摘要:
Resonances in bimolecular reactions Kopin Liu,*a Rex T. Skodjeb and David E. Manolopoulosc aInstitute of Atomic and Molecular Sciences (IAMS), Academia Sinica, Taipei, Taiwan, R.O.C., 106. E-mail: kpliu@gate.sinica.edu.tw bDepartment of Chemistry, University of Colorado, Boulder, CO 80309, USA cPhysical and Theoretical Chemistry Laboratory, South Parks Road, Oxford, UK OX1 3QZ Received 19th November 2001, Accepted 4th January 2002 Published on the Web 18th January 2002 In this Perspective we briefly review our recent studies which prove unequivocally the existence of a quantum dynamical resonance in the F z HD A HF z D reaction. The signatures of the resonance in the integral and differential cross sections of this reaction are elucidated. The interplay between experiment and theory is crucial in establishing the existence of a resonance in a bimolecular reaction and in revealing its physical characteristics. 1 Introduction The F z H2 A HF z H reaction is one of the most studied chemical reactions in science, and interest in this reaction dates back to the discovery of the chemical laser.1 In the early 1970s, a collinear quantum scattering treatment of the reaction predicted the existence of isolated resonances.2 Subsequent theoretical investigations, using various dynamical approximations on several different potential energy surfaces (PESs), essentially all confirmed this prediction.The term ‘‘resonance’’ in this context refers to a transient metastable species produced as the reaction occurs. Transient intermediates are well known in many kinds of atomic and molecular processes, as well as in nuclear and particle physics.What makes reactive resonances unique is that they are not necessarily associated with trapping in potential energy wells, but can be localized near the repulsive barrier (or transition state) region of the Born–Oppenheimer PES. There have been numerous attempts to observe this intriguing phenomenon, beginning with the groundbreaking molecular beam studies of Lee and coworkers in the mid 1980s.3–5 More recently, Neumark and coworkers have spectroscopically probed the dynamics of the transition state of the F z H2 reaction through photodetachment of the FH22 anion.6 Unfortunately, the experimental search for quasibound or trapped states in chemical reactions has shown that they are particularly elusive to direct observation.Because of this, several of the reported ‘‘sightings’’ of dynamical resonances in the F z H2 system have since been shown theoretically to be ambiguous.7–9 The situation is somewhat better for the heavy– light–heavy I z HI reaction, since a zero electron kinetic energy (ZEKE) photodetachment experiment on the IHI2 anion10 has provided unambiguous evidence for the existence of symmetric stretch resonances associated with the I z HI transition state region.11,12 However, the effect of these resonances on the I z HI reaction dynamics has not yet been ascertained in a full collision experiment and it seems unlikely, given the large number of partial waves that contribute to this reaction, that these resonances will survive the effect of angular momentum averaging.In this Perspective, we present a pedagogical account of our combined molecular beam experiments and theoretical investigations which prove for the first time the existence of a quantum dynamical resonance in the F z HD reaction.13–15 Particular emphasis is placed on the fingerprints of the resonance in experimental observables. The choice of the F z HD reaction for this study is based on the anticipated resonance hierarchy:4 (F z HD A HF z D) w (F z H HF z H) w (F z D2 A DF z D) w (F z HD A DF z H). In other words, resonance effects are expected to be most (least) pronounced for the HFzD (DFzH) product channel in the very same F z HD reaction, because these product channels are kinematically the most (least) constrained.4 The interrogation of both isotopic product channels therefore provides a convenient internal ‘‘calibration’’ in the search for the experimental signatures of a resonance.2 Signatures of and evidence for a resonance DOI: 10.1039/b110570a PhysChemComm , 2002, 5(4), 27-33 27 2.1 Excitation functions s(E Fig. 1 shows the experimental F z HD excitation functions, c), into both isotopic product channels, along with the predictions of a quasiclassical trajectory (QCT) simulation,16 a quantum mechanical (QM) scattering calculation,13 and a resonance model.13 Quite apparent in the FzHDAHFzD excitation function is a distinct step near Ec ~ 20 meV.This feature is not reproduced by the QCT calculation, and hence has a quantum mechanical origin. Indeed, theoretical considerations indicate that the classical barrier to reaction lies around Ec ~ 45 meV, which situates this feature in the tunneling energy regime. A much more gradual increase in the experimental excitation function is observed for Ec ¢ 45 meV, indicative of the onset of direct, over-the-barrier reaction. By contrast, there is no analogous step-like feature in the FzHD A DF z H excitation function, which instead is apparently dominated by direct reaction. The QM excitation functions shown in Fig. 1 were computed using quantum scattering theory on the now standard SW (Stark and Werner) PES.17 The prediction for the DF z H channel is seen to perfectly reproduce the experiment.In the HF z D channel, the step-like feature is also well reproduced in position and shape, but is too high by about a factor of two. The good predictions of the thresholds in both product channels, and also in the F z H2 and F z D2 reactions,15 are evidence that the barrier height on the SW PES is fairly accurate. We suspect that the error in the computed height of the step feature may therefore be due to too small a barrier width on the SW PES which gives rise to excess tunneling and, hence, to too large an excitation function. This journal is # The Royal Society of Chemistry 2002 Perspective2 AFig. 1 Normalized excitation functions of the two isotopic product channels in the F zHD(v~ 0; 80% j~0z20% j~1) reaction.The black dots are the experimental results, which have been normalized to the QM curves using a single scaling factor for both channels (modified from ref. 13.) To understand the physical origin of the step-like feature in s(Ec), we first performed a partial wave analysis of the time-independent QM results. In this approach, the fundamental quantity is the quantum scattering matrix element SJv0j0m0;vjm(Ec), where J is the total angular momentum quantum number of the triatomic system and Ec is the collision energy. The indices vjm and v’j’m’ denote the complete state specifications of the reactant and product molecules. The state-to-state integral cross section is obtained as (1) c J sv’j’m’/vjm(Ec)~ pB2 X(2Jz1)SvJ’j’m’,vjm(Ec) 2, 2E where m is the reduced mass of the reactants.As shown in Fig. 2, the contribution to the total F z HD A HF z D reactive cross section from each partial wave (i.e., each J value) exhibits a clear Lorentzian-like peak followed by a threshold. The position and width of the Lorentzian peak are plotted as a function of J in the lower panel of Fig. 2. While the peak position scales linearly with J(J z 1), the peak width varies with J in a more complicated manner. It can also be seen from the upper panel of Fig. 2 that the step-like feature in the excitation function s(Ec) arises from an overlap of the Lorentzian peaks in several consecutive partial waves.The remaining question is the physical origin of the Lorentzian peaks in each partial wave. Using the timedependent wavepacket method and the spectral quantization method detailed elsewhere,18 we were able to identify a quasibound resonance state at Ec ~ 22 meV for total angular momentum J ~ 0, precisely the location of the peak observed in the J ~ 0 QM scattering calculation. The average geometry 28 PhysChemComm , 2002, 5(4), 27-33 Fig. 2 Upper panel: computed partial cross sections as a function of the collision energy Ec (adapted from ref. 13.) Lower panel: J-dependence of the characteristics of the Lorentzian-like resonance peak. of this resonant state was found to be linear, consistent with linear dependence of the Lorentzian peak position on J(J z 1) in Fig.2. The fact that the peak width increases with J in a more complicated manner can be attributed to the role of overall rotation in the resonance decay mechanism.13 Based on the Lorentzian parameters shown in Fig. 2, and using a Breit–Wigner form for the S-matrix near a resonance, eqn. (1) can be used to calculate the total resonance contribution to the excitation function.13 The result, which is shown as a dotted line in Fig. 1, clearly confirms that the steplike feature is, in fact, a resonance signature in the integral cross section. The difference between the total cross section and the resonant component, shown as a dashed line in Fig. 1, can be viewed as the contribution from direct (non-resonant) reaction and shows a more typical threshold behavior.2.2 Product vibrational branching ratios 1/2). At low energies, Fig. 3 presents the collision energy dependence of the experimental product vibrational branching ratios in the HF(v’) z D arrangement channel. Also indicated as the shaded area in the upper panel is the small contribution from the spin-orbit excited reagent, F*(2P Ec v 45 meV, most of the reactive flux (w90%) is channeled into HF(v’ ~ 2). However, as soon as the v’ ~ 3 channel becomes energetically accessible, its contribution rises sharply with a concurrent drop in v’ ~ 2. At even higher energies, the production of v’~3 declines gradually whereas v’~2 stays more-or-less constant. The v’ ~ 0 and v’ ~ 1 populations both increase slowly with increasing collision energy.An analogous plot (not shown here, see ref. 15) for the DF(v’) z H arrangement channel shows slowly varying populations in all the final vibrational states as a function of Ec. We speculate that the unusual vibrational branching behavior seen in Fig. 3 could be another ramification of theFig. 3 Collision energy dependence of the HF product vibrational branching ratio (upper panel) and the vibration-specific excitation function for F z HD A HF(v’) z D (lower panel). The arrow marks the threshold for the production of HF(v’ ~ 3) z D (adapted from ref. 15.) resonance in the integral cross section. From the discussion above, we know that the low energy F z HD A HF z D reaction is almost entirely governed by resonant scattering.As will be shown below, the resonance state involved can be identified as (003), with three quanta in the HF stretch in the local mode picture. Thus the dominance of the HF(v’ ~ 2) product at low energies follows the usual Dv propensity rule in a vibrational predissociation process, i.e., the larger the value of Dv ~ vi 2 vf, the less probable the dissociation.19 We will loosely term a decay process with Dv w 0 a Feshbach-resonance decay.20 At higher energies the decay of the resonance state into HF(v’~3) via a shape resonance decay process20,21 becomes energetically feasible. Hence, there is a competition between the Feshbach and shape resonance decay mechanisms, which may explain the abrupt rise in the v’ ~ 3 product around the energy where HF(v’ ~ 3) production first becomes feasible.At higher collision energies the resonance will be associated with higher partial waves and hence the centrifugal barrier for shape resonance decay will increase, thereby lowering the probability of this decay mechanism. This could therefore be the origin of the gradual decline in HF(v’ ~ 3) production after 90 meV, which results in a prominent peak in both the state-specific excitation function and the collision energy dependence of the vibrational branching into Fig. 4 Comparison of experimental and theoretical FzHD(v~0,j~ 0 and 1)AHFzD differential cross sections in the low energy regime (adapted from ref. 14.) HF(v’ ~ 3). However, there is of course an additional direct scattering component for Ec w 45 meV, and its interference with the resonant scattering may also play a role.2.3 Product angular distributions Fig. 4 shows the total experimental F z HD A HF z D reaction angular distribution s(h,Ec), obtained by summing over all the HF(v’,j’) product states, in the vicinity of the 20 meV step feature in the excitation function. The angular distribution starts as backward-peaked near threshold. With increasing collision energy, a dramatic and systematic shift toward the sideways direction is observed. As a result, a ridge is clearly visible in the angle-energy plot. Similar behavior is seen in the QM simulation, as shown in the lower panel of the figure. Based on theoretical considerations for the D z H2 reaction, Miller and Zhang22 predicted such a ridge to be the signature of a resonance.However, Aoiz et al.23 subsequently showed that a PhysChemComm , 2002, 5(4), 27-33 29Fig. 5 Experimental total differential cross sections of the two isotopic product channels over an extended range of collision energies (adapted from ref. 14.) similar ridge was also seen in a QCT simulation and thus was not conclusive evidence for a resonance. In any event, the striking variation of the product angular distribution over such narrow energy range is unprecedented in any previous molecular beam scattering experiment, and it is entirely consistent with what one would expect from a resonant mechanism as discussed in more detail in the Appendix.In Fig. 5 the experimental differential cross section s(h,Ec) is displayed over an extended energy range for both isotopic product channels. An oscillatory forward–backward peaking in the HF z D distribution is quite apparent. These peaks actually show a rapid exchange of flux as a function of collision energy. In the DF z H channel, the distribution is smoother and localized in a broad swathe in the backward hemisphere. An analogous swathe is also observed in the HF z D channel. From the excitation function results, we know that the DF z H channel is dominated by direct reaction. In terms of the differential cross section, it is therefore reasonable to conclude that the broad swathe is characteristic of direct reaction.If this 30 PhysChemComm , 2002, 5(4), 27-33 Fig. 6 Computed differential cross sections for F z HD A HF z D, over the same extended energy range as in Fig. 5. The upper panel is the result of theQMcalculation, while the lower panel shows the resonance contribution (adapted from ref. 14.) conclusion is extended to the HF z D channel, the swathe in this channel is also direct. It is then conjectured that the remaining ridge structure, at low energy, and the highly oscillatory forward–backward peaking, at higher energy, are the fingerprints of the resonant scattering. Fig. 6 shows the QM simulation of the differential cross section (DCS) in the HF z D channel (the upper panel), over the same extended energy range as in Fig. 5. The agreement with experiment is seen to be qualitatively reasonable.The forward–backward peaking and direct reaction swathe observed in the experiment also occur in the QM calculation, although the relative magnitudes are not consistent. Thus fully quantitative agreement between QM calculations and experiment in all of the reaction attributes must await further refinements of the PES, and/or a more rigorous treatment of the open-shell character of the F(2P) atom.24 An analysis of the resonance contribution to the differential cross section requires a more detailed calculation than forthe integral cross section since the partial wave contributions add as amplitudes (i.e., with phases) in this case rather than as probabilities. In the QM calculation, the DCS is expressed as c sv’j’m’/vjm(h,Ec)~ B2 X(2Jz1)dmJ ’m(p{h)SvJ’j’m’,vjm(Ec) , (2) 2 8E J where dJm0m(p 2 h) is a Wigner rotation matrix element and h is the angle between the initial relative velocity vector of the F atom and the final relative velocity vector of the HF (or DF) molecule.The resonance contribution to the DCS can be estimated from eqn. (2) by fitting each S-matrix element to a Breit– Wigner pole plus background.14 The pole contribution is then used as the resonance component. The results are shown in the lower panel of Fig. 6. As is seen, the resonance ridge prevails at low energies. At higher energies, the resonance contribution shows a highly peaked forward–backward distribution.The direct scattering swathe disappears completely. This confirms the experimental conjecture that the low-energy ridge and the highly oscillatory forward–backward peaking at higher energies are the signatures of a quantum mechanical resonance in the differential cross section. 3 Nature of the resonance We now know how a resonance manifests itself in both the integral and the differential cross sections of the F z HD A HF z D reaction. But what is the nature of this resonance? Perhaps the most illuminating way to reveal it is to examine its probability density, i.e., the square modulus of the resonance wavefunction, |y|2. This was achieved using a time-dependent wavepacket approach from which the time-independent resonance wavefunction was extracted through a Fourier transform of a variationally optimized wavepacket.13 Fig.7 shows the result. In Fig. 7(a) the FHD collinear subspace is shown using the coordinates R,r, where R is the distance between the F atom and the center-of-mass of the HD molecule, and r is the distance between the H and D atoms. The dotted lines are the PES contours and the solid lines are contours of |y|2. In Fig. 7(b), the probability density is sliced at 0 (a0 ~ bohr) and is shown as a function of R and c, r ~ 2 a where c is the angle between the vectors R and r. The plot clearly shows a quantum state with 3 nodes along the asymmetric stretch and no nodes in either the symmetric stretch or the bend. Thus, the resonance is assigned as a (003) state.This quasibound state is localized in the strong interaction region and is dynamically trapped in a well on the vibrationally adiabatic potential surface.25,26 Quantum dynamics studies yield a lifetime of about 110 fs for J ~ 0. In simple terms, the reaction is envisioned to proceed as follows. Initially, the two reagents approach each other preferentially along the bent geometry of the reaction coordinate. The internal rotation of the HD moiety then induces tunneling through the reaction barrier via a resonant tunneling mechanism to form the trapped resonance state. The system then vibrates 5–10 times in stretching motions before decaying into the reaction products, HF z D. 4 Conclusions In summary, our recent studies of the F z HD reaction have provided the first unequivocal evidence for a reactive resonance in a molecular beam scattering experiment on a bimolecular chemical reaction.The central questions concerning any resonance phenomenon are the nature of the trapping mechanism, the mode of c ~ Fig. 7 The probability density of the resonance wavefunction at E 22 meV (red contours), superimposed on a potential energy contour diagram (blue lines). The ‘‘X’’ indicates the location of the barrier in linear geometry to reaction (adapted from ref. 13.) the internal excitation or the quantum state assignment of the metastable species, and the mechanism of resonance decay. It does not seem possible in this field to answer these questions purely experimentally, but it is possible to do so by comparison with a theoretical simulation.Provided the features seen in the experiment are at least qualitatively reproduced by the simulation, as they are in the case of FzHD, one can ‘unravel’ the simulation to find the origin of the experimental observations. The acid test for identifying a quantum mechanical resonance is whether or not one can tie the observations to a metastable quantum state that is genuinely trapped (and therefore quantized) in all of the available degrees of freedom. In the case of the FzHD reaction we have shown that the observed features in the differential and integral cross sections can indeed be tied to such a quantum state, and we have provided a picture of it in Fig.7. The observable consequences of the resonance are then intimately associated with how this metastable state forms and decays, as we have discussed in the previous section. As far as these consequences are concerned, several clear resonance fingerprints have been identified and documented here, despite the impact parameter and quantum state averaging which tends to obscure the Lorentzian profile of a resonance from experimental observables. These fingerprints should provide useful guidance for future studies of resonance phenomena in other systems. From an experimental point of view, a resonance manifests itself most clearly in the evolution of reaction attributes with collision energy. The appearance of the collision energy as at least one of the variables in –6 underlines this and makes it clear that studying the collision energy dependence of the reaction attributes is absolutely vital for a convincing identification of a resonance in a bimolecular chemical reaction.This message should be loud and clear. PhysChemComm , 2002, 5(4), 27-33 312 (A1) (2Jz1)P : J ( cos½p{h)SJv’j’,vj (Ec) (A2) 2 p(Jz1=2) sin h |cos½(Jz1=2)(p{h){p=4, Acknowledgement This work was supported by grants from the National Science Council of Taiwan (KL), the National Science Foundation (RTS), and the European Union (DEM). c J~0 X ? Legendre polynomial,27 PJ ( cos½p{h)^ v’j’,vj(Ec): c{E0{BJ(Jz1)ziC=2 : The restriction to m’ ~ m ~ 0 in eqn. (A1) is justified under the conditions of Fig.4 because 80% of the HD molecules are in j ~ 0 and the dominant reaction probabilities at low collision energies are to product channels with m’ ~ 0. The neglect of any background contribution to the S-matrix element in eqn. (A3) is also justified, since at collision energies c v 45 meV the background contribution to the reactive scattering is negligible. However, the assumption that the resonance width C in eqn. (A3) is independent of J is clearly an oversimplification (see Fig. 2), as is the neglect of any J-dependence of the resonance amplitude Av’j’,vj(Ec). Substituting eqn. (A2) and (A3) into eqn. (A1) gives the differential cross section as Av’j’,vj (Ec) f (h,Ec) , Ec v’j’,vj (h,Ec)~ B2 5 Appendix We shall now present a simple explanation for why the experimental F z HD A HF z D differential cross section evolves from backward-peaked below the J ~ 0 resonance energy at E0c22 meV to forward–backward symmetric above the resonance energy, as seen in Fig. 4 and 5.The model we shall use to do this is too crude to give a quantitative fit to the experiment (it does not contain enough parameters), but it does nevertheless provide some useful qualitative insight into what is going on. The model has three ingredients. The first is the formula for the differential cross section, eqn. (2), in the case where m’ ~ m ~ 0 and the rotation matrix element dJm0m(p 2 h) becomes a Legendre polynomial: s The second is the standard semiclassical approximation to the v’j’/vj (h,Ec)~ B2 8E Ef (h,Ec)~ 1=2 and the third is a single pole approximation to the S-matrix element SJ Av’j’,vj (Ec) (A3) (A4) (A5) |SJv’j’,vj (Ec)^E 1=2 s distribution is given by 1 p sin h J~0 X ? 2j j2 where the amplitude f(h,Ec) that determines the angular (Jz1=2)1=2 cos½(Jz1=2)(p{h){p=4: Ec{E0{BJ(Jz1)ziC=2 Now the fact that several values of J contribute to the reaction cross section at each value of the collision energy (see Fig.2) allows us to approximate the sum over J in eqn. (A5) with an 32 PhysChemComm , 2002, 5(4), 27-33 Fig. 8 Behavior of c(Ec) in the complex plane as the collision energy Ec sweeps through the J ~ 0 resonance energy E0.(See eqn. (A7) in the Appendix.) integral to obtain 1=2 f (h,E (A6) dx, c)^{1 1 0 � B sin h ? x1=2 cos½(p{h)x{p=4 c2(Ec){x2 | where x ~ J z 1/2 and (A7) �0 dx~p e{2Re½c(Ec)(p{h) (A9) : c2(Ec)~½E0{B=4{iC=2{Ec=B: The integrn eqn. (A6) can be evaluated exactly to give28 ? x1=2 cos½(p{h)x{p=4 2 c{1=2e{c(p{h), (A8) 2p c) c2zx2 and hence the final expression for the angular distribution is found to be f (h,Ec) ^B2 c(E j j2 j j sin h The behaviour of this angular distribution as the collision energy Ec increases through the resonance energy E0 can be deduced from the expression for c2(Ec) in eqn. (A7). As shown in Fig. 8, at collision energies well below the resonance energy the real part of c(Ec) is large and positive and hence the angular distribution is exponentially localized in the backward direction (h A p). However, as the collision energy increases through the resonance energy, Re[c(Ec)] decreases monotonically, and in the limit as (Ec 2 E0)/B A ‘ the real part of c(Ec) tends to zero.In this limit we therefore recover the classic forward–backward symmetric 1/sinh angular distribution that has been known for many years to be the signature of a longlived collision complex.29 The present analysis shows that this symmetric angular distribution is only to be expected at collision energies above the J ~ 0 resonance energy, in accordance with the experimental result in Fig. 4 and 5. References 1 P. H. Corneil and G.C. Pimentel, J. Chem. Phys., 1968, 49, 1379. 2 S.-F. Wu, B. R. Johnson and R. D. Levine, Mol. Phys., 1973, 25, 839. 3 D. M. Neumark, A. M. Wodtke, G. N. Robinson, C. C. Hayden and Y. T. Lee, J. Chem. Phys., 1985, 82, 3045. ƒƒ A groundbreaking (and, for 1985, state-of-the-art) molecular beam experiment on the F z H2 reaction. This paper is responsible more than any other for the intense interest in the F z H2 system in the last fifteen years.4 D. M. Neumark, A. M. Wodtke, G. N. Robinson, C. C. Hayden, K. Shobatake, R. K. Sparks, T. P. Schafer and Y. T. Lee, J. Chem. Phys., 1985, 82, 3067. ƒ As above, but for the FzD2 and F z HD isotopomers of the reaction. 5 Y. T. Lee, Science, 1987, 236, 793. 6 D. E. Manolopoulos, K. Stark, H.-J. Werner, D.W. Arnold, S. E. Bradforth and D. M. Neumark, Science, 1993, 262, 1852. ƒ A J. Chem. Phys., 1996, 104, 6531. novel transition state spectroscopy experiment which provided the first experimental confirmation of a non-linear transition state for F z H2, and also the first indication as to the accuracy of the Stark–Werner PES (ref. 17). 7 F. J. Aoiz, L. Banares, V. J. Herrero, V. Sayez Rabanos, K. Stark and H.-J. Werner, Chem. Phys. Lett., 1994, 223, 215. 8 J. F. Castillo, D. E. Manolopoulos, K. Stark and H.-J. Werner, 9 C. L. Russell and D. E. Manolopoulos, Chem. Phys. Lett., 1996, 256, 465. 10 I. M. Waller, T. N. Kitsopoulos and D. M. Neumark, J. Phys. Chem., 1990, 94, 2240. ƒ The ZEKE photoelectron spectrum of IHI2 is shown to provide unambiguous evidence for the I z HI reactive scattering resonances predicted in refs.11 and 12. As far as we are aware, this is the only previous example in which conclusive experimental evidence has been provided for resonances in a bimolecular reaction. 11 G. C. Schatz, J. Chem. Phys., 1989, 90, 4847. 12 B. Gazdy and J. M. Bowman, J. Chem. Phys., 1989, 91, 4615. 13 R. T. Skodje, D. Skouteris, D. E. Manolopoulos, S.-H. Lee, F. Dong and K. Liu, J. Chem. Phys., 2000, 112, 4536. ƒ The first conclusive experimental evidence for a transition state resonance in the integral cross section of the F z HD A HF z D reaction (15 years on from ref. 4). 14 R. T. Skodje, D. Skouteris, D. E. Manolopoulos, S.-H. Lee, F. Dong and K. Liu, Phys.Rev. Lett., 2000, 85, 1206. ƒ As above, but showing the signatures of the resonance in the differential cross section. 15 F. Dong, S. H. Lee and K. Liu, J. Chem. Phys., 2000, 113, 3633. ƒ And in the product vibrational distribution. 16 F. J. Aoiz, L. Banares, V. J. Herrero, V. Saez Rabanos, K. Stark, I. Tanarro and H. J. Werner, Chem. Phys. Lett., 1996, 262, 175. 17 K. Stark and H.-J. Werner, J. Chem. Phys., 1996, 104, 6515. ƒ A fully ab initio potential energy surface for the F z H2 system with the correct topology in the reactant valley, the product valley, and the transition state region (see also the comments in ref. 6). 18 (a) R. SadeghiR. T. Skodje, J. Chem. Phys., 1993, 99, 5126; (b) J. Chem. Phys., 1995, 102, 193; (c) J. Chem. Phys., 1996, 105, 7504. 19 G.E. Ewing, J. Phys. Chem., 1987, 91, 4662. 20 M. S. Child, Molecular Collision Theory, Academic Press, New York, 1974, p. 53. 21 Strictly speaking, shape resonances are only properly defined for single-channel processes (see ref. 20). For multichannel scattering things are more complicated, as multichannel coupling can often play a role in the resonance decay mechanism (in addition to the single-channel tunnelling through an angular momentum barrier that characterizes shape resonance decay). Nevertheless, we have decided to use the term ‘‘shape resonance’’ to describe the decay of the present resonance into HF(v’ ~ 3), to contrast this with the ‘‘Feshbach resonance’’ situation in which the resonance state predissociates into HF(v’ ¡ 2). 22 W. H. Miller and J. Z. H. Zhang, J. Phys. Chem., 1991, 95, 12. ƒ This paper contains an illuminating discussion of the effect of resonances on differential cross sections that is complementary to the analysis in the present Appendix.(See, however, the caveat in ref. 23 concerning the implications of the predicted ‘‘resonance ridge’’ for the D z H2 reaction). 23 F. J. Aoiz, V. J. Herrero and V. Sayez Rabanos, J. Chem. Phys., 1992, 97, 7423. 24 M. H. Alexander, D. E. Manolopoulos and H.-J. Werner, J. Chem. Phys., 2000, 113, 11084. 25 D. G. Truhlar and A. Kuppermann, J. Chem. Phys., 1972, 56, 2232. 26 J. M. Launay and M. LeDourneuf, J. Phys. B, 1982, 15, L455. 27 M. S. Child, Semiclassical Mechanics with Molecular Applications, Clarendon Press, Oxford, 1991, p. 33. 28 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th edn., Academic Press, San Diego, 1994, p. 423. 29 See, e.g.: R. D. Levine and R. B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity, Oxford University Press, New York, 1987, p. 413. PhysChemComm , 2002, 5(4), 27-33 33
ISSN:1460-2733
DOI:10.1039/b110570a
出版商:RSC
年代:2002
数据来源: RSC