced excitation Although the correlation function formalism just sketched was of particular molecular normal modes under the assumption there is no coupling of the vibrational normal modes of the polyatomic designed to describe very low energy collision-induced vibrational relaxation, it can, with a few extensions, also describe arbitrary molecule. The cross sections for these several excitations are different enough to imply considerable selectivity in the collienergy collision-induced relaxation processes. However, it is necessary to recognize that, for the purpose of describing "higher sion-induced energy transfer. energy" collision-induced relaxation, the Cerjan-Lipkin-Rice Despite the qualitative successes of such detailed computational model has two unsatisfactory features. These are, first, that despite approaches, the results obtained to date are not suggestive of the simplifications introduced by use of the distorted wave Born general rules, say incorporating the molecular symmetry and approximation the analysis still requires the solution of a difficult particular mode-mode couplings, which permit prediction of the branching pattern for collision-induced energy transfer. Cerjan, scattering problem and, second, the form of the wave function makes it unlikely that any analytical predictions can be obtained. Lipkin, and Rice7'have attempted to isolate, from a formal theory, those general rules. Their starting point is the set of observations Gray and Rice73attempt to remove these unsatisfactory features v-vii of section 11. They have proposed a formalism based upon by developing a semiclassical version of the correlation function formalism. They show that, for the case that the atom-molecule an analogy with the analysis of neutron diffraction by molecules coupling is linear in the amplitudes of the molecule's normal as a replacement for the exact close-coupling formalism of scattering theory. It may seem that the theory of neutron difcoordinates, the correlation function, whose Fourier transform gives the energy-transfer spectrum, is a product of Bessel functions fraction can be immediately applied to describe very low energy (one for each normal mode).74 Furthermore, when the atomatom-polyatomic molecule scattering, since the atomic wave molecule potential is representable as a sum of atom-atom confunction is primarily influenced by the geometry of the molecule and does not modify the internal structure of the molecule. tributions, the correlation function factors into a product of Unfortunately, this is not the case. In neutron scattering, the atom-atom terms. The details of this analysis can be found in ref 13. neutron-nucleus interaction is of such short range that each nucleus is a distinct and isolated scattering center and the scattering Freed75has presented an interesting qualitative analysis which is weak enough to be accurately described in the Born approxisuggests that the patterns of collision-induced vibrational relaxation mation. For our purposes this means the initial and final state are mostly determined by intramolecular processes, in particular neutron wave functions are accurately described by plane waves. mode mixing in the isolated molecule. His argument starts with In atom-molecule scattering, the potential interaction is always the expansion of the isolated molecule rovibrational eigenstates strong enough to distort the wave function of the atom, invalidating in a convenient zero-order set of rovibrational states, say the the plane wave (Born) description. In order to retain the simplicity conventional rigid rotator-normal modes. The coefficients in this and physical content of the correlation function representation expansion can, but need not, be functions of time when an atom which describes neutron scattering, it is necessary to include the collides with the molecule. Freed then approximates the matrix effects of the atom-molecule interaction, and the most direct element connecting two vibrational states of the molecule in terms method for doing so is to introduce the distorted wave Born of diagonal matrix elements of the same potential in the basis set approximation (DWBA). Furthermore, in neutron scattering the of zero-order modes. It is argued that the effective potential is representation of the molecule-neutron interaction as a sum of long-ranged, thereby leading to a relaxation cross section for atom-neutron interactions (or better, nucleus-neutron interactions) vibrations comparable to the rotational relaxation cross section, is very accurate. The corresponding representation of the atomand that propensity rules are defined by the characteristics of the molecule interaction as a sum of atom-atom interactions may be coefficients that mix the zero-order modes. For these features a good or a poor approximation, depending on the system studied. (70) D. C. Clary, J . Am. Chem. SOC.,106, 970 (1984); Mol. Phys., 51, 1299 (1984). (71) C . Cerjan. M. Lipkin, and S. A. Rice, J. Chem. Phys., 78, 4949 (1983).
(72) D. A. Micha, J . Chem. Phys., 70, 5658 3165 (1979). (73) S. K. Gray and S.A. Rice, to be submitted for publication. (74) A result very similar to that obtained by Gray and Rice was found, in a different Context, by C. Cerjan, X. Shi, and W. H. Miller, J . Phys. Chem., 86, 2244 (1982). (75) K. F. Freed, Chem. Phys. Letl., 106, 1 (1984).
Feature Article
Figure 1. Model ground electronic state potential energy surface. This surface supports a stable triatomic, ABC, which may dissociate to A + BC (channel i) or AB + C (channel ii). From ref 79.
to be valid it is necessary that Coriolis effects dominate the mode mixing: it is not clear that such will be the case for all polyatomic molecules. Mixed states can be included in the Gray-Rice semiclassical correlation function theory of collision-induced vibrational relaxation, but the results are very much more complicated than in the absence of mode mixing. Moreover, a t least for the decomposition of the atom-molecule potential they consider, the effective potential does not appear to be long-ranged, thereby leaving unexplained the very large magnitude of the vibrational relaxation cross section. A pithy summary of the current situation is simple to state: the principal features of collision-induced vibrational relaxation, in particular the magnitude of the cross section and the selectivity of the relaxation pathways, are very poorly understood.
VI. Control of Selectivity of Chemical Reaction via Control of Wave Packet Evolution The preceding parts of this paper discuss briefly some of the evidence for selectivity in two elementary molecular processes, namely collision-induced vibrational relaxation and van der Waals molecule fragmentation. Consider, by extension, the following question: Is it possible, by control of the nature of the excitation process, to control the selectivity of a chemical reaction? Previous attempts to answer this question have focused attention on the free evolution of an excited molecule, treating the excitation and the evolution processes as separable. These analyses, and the various experiments carried out to date, lead to the conclusion that, because of rapid intramolecular vibrational redistribution, laser-induced selectivity of reaction is not, in general, feasible. Tannor and Rice76 show that, despite dephasing of a prepared vibrational-state distribution, selectivity of chemical reactivity can (76) D. Tannor and S.A. Rice, J . Chern. Phys., 83, 5013 (1985).
The Journal of Physical Chemistry, Vol. 90, No. 14, 1986 3069
Figure 2. Harmonic excited-state Born-Oppenheimer potential energy surface. The classical trajectory that originates at rest from the ground-state equilibrium geometry is shown superimposed.
be achieved by use of coherent two-photon (or multiphoton) processes. Originally, the Tannor-Rice procedure for controlling reaction selectivity was based on an inversion of the usual interpretation of two-photon spectroscopy. Although the theory has since been extended, it is pedagogically sound to examine the original ideas. In second-order perturbation theory, the probability of a twophoton transition between states & and & is proportional to the forms of the first and second electromagnetic field pulses, call them a(tJ and b(t2). In the conventional interpretation, a(tJ and b(t,) are thought of as “given”. The probability of transition determines the distribution of intensity observed in, e.g., an absorptionfluorescence or absorption-stimulated fluorescence experiment, the result being an accepted consequence of selection rules and molecular parameters. Tannor and Rice, instead, ask the following question: What are the wave forms and/or pulse sequences that optimize the probability of transition to a specified final state from a specified initial state? This question can be formulated as a variational problem for the functions a(tJ and b(t2). That formulation leads to equations that can be solved analytically in simple cases, e.g., stimulated fluorescence emission, but in general must be solved numerically. Tannor and Rice show that optimizing the transition probability between and 4r leads to
In (l), H is the Hamiltonian for propagation of the coupled molecule and radiation field, I&) = plxi) and I&) = pixf) where p is the transition dipole moment, and xi and Xr are the initial and final vibrational-state functions. The interpretation of (1) is straightforward: The stimulating wave form should be “matched” to the convolution of the excitation pulse shape with the wave packet as it evolves subject to the dynamics on the
3070 The Journal of Physical Chemistry, Vol. 90, No. 14. 1986
Rice
b
Figure 3. Classical trajectories on the ground-state surface that arise from a vertical transition down (coordinates and momenta unchanged) after propagation time t2 - 2 , on the excited-state potential energy surface. (a) t 2 - ti = 600 au. (b) t2 - t , = 2100 au.
excited-state potential surface. This result has analogues in electrical engineering and information theory, where it is part of matched filter theory.77 The preceding language refers to the case of transition between a specified initial state and a specified final state; we are interested in the case when transitions to several final states or a continuum of final states must be considered, as when only the chemical identities and not the internal levels of a set of products of a reaction define the result of interest. It is easier to understand the concepts involved by examining a simple example. Consider the hypothetical ground electronic state Born-Oppenheimer potential surface shown in Figure 1 . The surface has a central minimum and two different exit channels, each separated from the minimum by a saddle point. The minimum corresponds to a stable form for the reactant molecule ABC, whereas the exit channels correspond to formation of products via the reactions (i) ABC AB + C and (ii) ABC A BC. There are several ways, in general, to promote one or both of these reactions. We explore whether one can maximize the probability of a desired pathway, e.g. favoring ABC AB + C over ABC A BC. For the potential surface shown in Figure 1, there is a qualitative and quantitative distinction between the two exit channels. Tannor and Rice define projection operators ( P I and P2) that include the coordinate space associated with each exit channel and examine the two-photon transition probability between c#+ and the states of that exit channel. They then show that the optimization of this transition probability yields the following result
-
-
-
1300
1
+
-
+
3000
-1
N -J
w
z 2 i r
-
1000
where
2000
TI ME ( A U )
2000
3000
TIME (AU) Figure 4. (a) Probability of exit (0 or 1) from channel i as a function of propagation time on the excited-state potential energy surface. (b) Probability of exit (0 or 1) from channel ii as a function of propagation time on the excited-state potential energy surface.
(77) See, for example, R. M. Fano, Transmission ofInformation, MIT Press, Cambridge, MA, 1961.
It can now be shown that selectivity of reactivity is possible if an excited electronic state is used as an intermediary to “assist” chemistry on the ground electronic state surface. The Tannor-Rice scheme supposes that the excited-state surface that plays the role of an intermediary has both a displaced minimum and normal coordinates which are rotated relative to those of the ground-state surface. Consider, first, the classical trajectory that originates at the minimum of the ground-state surface (x = 0, y = 0, p x = 0, pv
The JourMI of Physical Chemistry, Vol. 90. No. 14. 1986 3071
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a
Fiyre 5. Ground-state wave function following propagation on the excited-state potential energy surface for 600 au. Times of propagation on the ground state surface are (a) 0 au. (b) 800 au, and (c) 1000 au. Note the agrccment with Figure 3a.
a
b
C
Ground-state wave function following propagation on the excited-state potential energy surface for 800 au. Times of propagation on the i d state surface are (a) 0 au, (b) loo0 au, and (c) 1200 au. The amplitude which does exist does so exclusively from channel ii. iwe 6.
= 0). and imagine this trajectory projected vertically onto the excited-state surface. The trajectory now propagates along x as a result of a large Franck-Condon displacement in this coordinate. After several vibrational periods, because the excited-statenormal coordinates are not parallel to those of the ground state, the trajectory begins to wind around the oscillate along y (Lissajous motion). If the trajectory is now projected vertically down, back to the ground-state surface, depending on when and where along the Lissajous motion on the excited-state surface the projection starts, the new trajectory on the ground-state surface may exit from either channel i or channel ii or remain trapped in the well. The quantum-mechanical theory of two-photon processes is amenable to an interpretation very similar to the classical mechanical description given above. Then the 'instant" of arrival of the first photon, tl, marks the vertical transition to the excited-state surface and the instant of emission of the second photon, t2,marks the vertical transition back to the ground-state surface, in conformity with the Franck-Condon principle. Quantum mechanically, the initial state, 6,.is a localized wave packet: the motion of the center of this wave packet, @ , ( I ) = e""/"q5,(0), is
the analogue of the classical trajectory discussed above. Under favorable conditions the wave packet remains narrow and will track its corresponding classical trajectory for many vibrational periods. This description has been advocated by Heller and cc-~orkers'~ in discussing a variety of molecular spectroscopies. However, the available analyses refer to the case of continuous wave light, Le., the use of one or two photons with &function frequency distributions. Our interest is in arbitrary wave forms and, in particular, in coherent pulse sequences that will enhance the fraction of a desired reaction product. The Tannor-Rice scheme posits a shift in outlook from passive to active. Indeed, we are no longer referring to spontaneous emission, but to stimulated emission with carefully tailored wave forms. One must distinguish clearly between the achievable control of wave packet propagation, phase of the wave packet, and production of the composite wave function. The wave packet (78) S. Y.Leeand E.I. Hellcr. I.Chem. Phys., 71,4777 (1979); 16,3035 (1982); E. I. Hcller, Am. Chem. Res., 14. 368 (1981); I.Chcm. Phyr., 68. 2066 (1978).
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propagation is only partially under experimental control. It is governed, first and foremost, by the forms of the ground- and excited-state potential surfaces; the experimenter can affect the wave packet propagation insofar as he/she selects the instant to change surfaces. Similarly, the phase of the wave packet is only partially under experimental control. There is a coordinate-dependent phase factor (which reflects the momentum of the wave packet) which is determined by the properties of the potential surfaces. However, there is also a coordinate-independent phase factor, which plays a vital role in determining the nodal patterns set up by several interfering wave packets and which can be manipulated via control of the phase of the pulse. Finally, the composite wave form produced may be thought of as a superposition of some number of individual wave packets. The extension of the composite wave function, its component segments, and the relative phases of these component segments is under experimental control, since these features are determined by the overall duration of the wave form of the light, the component pulse sequences, and the relative phases of the component pulses, respectively. It is important to note that if strong fields are used to implement the Tannor-Rice scheme, then the perturbation theory analysis must be extended. Tannor, Kosloff, and Rice79have generalized the theory of selectivity of reactivity sketched above, for the case of two electronic potential energy surfaces, to be valid for arbitrary field strength. In essence, using a matrix formulation of the problem, they obtain exact (although numerical) solutions to the SchrGdinger equation describing the strongly coupled matterradiation system for several cases with different potential energy surfaces, pulse widths, and separations, etc. The results of one set of calculations, and the potential energy surfaces used, are shown in Figures 1-6; the reader is referred to ref 79 for details of the calculations. Consider the ground-state potential energy surface shown in Figure 1 and the excited-state potential energy surface shown in Figure 2. We also show in Figure 2 a classical trajectory on the excited-state surface. If that classical trajectory on the excited surface is terminated after propagating, say 600 au in time, and projected to the ground-state surface, the new trajectory is that shown in Figure 3a. Similarly, if the projection to the ground-state surface occurs after 2100 au of time of propagation on the excited-state surface, we find the result shown in Figure 3b. Clearly, there is preferential formation of one or the other product depending on the time delay between exciting and stimulating pulses. A time spectrum of the “windows” for forming one or the other product is shown in Figure 4a,b. We now use the classical windows for selective product formation to select a time delay in the full qauntum-mechanical calculations. Typical results are shown in Figures 5a-c and 6a-c. (79) D.Tannor, S. A. Rice, and R. Kosloff, J . G e m . Phys., in press.
Rice Clearly, the classically predicted selectivity of product formation is preserved in the quantum-mechanical description of selectivity of reactivity. The Tannor-Rice method of controlling the selectivity of a reaction depends on the coherence of the dephasing of the wave packet prepared on the excited-state potential energy surface. By virtue of its preparation, that wave packet can be tought of as a superposition of component eigenstates each of which has a nonvanishing transition dipole matrix element with the initial state on the ground-state surface. Of course, each of the eigenstate components of the wave packet evolves deterministically, under an evolution operator defined by its energy, so at any time following absorption of the first photon the relative phases of the components of the wave packet are well-defined and known. Then optimization of the second pulse wave form achieves the best stimulated emission from all of the components of the prepared wave packet; Le., the effects of intramolecular vibrational redistribution on the excited-state potential surface are used to best advantage. The time scale available for selectivity using the Tannor-Rice scheme is of the order of a vibrational period, or perhaps a few periods. For light-atom systems this implies femtosecond duration pulses, while for heavy-atom systems the time scale might stretch to 100 fs. The shortest pulses reported are -8 fs, but it is reasonable to expect improvement in technology will make even shorter pulses available. The importance of the Tannor-Rice scheme for controlling selectivity of reactivity is that it defines a theoretical paradigm, free of arbitrary assumptions concerning localization of energy and the chemical activation thereby induced. Rather, the scheme employs the features of the potential energy surface and accounts for all effects of anharmonicity, etc. The sensitivity of the selectivity to details of the potential energy surface is currently being investigated.80
Acknowledgment. None of the work described in this paper could have been carried out except for the contributions of the gifted students and postdoctorals who have worked with me; the many individuals involved are identified in the text and the references. The program of research described has been supported by grants from the NSF and the AFOSR. Supplementary Material Available: Diagrams of energy pathways of vibrational excitation and relaxation, plots of intensity ratio vs. distance downstream of nozzle and cross section vs. energy, and scattering resonances for four potential energy surfaces of the HeIz system (Figures S1-S10) (17 pages). Ordering information is given on any current masthead page. (80)D.Tannor, S. A. Rice, and R. Kosloff, work in progress.
