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Faraday Discuss. 1998 110 1»21 Spiers Memorial Lecture Quantum and semiclassical theory of chemical reaction rates William H. Miller Department of Chemistry University of California Berkeley and Chemical Sciences Division L awrence Berkeley National L aboratory Berkeley California 94720-1460 USA Transition state theory (TST) has provided the qualitative picture of chemical reaction rates for over sixty years. Recent theoretical developments however have made it possible to calculate rate constants fully quantum mechanically and efficiently at least for small molecular systems; vestiges of TST can be seen both in the resulting —ux correlation functions and in the algorithmic structure of the methodology itself. One approach for dealing with more complex molecular systems is the semiclassical (SC) initial value representation (IVR) which is essentially a way of generalizing classical molecular dynamics simulations to include quantum interference ; electronic degrees of freedom in an electronically non-adiabatic process can also be included on a dynamically equivalent basis.Application of the SC-IVR to models of unimolecular isomerization and of electronically non-adiabatic transitions both coupled to an in–nite bath of harmonic oscillators gives excellent agreement with (essentially exact) quantum path integral calculations for these systems over the entire range of coupling strength. I Reaction rate theory Retrospective and present directions In mid-September of 1937 the 67th General Discussion1 of the Faraday Society was held at the University of Manchester on the subject Reaction Kinetics the purpose being ì .. . to clarify certain aspects of the reaction velocity problemœ. That clari–cation was needed is evidenced by the Presidentœs opening remarks ìAs to whether these methods are fundamentally sound or unsound is a question the consideration of which belongs rather to the domain of philosophy than to that of chemistry . . . œ ! ìThese methodsœ of course refers to the theory of reaction rates (4ì velocities œ) that had evolved over the preceding –ve years or so variously called the transition state (TS) method activated complex theory or the theory of absolute reaction rates and now usually referred to as ì transition state theoryœ (TST). The founding Trinity of TST» M.Polanyi Eyring and Wigner»were there as were most of the other notables in the –eld of kinetics. The excitement even passion of the moment still comes through on reading the discussion comments. One participant for example felt that ì . . . arguments used by Evans and Polanyi . . . seem to us particularly dangerous because their logical signi–cance is concealed by the method of presentation,œ and of Eyringœs paper that ì . . . I do not believe the treatment described can be of any use,œ while another more experimentally inclined participant felt the entire matter of dubious merit since ì . . . it would take much longer to estimate the partition function of the cluster (if at all possible) than to –nd new and enlightening experimental information on the mechanism of reactionœ.Eyring opened the Discussion with a paper2 on calculating potential energy surfaces (PESs)»just as our Discussion begins»and their centrality was especially clear to Wigner who noted that ì . . . every calculation of reaction rates is bound to founder 1 2 Spiers Memorial L ecture unless we have more detailed energy surfaces than are available at present . . . œ. Some participants were pessimistic on this issue one noting that ìNot even for the simplest bimolecular process is it possible to calculate the interaction between two molecules with an accuracy adequate to give even a useful approximation to the energy of activationœ but Polanyi had a more philosophical attitude ìPersonally I attach no importance at the present stage to a precise numerical agreement between theory and experiment .. . œ being content with the qualitative insight TST provided. Eyringœs second paper,3 and one by Evans,4 dealt with the thermodynamic version of TST and some applications but the most perceptive paper was Wignerœs.5 His ìthree threesœ (cf. Table 1) gave a clear outline of the whole subject. First were the three ìsteps œ in the theory of kinetics (1) constructing the PESs (2) calculating the rates of the elementary reactions and (3) combining the many elementary reactions into a complex reaction mechanism. Next were the three ìgroupsœ of elementary reactions (1) vibrationally/rotationally inelastic collisions (2) reactive collisions on a single PES and (3) electronically non-adiabatic (4ìdiabaticœ in Polanyiœs6 terminology) reactions involving several PESs.And –nally were the three ìassumptionsœ of TST [which he noted was only applicable to reactions of type (2) above] (1) no electronically non-adiabatic transitions (2) validity of classical mechanics for the nuclear motion (with perhaps small quantum corrections) and (3) the existence of a dividing surface separating reactants and products that no classical trajectory passes through more than once. This overall picture of reaction dynamics is still quite appropriate today. Wigner then discussed the three assumptions of TST in some detail particularly (2) and (3). He noted that it is ì . . . much more difficult to apply the transition state method in quantum theory than it is in classical theory .. . œ and concluded that ìAll that appears to be possible in general is to start out from the classical expression and develop the quantum corrections into a power series in + . . . œ. He noted that assumption (3) of no re-crossing trajectories (i.e. a transmission coefficient i\1) ì . . . will lead in general to too high values of the reaction rate . . . œ (from which follows the variational aspect of classical TST) though the assumption does in fact become exact at sufficiently low temperature. ìHowever at higher temperature i will decrease . . . [and in general] it may be quite difficult to –nd its accurate value.œ What has happened since 1937? First with regard to Wignerœs ì –rst stepœ in Table 1 one is –nally beginning to have reliable PESs for a wide variety of molecular systems.The increase in computational power7 (a factor of ca. 106) in only my ca. 30 year scienti- –c lifetime has been staggering and together with the methodological and algorithmic advances8 over this same period (maybe a factor of 102) the pessimism of 1937 is Table 1 Wignerœs ìthree threesœ of reaction kinetics Three steps in theory of kinetics (1) Determine potential energy surfaces (2) Determine elementary reaction rates* (3) Solve rate equations for complex reaction mechanism *Three groups of elementary reactions (1) Vibrationally/rotationally inelastic collisions (2) Reactive collisions on a single PES** (3) Electronically non-adiabatic reactive collisions **Three assumptions of TST (1) Electronic adiabaticity (2) Validity of classical mechanics (3) Existence of a dividing surface that trajectories do not re-cross 3 W .H. Miller abating. One is even beginning to think of being able routinely to evaluate the PES ìon the —yœ during a classical trajectory or quantum dynamics calculation. This will –nally eliminate the dreadful task of having to –t some complicated algebraic function to a small number of ab initio electronic structure calculations in order to obtain a global PES. Another major change since 1937 is that one can now study the dynamics of chemical reactions at a much greater level of detail than just via their thermal rate constant. Beginning in the late 1950s molecular beam (and later a host of laser-based) methods9 made it possible to determine the dependence of reaction cross-sections on collision energy scattering angle and initial/–nal quantum states of reactants/products and most recently ultrafast laser technology has made it possible to study chemical reactions in real time10 by various pump»probe schemes.These experimental advances in turn stimulated an explosion of theoretical development,11 starting in the early 1960s and continuing to the present as one tries to develop new models and theoretical methods to describe these new experimental phenomena. Indeed some of the most signi–cant advances in reactive scattering theory have taken place in only the last decade.12 Several contributions to this Discussion are excellent examples of the state-of-the-art in state-tostate reactive scattering calculations for small molecular systems.Reactive scattering methodology has also been adapted to the scattering of small molecules from solid surfaces and this area is also well represented at this meeting. Finally the original goal of theoretical reaction kinetics»the determination of the thermal rate constant which is the average of the reactive cross-section over energy scattering angle and initial/–nal quantum states»had a re-birth of interest and focus in the 1970s that has continued to the present. This is due in part to practical considerations»a realistic modeling of combustion and of atmospheric phenomena requires rate constants for thousands of elementary gas-phase reactions»and also due to the intellectual challenge i.e.is it possible even in principle to obtain the thermal rate constant absolutely correctly without –rst calculating the state-to-state reactive crosssections explicitly and then averaging them appropriately ? Also for reactions in complex environments (e.g. solution) it is still often the case that only the rate constant (or some other highly averaged quantity) is of relevance. A paper by Pechukas and McLaÜerty13 was particularly in—uential in this revival of interest in TST. Though dealing only with classical TST they emphasized the Wignerian dynamical perspective (i.e. no re-crossing trajectories) that had essentially been buried by the enormous number of applications using Eyringœs thermodynamic picture. The thermodynamic picture does of course lead to useful phenomenology (entropy of activation etc.) but it masks the dynamical understanding of TST that is necessary to extend and generalize the theory.In particular this re-focus on the dynamical perspective of TST stimulated the search14,15 for a rigorous quantum version of TST i.e. a way to implement the assumption of no re-crossing dynamics quantum mechanically without incorporating any further approximations (such as assuming separability of a reaction coordinate from the other degrees of freedom of the molecular system). The rationale was that since the TST assumption (of no re-crossing dynamics) is accurate at low energy which is the most important region for determining thermal rate constants but at low energies tunneling and other quantum eÜects become more important a quantum theory built on the TST assumption would be quite accurate for thermal rate constants (at least those with a simple activation barrier).Nature turned out to be not quite so simple. Though some very interesting and useful approaches have emerged from this quest for a quantum TST (QTST)»e.g. the instanton model15b (a periodic orbit in pure imaginary time that describes tunneling through the TS region of the PES) a semiclassical TST involving the conserved classical action variables at the TS,15f path integral centroid approaches,16,17 and a variety of other semiclassical models for including the eÜects of reaction path curvature on TS 4 Spiers Memorial L ecture tunneling probabilities18h20»the conclusion of it all is that there is no uniquely well de–ned quantum version of TST in the sense that there is in classical mechanics.This is because tunneling along the reaction coordinate necessarily requires one to solve the (quantum) dynamics for some –nite region about the TS dividing surface and if one does this fully quantum mechanically there is no ìtheoryœ left i.e. one has a full dimensional quantum dynamics treatment which is ipso facto exact a quantum simulation. The frustrated quest for a rigorous QTST was not for naught however for the theoretical formulation it engendered has led in recent years to a very useful procedure for calculating rate constants ì directlyœ i.e. without having to solve the state-to-state reactive scattering problem explicitly but also ì correctlyœ i.e.without any inherent dynamical approximations. This approach is summarized brie—y in the next section with some illustratory examples presented in Section III. A more comprehensive review of this work has recently been published elsewhere.21 Though this ì direct œ and ìcorrectœ approach is a signi–cant step in reaction rate theory it can at present be applied without approximation only for relatively small molecular systems. It is of course possible»and often quite useful»to divide a complex molecular system of many degrees of freedom into a few that are most central to the reaction and treat them by rigorous quantum methods with the remaining degrees of freedom treated approximately. Particularly useful in this regard are the ìreduced dimensionalityœ models that Bowman and co-workers22 have described and applied the time-dependent self-consistent –eld approximation23 that Gerber and others have used extensively and the multi-con–guration time-dependent Hartree (MC-TDH) methodology24 that Manthe et al.have applied to this problem. The many semiclassical approximations18h20 for making tunneling corrections to TST can also be thought of as approximate versions of this strategy as can the many kinds of ìmixed quantumclassical œ approaches25 that treat the (many) less important degrees of freedom by classical mechanics. Another approach for dealing with more complex molecular systems is to use the semiclassical (SC) initial value representation (IVR)26h30 to evaluate the rigorous quantum rate expressions.This has the advantage that all degrees of freedom»even the electronic degrees of freedom in electronically non-adiabatic processes31h32»are treated equivalently. The goal here is to generalize classical molecular dynamics (MD) simulation methods to include quantum interference and tunneling eÜects. Section IV describes this approach and some of its applications that have been carried out to date. Since classical trajectory simulations can be performed for truly complex molecular systems» e.g. reactions in solution clusters biomolecular environments or on surfaces»these SC-IVR approaches have the potential of extending rigorous quantum rate theory into such areas. (1) t? is = (2) II Rigorous quantum rate theory I have reviewed this topic in several places21,33 and thus give only a brief synopsis here.Assuming that the reactants are in Boltzmann equilibrium (and the validity of nonrelativistic quantum mechanics) one can show15a that the proper average of the state-tostate reactive scattering cross-section over energy scattering angle and initial/–nal quantum states gives the following relatively simple trace expression for the rate constant of a bimolecular reaction k(T )\ lim Cfs(t) where the —ux-side correlation function Cfs Cfs(t)\Qr(T )~1 tr [e~bHv Få På (t)] 5 W . H. Miller Here H F å is the Hamiltonian operator of the molecular system å the —ux operator with respect to a dividing surface that separates reactants and products På (t) the projector onto all states that are in the product region at time t and b\(kB T )~1; Qr(T ) is the reactant partition function per unit volume.If s(q)\0 is the equation which de–nes the dividing surface»s(q)[0 being the product region and s(q)\0 the reactant region» then the projection operator På (t) can be expressed as34 (3) På (t)\eiHv t@ä h[s(q)]e~iHv t@ä where h(s) is the standard Heaviside function (4) h(s)\G1 s[0 0 s\0 This makes the classical limit of eqn. (2) transparent a classical phase space average (5) replaces the C trace fs CL(t)\ giving Qr(T )~1(2n+)~F Pdp0 dq0 e~bH(p0 q0) F(p0 q0)h[s(qt)] P qt4qt(p0 q0) is the F-dimensional coordinate at time t that has evolved from where initial conditions (p0 q0). Sometimes it is useful to diÜerentiate eqn.(2) and then integrate the result to obtain the rate,34 (6a) k(T )\P0 =dtC f (t) where (6b) Cf(t)\ d d t Cfs(t)\Qr(T )~1 tr [e~bHv Få eiHv t@ä Få e~iHv t@ä] the fact having been used that (7) dt eiHv t@ä h(s)e~iHv t@ä\eiHv t@ä Få e~iHv t@ä d fs(t) or as the time integral of Cf(t) is a matter of convenience. It has also Cf(t) is a —ux»—ux autocorrelation function ; whether one expresses the rate as the long time limit of C been noted34 that the rate constant (though not the correlation functions) is unchanged if one makes the following modi–cation to the above expressions (8) e~bHv Få ]e~jHv Få e~(b~j)Hv for any value of j between 0 and b. In most applications we have made the choice j\b/2 initially so that the Boltzmann operator could be combined with the time evolution operators and have just one (complex time) propagator i.e.(9) f(t) by analytic Cf(t)\Qr(T )~1 tr [Få eiHv tC*@ä Få e~iHv tC@ä] where tC\t[i+b/2. [This was particularly useful in trying to compute C continuation methods;35,36 i.e. one computes (10) Cf([iq)4Q r (T )~1 tr [Få e~(b@2~q@ä)Hv Få e~(b@2`q@ä)Hv ] for a range of real q values in the interval [+b/2\q\+b/2 and then analytically C continues f([iq) to the complex value q\it to obtain Cf(t).] Much earlier Yamamoto, 37 using Kubo linear response theory expressed the rate as the time integral of a 6 Spiers Memorial L ecture —ux»—ux autocorrelation function that corresponds to averaging eqn. (8) over j. Though the rate constant is formally the same as that given by eqn.(9) this is less useful in practice because the resulting correlation function is singular at t\0. Rather than computing these time correlation functions in some applications it is useful rather to compute the cumulative reaction probability15b,34,38 (CRP) N(E) as a function of total energy E in terms of which the thermal rate constant is given by (11) k(T )\[2n+Qr(T )]~1 P= dE e~bE N(E) ~= Furthermore in some applications the microcanonical rate constant k(E) is the object of interest (typically for unimolecular reactions) and it is given in terms of the CRP where o expression for N(E) that corresponds to the above time correlation functions is34 N(E)\1 k(E)\[2n+ or(E)]~1 N(E) (12) r(E) is the reactant density of states per unit energy.The ì direct œ and ìcorrectœ 2(2n+)2 tr [Få d(E[Hå )Få d(E[Hå )] (13) where d(E[Hå ) is the microcanonical density operator that can be represented in terms of the outgoing wave Greenœs function (14a) d(E[Hå )\[1 Im Gå (`)(E) n (14b) Gå (`)(E)\(E]ie[Hå )~1 or in terms of the time evolution operator (15) d(E[Hå )\(2n+)~1 P= dt eiEt@ä e~iHv t@ä ~= Note that eqn. (12) looks super–cially like the RRKM expression for the microcanonical rate constant with the CRP replacing the ì integral density of states of the activated complexœ of the RRKM (4microcanonical TST) approximation. Though some of these formally exact expressions for the rate constant have been known for many years it was less than ten years ago that practical ways began to be developed for evaluating them.The work of my research group in this regard is reviewed elsewhere,21,33 and I also want to note especially the important contributions of Light and co-workers39 and Manthe and co-workers40 in developing the methodology for these calculations. Applications have been made to a number of reactions all in their full dimensionality H]H2 ]H2]H,38b,39a D]H2 ]HD]H,39a H]D2 ]HD ]D,39a O]HD]OD]H,OH]D,41 Cl]H2 ]HCl]H,42 F]H2 ]HF]H,43 O]HCl]OH]Cl,44 H]O2 ]OH]O,45 H2]OH]H2O]H,39d,40c,46 D2 ]OH]HOD]D,40c and H2]CN]HCN]H.40b,39e The methodology is now sufficiently well developed that it should be applicable to essentially any small molecular system. To close this summary of formal rate expressions I note another that has not yet found practical use but has interesting possibilities.The derivation (omitted here) is a rather straight-forward route from eqn. (9) the canonical and microcanonical rate constants can be expressed in terms of the following correlation function (16) C(t)\ lim tr [Få eiHv (t{`t@2)ä h(s)e~iHv (t{~t@2)@ä] t{?= 7 W . H. Miller which is a function neither of E nor T . The CRP is the Fourier transform of this correlation function (17) N(E)\P= dt e~iEt@ä C(t) ~= and the thermal constant given by its analytic continuation to the imaginary value t\i+b (18) k(T )\Qr(T )~1C(i+b) Calculation of this one correlation function would thus allow one to obtain N(E) and k(T ) for all (or at least some range of) values of E and T respectively.It remains to be seen if this will lead to practically useful approaches. (19) Cf(t)\2 d(t) kB h T Q Q t(T ) A+bB2 C (20) 2 d(t)] 2 t2]A2 +bB2D3@2 III Some examples In the classical limit if the TST assumption of no re-crossing trajectories is correct»i.e. a direct reaction»it is not hard to show that the classical —ux»—ux autocorrelation function is proportional to a Dirac delta function at t\0 i.e. r(T ) 0= dt d(t)\12 this integrates [cf. eqn. (6a)] to give the TST rate constant and it Cf(t) only for very short (in fact in–nitesimal) Since / also shows that one needs to evaluate times in order to be able to integrate it to obtain the rate constant. Quantum mechanics is not quite so simple even for a direct reaction.This can be seen explicitly for the case of a one dimensional reaction coordinate separable from the other (F[1) degrees of freedom; if the reaction coordinate is treated as a free particle eqn. (19) is modi–ed as follows34 The right hand side of eqn. (20) also integrates to unity so that the TST rate expression is maintained (though with quantum partition functions) but here it is necessary to C from 0 to a time of at least ca. +b in order to obtain the quantum rate integrate f(t) constant. This is still a short time (+bB27 fs at T \300 K) but not zero. One sees this same qualitative behaviour in the quantum correlation function for direct reactions of real molecular systems»for which all the degrees of freedom are coupled»i.e. Cf(t) falls eÜectively to zero in a time of ca.+b. This is the ultimate source of the efficiency and usefulness of this ì direct œ and ìcorrectœ way of computing rate constants it is necessary to generate the full quantum dynamics of the coupled system but only for short time. It is during this time from 0 to +b that all the tunneling corner-cutting etc. phenomena take place. Fig. 1 shows the —ux correlation function for a prototypical direct reaction,42 Cl ]H2 ]HCl]H. The lower temperature (300 K) in Fig. 1(a) shows the classic behaviour for a direct reaction but at the higher temperature (T \1500 K +bB5 fs) in Fig. C 1(b) one sees a small negative region of f(t) at longer time; this is the manifestation of re-crossing dynamics i.e. a breakdown of the no re-crossing trajectory TST picture that Wigner noted would arise at high temperature/energy.Fig. 2 shows Cf(t) for a 8 Spiers Memorial L ecture Cl]H2 ]HCl]H reaction (in 3D J\0) for Fig. 1 Flux»—ux autocorrelation function for the (a) T \300 K and (b) T \1500 K Fig. 2 Flux»—ux autocorrelation function for the O]HCl]OH]Cl reaction for T \300 K 9 W . H. Miller Fig. 3 Flux»—ux autocorrelation functions for the OH]O]H]O reaction at T \1200 K 2 heavy]light-heavy reaction,44 O]HCl]OH]Cl for which it is well known that re-crossing dynamics is common. for the reaction45b C ]OH]O2]H which involves the forma- f(t) O Fig. 3 shows C at t\0 is barely tion and decay of a long lived collision complex. The peak in f(t) discernible in the –gure since +bB7 fs\0.007 ps at T \1200 K and the negative lobe of re-crossing —ux extends to ca.0.5 ps. The area under the negative lobe is about 70% of that under the positive peak so that the transmission coefficient in TST language is ca. 0.3. This example illustrates the fact that complex-forming reactions are more difficult to treat without approximation since one must follow the quantum dynamics for long enough time for the quantum wavepackets to ìdecideœ whether they will be products or reactants (analogous to the situation in a classical trajectory calculation). Other aspects of these examples and others are discussed more fully elsewhere.21 Even with this very brief discussion of the nature of —ux correlation functions and their applications one sees how vestiges of TST emerge in these rigorous calculations.This is also evident in the algorithmic details of the methodology.38 Thus the TST picture survives as an important tool for understanding the qualitative behaviour of —ux correlation functions even when they are computed by rigorous quantum simulation methods. IV Semiclassical approximations for complex systems a The semiclassical initial value representation The rigorous quantum approaches summarized in the previous section are at present only feasible without introducing approximations for small molecular systems. For more complex systems most approximations involve dividing the overall molecular system into two parts and treating the (small) most relevant part rigorously and the remainder approximately as discussed in the Introduction.Here I describe another approach one which treats all degrees of freedom equivalently ; it is based on using the semiclassical (SC) initial value representation (IVR)26h30 to evaluate the quantum rate expressions of the previous section. The SC-IVR is a classical trajectory-based theory that had its origins in the early 1970s but which has recently had a re-birth of interest as a way of building quantum interference and tunneling eÜects into classical molecular dynamics (MD) simulations. The primary diÜerence in the recent IVR approaches from the original version is that they are now implemented in the cartesian coordinate (or 10 Spiers Memorial L ecture coherent state) representation rather than in action-angle variables and this is more general better behaved numerically and also more accurate.The coordinate space IVR for the time evolution operator involves a phase space (21a) average over initial d e conditions ~iHv t@ä\Pfor dp classical q trajectories 0 0Ct(p0 q0)eiSt(p0 q0)@ä o qtTSq0 o P t where qt4qt(p0 q0) is the coordinate at time t that evolves from these initial conditions and S is the classical action integral (the time integral of the Lagrangian) along the trajectory. The pre-exponential factor involves the Jacobian relating the coordinates at (21b) KN 0 time t to the initial momenta Ct(p0 q0)\CKLqt( L p p 0 q0) (2ni+)FD1@2 The states Mo qTN in eqn. (21a) are the usual Dirac coordinate eigenstates. The momentum space IVR is similar to eqn.(21) but diÜerent. Herman and Kluk (HK)29 introduced a hybrid IVR that eÜectively interpolates between the coordinate and momentum versions using coherent states (minimum uncertainty wavepackets). The HK expression for the propagator is the same as eqn. (21a) with the Dirac states replaced by coherent states (22a) o qtT]o pt qtT (22b) (23) ]ei*St(p0 q0)~St(p0{ q0)+@ä and where qt\qt(p0 q0) (24) o q0T]o p0 q0T pBF@4 expC[ c whose coordinate S space q@ o p q wavefunction T\Ac is 2 o q@[q o2]ip … (q@[q)/+D and with a modi–ed CtHK(p0 q pre-exponential 0)\(2p+)~FCdet factor 1 2 AL L q qt ]L L p pt ]+ i c L L q pt ] + i c L L p qtBD1@2 (22c) 0 0 0 0 If the coherent state parameter c]O the coordinate space IVR eqn.(21) is recovered and c]0 gives the momentum space IVR. of eqn. (2)»(4) ; with the SC-IVR of eqn. (21) for the propagators this Since eqn. (21) and (22) provide a way to evaluate the time evolution operator an SC approximation can be immediately written down for any formal quantum expression involving it. A number of applications27h30,47h51 have shown the SC-IVR to be capable of providing an accurate description of quantum eÜects in a variety of molecular phenomena. Our present interest is to use it to obtain an SC approximation for the correlation C function fs(t) becomes Cfs(t)\Qr(T )~1 Pdq0 dp0 dp0 @ h[s(q0)]Sqt oFå (b) o qt@TCt(p0 q0)Ct(p0 @ q0)* P P and Få (b) is the Boltzmannized —ux operator t @ \qt(p0 @ q0) Få (b)\e~bHv @2 Få e~bHv @2 q Here there is a double phase space average (the initial coordinates for the two trajectories are the same because h[s(q)] is a local operator) in contrast to the single phase space average of the classical expression eqn.(5). More troublesome however is the oscillatory character of the integrand in eqn. (23) which arises from the interference 11 W . H. Miller between the two trajectories and makes the phase space integrals considerably more difficult to carry out than in the classical case. Applications to date have used various –ltering methods27a,52h54 to dampen the oscillatory character and make Monte Carlo calculations feasible for simple systems but a more general solution to this problem is a matter of intense research activity.b The linearization approximation In lieu of carrying out the full SC-IVR evaluation in eqn. (23) we have considered a rather drastic approximation that leads to a much simpli–ed approach and rather surprisingly (at least to me) has been seen to give very good results for some non-trivial applications. The idea is to expand the integrand of eqn. (23) to –rst order in the diÜer- 0 @ which linearizes the diÜerence in the actions,55h58 ence between p and p 0 (25) St(p0 q0)[St(p0 @ q0)B LSt( L p p 0 q0) … (p0[p0 @ ) with 0 p0\12 (p0]p0 @ ). Carrying out this approximation leads (after some manipulation) (26a) to the following very C simple fs(t)\ result Qr(T 59,60 )~1 Pdp0 dq0FW b (p0 q0)h[s(qt)] P where is the Wigner transform61 of the operator Få (b) FW b FW b (p0 q0)\(2p+)~F Pdq e~ip0 ’ q@äSq0]q/2 oFå (b) o q0[q/2T (26b) Eqn.(26a) is seen to be eÜectively the same as the classical expression [eqn. (5)] with the Wigner transform of Få (b) replacing the classical Boltzmann-—ux function ; one thus computes classical trajectories»to see if h[s(qt)] is 1 or 0 i.e. on the product or reactant side of the dividing surface at time t»simply with a modi–ed distribution of initial conditions. This linearized SC-IVR (LSC-IVR) has been applied59 to a popular model for chemical reactions in a condensed phase environment namely a double well potential coupled juj2B2 (27a) to an in–nite bath H(p 2m]V0(s)]; 2 P m j s P s s , of , harmonic Q)\ p oscillators 2 for which 2 ] the 2 1 m Hamiltonian juj2AQj[m cj is s j j The mass m is that of an H atom the barrier height of the double well potential V0(s) is ca.6 kcal mol~1 (typical of H atom transfer reactions) and the spectral density of the harmonic bath is of the usual Ohmic form with an exponential cut-oÜ (27b) J(u)4 p c 2 ;j mjuj d(u[uj) j2 (27c) \gu e~u@uC the same model for which Topaler and Makri62 have carried out (numerically exact) Feynman path integral calculations. Fig. 4 shows the rate constant (at T \300 K) as a function of the coupling strength C g and one sees that the results of the LSC-IVR approximation are in essentially quantitative agreement with the accurate quantum values over the entire range.More insight C into the dynamics is revealed by examining the correlation function fs(t) itself to see how the t]O limit»the rate constant»is approached. Fig. 5(a) shows Cfs(t) for a case of strong coupling (large g) and it is seen to behave just as for the direct reaction in Fig. 1 [remember that fs(t)\/0 t dt@Cf(t@)] reaching its ìplateauœ value in a time of ca. 12 Spiers Memorial L ecture Fig. 4 Rate constant for isomerization in a double well potential coupled to a harmonic bath; cf. eqn. (27). The quantity shown is the rate k(T ) relative to the classical TST rate i.e. the transmission coefficient i4k(T )/kCLTST for T \300 K as a function of the coupling strength to the bath.The solid line is the result of the linearized SC-IVR approximation described in Section IVb and the solid points the accurate quantum path integral calculations of Topaler and Makri ref. 62. still reaches its TST plateau value in a time of ca. 27 fs but now C C +b\27 fs ; i.e. this is a case of TST-like dynamics trajectories leave the region of the dividing surface and do not re-cross it because they rapidly lose energy to the bath and are trapped in one of the potential wells. In contrast for a weak coupling (small g) case C shown in Fig. 5(b) fs(t) coupling to the bath is too weak to prevent re-crossings which cause fs(t) to decrease then increase etc. ; one can identify an average of about three re-crossings before it reaches its –nal value (which gives the rate constant).The ìtransmission coefficient œ in this case»the ratio fs(O)/Cfs TST»is seen to be about 1/2. c Electronically non-adiabatic dynamics It is also possible to treat electronically non-adiabatic processes via the SC-IVR by using the Meyer»Miller (MM) approach63 to model the electronic degrees of freedom. In the Cartesian representation this gives the following classical Hamiltonian for the nuclear (P Q) and ìelectronic œ (p x) degrees of freedom,31,32 N 1 N 2 (pi2]xi2[1)Hii(Q)] ; (pi pj]xi xj)Hij(Q) (28) 2k] ; i/i i:j/1 MHij(Q)N is an N]N diabatic electronic PES provided for example by ab initio H(P Q p x)\P2 where electronic structure calculations. (There is also an adiabatic electronic representation of similar form.63,31) A number of applications were made some years ago64 using this Hamiltonian to carry out quasiclassical trajectory calculations (sampling initial conditions via action-angle variables histograming –nal action variables etc.) treating electronic and nuclear degrees of freedom equivalently but it is clear that one can ìup-gradeœ the treatment to the SC level by using the IVR.(MM also discussed the semiclassical implementation of the model.) If one up-grades the description to the quantum level»i.e. takes eqn. (28) to be a Hamiltonian operator and solves the Schroé dinger equation with it»then one has an exact treatment of the nuclear-electronic dynamics. 13 W . H. Miller (g/mub\2.8) (b) weak coupling (g/mub\0.2).quantity shown is Fig. 5 Flux»side correlation function Cfs(t) corresponding to the rate constants of Fig. 4. The so that the t]O limit is i of Fig. 4. (a) Strong coupling i(t)4Cfs(t)/kCLTST Application of the SC-IVR to the MM Hamiltonian [eqn. (28)] has been made31 to the suite of two-state one (nuclear) dimensional scattering problems used by Tully65 for testing surface-hopping approximations for non-adiabatic dynamics and yielded very good agreement with the correct quantum results for these examples. Stock and Thoss32 also applied it to the two-state problem coupled to a harmonic oscillator also with excellent results. Perhaps more impressive is recent work66 applying the linearized (LSC-IVR) version of the SC-IVR (described in Section IVb above) to the popular spinboson problem.67 This is a two-electronic state model often used to describe radiationless transitions electron transfer etc.in large polyatomic molecules or to model the eÜect of condensed phase environments. The two diabatic PESs are in–nite dimensional harmonic oscillators with shifted equilibrium positions and the oÜ-diagonal PES is constant ; eqn. (28) becomes 2 mjuj2Qj2B]1 2 (p12]x12[p22[x2 2) ; cjQj AP ]1 2m j 2 j j (29) H(P Q p x)\;j ]*(p1p2]x1x2) 14 Spiers Memorial L ecture H where D4H12(Q) is constant. The spectral density of the harmonic bath is the same as in eqn. (27) above. Fig. 6 shows the population relaxation from electronic state 1 [with the bath initially 11(Q)] as a function of time in a Boltzmann distribution on PES D(t)4P1H1(T )[P2H1(T )\Q1(T )~1 tr [e~bHv 11 o 1TS1 o eiHv t@äp� z e~iHv t@ä] (30a) where Q1(T )\tr [o 1TS1 o e~bHv 11] (30b) and p� z\o 1TS1 o[o 2TS2 o (30c) One sees excellent agreement with (numerically exact) Feynmann path integral calculations by Makarov and Makri68 for both large electronic coupling [Fig.6(a)] which Fig. 6 Electronic population decay [D(t) of eqn. (30)] for the two-state system coupled to a harmonic bath (spin-boson system) ; cf. eqn. (29) of Section IVc. The solid line is the result of the linearized SC-IVR approximation and the points the path integral calculations by Makarov and Makri ref. 68. (a) Large electronic coupling bD\5 leading to oscillatory behaviour (b) small electronic coupling bD\0.1 leading to monotonic decay.15 W . H. Miller leads to oscillatory (coherent) structure in the decay and also weak coupling [Fig. 6(b)] that has little coherent structure. Fig. 7 shows the spin»spin correlation function (which is closely related to the ì side»side œ correlation function34 for a continuous potential), (31a) C(t)\Q(T )~1 tr [e~bHv p� z eiHv t@äp� z e~iHv t@ä] where here Q(T ) is the total partition function (31b) C(t)\tr [e~bHv ] Again there is excellent agreement with Feynmann path integral calculations (here by Mak and Chandler69) for both the coherent and incoherent regimes. The simple linearized version of the SC-IVR thus captures essentially all of the dynamical features in the ìcondensed phaseœ processes here and in Section IVb above.This is surprising since the real time dynamics in this approximation is essentially that of classical mechanics the only true quantum aspects of it being the Wigner transform of the Boltzmannized —ux operator. This produces the correct quantum dynamics Fig. 7 Spin correlation function C(t) of eqn. (31) for the spin-boson system of Section IVc. The solid line is the result of the linearized SC-IVR approximation and the points the path integral calculations by Mak and Chandler ref. 69. (a) Weak coupling to the bath (the coherent regime) (b) strong coupling to the bath (the incoherent regime). 16 Spiers Memorial L ecture (within the SC-IVR description) for times of order +b»which is sufficient to describe quantum eÜects in TST-like dynamics»but the longer time dynamics is basically that of classical mechanics.Other applications59b have indeed shown this namely that the longer time dynamics given by the LSC-IVR even the coherent features seen in Fig. 5»7 are identical to that of classical mechanics. A full SC-IVR treatment,59b without the linearization approximation is able to describe true quantum eÜects in the longer time dynamics but these calculations are considerably more diÜerent. d Forwardñbackward IVR Though the above results using the linearized approximation to the SC-IVR are very impressive and encouraging one would like to be able to implement the SC-IVR without having to make this approximation. Otherwise one wr know whether quantum eÜects are important for t?+b or not.Therefore I conclude this section with a suggestion for how to simplify the SC-IVR calculation of time correlation functions (32) CAB(t)\tr [Aå eiHv t@äBå e~iHv t@ä] which are typically the objects of interest in a complex dynamical system. As noted above since there are two time evolution operators involved in eqn. (32) straightforward use of the SC-IVR [eqn. (31) or (32)] for each propagator leads to a double phase space integral which is not only twice the dimensionality of the corresponding classical expression but also has an oscillatory integrand because of the interference between the two trajectories. The basic idea for simplifying matters is to combine the two time evolution operators into one IVR.70,71 To do this suppose –rst that operator Bå in eqn.(32) is a multiplicative operator of the form (33) Bå \ei’(q)@ä where /(q) is a sufficiently smooth function of q. The operator Uå (34a) Uå 4eiHv t@ä ei’(q)@ä e~iHv t@ä is unitary and can be thought of as the time evolution operator for a time increment 0]t and then t]0 with the time-dependent Hamiltonian (34b) Hå (t@)\Hå [d(t@[t)/(q) Since the general SC expression for a time evolution operator has the same form also for a time-dependent potential energy function the SC-IVR for operator Uå is given by eqn. (21) or (22) above with the trajectories (and action integral) computed from the Hamiltonian of eqn. (34b). It is not hard to show that these trajectories are as follows starting with initial condition (p0 q0) at time 0 one integrates to time t with the molecular Hamiltonian H(p q) yielding coordinates and momenta (pt qt) ; at this point one makes the following change in the momentum (34c) B t ]pt]AL/ Lq (q) q/qt p and then integrates from time t back to 0 yielding the –nal values and 0 @ (p0 0 @ (p0 q0) q0).p (34d) q The action integral S ( t 0(p0 q along 0)\Pthis dt@ trajectory p … q 5 [H is )]/(qt)]P0dt@(p … q 5 [H) t 0 17 W . H. Miller (35a) so that eqn. (21) then U gives å \Pdp dq0C0(p0 q0) eiS0(p0 q0)@äo q0@ TSq0 o 0 P with (35b) C0(p0 q0)\CKLq0 @ ( L p p 0 q0) (2pi+)FD1@2 KN 0 The HK-IVR indicated by eqn. (22) is obtained similarly. With the SC-IVR for operator Uå [eqn. (35)] the expression for the correlation func- (36) tion of eqn.(32) thus CAB becomes (t)\Pdp0 dq0C0(p0 q0) eiS0(p0 q0)@äSq0 oAå o q0@ T P which involves only a single phase space integral. Furthermore since the action integral S [eqn. (34d)] is for a forward and backward time increment»though with a momentum jump [eqn. (34c)] in the middle»one expects the integrand to be much less oscil- 0 latory than the double phase space integral. For the same reason the pre-exponential factor should also be better behaved (i.e. closer to unity). Cfs B (t) of interest however the operator å in For the —ux-side correlation function eqn. (32) is not of the form of eqn. (33) but rather corresponds to Bå \h[s(q)]. By using the Fourier transform of the Heaviside function though it can be written as a onedimensional integral over operators of this form (37) s[2pi(ps[ie)]~1 eipss(q)@ä h[s(q)]\P~= = dp where e is a small positive constant.(In practice one can set e40 since other factors in the integrand are zero when ps\0.) Thus the forward»backward SC-IVR for Cfs(t) is given C by fs(t)\Qr(T )~1 P= dps(2pips)~1 Pdp0 dq0C0(p0 q0) eiS0(p0 q0)@äSq0 oFå (b) o q0@ T P ~= (38a) (38b) B where the ìmomentum jumpœ of eqn. (34c) at time t is t ]pt]psAL L s( q q) q/qt and the action (38c) p S S 0 0 is (p0 q0)\Pt dt@(p … q 5 [H)]ps s(qt)]P0dt@(p … q 5 [H) t 0 Eqn. (38) expresses Cfs(t) as a single phase space integral over initial conditions plus a one-dimensional integral (over the ìmomentum jumpœ ps) and the oscillatory character of the integrand is expected to be much less than for the double phase space integral.This is therefore only slightly more involved than the corresponding classical expression and probably about as simple and efficient as one can hope for. Preliminary applications indicate that these optimistic features are indeed borne out.71 Finally one can use this forward»backward idea also to simplify applications other than time correlation functions. The SC eigenvalues for a molecular system for example, 18 Spiers Memorial L ecture are typically determined by computing a matrix element of the microcanonical density operator with respect to some reference wavefunction o sT (39a) I(E)4Ss o d(E[H) o sT\; o Ss otkT o2 d(E[Ek) k Motk and TN MEkN are the eigenfunctions and eigenvalues.Using the Fourier repre- where sentation of the delta function gives the following expression for I(E) (39b) p+ P I(E)\Re = dt eiEt@äSs o e~iHv t@ä o sT 0 (39c) and the SC-IVR d Ss is o used e~iHv t@ for ä o sT the \ matrix Pdp element q of the propagator 0 0Ct(p0 q0) eiSt(p0 q0)@äs(qt)*s(q0) P Hå (or the HK version). This expression involves only one time evolution operator and thus only a single phase space integral but it can nevertheless be difficult to evaluate because of the oscillatory character of the integrand. This can be reduced by taking os[ to be the eigenfunction of some reference Hamiltonian Hå that approximates Hå as 0 o sT\E0 o sT one has closely as possible.Then since 0 (40) o sT\e~iE0t@ä eiHv 0t@ä o sT (41a) p+ P so that eqn. (39b) can be written as I(E)\Re = dt ei(E~E0t@äSs o eiHv 0t@ä e~iHv t@ä o sT 0 One now applies the SC-IVR to the operator Uå (41b) Uå 4eiHv 0t@ä e~iHv t@ä The SC-IVR for operator which corresponds to the time evolution operator for the time increment 0]t with Hamiltonian Uå is Hå and then t]0 with Hamiltonian Hå 0 . (41c) therefore of standard U form å \Pdp dq0C0(p0 q0) eiS0(p0 q0)@ä o q0@ TSq0 o 0 P so that Ss o eiHv 0t@ä e~iHv t@ä o sT4Ss oUå o sT\Pdp dq0C0(p0 q0) eiS0(p0 q0)@äs(q0 @ )*s(q0) (41d) P 0 (p0 @ q0 @ ) and the action integral S result from the trajectory that starts with initial 0 (p0 q0) ( and is integrated to time t with Hamiltonian H and then»with qt where conditions p continuous»is integrated back to time 0 with Hamiltonian H0 .Application of eqn. t) (41) to the vibrational»rotational eigenvalues of (HCl) »treated earlier72 via eqn. (39)» shows that using the forward»backward IVR with a reference Hamiltonian H does 2 0 indeed make the calculation better behaved.73 V Concluding remarks As has often been noted science makes progress by each generation standing on the shoulders of its predecessors and nothing could more accurately describe the subject of this Faraday Discussion. Transition state theory provided the essential qualitative picture for understanding thermal reaction rates and long after more rigorous quantum 19 W . H. Miller methods will have eliminated the need for the speci–c approximations it entails the insight it aÜords will survive.Yet the present paradigm of chemical reaction theory is simulation engendered by the computer revolution. One now fully appreciates that understanding and computing/ simulation go hand in hand understanding a particular process theoretically allows one to carry out calculations to model it quantitatively and simulation of a complex process often leads to understanding in terms of simple pictures and models. The papers presented at this Discussion are an excellent example of this interaction between computing and understanding. For the simplest chemical reactions»gas phase reactions of small molecular systems»the theoretical methodology is essentially in hand to permit rigorous quantum calculations for any quantity of interest from the thermal rate constant to the most detailed state-speci–c quantity.Yet even here there is a rich variety of dynamical phenomena that continue to surprise us it seems with each new system that is studied. At the opposite end of the spectrum it is not even possible to carry out classical simulations for the most complex biomolecular processes and in between there are many complex processes that can be treated classically but for which quantum eÜects are signi–cant and must be included approximately. This range from the simple to the complex is well represented at this Discussion. It is also interesting to see the extent to which the rigorous methodology presently applicable to simple systems is working its way upwards to address more complex reactions.It would be interesting though unlikely at least for me to be present at a Faraday Discussion on this topic sixty years hence to see what directions it had taken»and whether (or not) the participants would be musing over any of the comments during this Discussion. This work has been supported by the Director Office of Energy Research Office of Basic Energy Sciences Chemical Sciences Division of the U.S. Department of Energy under contract No. DE-AC03-76SF00098 by the Laboratory Directed Research and Development (LDRD) project from the National Energy Research Scienti–c Computing (NERSC) Center Lawrence Berkeley National Laboratory and also by the National Science Foundation under Grant No.CHE97-32758. References 1 T rans. Faraday Soc. 1938 34. 2 H. Eyring ref. 1 p. 3. 3 H. Eyring ref. 1 p. 41. 4 M. G. Evans ref. 1 p. 49. 5 E. Wigner ref. 1 p. 29. 6 R. A. Ogg Jr. and M. Polanyi T rans. Faraday Soc. 1935 31 1375. 7 From the IBM 7094 of the mid 1960s to the present 10»100 giga—op computers. 8 Estimate by M. Head-Gordon personal communication. 9 See for example (a) Adv. Chem. Phys. 1966 10; (b) Discuss. Faraday Soc. 1967 44. 10 See for example (a) Femtosecond Chemistry ed. J. Manz and L. Woé ste VCH Weinheim 1995; (b) Adv. Chem. Phys. 1997 101. 11 See for example (a) Modern T heoretical Chemistry ed. H. F. Schaefer G. A. Segal W. H. Miller and B. J. Berne Plenum New York 1976; (b) T heory of Chemical Reaction Dynamics ed.M. Baer CRC Press Boca Raton (FL) vol. I»IV 1985. 12 (a) D. E. Manolopoulos and D. C. Clary Ann. Rep. Prog. Chem. C 1989 86 95; (b) W. H. Miller Ann. Rev. Phys. Chem. 1990 41 245. (c) R. KosloÜ Ann. Rev. Phys. Chem. 1994 45 145. 13 P. Pechukas and F. J. McLaÜerty J. Chem. Phys. 1973 58 1622. 14 F. J. McLaÜerty and P. Pechukas Chem. Phys. L ett. 1974 27 511. 15 (a) W. H. Miller J. Chem. Phys. 1974 61 1823; (b) W. H. Miller J. Chem. Phys. 1975 62 1899; (c) W. H. Miller J. Chem. Phys. 1975 63 1166; (d) S. Chapman B. C. Garrett and W. H. Miller J. Chem. Phys. 1975 63 2710; (e) W. H. Miller Acc. Chem. Res. 1976 9 309; ( f ) W. H. Miller Faraday Discuss. Chem. Soc. 1977 62 40; (g) W. H. Miller R. Hernandez N. C. Handy D. Jayatilaka and A.Willetts Chem. Phys. L ett. 1990 62 172. 20 Spiers Memorial L ecture 16 G. A. Voth D. Chandler W. H. Miller J. Chem. Phys. 1989 91 7749. 17 J. Cao and J. A. Voth J. Chem. Phys. 1994 100 5093 5106; 1994 101 6157 6168 6184. 18 R. A. Marcus and M. E. Coltrin J. Chem. Phys. 1977 67 2609. 19 (a) D. G. Truhlar and B. C. Garrett Acc. Chem. Res. 1980 13 440; (b) D. G. Truhlar and B. C. Garrett Ann. Rev. Phys. Chem. 1984 35 159; (c) D. G. Truhlar A. D. Issaacson and B. C. Garrett ref. 11b vol. IV p. 65. 20 (a) W. H. Miller N. C. Handy and J. E. Adams J. Chem. Phys. 1980 72 99; (b) S. K. Gray W. H. Miller Y. Yamaguchi and H. F. Schafer J. Chem. Phys. 1980 73 2733; J. Am Chem. Soc. 1981 103 1900. 21 W. H. Miller J. Phys. Chem. 1998 102 793. 22 J. M. Bowman J.Phys. Chem. 1991 95 4960. 23 (a) R. B. Gerber V. Buch and M. A. Ratner J. Chem. Phys. 1982 77 3022; (b) V. Buch R. B. Gerber and M. A. Ratner Chem. Phys. L ett. 1983 101 44. 24 (a) H. D. Meyer U. Manthe and L. S. Cederbaum Chem. Phys. L ett. 1990 165 73; (b) U. Manthe H. D. Meyer and L. S. Cederbaum J. Chem. Phys. 1992 97 3199; (c) F. Matzkies and U. Manthe J. Chem. Phys. 1997 106 2646; 1998 108 4828. 25 Some recent examples include (a) N. P. Blake and H. Metiu J. Chem. Phys. 1994 101 223; (b) M. Ben-Nun and R. D. Levine Chem. Phys. 1995 201 163; (c) Z. Li and R. B. Gerber J. Chem. Phys. 1995 102 4056; (d) J. Cao C. Minichino and G. A. Voth J. Chem. Phys. 1995 103 1391; (e) L. Liu and H. Guo J. Chem. Phys. 1995 103 7851; ( f ) C. Scheurer and P. Saalfrank J.Chem. Phys. 1995 104 2869; (g) J. Fang and C. C. Martens J. Chem. Phys. 1996 104 3684; (h) S. Consta and R. Kapral J. Chem. Phys. 1996 104 4581; (i) P. Bala P. Grochowski B. Lesyng and J. A. McCammon J. Phys. Chem. 1996 100 2535; ( j) A. B. McCoy J. Chem. Phys. 1995 103 986; (k) H. J. C. Berendsen and J. Mavri Int. J. Quantum Chem. 1996 57 975; (l) S. Hammes-SchiÜer and J. C. Tully J. Chem. Phys. 1994 101 4657; (m) J-Y. Fang and S. Hammes-SchiÜer J. Chem. Phys. 1997 106 8442; 1998 108 7085; (n) J. C. Tully Faraday Discuss. 1998 110 407. 26 (a) W. H. Miller J. Chem. Phys. 1970 53 3578; (b) W. H. Miller and T. F. George J. Chem. Phys. 1972 56 5668 Appendix D. 27 (a) E. J. Heller J. Chem. Phys. 1991 94 2723; (b) W. H. Miller J. Chem. Phys. 1991 95 9428; (c) E.J. Heller J. Chem. Phys. 1991 95 9431; (d) M. A. Sepulveda S. Tomsovic and E. J. Heller Phys. Rev. L ett. 1992 69 402; (e) M. A. Sepulveda and E. J. Heller J. Chem. Phys. 1994 101 8004. 28 K. G. Kay J. Chem. Phys. 1994 100 4377; 1994 100 4432; 1994 101 2250. 29 (a) M. F. Herman and E. Kluk Chem. Phys. 1984 91 27; (b) E. Kluk M. F. Herman and H. L. Davis J. Chem. Phys. 1986 84 326; (c) M. F. Herman J. Chem. Phys. 1986 85 2069; (d) M. F. Herman Chem. Phys. L ett 1997 275 445; (e) B. E. Guerin and M. F. Herman Chem. Phys. L ett 1998 286 361. 30 (a) G. Campolieti and P. Brumer J. Chem. Phys. 1992 96 5969; (b) G. Campolieti and P. Brumer Phys. Rev. A 1994 50 997; (c) D. Provost and P. Brumer Phys. Rev. L ett. 1995 74 250; (d) G. Campolieti and P. Brumer J.Chem. Phys. 1997 107 791. 31 X. Sun and W. H. Miller J. Chem. Phys. 1997 106 6346. 32 G. Stock and M. Thoss Phys. Rev. L ett. 1997 78 578. 33 (a) W. H. Miller Acc. Chem. Res. 1993 26 174; (b) W. H. Miller in New T rends in Reaction Rate T heory ed. P. Talkner and P. Haé nggi Kluwer Academic Dordrecht 1995 pp. 225»246; (c) W. H. Miller in Proceedings of the Robert A. W elch Foundation 38th Conference on Chemical Research Chemical Dynamics of T ransient Species Robert A. Welch Foundation Houston TX 1994 pp. 17»27; (d) W. H. Miller in Dynamics of Molecules and Chemical Reactions ed. J. Zhang and R. Wyatt Marcel Dekker NY 1995 pp. 387»410; (e) W. H. Miller Adv. Chem. Phys. 1997 101 853. 34 W. H. Miller S. D. Schwartz and J. W. Tromp J. Chem. Phys. 1983 79 4889.35 D. Thirumalai and B. J. Berne J. Chem. Phys. 1983 79 5029. 36 K. Y. Yamashita and W. H. Miller J. Chem. Phys. 1985 82 5475. 37 T. Yamamoto J. Chem. Phys. 1960 33 281. 38 (a) T. Seideman and W. H. Miller J. Chem. Phys. 1992 96 4412; (b) T. Seideman and W. H. Miller J. Chem. Phys. 1992 97 2499; (c) U. Manthe and W. H. Miller J. Chem. Phys. 1993 99 3411. 39 (a) T. P. Park and J. C. Light J. Chem. Phys. 1986 85 5870; 1988 88 4897; 1989 91 974; 1991 94 2946; 1992 96 8853; (b) M. Founargiotakis and J. C. Light J. Chem. Phys. 1990 93 633; (c) D. Brown and J. C. Light J. Chem. Phys. 1992 97 5465; (d) D. H. Zhang and J. C. Light J. Chem. Phys. 1996 104 6184; 1997 106 551; (e) J. C. Light and D. H. Zhang Faraday Discuss. 1998 110 105. 40 (a) U. 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Chem. Phys. 1998 108 8870. 48 (a) A. R. Walton and D. E. Manolopoulos Mol. Phys. 1996 87 961; (b) A. R. Walton and D. E. Manolopoulos Chem. Phys. L ett. 1995 244 448; (c) M. L. Brewer J. S. Hulme and D. E. Manolopoulos J. Chem. Phys. 1997 106 4832. 49 (a) S. Garashchuk and D. J. Tannor Chem. Phys. L ett. 1996 262 477; (b) S. Garashohuk F. Grossman and D. Tannor J. Chem. Soc. Faraday T rans. 1997 93 781. 50 F. Grossman Chem. Phys. L ett. 1996 262 470. 51 M. Ovchinnikov and V. A. Apkarian J. Chem. Phys. 1996 105 10 312; 1997 106 5775; 1998 108 2277.52 V. S. Filinov Nucl. Phys. B 1986 271 717. 53 N. Makri and W. H. Miller Chem. Phys. L ett. 1987 139 10. 54 J. D. Doll D. L. Freeman and T. L. Beck Adv. Chem. Phys. 1994 78 61. 55 R. E. Cline Jr. and P. G. Wolynes J. Chem. Phys. 1988 88 4334. 56 V. Khidekel V. Chernyak and S. Mukamel in Femtochemistry Ultrafast Chemical and Physical Processes in Molecular Systems ed. M. Chergui World Scienti–c Singapore 1996 p. 507. 57 J. S. Cao and G. A. Voth J. Chem. Phys. 1996 104 273. 58 X. Sun and W. H. Miller J. Chem. Phys. 1997 106 916. 59 (a) H. Wang X. Sun and W. H. Miller J. Chem. Phys. 1998 108 9726; (b) X. Sun H. Wang and W. H. Miller J. Chem. Phys. 1998 109 4190. 60 (a) E. Pollak and J. L. Liao J. Chem. Phys. 1998 108 2733; (b) J. Shao J. L. Liao and E. Pollak J. Chem. Phys. 1998 108 9711. 61 E. Wigner Phys. Rev. 1932 40 749. 62 M. Topaler and N. Makri J. Chem. Phys. 1994 101 7500. 63 (a) H. D. Meyer and W. H. Miller J. Chem. Phys. 1979 70 3214; (b) H. D. Meyer and W. H. Miller J. Chem. Phys. 1979 71 2156. 64 (a) H. D. Meyer and W. H. Miller J. Chem. Phys. 1980 72 2272; (b) A. E. Orel and W. H. Miller J. Chem. Phys. 1980 73 241; (c) A. E. Orel D. P. Ali and W. H. Miller Chem. Phys. L ett. 1981 79 137; (d) W. H. Miller and A. E. Orel J. Chem. Phys. 1981 74 6075; (e) S. K. Gray and W. H. Miller Chem. Phys. L ett. 1982 93 341; ( f ) D. P. Ali and W. H. Miller Chem. Phys. L ett. 1984 103 470. 65 J. C. Tully J. Chem. Phys. 1990 93 1061. 66 X. Sun H. Wang and W. H. Miller J. Chem. Phys. 1998 109. 67 A. J. Leggett S. Chakravarty A. T. Dorsey M. P. Fisher A. Garg and W. Zwerger Rev. Mod. Phys. 1987 59 1. 68 D. E. Makarov and N. Makri Chem. Phys. L ett. 1994 221 482. 69 C. H. Mak and D. Chandler Phys. Rev. A 1991 44 2352. 70 N. Makri and K. Thompson Chem. Phys. L ett. 1998 291 101. 71 X. Sun and W. H. Miller work in progress. 72 X. Sun and W. H. Miller J. Chem. Phys. 1998 108 8870. 73 X. Sun unpublished results. Paper 8/05196H; Received 6th July 1998

ISSN:1359-6640    

DOI:10.1039/a805196h

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 23»50 Fragmentation surface of triplet ketene Rollin A. King Wesley D. Allen,* Buyong Ma and Henry F. Schaefer III Center for Computational Quantum Chemistry Department of Chemistry University of Georgia Athens GA 30602[2525 USA The photofragmentation of ketene to triplet methylene and carbon monoxide is a paradigm for unimolecular dissociation over an exit channel barrier. The geometric structures quadratic force –elds and harmonic vibrational frequencies of the triplet ketene reactant the 3B1 CH2]1&` CO products and both in-plane (CsII) ( and out-of-plane Cs I ) transition states have been determined at the TZ(2d1f,2p) coupled-cluster singles and doubles (CCSD) level of theory. An unusual shallow minimum at long range [R(CwC)\4.0 ”] has also been discovered and characterized.A rigorous mapping and analytic parametrization has been performed of the TZ(2d1f 2p) CCSD intrinsic reaction paths connecting the CsII transition state to both the reactant and products. Final potential-energy functions along the entire reaction path have been determined with the aid of [(C,O)/H] atomicorbital basis sets as large as [6s5p4d3f2g1h/5s4p3d2f1g] and electron correlation treatments as extensive as the coupled-cluster method through triple excitations [CCSDT or CCSD(T)]. The –nal theoretical curve is highly anharmonic in the transition-state region displaying a classical barrier of 1045 cm~1 a critical CwC distance of 2.257 ” and a barrier frequency of 321i cm~1. EÜective barrier frequencies in the 100i cm~1 range which result from RRKM modelling with tunnelling corrections of the observed steplike structure in the triplet ketene dissociation rate constant are thus shown to be physically untenable.Various implications of such ab initio predictions on unravelling the intricacies of the fragmentation dynamics are discussed. 1 (A3 1AA) state which exhibits a weakened 23 1 Introduction The cover of the 12 June 1992 issue of Science and a perspective by Marcus1 appearing therein entitled ìSkiing the Reaction Rate Slopes,œ heralded a breakthrough test in which novel experimental evidence was advanced in support of one of chemistryœs most fundamental theories viz. the Rice»Ramsperger»Kassel»Marcus (RRKM)2h7 microcanonical form of transition-state theory.8 The experiments in focus were those of Lovejoy et al.,9 in which rate constants for the dissociation of highly vibrationally excited ketene (CH CO) were measured in a supersonic jet expansion at the threshold 2 X3 3B for production of and X1&` CO.By means of tunable pulsed laser excitation of ketene cooled to ca. 5 K high resolution of the energy dependence of the triplet 1 CH2 photofragmentation rate constant [k(E)] was achieved revealing clear steplike structures in accord with the long-standing RRKM premise that the rate of such a reaction is controlled by —ux through a quantized transition state. These experiments which have been described by Smith10 as a tour de force were in fact only part of a continuing series of investigations on both singlet and triplet ketene fragmentation performed over several years9,11h30 to test contemporary unimolecular reaction theories.A comprehensive collection of ab initio theoretical studies on ketene,31h45 dating back to the early 1970s has elucidated the photofragmentation process. Absorption of a near-UV photon excites the molecule to the S 24 Fragmentation surface of triplet ketene carbon»carbon bond and a strongly bent CwCwO skeleton.40,41,44 Intersections of the S and S potential-energy surfaces in the vicinity of the A3 1AA equilibrium promote 0 1 rapid internal conversion to the ground electronic state.39,45 For energies above 30 116 cm~1,15 the resulting highly vibrationally excited ketene can dissociate directly to a8 1A1 CH2]1&` CO via paths constrained by orbital symmetry to be out-of-plane bent Cs I .36,41,43 However a lower-energy channel can be accessed by intersystem 1 C stationary s I 2 (3A1) state generated by 2v) ketene but is also smoothly connected to the denoted as crossing of S to the T (a8 3AA) state of ketene whose electronic structure permits a 0 Woodward»HoÜmann-allowed fragmentation to 3B1 CH2]1&` CO along in-plane bent or CsII paths.41,43 Notwithstanding the ful–llment of orbital-symmetry requirements the CwC r-bonding electron pair of a8 3AA ketene must be completely released into the 5r orbital of CO and the in-plane unpaired electron must relocalize into the 3a orbital of CH in order to eÜect dissociation.41 Repulsive interactions within these 1 2 in-plane frontier orbitals are diminished by bending of the CwCwO backbone in CsII conformations but not sufficiently to avert a small barrier (ca.15 kJ mol~1) in the exit channel.17,27 The fragmentation of triplet ketene is thus mediated by a classic transition state which dynamically controls energy release and which has been the target of the intense spectroscopic interrogations cited above. In 1988 two of us reported an ab initio characterization41 of the CsII fragmentation transition state of a8 3AA ketene executed with the restricted-Hartree»Fock (RHF) and con–guration interaction singles and doubles (CISD) methods in a double-f plus polarization (DZP) basis set. Concurrently investigated was an accompanying point of 3A@ CH CO which correlates backward to the T vertical excitation of ground-state (C 2 a8 3AA (CsII) H transition state by torsional motion.40 In 1990 Chen and Moore18 found that the rotational-state distributions of CO produced in ketene photolysis near 2CwCO the triplet threshold were remarkably well reproduced by a simple impulse model applied to the theoretical geometric structure and harmonic vibrational frequencies of C the sII col con–rming especially the predicted extent of CwCwO deformation from linearity.The later groundbreaking work9 of 1992 pushed the synergy between theory and experiment further as the early intervals of the fascinating jumps observed in k(E) were rationalized on the basis of the ab initio transition-state vibrational levels.41 In brief the steplike structure in the dissociation rate constant appeared to be explainable within the RRKM formalism in terms of new vibrational states orthogonal to the reaction coordinate which become accessible in the transition state as the energy is increased.A full analysis of the observed energy pro–le of k(E) was completed by Kim et al. in 1995.27 The –rst steps in k(E) were indeed successfully assigned to the three lowest-energy modes in the transition state the H2CwCO torsion the CwCwO bend and the CH2 wag treated in the –rst instance as a hindered rotor with an internal rotation barrier of 240 cm~1 and in the last two cases as small-amplitude normal vibrations with frequencies of 250 and 290 cm~1 respectively. By comparison the best theoretical values41 for the transition-state features were 384 cm~1 for the torsional barrier and 252 and 366 cm~1 for the corresponding harmonic vibrational frequencies.The threshold for triplet ketene fragmentation was pinpointed by the same analysis27 of the k(E) experiments yielding a vibrationally adiabatic barrier of 1281^15 cm~1 for the exit channel or ca. 40% less than the best ab initio prediction.41 In summary if the anticipated accuracy of the various pre-1990 ab initio quantities is considered the central interpretations of the breakthrough experiments on triplet ketene fragmentation seemed to be nicely con- –rmed by electronic structure theory. The appealing con—uence of theory and experiment has not withstood detailed scrutiny however. First Kim et al.27,28 were frustrated in their attempts to explain several details of the steplike energy-dependence of the reaction rate by standard RRKM theory.Second when one-dimensional tunnelling corrections were incorporated into the 25 R. A. King et al. t C RRKM modelling of triplet ketene fragmentation through the sII transition state empirical barrier frequencies in the (100^40)i cm~1 range were found,17,27 lower than either the direct ab initio results,41 or reasonable extrapolations therefrom by a factor of at least 3. Third the RRKM –ts27,28 of k(E) near the triplet ketene threshold gave a density of CH2CO reactant states equal to 1.11g times the Whitten»Rabinovich estimate where g is the number of spin sublevels in the T manifold strongly coupled by t 1 Both experiments30 on the singlet dissociation channel of intersystem crossing to S0 .ketene and direct state counts43 achieved by an extensive theoretical anharmonic vibrational analysis42 of the X3 1A1 CH2CO surface have shown recently that g is very close to 1 rather than 3 a discovery which requires explanation if the interpretation of the t triplet k(E) pro–le is to stand. Such lacunae attracted dynamical investigation of the problem by Gezelter and Miller46 in 1996. They utilized a quantum reactive scattering methodology of absorbing boundary conditions (ABC) with a discrete variable representation (DVR) to obtain the cumulative reaction probability for dissociation over the barrier on the triplet surface. The dissociation rates arising from reduced-dimensionality computations accounting for one or two internal modes gave good overall agreement with experiment but the steplike features recorded by Kim et al.27 were washed out by tunnelling through the relatively narrow barrier predicted by improved ab initio theory.47 Model calculations con–rmed that a barrier frequency below 100i cm~1 is required to recover the observed jumps in k(E).Gezelter and Miller46 were led to suggest three possibilities (a) there is another transition-state region on the T surface farther out towards the product channel; (b) there is surface-hopping dynamics taking place between the T and S 1 1 potential-energy surfaces ; or (c) the ab initio barrier frequency is simply too large. To0 quote these authors ììWhich scenario is a most likely explanation for the observed features awaits a more comprehensive exploration of both the triplet and singlet potential energy surfaces by ab initio methods.œœ Cui and Morokuma did not view the third possibility as a viable explanation and thus pursued a very recently reported ab initio study45 of non-adiabatic interactions in the photodissociation of ketene.Seams of crossing among the S0 S1 T1 T2 and surfaces and associated matrix elements of spin»orbit coupling were computed at the 6- 31G(d,p) CASSCF and EOM-CCSD levels of theory. It was shown inter alia that C crossing of S with T sII (Cs I ) symmetry occurs both at the triplet equilibrium 1 (T2) in 0 and all along possible dissociation pathways; moreover as the CwC separation increases the energy of the crossing increases and the crossing structures deviate substantially from the intrinsic reaction paths through the transition states.The question of whether intersystem crossing plays a predominant role in the triplet fragmentation dynamics thus appears to rest on the relative importance of crossings at lower versus higher energies.45 Finally new insight into this problem has been achieved recently by Morgan et al.,48 who used metastable time-of-—ight spectroscopy to measure the translational energy distribution of speci–c rotational states of CO formed from ketene photodissociation at 351 nm. In this manner the correlated internal energy distribution of the 3B ment was obtained and found to exhibit some marked deviations from the predictions of 1 CH2 fragthe simple impulse model which previously simulated18,27 the CO rotational distribution so well.It was suggested that strong vibrational coupling in triplet ketene between internal rotation about the CwC bond and CH wagging motion operative in the inter- (CsII Cs I ) triplet ketene transition-state region is responsible for preferential connected 2 a-axis rotational excitation of the 3B tive scattering calculations seeking to elucidate the mystery of the steplike structures in 1 CH2 product. Presumably future quantum reacthe k(E) pro–le should treat this eÜect appropriately. The decade which has passed since our –rst ab initio examination41 of triplet ketene fragmentation paths has brought dramatic improvements both in electronic structure 26 Fragmentation surface of triplet ketene methodologies and in the computational platforms for their execution increasing the achievable level of predictive certainty by perhaps an order of magnitude.In this paper the highest possible levels of ab initio theory are applied to the triplet ketene problem in order to exclude the aforementioned possibilities (a) and (c) of Gezelter and Miller,46 to suggest yet another issue to be considered and to map conclusively critical aspects of the fragmentation surface thus laying the groundwork for improved dynamical computations. After specifying computational methods (Section 2) a detailed examination is made of optimized geometric structures for the pertinent stationary points of triplet ketene fragmentation (Section 3.1) vibrational force –elds for these structures are presented (Section 3.2) and an analytic parametrization is achieved for the entire (CsII) intrinsic reaction path (Section 3.3).An energetic analysis is then provided which dissects the approach of the electronic structure hierarchy to the one- and n-particle ab initio limits of the dissociation energy and exit (or association) barrier (Section 3.4). With this information one-dimensional analytic potential-energy functions are developed for the (CsII) reaction path at very high levels of theory and the position height and shape of the exit barrier are thus revealed (Section 3.4). Finally some implications of the theoretical results are highlighted along with contextual remarks (Section 4). d (O)\1.428 ; a (H)\0.388 1.407] 2 Computational methods The reaction path for the dissociation of a8 3AA ketene was investigated by means of carefully selected one-particle Gaussian basis sets developed by Dunning and coworkers.49h53 Most analytic gradient computations utilized a TZ(2d1f,2p) basis comprised of established C,O(10s6p/5s3p) and (unscaled) H(5s/3s) segmented sp contractions49 augmented with correlation-optimized polarization manifolds50 [a (C)\ af(C)\0.761 ; a (O)\0.645 0.318 1.097 ; af 2.314 ; d giving a total of 111 orbitals. For single-point computations along the reaction path the p cc-pVXZ family of basis sets50h53 was employed to eÜectuate a systematic approach of energetic predictions to the complete basis set limit. In particular correlated wavefunctions were determined for the (X\D T Q 5) series whose contracted Gaussian sets for the [(C,O)/H] atoms range in extent from [3s2p1d/2s1p] to [6s5p4d3f2g1h/ 5s4p3d2f1g] or from 52 to 383 independent functions.In all work reported here supernumerary components of the Cartesian polarization sets were removed to create manifolds spanning only pure spherical harmonics.54 Reference electronic wavefunctions were determined by the single-con–guration selfconsistent-–eld restricted and unrestricted Hartree»Fock methods (RHF and UHF).55h58 Dynamical electron correlation was accounted for by second-order M‘ller»Plesset perturbation theory (MP2),57h60 by the coupled-cluster approach61h63 including all single double (CCSD)62h65 and in cases triple substitutions (CCSDT),66h68 or by CCSD theory appended with a perturbative contribution from connected triple excitations [CCSD(T)].69,70 In mapping the stationary points and fragmentation path of triplet ketene TZ(2d1f,2p) open-shell coupled-cluster energies and gradients were determined by application of the ordinary spin»orbital formalism of the theory with RHF orbitals71,72 and with no restrictions on the active space.In contrast –nal energetic computations along the reaction path employed UHF reference wavefunctions in the correlation treatments denoted here as UMP2 UCCSD UCCSD(T) and UCCSDT for clarity ; the carbon and oxygen 1s core orbitals were then kept doubly occupied but no virtual counterparts were frozen. The electronic structure computations were executed by the program packages ACESII,73 GAUSSIAN94,74 and PSI.75 Analytic gradient techniques76 at the TZ(2d1f,2p) CCSD level of theory were used to tightly optimize geometric structures and evaluate harmonic vibrational frequencies for the reactant transition states long-range complex and products of triplet ketene fragmentation.In the –nite-diÜerence procedures for the quadratic force –elds internal coor-27 R. A. King et al. dinate displacements of ” ^0.005 or ^0.02 rad were applied and the numerical error in the resulting frequencies is gauged to be less than 0.5 cm~1. The TZ(2d1f,2p) CCSD transition state (CsII) structure and force –eld provided a starting point for a detailed mapping of the intrinsic reaction path77,78 both forward to the 3B1 CH2]1&` CO products and backward to the a8 3AA CH2CO equilibrium.In particular steepest descent paths in mass-weighted Cartesian coordinates were meticulously followed with a step size of 0.2 au via the algorithm of Gonzalez and Schlegel.79 By means of extensive testing with the DZP RHF and DZP CISD levels of theory employed in earlier work on triplet ketene fragmentation,41 algorithm step sizes and microiteration tolerances were selected to yield 5 signi–cant digits of accuracy in the internal coordinates over accumulated arc lengths of 3 au. After myriad TZ(2d1f,2p) CCSD analytic gradient runs the intrinsic reaction coordinate (s) was propagated 3.4 (2.2) au of arc length toward reactant (products) bringing the total energy within 220 (650) cm~1 of the endpoint. Explicit generation of the intrinsic reaction coordinate past these points proved both prohibitive and unnecessary owing to the —atness of the potential-energy surface.However it was observed that the –nal backward tail of the reaction path into the a8 3AA CH2CO equilibrium could be efficaciously determined by running the path-following algorithm on the reactant quadratic force –eld as augmented with the semidiagonal cubic force constants appearing as byproducts of its numerical evaluation. Moreover classical electrostatic considerations (vide infra) provide asymptotic constraints for variation of the interfragment angles in the product tail of the reaction path. With such auxiliary input the explicit TZ(2d1f,2p) CCSD data were sufficiently supplemented to allow the entire intrinsic reaction path for triplet ketene fragmentation to be parametrized.3 Theoretical results 3.1 Stationary structures Ab initio geometric structures for the stationary points along the C dissociation path of sII a8 3AA ketene are speci–ed in Table 1 and depicted in Fig. 1. The characteristic feature of the a8 3AA structures is the strongly bent CwCwO skeleton a consequence of the massive stabilization of the singly occupied HOMO in this excited state ensuing from C structures are distortion of the cumulenone moiety from linearity. In this manner the sII in accord with the principles of Walsh,80,81 conservative of orbital symmetry along the fragmentation path,41,82 and analogous to the lowest triplet excited states of CO The TZ(2d1f,2p) CCSD geometric predictions constitute substantial improvements over 2 .83 Fig.1 CsII transition state and internal coordinates for the dissociation of a8 3AA ketene dissociation path of a 8 3AA ketenea Table 1 Stationary structures along the CsII fragmentation transition state triplet CH2CO equilibrium DZP CISD DZP RHF DZP CISD DZP RHF TZ(2d1f,2p) CCSDb r(CwO) R(CwC) tanti(CwH) 1.1481 2.0708 1.0813 1.1315 1.9472 1.0742 1.1863 (1.1865) 1.4464 (1.4423) 1.0766 (1.0727) 1.1930 1.4658 1.0804 1.1716 1.4660 1.0735 tsyn(CwH) d(CwCwO) a(HwCwH) 1.0843 116.26 128.21 1.71 3159 1.0781 117.67 126.98 0.78 6465 1.0855 127.65 120.11 0.23 1.0831 (1.0790) 128.13 (128.32) 120.13 (120.23) 1.0780 129.25 120.46 [0.34 [2521 *b(HwCwC) 0.01 (0.17) [8104 ([8537) *Ee d [6369 a Bond distances in ” bond angles in degrees and relative energies *Ee) in cm~1.The geometric parameters are ( depicted in Fig. 1; *b4bsyn[banti is the methylene rocking angle. The DZP RHF and DZP CISD data are taken from earlier work.41 b Corresponding cc-pVTZ CCSD results are given in parentheses. c Ref. 84 and 85. d Referenced to the following absolute total energies of the products [151.687 022 (DZP RHF) [152.050 111 (DZP CISD) [152.266 017 [TZ(2d1f,2p) CCSD] and [152.251 618 (cc-pVTZ CCSD). Fragmentation surface of triplet ketene TZ(2d1f,2p) CCSDb 1.0812 » 131.14 »0 1.0788 (1.0743) 115.38 (115.42) 130.52 (130.88) 129.36 »0 0.98 (0.87) 1659 (1646) 28 3B1 CH2]1&` CO products TZ(2d1f,2p) CCSD DZP CISD DZP RHF expt.c 1.1337 (1.1340) 2.1799 (2.1835) 1.0765 (1.0722) 1.1174 1.1386 1.1252 1.1283 O O O O 1.0812 1.0759 1.0753 1.0756 1.0756 » 1.0753 » 133.93 »0 1.0759 » 133.48 »0 29 R.A. King et al. earlier DZP RHF and DZP CISD results,41 as well as partial 6-31G(d,p) CASSCF and EOM-CCSD structures reported recently.45 For the 3B1 CH2]1&` CO products the TZ(2d1f,2p) CCSD optimum r(CwO) t(CwH) and a(HwCwH) values deviate from experiment84,85 by only [0.003 ” ]0.0006 ” and [0.45° respectively indicating that this level of theory achieves an excellent balance between basis set quality and degree of electron correlation.86,87 The TZ(2d1f,2p) CCSD geometric parameters are generally in close agreement with the auxiliary cc-pVTZ CCSD results of Table 1 and are probably superior in quality for the CwH bond lengths.Moreover the TZ(2d1f,2p) CCSD total C stationary points is always over 10 mh (10 mE energy at the sII h) lower than its cc-pVTZ counterpart despite being computed with seven fewer contracted Gaussian functions (111 vs. 118). As a –nal merit the TZ(2d1f,2p) CCSD level provides a feasible dynamically correlated method for computing analytic gradients which simultaneously yields a fully optimized binding energy (8104 cm~1) and barrier height (1659 cm~1) commensurate with the best previous single-point energy data.41 C transition state sII equilibrium requires elongation of the CwC bond from 1.446 ” According to the TZ(2d1f,2p) CCSD method ascending to the 2CO from the a8 3AA CH which is halfway between prototypical double- and single-bond distances,40 to 2.180 ”.Attendant structural changes of signi–cance are the 0.053 ” contraction of r(CwO) the 12.8° increase in the deviation of d(CwCwO) from linearity and the 10.4° widening of a(HwCwH). The CsII col is thus a classic late transition state as r(CwO) and a(HwCwH) have already completed 86% and 78% respectively of their egress toward the product structure. It is clear from Table 1 that the transition state systematically migrates toward products and the barrier height is concomitantly lowered as the level of theory is improved; n.b.the [DZP RHF]DZP CISD]TZ(2d1f,2p) CCSD] series yields R(CwC)*\(1.947 2.071 2.180) ” and the classical barriers (6465 3159 1659) cm~1. The lack of convergence in these predictions is problematic and warrants further theoretical eÜorts. In brief while the TZ(2d1f,2p) CCSD method seems entirely adequate for mapping the multivariate geometric path for fragmentation de–nitive determinination of the energy pro–le along this path especially the position and height of the barrier cannot be achieved until much higher levels of theory are invoked as reported in Section 3.4. Cs I dissociation path of 3A@ 1 3A@(C con–guration space to the 8 3AA(C s I ) asII) surface despite the 1 ]3b1) (C of ground-state ketene.40,44 Starting from the TZ(2d1f,2p) Stationary points on the companion out-of-plane bent ketene are characterized in Table 2 and Fig.2. This path de–nes the torsional ridge separating equivalent CsII routes and regulating z-axis internal rotation dynamics during triplet ketene fragmentation.41 A widely depicted88 conical intersection is responsible for smoothly connecting the correlation of the two electronic states to diÜerent vertical excitations [3A (2b 3A (2b and 1 ]3b2)] 2 2v) CCSD equilibrium structure of a8 3AA ketene execution of HwCwCwO torsional Fig. 2 Cs I transition state and internal coordinates for the dissociation of 3A@ ketene triplet CH2CO DZP CISD DZP RHF r(CwO) R(CwC) t(CwH) 1.1861 1.5180 1.0841 1.1652 1.5163 1.0788 d(CwCwO) a(HwCwH) c(H2CwC) 125.47 117.66 32.62 412 123.25 118.64 28.17 *Ee d [3266 a Bond distances in ” angles in degrees and relative energies *Ee) in cm~1.The geometric ( parameters are depicted in Fig. 2; c is the methylene wagging angle. The DZP RHF and DZP CISD data are taken from earlier work.41 b Ref. 84 and 85. c Table VIII of ref. 41 contains a misprint of this value. d See footnote d of Table 1. Table 2 Stationary structures along the CIs fragmentation transition state DZP CISD DZP RHF TZ(2d1f,2p) CCSD 1.1488 2.0544 1.0833 1.1325 1.9444 1.0775 1.1765 1.5073 1.0800 117.74 126.98 23.06 3669 119.00 124.90c 30.44 123.19 119.57 24.93 [4711 7076 Fragmentation surface of triplet ketene dissociation path of 3A@ ketenea TZ(2d1f,2p) CCSD DZP CISD DZP RHF TZ(2d1f,2p) CCSD exptb 1.1347 2.1474 1.0780 1.1174 1.1386 1.1252 1.1283 O O O O 1.0812 1.0759 1.0753 1.0756 » 129.36 »0 » 133.93 »0 » 133.48 »0 » 131.14 »0 116.66 129.49 19.98 2048 30 3B1 CH2]1&` CO products 31 R.A. King et al. 3A@(C torsional maximum which actually has a Hessian index of 2. Because the s I ) CsII fragmentation transition states,41 3A@(C stationary points 8 3B1 CH2]1&` CO and e\208 cm~1. Computation of the quadratic force constant matrix u (CH wag)\154 cm~1 and u9(HwCwCwO 2C motion requires a 0.061 ” elongation of R(CwC) a 4.9° tightening of d(CwCwO) and a 3393 cm~1 increase in energy to arrive at the corresponding 3A@(Cs I ) torsional maximum.By comparison HwCwCwO torsional excursion from the TZ(2d1f,2p) CCSD a ” 8 3AA fragmentation transition state is accompanied by a 0.032 shortening of R(CwC) a 1.3° widening of d(CwCwO) and a 389 cm~1 increase in energy in reaching its DZP RHF and DZP CISD methods give tighter their torsional barrier predictions 611 and 510 cm~1 respectively are substantially higher than the superior TZ(2d1f,2p) CCSD result. Finally the s I ) all exhibit modest methylene wagging angles (20°»35°) which systematically decrease both with CwC stretching and improvements in the level of theory. No previous studies have addressed the question of a long-range minimum on the triplet ketene fragmentation surface which becomes an issue in the current investigation because of the possibility that the reaction path leading from the CsII transition state might terminate on such a complex.The TZ(2d1f,2p) CCSD level of theory provides a proper description of the small troublesome dipole moment of carbon monoxide yielding ke\0.0879 D (cf. k0\0.10980 D from experiment89) with the correct ChO` orientation ; the corresponding prediction for triplet methylene a less problematic species is ly robust for searches at large separations for shallow minima. Starting from very large 0.611 D oriented as ChH2 `. Therefore the TZ(2d1f,2p) CCSD method appears sufficient- CwC distances with the dipole moments of the fragments favourably oriented sampling of the interaction energy was performed to identify a reasonable geometry for beginning a structural optimization.It was discovered that a driving force exists for removing head-to-tail alignments of the fragment dipoles. After a tedious unconstrained and tight optimization in the CsII space with analytic TZ(2d1f,2p) CCSD gradients the structure shown in Fig. 3 was found. The complex exhibits a CwC distance very near 4.0 ” intrafragment coordinates barely disturbed from those of free a binding energy D for out-of-plane distortions provided torsion)\13 cm~1 indicating that the loose harmonic approximation for zero-point vibrational energy (ZPVE) is likely to cancel out sII structure is a local minimum. While the most of the potential energy of binding for the complex we speculate that a proper e) Fig.3 Long-range TZ(2d1f,2p) CCSD minimum of a8 3AA ketene with depiction of optimum geometric coordinates and the orientation of fragment dipole moments. The binding energy (D with respect to 3B1 CH2]1&` CO is 208 cm~1 whereas the out-of-plane (aA) harmonic vibrational frequencies are 154 (CH wag) and 13 (HwCwCwO torsion) cm~1. 2 32 Fragmentation surface of triplet ketene 2 anharmonic accounting of ZPVE would unambiguously show that the long-range minimum does support bound vibrational states. The most fascinating feature of the CH … … …CO complex is the nearly orthogonal orientation of the fragment dipole vectors suggesting that its physical origin is some convolution of higher-order (quadrupolar) nascent electrostatics with dispersion and exchange repulsion phenomena.In fact the dipole moment of the complex is 0.688 D directed at [125.1° with respect to the z axis in Fig. 3 whereas the vector sum of the fragment dipoles gives a magnitude of 0.629 D with a bearing of [114.1°. Polarization and charge transfer eÜects may thus be operative to a small extent as well. The multipole expansionao for the electrostatic energy of interaction (W ) between two uncharged separated fragments (A B) with dipole moments l and traceless quadrupole tensors Qis W \ 1 R3 [lA Æ lB[3(lA Æ n)(lB Æ n)] ] 1 R4 nTQAlB[nTQBlA] 2 5 (lA Æ n)(nTQBn)[ 2 5 (lB Æ n)(nTQAn)D] . . . (1) (3) 2RCC 4 2R C where the moments are computed with a common orientation of Cartesian axes but about distinct intrafragment origins a and b separated by a distance R while n is the unit vector directed from a to b.In the respective principal axis systems with carbonatom 0 au) origins the 0 0 [ 1 2 11.500 TZ(2d1f,2p) CCSD method yields for 2.200 the CO and CH2 0 fragments 2 (in QCO\ ; QCH2\ 0 (2) 1.500 0 0 3.021 0 [3.000 0 [0.821 0 0 0 If / and s denote the angles between n (i.e. the z axis in Fig. 3) and the local principal axes of CO and CH respectively then eqn. (1) reduces to 2 W \ kCOkCH2 [3 cos / cos s[cos(/[s)] RCC 3 ] kCH2Qzz CO [eCO sin 2/ sin s]3 cos s(1]eCO sin2 /)] ] kCOQzz CH2 [eCH2 sin 2s sin /]3 cos /(1]eCH2 sin2 s)]]. . . CC 4 e4(Qyy[Qzz)/Qzz where or (eCO eCH2)\([1.500 [4.681) in particular.At the carbon»carbon separation of the complex (3.9946 ”) the global minimum of eqn. (3) occurs as expected when both fragment dipoles are aligned in the [z direction of Fig. 3. The resulting electrostatic energy is W \[79 cm~1 of which only [8 cm~1 is due to the dipole»dipole term. Subsequently comparing W (/ s) to explicitly computed TZ(2d1f,2p) CCSD binding energies for rigid internal rotation over a full period of the *b(HwCwC) rocking angle provided additional insight into the nature of the long-range complex. With the fragment dipoles aligned in the [z direction the actual quantum mechanical interaction energy is [83 cm~1 almost identical to the optimum W (/ s) value. However such a structure proves to be one of two maxima for internal rotation the other occurring at ]26 cm~1 when the fragment dipoles orient tail-to-tail along the z axis.In between these maxima the quadrupole»quadrupole and/or non-classical interactions inherent in the TZ(2d1f,2p) CCSD approach overtake the lower-order electrostatic components and drive the system to the [208 cm~1 minimum of Fig. 3 where eqn. (3) yields an interaction energy of only [42 cm~1. This analysis demonstrates that the precise physical C complex appears to be sensitive to origin of the binding in the intriguing long-range sII the orientation of the fragments. 3.2 Harmonic force –elds Quadratic force constants for the a8 3AA CH2CO equilibrium its fragmentation transition 3B state and the 1 CH2]1&` CO products are presented in Table 3 and the corresponding harmonic vibrational frequencies are listed in Table 4 together with associated results for the s I ) Table 3 TZ(2d1f,2p) CCSD force constants (F ciation path of a8 3AA ketenea Fij 11 21 22 31 32 33 41 42 43 44 51 52 53 54 55 61 62 63 64 65 66 71 72 73 74 75 77 76 88 98 99 S S S S 1\2~1@2[tsyn(CwH)]tanti(CwH)] 2\a(HwCwH) 3\2~1@2[tsyn(CwH)[tanti(CwH)] 4\r(CwO) S5\R(CwC) a Units consistent with energy in aJ distances in ” and angles in rad.b The methylene wagging angle is de–ned to be positive for hydrogen excursions in the [x direction as shown in Fig. 2. 3A@(C dissociation col.These TZ(2d1f,2p) CCSD results not only provide CH equilibrium 2CO 5.6696 0.0900 0.6666 [0.1376 0.0052 5.6526 0.0407 [0.0390 0.0583 14.2206 0.0983 [0.2253 0.0266 1.3979 4.6109 [0.0317 0.0095 [0.0907 0.3266 0.3346 0.9879 [0.0051 0.0209 0.1080 [0.0420 [0.0706 [0.1520 0.5057 0.1542 0.0437 0.1126 R. A. King et al. S along the C disso- ij) (i) sII and internal coordinates fragmentation transition state 5.6894 0.1571 0.4257 [0.0465 0.0028 5.8249 0.0325 [0.0817 0.0186 18.8436 [0.0132 [0.0610 0.0121 0.6759 [0.4922 [0.0035 0.0255 [0.0135 0.0926 0.1052 0.3124 0.0019 [0.0002 [0.0114 0.0013 [0.0255 [0.0447 0.0915 0.0206 0.0029 0.0161 S S S S 6\d(CwCwO) 7\2~1@2[bsyn(CwCwH)[banti(CwCwH)] 8\c(H2CwC)b 9\qsyn(HwCwCwO) 33 3B1 CH2]1&` CO products 5.7634 0.1567 0.3756 0 0 5.9204 0 0 0 20.2232 0 0 00 0 0 0 0 000 0 00 0 0 0 0 000 34 Fragmentation surface of triplet ketene Table 4 TZ(2d1f,2p) CCSD harmonic vibrational frequencies (cm~1) for stationary structures of triplet ketene 3B1 CH2]1&` CO products 3AA (CsII) 3A@ (C transition state transition state s I ) 3AA (C equilibrium sII) asym CwH stretch sym CwH stretch u3(b2) 3374 u1(a1) 3152 2 ue(r) 2238 u2(a1) 1110 » » u1(a@) 3279 u1(a@) 3344 u2(a@) 3139 u2(a@) 3135 u3(a@) 1859 u3(a@) 2141 u4(a@) 1460 u4(a@) 1167 u5(a@) 1079 u5(a@) 421 u6(a@) 988 u7(a@) 379i u6(a@) 394i CwO stretch CH scissor CH rock 2 CwC stretch CwCwO bend » » » u7(aA) 3338 u1(a@) 3137 u2(a@) 2129 u3(a@) 1180 u8(aA) 123 u u5(a@) 267 4(a@) 352 u9(aA) 43i 4937 5263 harmonic ZPVE u7(a@) 474 u6(a@) 228 CH wag u8(aA) 761 u8(aA) 338 HwCwCwO torsion u9(aA) 389 u9(aA) 132 2 6714 5453 de–nitive characterization of the salient stationary structures and zero-point vibrational energy corrections for the reaction energy pro–le but also constitute key reference points in the generation of the quadratic force –eld along the entire reaction path a task currently in progress.Indeed force –elds for the orthogonal complement space of vibrations will be central to the application of reaction-path Hamiltonians to the dissociation dynamics of triplet ketene.Calibration of the TZ(2d1f,2p) CCSD vibrational frequencies can be carried out on the 3B1 CH2]1&` CO products. In 1988 Jensen and Bunker85 employed a Morse oscillator-rigid bender internal dynamics (MORBID) Hamiltonian in a –tting to all extant rotation»vibration spectroscopic data for 3B1 CH2. We have taken their analytic molecular potential energy function and diÜerentiated it to obtain the quadratic force –eld for the equilibrium structure of triplet methylene. The purely harmonic empirically based normal vibration frequencies computed therefrom are u1(a1)\3141 u2(a1)\ u 1081 and 3(b2)\3317 cm~1 which indicate that the corresponding TZ(2d1f,2p) CCSD frequencies are in error by only ]0.4% ]2.7% and ]1.7% respectively.Furthermore for carbon monoxide the error in u compared to the standard experimental e value (2170 cm~1)84 is ]3.1%. This high level of accuracy obviates the need for the mode-speci–c scale factors invoked previously,41 which suÜer from some degree of unreliability and spawn confusion over anharmonic eÜects. u data in Table 4 con–rm that the 3AA(C i 3A@(C (*Ee*) and critical vibrational frequencies (u*) predicted for 3AA(CsII) ( fragmentation transition state at various levels of theory *Ee* u*) vary as The TZ(2d1f,2p) CCSD sII) equilibrium 3AA(CsII) transition state and Hessian indices of 0 1 and 2 respectively. These new results delineate a clear trend s I ) transition state are stationary structures with between the barrier heights the (6465 871i)](3159 560i)](1659 379i) cm~1 as DZP RHF]DZP CISD]TZ(2d1f 2p) CCSD.Accordingly the magnitude of the critical frequency at the TZ(2d1f,2p) CCSD level is much smaller than both those found in earlier work41 and the DZ(d p) (10e/9MO) CASSCF value of 552i cm~1 obtained by Cui and Morokuma.45 Because the model dynamical calculations of Gezelter and Miller46 reveals that a barrier frequency below 100i cm~1 is required to recover the step structure seen in the experimental dissociation rates the question of whether even more rigorous ab initio computations would reduce u* further by a factor of 4 has become controversial. Again the concerted theoretical analysis described in Section 3.4 is necessary to resolve this issue.For the modes complementary to the reaction path the trek from the 3AA equilibrium to its transition state for dissociation engenders expected changes in the harmonic frequencies. In particular the intrafragment frequencies migrate directly toward the 35 R. A. King et al. sII)]3A@(Cs I ) C traverse in the fragmentation transition state reduces theH2 rocking i 3A@(C torsional frequency u analogous product values with the two most aÜected modes the CwO stretch and CH scissor undergoing 74% and 80% of their overall shift respectively. Meanwhile the2 interfragment frequencies diminish greatly by factors of 3 to 4. Note also that the 3AA(C frequency by an additional factor of 3.4 while hardly aÜecting u for the other modes.Finally the TZ(2d1f,2p) CCSD prediction for the s I ) cm~1 is almost 4 times smaller in magnitude than earlier DZP RHF and DZP CISD values,41 in accord with the aforementioned reduction of the torsional barrier with enhancement of the electronic structure method. 9\43i In almost all cases the TZ(2d1f,2p) CCSD harmonic frequencies for the complemen- 3AA(CsII) equilibrium the TZ(2d1f,2p) CCSD tary modes are consistent with both the scaled and unscaled results reported before,41 especially considering the anharmonic eÜects embedded in the mode-speci–c scale factors used earlier. To wit for the approach uniformly reduces the unscaled DZP CISD (DZP RHF) in-plane u values by i 2.7% (6.4%) whereas the aA torsional frequency is changed by ]1.8% ([1.8%).However u (CH wag) of the equilibrium structure is an exception in this comparison 8 as the TZ(2d1f,2p) CCSD method shifts it upward by a sizable 6.7% (11.9%) necessi- 2 tating substantial revision of the –nal theoretical u values which were previously scaled 8 downward by 2.5% (15.9%).41 For the 3AA(C theory reduces all of the earlier DZP CISD (DZP RHF) frequencies by a modest 2.0% sII) transition state TZ(2d1f,2p) CCSD (4.4%) for the stretching modes but by a noteworthy 12.1% (21.9%) for the angular degrees of freedom. 3.3 Reaction path parametrization In Fig. 4»9 the evolution is shown of all seven internal coordinates along the C frag- sII mentation path of a8 3AA ketene. In Table 5 the functional forms which –t the TZ(2d1f,2p) CCSD data points are speci–ed and in Table 6 the requisite parameters are tabulated.de–ned to be zero at the Fig. 4 Variation of the CwC distance along the fragmentation path of a8 3AA ketene. The data points were generated at the TZ(2d1f,2p) CCSD level of theory and the solid curve displays the parametrization of the path via the functions of Tables 5 and 6. The arc length variable (s) is CsII transition state. 36 Fragmentation surface of triplet ketene sII fragmentation path of a8 3AA ketene Fig. 5 Variation of the CwO bond distance along the C (as Fig. 3) The non-linear –tting procedure for the parametrization of the reaction path was motivated by past experience90,91 and made possible by a powerful symbolic manipulation package.92 Speci–cally explorations were made of functional forms which characteristically vary smoothly e.g.for the intrafragment coordinates the technique of embedpath of a8 3AA ketene (as Fig. 3) Fig. 6 Variation of the syn (top) and anti (bottom) CwH distances along the C fragmentation sII R. A. King et al. C Fig. 7 Variation of the methylene bond angle along the (as Fig. 3) siderations and is imposed in the functional form of the –t. Fig. 8 Variation of the CwCwO angle along the CsII fragmentation path of a8 3AA ketene (as Fig. 3). A slow asymptotic decay to a linear C… … …OwC arrangement is predicted by electrostatic con- 37 sII fragmentation path of a8 3AA ketene 38 Fragmentation surface of triplet ketene considerations is imposed in the –t. Fig.9 Variation of the methylene rocking angle along the CsII fragmentation path of a8 3AA ketene (as Fig. 3). In the asymptotic region past s\5 au a decay to zero as predicted by electrostatic ding modulation functions within exponential functions proved useful. The reactant and product limits of Table 1 were imposed and careful consideration was given to ensuring proper asymptotic behaviour of the interfragment coordinates. The arc length (s) of the steepest-descent trajectories (in mass-weighted Cartesian coordinates for the parent C Table 5 Functional forms for parametrization of the sII dissociation path of a8 3AA ketene general de–nitions p f (s ; a b)41]a exp([bs) cm(s[s°)m n n(s ; c s°)4 ; h(s ; a b s*)4aM1[tanh[b(s[s*)]N r(s)\r R(s)\R°]p6(s ; c s°)[1]p5(s ; d s°)]~1 =][r°[r=]p4(s ; c s°)]exp[[(s[s°)g(s ; d s*)] m/1 g(s ; d s*)4d1]d2 sech[d3(s[s*)] distance functions CwC CwO CwH t(s)\t=][t°[t=]p4(s ; c s°)]exp[[(s[s°)g(s ; d s*)] angle functions HwCwH a(s)\a=][a°[a=]p6(s ; c s°)]exp[[(s[s°)g(s ; d s*)] CwCwO 4 d(s)\h(s ; d1 d2 s*) < f (s ; c2m~1 c2m) m/1 *b(s)\[*b°]p8(s ; c s°)]exp[[(s[s°)] CH rock 2 energy functions V [R(s)]\[d1]d1M1[exp[[c1(R[R1)]N2]d2 sech[c2(R[R2)]]d3 sech[c3(R[R3)] c1 c2 c3 c4 c5 c6 c7 c8 d1 d2 d3 d4 d5 s* asymptotic parameters R°\1.44637 r=\1.1252 a The units of the parameters are consistent with bond distances in ” angles in degrees and arc length in au.Suffixes in parentheses indicate powers of 10. r(CwO) R(CwC) 9.11([2) 9.93([2) 2.38([2) [7.97([3) 1.341([1) [3.410([1) 3.174([1) [4.597([3) 5.739([4) 1.48([2) [1.6785 6.585([1) 1.0383 1.555 1.12([1) 7.77([1) 5.269([2) [1.224([3) [4.12 0.319 r°\1.18627 =\1.0759 t° \1.07661 anti a=\133.48 t Table 6 Parameters for the anti(CwH) t 4.52([3) 2.38([3) [8.23([4) 1.92([3) 1.95([3) 6.11([3) [7.8([4) 1.334 1.85([1) 1.0 1.0 t° \1.08308 syn s°\[4.12 dissociation path of a 8 3AA ketenea CsII a(HwCwH) syn(CwH) t [8.850 [4.662 2.692 [6.288([1) 1.003([1) [6.88([3) 6.63([5) 6.290([1) [3.529([2) 5.254([1) 3.13([2) 1.0 1.0 [1.2056 1.0 *b°\0.012 a°\120.129 4.028([9) 4.196 107.776 39 *b(HwCwC) d(CwCwO) 5.72([2) 3.371([1) [1.312([2) 9.791 [1.040(1) 5.365 9.161([1) 2.27([3) 1.3918 5.089([2) [1.390 6.023([1) 3.695([2) 0.64 R.A. King et al. [1.034([1) 7.176([3) 40 Fragmentation surface of triplet ketene a8 3AA(CsII) minimum but long before the transitiona 8 3AA(C isotopomer) from the sII) transition state was chosen as the natural variable for the internal coordinate parametrizations. As such negative values of s refer to the backward descent to the reactant which was found via integration of the complete path (vide supra) to lie at s°\[4.12 au. For the positive direction of s the limiting form of the in–nitely long approach to products was taken to be a linear dependence on the CwC separation. As described in Section 2 explicit determination of the TZ(2d1f,2p) CCSD path between s\[3.4 and ]2.2 au provided the primary data set for –tting whereas a a8 3AA(C collection of ancillary points leading into the sII) minimum was generated from the corresponding semidiagonal cubic force –eld.For the most important primary data the rms errors of the –t were excellent erms\(0.00025 ” 0.00010 ” 0.00003 ” 0.04° 0.01° 0.02°) for [R(CwC) r(CwO) t 0.00002 anti(CwH) tsyn(CwH) a(HwCwH) d(CwCwO) *b(HwCwC)]. The interpolation errors of the CsII transition state structure (Table 1) were even more minuscule e(s\0)\(0.00002 ” 0.00004 ” 0.00001 ” ” 0.00002 0.002° 0.01° 0.01°) respectively. In Fig. 4 the functional variation of the CwC distance is shown to be almost perfectly linear for all values of s greater than [3.4 justifying the asymptotic form of the R(s) function in Table 5.For s\[3.4 R(CwC) rapidly levels oÜ; virtually all the energy of bond formation has been recovered and the –nal approach into equilibrium mainly entails relaxation of the bond angles. In Fig. 5 r(CwO) executes a barely perceptible rise before s\[3.4 and then undergoes most of its overall contraction prior to arriving at the transition state (s\0). As displayed in Fig. 6 the path dependence of the CwH bond distances is intriguing even though the range of variation is well below 0.01 ” in both cases. Each bond experiences maximum elongation between s\[2 and [3 i.e. outside the harmonic well of the state region is encountered. Note also that the longer bond (tsyn) decays toward the triplet methylene limit much more slowly than the shorter bond (tanti) presumably because of perturbations from the increasing deformation of the CwCwO skeleton.In Fig. 7 a(HwCwH) mirroring r(CwO) –rst dips slightly below 120° in the harmonic well and subsequently achieves most of its ascent before the transition state is traversed. a8 3AA(C a8 3AA(CsII) transition state decreases ” The routes taken by d(CwCwO) and *b(HwCwC) from reactant to products are more tortuous (Fig. 8 and 9). In exiting the sII) minimum the CwCwO skeleton –rst de—ects by ca. 2° back toward linearity but then a remarkably linear decrease with respect to s ensues which takes d(CwCwO) through the transition state and onward toward products. Concurrently the methylene rocking angle jumps sharply from zero at equilibrium to 1.8° near s\[3.5 oscillates back to ]0.3° at s\[1.3° and then pursues a steady acclivity through the transition state and to the endpoint of its sampling.The ultimate disposition of d(CwCwO) and *b(HwCwC) can be predicted from eqn. (3). As R]O the preferred fragment orientations occur at (/ s)\(0° 180°) and (180° 0°). Given the shape of the curve in Fig. 8 the –rst choice [/\d(CwCwO)]0] seems obvious and in either case *b becomes 0°. Moreover the explicit TZ(2d1f,2p) CCSD points for sP1.8 exhibit (positive negative) curvature in the (d *b) pro–les a fact which became much more obvious in tests which approximately extended the reaction path past s\2.2 by means of lower levels of theory.For these reasons the functional forms used to –t both d and *b were constrained to asymptotically approach 0° in the 3B1 CH2]1&` CO limit. The implication of the surmised long-range behaviour of the interfragment orientational angles is that the reaction path does not terminate on the minimum of Fig. 3 and that the attendant potential energy curve past the monotonically to the product limit. Because the loose CH … … …CO complex has 2 d(CwCwO)\171.4° and *b(HwCwC)\133.8° a sharp de—ection of the reaction path would be necessary to connect this minimum to the fragmentation transition state. It seems more likely that a ridge on the potential-energy surface shields the reaction path from the region of the bound complex. These conclusions are of course tentative.41 R. A. King et al. We certainly cannot rule out the possibility that the reaction path terminates on either the long-range minimum located in our search or on an alternative one which we have yet to locate. In summary more tedious and expensive eÜorts to follow gradient trajectories in very shallow long-range regions of the triplet ketene surface are needed to fully establish the asymptotic behavior of the reaction path but the information gained therefrom would be moot vis-a` -vis the fragmentation dynamics of greatest concern. 3.4 Potential-energy functions The development of an explicit high-quality reaction path for triplet ketene fragmentation facilitates the application of unusually high levels of electronic structure theory to this problem by eÜectively reducing the dimensionality requirements for locating and quantifying the association barrier (V *).To assess the performance of theoretical methods on the reaction energetics the dissociation energy (D and association barrier were subjected to a focal-point analysis at the TZ(2d1f,2p) CCSD optimum structures of e) Table 1. The focal-point scheme43,93h97 has the following characteristics (a) the utilization of basis sets which systematically approach completeness in this case cc-pVXZ for X\2 3 4 and 5; (b) the application of low levels of theory with very large basis sets direct-UHF and -UMP2 computations with several hundred basis functions here ; (c) the inclusion of higher-order correlation eÜects using smaller basis sets via a series of treatments through the (cc-pVDZ) UCCSDT level in the present study; (d) the layout of a two-dimensional grid for the dual extrapolation to the one- and n-particle ab initio limits based on an assumed additivity of correlation increments to the reaction energy pro–le; and (e) an avoidance of empirical corrections.Previous work has amply justi–ed the assumptions of the focal-point approach and its relation to other predictive thermochemical methods is discussed elsewhere.97 Results of the focal-point analysis are presented in Table 7. Several decisions made in its construction warrant mention. First test computations through the UMP5 level revealed that high orders of M‘ller»Plesset perturbation theory do not provide a systematic approach to the correlation limit.Also given recent concerns98 about the convergence of MPn series even for electronic systems dominated by a single reference coupled-cluster methods were favoured here and development of a competing perturbation theory analysis past second order was not attempted. Second distinct extrapolation formulae were used to approximate by means of sequences of cc-pVXZ computations the complete basis set limits of the UHF total energies and UMP2 correlation energies (4) EUHF(X)\EUHF = ]a exp([bX) and (5) Ecorr(X)4EUMP2(X)[EUHF(X)\Ecorr = ]X b 3 These simple forms are based on both theoretical considerations and bountiful computational observations.97,99h103 Finally UHF rather than RHF reference wavefunctions were adopted in the electron correlation procedures for determining the –nal potentialenergy curve for a8 3AA ketene fragmentation.Observations supporting this choice are that the UHF spin contamination never exceeds 0.08 au along the reaction path the unrestricted coupled-cluster series eÜectively coalesce before the CCSDT level is reached. UHF references provide superior starting values for V * and De and the restricted and To wit with the cc-pVDZ basis the [HF MP2 CCSD CCSD(T) CCSDT] values for V * are (3991 2372 2152 1824 1762) cm~1 in the UHF (frozen-core) series and (4949 2348 2194 1826 1770) cm~1 in the analogous RHF case. Table 7 Valence focal-point analysis (in cm~1) of the dissociation energy and association barrier of a 8 3AA ketenea dissociation energy cc-pVDZ (52) cc-pVTZ (118) cc-pVQZ/TZ (193) cc-pVQZ (225) cc-pV5Z (383) extrapolation limit (O) association barrier cc-pVDZ (52) cc-pVTZ (118) cc-pVQZ/TZ (193) cc-pVQZ (225) cc-pV5Z (383) extrapolation limit (O) a The analysis is performed at TZ(2d1f,2p) CCSD optimum structures.The symbol d denotes the increment in the relative energy (*Ee) with respect to the preceding level of theory in the correlation series UHF]UMP2]UCCSD]UCCSD(T)]UCCSDT. The higher-order correlation increments listed in brackets are taken for the purpose of extrapolation from corresponding entries for smaller basis sets thus yielding the –nal bracketed *Ee estimates. For each one-particle basis set the total number of contracted Gaussian functions is given in parentheses.The basis-set extrapolation limits are surmised from –ts of X\(3 4 5) RHF and X\(4 5) UMP2 energies to eqn. (4) and (5) of the text respectively. *E [UHF] e 2656 2944 2972 2980 3040 3077 3991 3960 3979 3977 3976 3974 d[UCCSD] d[UMP2] [253 [356 [381 ]4010 ]5329 ]5779 [[381] [[381] ]5801 ]5982 [[381] ]6173 [220 [216 [227 [1619 [2032 [2167 [[227] [[227] [2175 [2237 [[227] [2302 d[UCCSDT] d[UCCSD(T)] ]76 []76] []76] ]385 ]577 ]642 []76] []76] []642] []642] []76] []642] [62 [[62] [[62] [328 [434 [475 [[62] [[62] [[475] [[475] [[62] [[475] 42 Fragmentation surface of triplet ketene *E [UCCSDT] e 6874 [8570] [9088] [9118] [9359] [9587] 1762 [1216] [1048] [1038] [975] [908] 43 R.A. King et al. e Examining the data for both D and V * in Table 7 it is apparent that the basis-set convergence of the reference UHF values is reasonably rapid whereas that of the e d[UMP2] correlation increments is characteristically torpid. For the dissociation energy the cc-pVQZ basis is necessary to bring *E [UHF] to within 100 cm~1 of the inferred Hartree»Fock limit but for the association barrier all basis sets yield e *E [UHF] values within 20 cm~1 of the extrapolated result. It is noteworthy that Hartree»Fock theory gives a barrier height larger in magnitude than the well-depth a e gross de–ciency which is corrected by the prominent second-order increments.After steady aggrandizement the d[UMP2] terms reach apparent D and V * limits of ]6173 e and [2302 cm~1 respectively. Accordingly at the cc-pVQZ benchmark the respective net UMP2 predictions for these critical energetic features of the fragmentation curve are indicated to be too small in size by 469 and 130 cm~1 i.e. modest but undeniably signi–cant amounts. With the cc-pVQZ/TZ compromise basis in which the hydrogenatom description is lowered to the triple-f level the cc-pVQZ net UMP2 values for De and V * are reproduced excellently to 30 and 10 cm~1 respectively. This comparison is propitious for the higher-order correlation treatments facilitated by the more economical cc-pVQZ/TZ basis. The d[UCCSD] and d[UCCSD(T)] increments in Table 7 are generally an order of magnitude smaller than the d[UMP2] terms.In the (D V *) case they are of the (opposite same) sign thus establishing (oscillatory monotonic) behaviour in the march toward the correlation limit. In this molecular system focal-point additivity must assume one-particle convergence of the d[UCCSD] and d[UCCSD(T)] quantities with the ccpVQZ/ TZ basis a tack which –nds some support in the antecedent cc-pVDZ and cc-pVTZ results. Finally the computation of full UCCSDT energies with the cc-pVDZ basis merely provides ca. 20% augmentations of the d[UCCSD(T)] shifts for the dissociation energy and association barrier. In Table 7 the –nal predictions of the focal-point analysis based on the TZ(2d1f,2p) De\9587 cm~1 and V *\908 cm~1. With the harmo- CCSD optimum geometries are nic zero-point vibrational energies given in Table 4 these results translate into D0\ 7810 cm~1 and V 0*\1424 cm~1.Both anharmonic eÜects and residual errors in the TZ(2d1f,2p) CCSD harmonic frequencies should cause the theoretical ZPVEs for the individual stationary structures to be too high. For example the aforementioned MORBID Hamiltonian –tting85 of myriad 3B1 CH2 rotation»vibration spectroscopic transitions has determined a ZPVE of 3689 cm~1 which upon addition with the well known quantity for diatomic carbon monoxide,84 yields an empirical ZPVE of 4771 cm~1 for the products of triplet ketene fragmentation. The harmonic ZPVE computed here thus appears to be ca. 3.5% too large. Small errors of this magnitude should be C reaction present in the harmonic ZPVEs for all of the stationary points along the sII path and in the absence of an empirical scaling of frequencies conjoined with laborious anharmonic vibrational analyses residual errors in ZPVE diÜerences are neglected here.No direct experimental values exist for the a8 3AA CH2CO dissociation energy. In X3 1A contrast the threshold for the 1 CH2CO]a8 1A1 CH2]1&` CO fragmentation (D0\30 116.2^0.4 cm~1) has been measured precisely by Chen et al.15 When com- 0\3147^5 cm~1),85 one 1 CH2CO]X3 3B1 CH2]1&` bined with the known singlet»triplet splitting in methylene (T arrives at 26 969 cm~1 for the threshold energy of X3 1A CO. This empirical value can be adjusted by the adiabatic excitation energy for a8 3AA CH2CO (T0\19 150 cm~1) surmised in our earlier theoretical investigation,41 resulting D in 0\7819 cm~1 for triplet ketene which is within 10 cm~1 of the dissociation energy given by the present focal-point analysis.This remarkable accord while slightly fortuitous given the uncertainty in T for a8 3AA ketene reveals the merit of the focal-point extrapolations which recover substantial portions of the carbon»carbon bond energy 0 arising from angular mometum manifolds past g functions in the heavy-atom oneparticle basis sets as well as higher-order correlation eÜects. 44 Fragmentation surface of triplet ketene The 1995 experimental study of Kim et al.27 examined the intriguing stepwise variations in the energy dependence of the rate constant for a8 3AA ketene photofragmentation V and determined 0*\1281^15 cm~1 from the –rst jump in the k(E) and/or photofragment excitation (PHOFEX) spectrum.This work re–ned the barrier height (1325^20 cm~1) reported in 1990 by Chen and Moore.17 The focal-point results from Table 7 (V *\908 cm~1 and V 0*\1424 cm~1) thus compare very favourably with experiment especially considering the unknown eÜect of interfragment-mode vibrational anharmonicity on the ZPVE correction. In resolving the remaining discrepancy (ostensibly less than 150 cm~1) between theory and experiment over the barrier height it is paramount to obtain very high-quality ab initio potential-energy functions along the parametrized reaction path rather than single-point predictions at –xed reference geometries.Such functions not only allow the transition state to shift outward properly as the barrier height predictions are reduced but also provide a de–nitive energy pro–le for addressing the central dynamical issues of photofragmentation. As suggested by the focal-point analysis the generation of improved potential-energy fragmentation began with the computation of cc-pVQZ/TZ sII curves for a8 3AA CH UCCSD(T) total energies at all of the primary TZ(2d1f,2p) CCSD points along the C 2CO reaction path which occur at intervals of 0.2 between s\[3.4 and ]2.2 au. To extend the energetic data to within 150 cm~1 of the product asymptote additional cc-pVQZ/ TZ UCCSD(T) computations were carried out at the geometric structures for s\]2.8 and ]3.4 as given by the analytic reaction path parametrization.A correlated basis-set correction to the cc-pVQZ/TZ UCCSD(T) reference curve was determined by evaluating cc-pVQZ/TZ]cc-pV5Z UMP2 interaction energy shifts in the s ½ [[3.2 ]2.2] range at intervals of 0.6. At the same sampling points a higher-order correlation correction was obtained by computing analogous cc-pVDZ UCCSD(T)]UCCSDT shifts. Of course the cc-pVQZ/TZ UCCSD(T) reference curve as well as its basis-set and correlation corrections were also evaluated at the equilibrium The cc-pVQZ/TZ]cc-pV5Z UMP2 basis set correction exhibits a monotonic SCsII structure where s\[4.12. shaped variation ranging from [271 cm~1 at s\[4.12 to [73 cm~1 at the transition state (s\0) and then to [29 cm~1 at s\]2.2. On the other hand the cc-pVDZ UCCSD(T)]UCCSDT correlation correction takes on values of ([76 [93 [61 [10) cm~1 at s\([4.12 [1.4 0 ]2.2) displaying its greatest magnitude near s\[1.4.The origin of this behaviour became apparent in monitoring all wavefunctions along the path for signs of multireference character. Most importantly the norm of the t amplitudes of the UCCSD wave functions never exceeds that of the equilibrium structure by more than 6% and the UHF spin contamination stays below 0.08 au through- 1 out the fragmentation process. These indicators evidence a persistent predominance of the Hartree»Fock reference con–guration. Nonetheless the region in which both the t1 norm and the expectation value of S2 are greatest is centred around s\[1.2 precisely where the cc-pVDZ UCCSD(T)]UCCSDT correlation correction is largest.Assessment of the key V [R(s)] results is facilitated by Fig. 10 in which V In total the ab initio procedures of this investigation provided four successive potential-energy curves for the triplet ketene CsII fragmentation path VA\TZ(2d1f,2p) CCSD; VB\cc-pVQZ/TZ UCCSD(T); VC\VB]cc-pV5Z UMP2 basis set correction ; V and D\VC ]cc-pVDZ UCCSDT correlation correction. The data sets for these four curves were employed in separate non-linear least-squares –ts of the nine-parameter functional form prescribed in Table 5 which consists of a Morse potential augmented by longer-range primary and secondary hyperbolic secant functions for describing the transition-state region. On physical and interpretive grounds the principal argument for the potential-energy functions was chosen to be the carbon»carbon distance which in turn depends parametrically on the arc-length variable according to the tabulated function R(s).The quality of the –ts was exceptionally good with rms errors around 3 cm~1. Final parameters for the four V [R(s)] dissociation curves are listed in Table 8. A VB and VD R. A. King et al. CsII dissociation curve Table 8 Parameters for the V [R(s)] of a8 3AA ketenea VA VD VC VB R R1 d R2 1.5176 1.8802 1.8997 17894 10435 1517 1.5184 1.8818 1.9911 17372 10260 1402 1.5185 1.8810 1.9126 17117 10067 1437 1.5088 1.8998 2.4370 13849 9311 347 3 d d12 1 1.8593 2.1767 1.7844 1.8672 2.2342 1.7425 2.0700 2.4952 2.6326 cc 3 c23 1.8963 2.2075 1.9029 features of stationary points 8106 9107 1.438 1156 DR e V *e 1.435 1659 2.179 9367 1.436 1045 2.257 321i 9291 1.436 1093 2.249 328i 2.242 332i 380i Ru ** a A\TZ(2d1f,2p) CCSD; B\cc-pVQZ/TZ UCCSD(T); C\cc-pVQZ/TZ UCCSD(T)]cc-pV5Z UMP2 basis set correction ; and D\cc-pVQZ/TZ UCCSD(T)]cc-pV5Z UMP2 basis set and cc-pVDZ UCCSDT correlation corrections.The units of the parameters are consistent with relative energy in cm~1 and bond distances in ”. De\8106 cm~1 and Re\1.435 ” ”). are plotted vs. R(CwC). The salient variations in the curves which occur as the theoretical treatment is elevated are (a) an aggrandizement of the well-depth (b) a diminution of the association barrier (c) an outward migration of the transition state and (d) a burgeoning of the anharmonic nature of the barrier pro–le.Critical features of the stationary points of the analytic V [R(s)] functions are appended to Table 8. First the equilibrium quantities D and R are in acceptable agreement with the data in Tables 1 e and 7 even though no eÜort was made to sample the potential well sufficiently to e actually pinpoint the depth and position of each minimum. For example the TZ(2d1f 2p) CCSD analytic potential function exhibits as compared to actual values of 8104 cm ” ~1 and 1.446 from Table 1. Second the V function almost perfectly reproduces the characteristic quantities of the TZ(2d1f,2p) A CCSD transition state providing con–rmation of the efficacy of the –tting procedure in the region of greatest importance.Speci–cally the discrepancies between actual and –tted values for the barrier position height and harmonic frequency are less than 0.0007 ” 1 cm~1 and 1 cm~1 respectively. Third the position of the transition state (R*) moves outward by a sizable 0.063 ” as the level of theory is improved from TZ(2d1f,2p) CCSD to cc-pVQZ/TZ UCCSD(T) and in the –nal V curve it is predicted to lie at an Fourth the V * values reveal that repositioning of the even larger separation (2.257 D transition state eÜectively raises the barriers given by the focal-point analysis at the optimum TZ(2d1f,2p) CCSD structures. Most notably in the V case the optimized D barrier is actually 1045 cm~1 or 70 cm~1 above the corresponding *E [cc-pV5Z e UCCSDT] entry in Table 7; consequently the dual focal-point limit of 908 cm~1 should be revised upward by 50»100 cm~1 somewhat diminishing its propinquity to experiment.27 Lastly the u* values from the analytic potential functions demonstrate that the actual (rather than eÜective) surface curvature in the transition state region is quite stable in the face of barrier migration and reduction. Although there is an initial fall-oÜ 45 46 Fragmentation surface of triplet ketene Fig. 10 Ab initio potential-energy curves for the CsII fragmentation of a8 3AA ketene represented in terms of the CwC distance along the path [R(s)]. Top TZ(2d1f,2p) CCSD; middle cc-pVQZ/TZ UCCSD(T); bottom cc-pVQZ/TZ UCCSD(T)]cc-pV5Z MP2 basis set and cc-pVDZ UCCSDT correlation corrections.Speci–cation of the –tted V [R(s)] functions and critical features of the predicted transition states is given in Tables 5 and 8. from the TZ(2d1f,2p) CCSD value of 380i cm~1 u* remains in the 320i cm~1 range even for the V potential function. As emphasized in our conclusions this discovery has important rami–cations on the modelling of triplet ketene photofragmentation D dynamics. 4 Conclusions In this investigation the mapping of the intrinsic reaction path and determination of the associated potential-energy function V [R(s)] for the fragmentation a8 3AA CH2CO]X3 3B1 CH2]X1&` CO provides resolution of some earlier proposals for explaining the fascinating intricacies of the dynamics in this system.The intrinsic path (C eÜected by the conservation of orbital symmetry to be in-plane bent sII) has been followed backward to the reactant equilibrium and then forward to the asymptotic regime of the products where the intrafragment coordinates have essentially converged and the energy has fallen to within 150 cm~1 of its limit. The monotonic decay observed in the product tail of the path excludes the possibility46 that a second longer-range transition state exists on the T surface in accord with qualitative frontier orbital con- 1 cepts.41 Therefore any appearance of transition states acting in series would have to be purely dynamical in origin. A long-range CH … … …CO minimum with a binding energy of 2 only a few kJ mol~1 has been discovered in this work but the intrinsic reaction path apparently does not lead to this endpoint and its eÜects on the fragmentation dynamics may not be signi–cant.Our –nal theoretical V [R(s)] curve is constructed from cc-pVQZ/TZ UCCSD(T) energy points adjusted by a cc-pVQZ/TZ]cc-pV5Z UMP2 basis set correction and a 47 R. A. King et al. cc-pVDZ UCCSD(T)]UCCSDT higher-order correlation shift. It exhibits a classical exit barrier of 1045 cm~1 a critical CwC distance of 2.257 ” and a barrier frequency of 321i cm ” ~1. The transition state is thus shifted outward in R(CwC) by almost 0.2 from the analogous DZP CISD result of 1988.41 An extensive focal-point extrapolation of the approach of systematic ab initio energetic predictions to the dual one- and n-particle limits suggests that basis-set augmentation past the [6s5p4d3f2g1h/5s4p3d2f1g] level would lower the barrier further by roughly 70 cm~1.While the reckoning of ZPVEs on the barrier is treacherous the TZ(2d1f,2p) CCSD harmonic frequencies obtained here predict a ZPVE shift of ]516 cm~1 placing the best theoretical barrier estimate ca. 200 cm~1 above experiment (1281^15 cm~1).27 Computations are underway to determine whether connected quadruple excitations in the coupled-cluster wave functions primarily account for this remaining disparity. It is clear that the systematic outward migration of C col with increasing levels of theory must be considered in order to achieve accu- the sII racy better than 1 kJ mol~1.Another central conclusion of the current study is that the third possibility of Gezelter and Miller46 is excluded i.e. we are con–dent that errors in the ab initio barrier frequencies are not responsible for the discord between quantum reactive scattering calculations and experiment9,27 concerning steplike structures in the dissociation rate constant [k(E)]. The strength of this conclusion is bolstered by examination of the expanded C transition state. First consider- view in Fig. 11 of the potential-energy pro–le of the sII ing the very high quality of the theoretical curve harmonic approximations to the barrier with frequencies as low as 100i cm~1 are seen to be physically untenable. Nonetheless the superposition of the exact curve with its own harmonic approximation (u*\321i cm~1) and with an intermediate representation (u*\200i cm~1) demonstrates that the anharmonicity of the transition state region could engender eÜective Fig.11 Expanded view of the potential-energy pro–le of the fragmentation. The –nal (»») theoretical curve is shown in comparison to (… … … … …) CsII transition state for a8 3AA ketene VD[R(s)] quadratic functions with critical frequencies (u*) of 100i 200i and 321i cm~1. 48 Fragmentation surface of triplet ketene barrier frequencies from RRKM modelling of the experiments (with one-dimensional tunnelling corrections) which are signi–cantly lower than the physically correct values. In brief the anharmonicity of the barrier pro–le substantially increases its width and should reduce the facility of tunneling which washes out steplike structures in k(E).Therefore we raise the possibility that anharmonic eÜects may help explain at least part of the k(E) mystery and we look forward to new dynamical calculations which account fully for the highly skewed shape of the exit barrier. The quantitative accounting of experimental k(E) measurements for the fragmentation of singlet ketene has already been achieved by a rigorous synthesis of state-of-the-art electronic structure techniques with the latest advances in statistical theories of unimolecular reactions.30,43 Achievement of a similar con—uence between theory and experiment for triplet ketene fragmentation which exhibits markedly diÜerent dynamics from its singlet counterpart would provide a vivid demonstration of the power of contemporary ab initio methods.This research was supported by the U. S. 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ISSN:1359-6640    

DOI:10.1039/a801187g

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 51»70 Potential-energy surfaces for ultrafast photochemistry Static and dynamic aspects Marco Garavelli,a Fernando Bernardi,a Massimo Olivucci,a* Thom Vreven,b Steç phane Klein,b Paolo Celanib and Michael A. Robbb* a Dipartimento di Chimica ììG. Ciamicianœœ dellœ Universita` di Bologna via Selmi 2 40126 Bologna Italy b Kingœs College L ondon L ondon UK W C2R 2L S The –rst singlet excited states (S which control the ultrafast (i.e. subpicosecond) photochemistry of 2-cis-penta-2,4-dieniminium cation (2-cis- 1) diene (CHD) have been investigated using ab initio MCSCF and multi- C5H6NH2 `) all-trans-hexa-1,3,5-triene (all-trans-HT) and cyclohexa-1,3- reference MP2 theories. The structure of the corresponding potential energy surfaces (PESs) has been characterized by computing novel unconstrained and symmetry-constrained minimum-energy paths (MEP) starting from Franck»Condon and S2/S1 conical intersection points on S1 .Furthermore analytical frequency computations have been used to produce quantitative information on the surface curvature. We show that the S energy surface is characterized by two domains region I and region II. Region I controls the initial acceleration of the 1 excited state molecule. In contrast region II is a low-lying region of S and controls the evolution towards fully efficient decay to the ground state. The 1 energy surface structure indicates that the double-bond isomerization of 2-cis-C5H6NH2 ` and all-trans-HT and the ring-opening of CHD are prototypes of three classes of barrierless reactions characterized by a diÜerent excited state dynamics.In 2-cis-C5H6NH2 ` and more loosely in all-trans- HT the initial relaxation results in the production of a totally symmetric S1 transient. The following triggering of the S1 ]S0 decay requires energy redistribution along a symmetry-breaking (torsional) mode leading to an S1/S0 conical intersection (CI). In contrast the shape of region I of CHD indicates that an almost direct (i.e. impulsive) motion towards an asymmetric S1/S0 CI occurs upon initial relaxation. Previously reported and novel semi-classical trajectory computations and the available experimental evidence seem to support these conclusions. 51 1 Introduction Ultrafast excited-state dynamics has been recently reported for linear and cyclic conjugated systems such as all-trans-hexa-1,3,5-triene1h5 (all-trans-HT) and cyclohexa-1,3- diene6h10 (CHD).These small molecules undergo a sub-picosecond decay to the ground state (S0) which has been assigned to the opening of an S1 ]S0 radiationless decay channel leading to 52 PESs for ultrafast photochemistry double-bond isomerization and ring-opening respectively. Picosecond excited-state dynamics has been also reported for the photoisomerization of larger systems such as the 11-cis retinal protonated SchiÜ base (PSB11),11 the chromophore of rhodopsin (the human retina visual pigment) and its isomers.12h14 These observations suggest the involvement of barrierless reactive processes where the molecules are not thermally equilibrated at the outset of the reaction.Thus the chemical change occurs on the same timescale as intramolecular vibrational relaxation (IVR). During the last –ve years we have demonstrated15 that the mechanism of several singlet photochemical reactions can be understood in terms of the evolution of an S excited-state species (generated directly or via decay from an upper excited state) 1 towards a conical intersection (CI) which acts as the photochemical decay channel.16,17 Thus in these cases information on the reaction rate and photoproduct distribution can be provided by computing the S energy barrier which controls the access to one or more CIs. However ultrafast reactions are basically barrierless. Thus the usual mecha- 1 nistic picture of a photochemical reactions,16,18 based upon the depletion of an equilibrated excited-state intermediate cannot apply to these processes.As a consequence the computational investigation of their mechanism cannot be carried out by simply determining the stationary points (i.e. minima and transition structures) on the S energy surface with standard methods. 1 The evolution of a photoexcited molecule along a barrierless energy surface which leads to S1 ]S0 decay can be investigated by using quantum or semi-classical dynamics computations. Regrettably for molecules of ìmediumœ size as those mentioned above the high computational cost renders the use of such methods difficult if not impossible. One is therefore seeking a diÜerent strategy capable of providing information on the excited state motion.In recent work,19h23 we have shown that qualitative information can be obtained without running expensive dynamics computations. In fact in certain conditions (see below) the shape of the S PES is expected to play a dominant role in 1S determining the short-lived dynamics on According to the Franck»Condon (FC) principle photoexcitation of a molecule produces a replica of its ground-state equi- 1. librium structure (i.e. of the corresponding vibrational wavepacket) on a non-stationary point (i.e. a point with non-zero gradient) on the excited-state energy surface (e.g. the FC point). Under these conditions the excited-state molecule has a negligible amount of initial kinetic energy directed along the reaction path and the excited-state dynamics will be determined by (i) the direction of the initial acceleration on the excited state surface (i.e.the direction of the gradient) which results in the population of only one or a few selected vibrational modes of the system and (ii) the structural details (i.e. the shape) of the excited-state surface which control the redistribution of the initially localised vibrational energy among other modes and thus to the reactive mode. Recently we have developed a new gradient-driven tool the initial relaxation direction (IRD) method,21,24 which facilitates the computation of the down-hill minimumenergy path (MEP) starting from a non-stationary point on the PES by supplying an initial direction for the reaction path in the space of all nuclear degrees of freedom of the molecule.This method has proved useful in the description of the path describing the relaxation from the FC point. We have recently applied this technique to investigate the S isomerization of the protonated SchiÜ base 2-cis-penta-2,4-dieniminium cation20 (2-cis-C 1 trated schematically in Scheme 1 on a model PES. 5H6NH2 `; see structure above) a simple model of PSB11. The results are illus- The computed MEP demonstrates that the relaxation from the FC structure occurs 1/S0 CI. The MEP coordinate suggests along a barrierless path which ends at a twisted S that the initial acceleration [see point (i) above] occurs along a stretching mode which has no component along the torsional deformation required to access the CI [see point (ii) above].These results suggest that IVR from the initially populated (stretching) mode to the reactive (torsional) mode must take place in order for the reaction to occur. In 53 M. Garavelli et al. Scheme 1 1/S0 CI (see the shaded area in Scheme 1) the understanding of the IVR 0 . other words the computation suggests that IVR is at least in part controlled by the structure of two ìorthogonalœ domains of the S surface labelled region I and region II in Scheme 1. The generality of these ideas has been reinforced by a recent study of 1 all-trans-HT.22 While standard unconstrained MEP computations provide quantitative information on the energy pro–le and reaction coordinate along the energy valley connecting FC and S process de–ned above requires a more detailed knowledge of the S surface structure.In particular population of the reactive mode may be determined by the shape of the 1 surface in regions which are not sampled by the MEP. Thus in the present work we use symmetry-constrained MEP and frequency computations to produce a detailed map of this region in 2-cis-C5H6NH2 ` all-trans-HT and CHD. As mentioned above unconstrained MEP computations on the –rst two molecules have already been reported.20,22 In what follows we will show that symmetry-constrained MEP unveils the detailed structure of the energy surface connecting regions I and II. Furthermore vibrational frequencies computed along a few selected MEP points yield a quantitative description of the force –eld that drives the system away from region I prompting IVR and population of region II and ultimately decay to S In subsequent discussion we will show that the S energy surfaces for 2-cis- C5H6NH2 ` and to a lesser extent for all-trans-HT conform to the tentative model given in Scheme 1.For this model surface region I has a valley-like structure and thus 1 the initial acceleration is expected to occur along this region. Consequently a rationalization of the relative magnitude of the excited-state lifetime in these two molecules is given on the basis of the surface curvature in region II in a direction orthogonal to that of the initial acceleration. In fact as mentioned above this curvature is expected to play a dominant role in determining the IVR rate.We will show how this conclusion is reinforced using the results of some previously published and new unreported S trajectory computations. 1 The structure of both regions I and II can be changed by changing the molecular structure of the reactant. In particular we shall discuss the investigation of the changes due to the inclusion of a strategically located methyl substituent in the 2-cis-C5H6NH2 ` 54 PESs for ultrafast photochemistry 1 length of the molecule from three to four conjugated double bonds by computing the S to yield 2-cis-a-methyl-C5H6NH2 `. Similarly we investigate the eÜect of increasing the MEP of all-trans-hepta-2,4,6-trieniminium cation (all-trans-C7H8NH2 `). In contrast with the case of parent and substituted 2-cis-C5H6NH2 ` all-trans- Scheme 1 cannot be used to describe the photochemical ring-opening reaction of CHD.C7H8NH2 ` and all-trans-HT photoisomerizations we –nd that the energy surface of In this case we will show that the results of novel MEP and frequency computations indicate that in CHD the reactive mode is populated almost ì directly œ during the initial relaxation. In fact in this molecule region I has the structure of a steep ridge rather than that of a valley. As a consequence the reaction coordinate in CHD describes a ìsidewaysœ acceleration populating region II directly. As a consequence this reaction should occur on a faster timescale. 2 Computational methods All the new MC-SCF energy gradient and frequency computations presented in this work have been carried out using a complete active space (CAS-SCF) with the doublef] polarization (6-31G*) basis sets available in Gaussian 94.25 The S minima transition states and MEPs have been computed using a six electrons in six orbitals active 1 space (i.e.the six valence p orbitals) at the CAS-SCF level. For the all-trans-C7H8NH2 ` we have used an eight electrons in eight orbitals active space. In order to improve the energetics by including the eÜect of dynamic electron correlation the energy of a set of selected points have been re-computed using the PT2F method26a included in MOLCAS-3.26b The level of theory used in each MEP computation is summarized in the corresponding –gure captions. The unconstrained and symmetry-constrained S relaxation paths are computed as MEP starting at non-stationary points of the corresponding PES.Such MEPs are unam- 1 biguously determined by using a new methodology24 to locate the initial direction of relaxation (IRD) from the starting points ( C i.e. the FC point of 2-cis- 2-cis- 5H6NH2 ` a-methyl-C5H6NH2 ` C all-trans- 7H8NH2 ` and all-trans-HT and the S2/S1 conical intersection structure of CHD). Brie—y an IRD corresponds to a local steepest-descent direction in mass-weighted cartesians from a given starting point. The IRD is calculated by locating the energy minimum on a hyperspherical [i.e. (n[1)-dimensional] cross-section of the n-dimensional PES centred on the starting point (n being the number of vibrational degrees of freedom of the molecule). The radius of this hypersphere is usually chosen to be small (typically ca.0.25»0.5 au in mass-weighted cartesians) in order to locate the steepest direction in the vicinity of the starting point (i.e. the hypersphere centre). The IRD is then de–ned as the vector joining the starting point (i.e. the centre of the hypersphere) to the energy minimum. Once one or more IRDs have been determined the associated MEP is computed as the steepest-descent line in mass-weighted cartesians using the IRD vector to de–ne the initial direction to follow. The vibrational frequency on stationary (energy minima and transition structures) and non-stationary points (i.e. the MEP points) have been computed analytically using the methodology available in Gaussian 94.25b As usual the frequency values have all 55 M.Garavelli et al. been scaled by 9%. In the case of non-stationary points (i.e. points with a non-zero energy gradient such as the FC point) the computed hessian is projected onto the (n[1)- dimensional space orthogonal to the energy gradient. The diagonalization of this hessian yields the normal modes and frequencies along the n[1 orthogonal space.27,28 These are used to determine the shape (e.g. valley-like or ridge-like) of the S surface along the computed MEP. 1 computations were performed using two methods that diÜer in the force –eld type 2-cis- 2 C5H6NH2 ` C -cis-a-methyl- 5H6NH2 ` all-trans-HT excited-state trajectory (CASSCF or MM-VB see ref. 29»31 for details). In each case the trajectories are computed by evaluation of the gradient on the —y32,33 with surface hopping,34 in the full space of coordinates.The surface hop algorithm of Tully and Preston35 was used to allow excited-state trajectories to transfer to the ground state at points where nonadiabatic coupling is strong (e.g. in the CI region). At each point on the excited state surface the ìhop probabilityœ is determined. When this oscillatory hop probability function has a maximum close to 1 the trajectory is continued on the ground state. The excess energy corresponding to the ìgapœ is then added to the momentum along the direction of the non-adiabatic derivative coupling vector to conserve the total energy. In the results on all-trans-HT (using the MM-VB force –eld) this algorithm has been re–ned36a to propagate the wavefunction on a mixed state through the region of strong interaction [for a recent comprehensive review of these methods see ref.36(b)]. This has the advantage that the dynamics ì feels œ the eÜect of the non-adiabatic coupling throughout the strong coupling region and the ambiguities of the surface hop algorithm are avoided. 2 5H6NH2 ` has 2 an ionic (1Bu-like) S1 excited 1 3 PES characterisation and all-trans-HT are both planar (C and C symmetry respectively) conjugated mol- The three molecules investigated here have diÜerent structures. While 2-cis-C5H6NH2 ` ecules with three double bonds CHD is a non-planar (C symmetry) molecule with only s two conjugated bonds. In addition 2-cis-C state while all-trans-HT and CHD have a covalent (2A -like) S state.Finally in contrast to g tions CHD undergoes a ring-opening reaction. cis-C5H6NH2 ` and all-trans-HT which undergo double-bond photoisomeriza- Photochemical processes in isolated organic molecules such as the double-bond cis-C5H6NH2 ` (or PSB11) and all-trans-HT and ring-opening in CHD 1 isomerization in are initiated by the photoexcitation of their conjugated p-systems. The equilibrium bond lengths of a p-system are changed upon promotion to the S excited state. Thus intuitively one expects that the initial acceleration on the excited-state surface results in the 1 production of a transient species with expanded p-bonds. Indeed the initial relaxation path computed for 2-cis-C5H6NH2 ` indicates that in order to prompt photoproduct formation the vibrational energy initially concentrated in totally symmetric stretching modes must be redistributed to a non-totally symmetric torsional mode.The computed20 S MEP for 2-cis-C5H6NH2 ` is reported in Fig. 1. photoproduct occurs via In this reaction the formation of the 2-trans-C5H6NH2 ` decay at an S0/S1 CI where the system has developed a ca. 80° twist about its central bond. The initial part of the MEP shows a rapid energy decrease associated with double-bond expansion and single-bond contraction. However after this initial progression the reaction coordinate changes direction and after a sort of in—ection enters a steep region which leads to the CI point. Following S1 ]S0 decay the reaction path leads to formation of the trans-photoproduct (or original cis-reactant).This behaviour can be interpreted in terms of the structural features of the model energy surface shown in Scheme 1. Thus the initial steep part of the MEP where the reaction coordinate has no torsional components is related to motion along the valley-shaped region I. On the 56 PESs for ultrafast photochemistry u- 1 S1/S0 CI decay point of 2-cis-C5H6NH2 ` (data from ref. 20). (K) S de–ne the (1B branch of the cis]trans photoisomerization path; ()) S give the energy of the covalent 2 Fig. 1 Energy pro–les along the MEP describing the excited state relaxation from the FC point (FC) to the like) (2A -like) state along the same path. (L) indicates the position of the CI. The energies of all points have been scaled to match the PT2F energies of the initial and –nal MEP points.The structures (geometrical parameters in ” and degrees) document the geometrical progression along the path. The MEP coordinate is in mass-weighted cartesians (au\u1@2 a0). 1 1 g other hand the in—ection of the potential energy along the MEP and the subsequent steep energy decrease towards the CI point can be related to the change in direction of the MEP coordinate which must now enter region II along the torsional mode. In this region the curvature must change from that of a valley to that of a ridge. In this section we report the results of additional MEP and analytical frequency PES of 2-cis-C computations on the S 5H6NH2 ` all-trans-HT and CHD. We will focus our attention on the structure of those regions of the S surface (region I and region II in Scheme 1) which are relevant to the excited-state dynamics of these systems.In Subsection 3.1 we investigate the double-bond photoisomerization in protonated SchiÜ surface for 2-cis-C bases. In 3.1.1 we describe the S 5H6NH2 `. In 3.1.2 we investigate the changes of the surface structure due to an a-methyl substituent. Finally in 3.1.3 we report the structure of the all-trans C7H8NH2 ` surface which illustrates the eÜect of increasing the length of the protonated SchiÜ base. In Subsection 3.2 we report the structure of the region I and II for all-trans-HT and compare it with that of 2-cis- 1 1 and II for the ring-opening of CHD. C 1 5H6NH2 `. S Finally in Subsection 3.3 we report the structure of the surface regions I 3.1 Double-bond photoisomerization in protonated SchiÜ bases 2-cis-C 3.1.1 The S energy surface of the isomerization.In Fig. 2(a) we 5H6NH2 ë show the initial part of the MEP given in Fig. 1 (open squares). The comparison of structure FC and of the slightly twisted structure located at 1.5 au reveals that the initial 2-cis-C5H6NH2 ` relaxation is dominated by double-bond expansion and single-bond contraction. To determine the structure of region I we have computed the relaxation 57 M. Garavelli et al. 5H6NH2 ` (K) cis-C5H6NH2 ` ()). (b) Comparison of the Cs 1 and a-methyl Fig. 2 (a) Comparison of the initial part (see framed region in Fig. 1) of the unconstrained S MEP for cis-C and a-methyl 5H6NH2 ` (K) cis-C symmetry-constrained S MEP (MEPplanar) for cis-C5H6NH2 ` The relevant geometrical parameters are given in ” and degrees.The MEP coordinate ()). 1 is in mass-weighted cartesians (au\u1@2 a0). path in the presence of a planarity constraint (MEPplanar) which keeps the system in its initial C symmetry. The resulting energy pro–le [see Fig. 2(b)] ends at a stationary s point (SP) located only ca. 5 kcal mol~1 below the FC point and featuring again expanded double-bonds and contracted single-bonds. The energy pro–le and reaction coordinate along the unconstrained and constrained paths are consistent with the energy-surface structures shown in Fig. 3. The shape of these surfaces changes from valley-like to ridge-like going down-hill along the relaxation path.A rigorous demonstration of this structural feature requires the evaluation of the vibrational frequencies along the (n[1)-coordinate space orthogonal to MEPplanar . The results of analytical frequency computations at 0.25 au and 0.75 au distance from FC and at SP are given in the same –gure. These data con–rm that a gradual change in surface shape occurs along the MEP. In particular the change from a valley shape to a ridge shape must be localised between 0.25 and 0.75 au along the path. In fact the frequencies corresponding to the torsional mode have a real value (212 cm~1) at 0.25 au distance which becomes imaginary (239 i cm~1) at 0.75 au indicating the presence of a ridge. The curvature becomes larger (frequency 264 i cm~1) at the stationary point located at 1.70 au.This stationary point is therefore a transision structure with respect to the central bond torsion. The frequency analysis reveals other details of the surface curvature along region I. As shown in Fig. 3(a) there are two small imaginary frequencies at 0.25 au distance. 58 PESs for ultrafast photochemistry 5H6NH2 ` cis-C Fig. 3 Structure of the S energy surface of (a) cis-C5H6NH2 `. and (b) a-methyl The initial part of MEP and MEP 1 are indicated by full and dashed lines respectively [see planar Fig. 2(a) and (b)]. The frequencies are computed at (K) and at the SP (=) and (+). The frequencies along the modes leading to decay (or towards the unconstrained MEP) are given on the surface. The direction of the initial gradient is represented by arrows on the top structure.The frequency along the SP modes which correlate with the initial gradient is given in parenthesis. Torsional coordinates are indicated by curly arrows on the right and left structures. These correspond to in-phase and out-of-phase rotation along the CwC single bonds. However as soon as relaxation is initiated these imaginary frequencies disappear and are replaced by real frequencies which indicate that the lower part of the S energy surface is ì stable œ with respect to deformation along these bonds. 1 3.1.2 EÜect of a-methyl substitution. Non-bonded interaction between an a-methyl group and the spatially close d-hydrogen (see structure in Fig. 2) may be important in in Fig. 2(a) and (b) the energy pro–le along the unconstrained and constrained MEP for triggering the isomerization process in 2-cis-C5H6NH2 `.37 For this reason we compare this unsubstituted (open square) and substituted (open diamond) protonated SchiÜ bases.Analysis of the corresponding reaction coordinates and the results of a frequency computation at point SP are consistent with the energy-surface structure shown in Fig. 3(b). It is clear from Fig. 2 that the a-methyl substituent increases the slope of both leads to a ca. two-fold increase in the initial slope along the MEP. In fact while the MEP and MEPplanar paths. In fact as reported in the same –gure a-methyl substitution molecular structure at 1.5 au is the same for both the substituted and unsubstituted PSB the former is two times more stable (relative to the FC point).Similarly while the structure of the relaxed planar-constrained stationary point SP does not signi–cantly change its stability with respect to the FC point is increased [see Fig. 2(b)]. The a-methyl substituent is also expected to destabilize the planar structure SP with respect to twisting. This means that the curvature along region II is expected to be larger for the substituted system thus prompting a larger acceleration along the torsional mode. However a frequency computation at the S planar stationary point reveals that the imaginary frequencies corresponding to the torsional mode (i.e. along 1 the transition vector at SP) are similar for the substituted and unsubstituted systems (264 i and 254 i cm~1).This can be explained by the increase in the reduced mass of the 59 M. Garavelli et al. substituted system 1.62 u cf. that of the parent 1.57 u. However remarkably the force constant does indeed show a small decrease going from the parent to the substituted compound (0.079 and 0.076 mdyn ”~1 respectively) thus contributing to the decrease in the imaginary frequency. This is due to the large expansion of the central bond in the SP structures which decreases the interaction between the a-methyl group and the dhydrogen. Indeed this length is slightly larger in the substituted system [see structures in Fig. 2(b)]. This and the data above support the S energy-surface structure given in Fig. 3(b). The most relevant diÜerence between the surfaces in Fig.3 is that the surface in 1 Fig. 3(b) shows a steeper initial slope. 3.1.3 EÜect of the chain length. In a recent study38 we have demonstrated that the S MEP of a ìlongœ protonated SchiÜ base (i.e. the 4-cis-c-methylnona-2,4,6,8-tetraenimminium cation) is similar to the MEP of Fig. 1. However this system has a much 1 —atter region II which is expected to exhibit slower excited-state dynamics. Given the size of this cation which has –ve conjugated double bonds a careful characterisation of the structure of its surface via analytical or numerical frequency computations is not feasible. For this reason in this work we have documented the S energy surface of the 7H8NH2 ` cation a shorter molecule with four conjugated double bonds. all-trans-C 1 This molecule may undergo trans]cis isomerization at either of the double-bonds in positions 2 and 4 as illustrated in Scheme 2.Scheme 2 The presence of competing channels that are barrierless or nearly barrierless is an interesting feature of longer protonated SchiÜ bases. Thus here we characterise the S MEPs of both these isomerization processes. 1 In Fig. 4 we report the two unconstrained MEPs. Again the structure of such paths is similar to that of Fig. 1 but with a longer energy plateau (from 1 to 3 au and from 1 to 5 au for the C2wC3 and C4wC5 isomerization respectively) consistent with the eÜect observed in 4-cis-c-methylnona-2,4,6,8-tetraenimminium cation.38 The two paths are nearly barrierless with only a \1 kcal mol~1 energy diÜerence along the long plateau region (in favour of the isomerization at position C2wC3).However the initial part of the MEP which is dominated by double-bond expansion is similar. Thus in contrast longer protonated SchiÜ base shows that evolution along the torsional coordinate (and with 2-cis-C5H6NH2 ` and 2-cis-a-methyl-C5H6NH2 ` the MEP coordinate of this therefore either of the two competing paths) begins only after relaxation in the vicinity of the planar SP. Thus in this system the unconstrained and planarly constrained MEPs are coincident. These –ndings suggest that the structure of the S energy surface has the form given in Fig. 5. 1 The results of analytical frequency computations con–rm that in a similar fashion to the short protonated SchiÜ base the FC structure evolves along a coordinate dominated by totally symmetric stretching modes and the structure of region I is that of a valley.This is also con–rmed by the direction of the gradient at the FC point and by the fact that at 0.73 au distance one has real (although extremely small) frequencies (8 and 59 cm~1) along the two relevant torsional modes. As seen in the case of the shorter 2-cis- C 1 5H6NH2 ` S there is one imaginary frequency at the SP. However the magnitude of this imaginary frequency and of the frequency corresponding to the alternative isomerization mode are only 45 i (C2wC3 torsional mode) and 62 cm~1 (C4wC5 torsional 60 PESs for ultrafast photochemistry Fig. 4 Energy pro–les along the MEPs describing the competing excited state isomerization paths from the FC point (FC) to the decay points S of all-trans- 1/S0 CIC2hC3 and S1/S0 CIC4hC5 C 1 (1B 7H8NH2 `.(K) S De–ne the u-like) energy. (L) Indicate the position of the CIs. The energies of all points have been scaled to match the PT2F energies of the initial and –nal MEP points. The structures (geometrical parameters in ” and degrees) document the geometrical progression along the path. The MEP coordinate is in mass-weighted cartesians (au\u1@2 a0). Fig. 5 Structure of the S1 energy surface of all-trans-C7H8NH2 `. The initial part of the MEPs (see framed region in Fig. 4) are indicated by full lines. The frequencies are computed at the points indicated (K) and at SP (=). The frequencies along the modes leading to decay are given on the surface.The frequency along the SP mode which correlates with the initial gradient is given in parentheses. The direction of the initial gradient is represented by arrows on the top structure. Torsional coordinates are indicated by curly arrows on the right and left structures. 61 M. Garavelli et al. mode) thus re—ecting the existence of the energy plateau and the general —atness of region II. 3.2 S energy surface of the all-trans-HT isomerization 1 1 2 1 1 1 In Fig. 6 we show the previously reported22 unconstrained MEP for the S relaxation of all-trans-HT. In this molecule the FC point is not located on the S state (see also Section 1) but on the S state. However a minor totally symmetric skeletal deformation of the FC structure is enough to enter an S2/S1 conical intersection which delivers the photoexcited system to the S state (see Fig.6). Despite the fact that all-trans-HT and the protonated SchiÜ bases belong to diÜerent classes of conjugated compounds (a neutral conjugated hydrocarbon and a charged conjugated immine) and have S states of diÜerent electronic nature (see above) both molecules undergo an initial totally symmetric relaxation dominated by double-bond expansion and in all-trans-HT this relaxation leads to a planar SP of C symmetry. Progression beyond this point occurs along a very —at region described by a non-totally symmetric path which connects SP to a conical intersection. However by comparing the S1/S0 CI structures in Fig. 1 and Fig. 6 it is apparent that the non-totally symmetric coordinate describing the deformation towards the CI points (i.e.the MEP co-ordinate along region II) is diÜerent. While in protonated SchiÜ bases this can be described as a simple torsional motion in all-trans- HT this appears to be a complex deformation involving at least two torsions and one bending. The detailed structure of region I in all-trans-HT has been investigated via analytical frequency computations. The results are shown in Fig. 7. Near FC (0.25 au) the system u 2h Fig. 6 Energy pro–les along the MEP describing the excited-state relaxation from the S2/S1 CI point to the S1/S0 CI decay point of all-trans-HT (data from ref. 22). ()) S1 (2Ag-like) branch of the cis]trans photoisomerization path; (K) S energy of the initial covalent (1B -like) branch; (L) the position of the conical intersection S1/S0 CI.The energies of all points have been scaled to match the PT2F energies of the initial and –nal MEP points. The structures (geometrical parameters in ” and degrees) document the geometrical progression along the path. The MEP coordi- 2 nate is in mass-weighted cartesians (au\u1@2 a0). 62 PESs for ultrafast photochemistry Fig. 7 Structure of the S energy surface of all-trans-HT. The initial part of the MEP is indicated 1 by a full line (see framed region in Fig. 6). The frequencies are computed at (K) and at C2h SP (=). The frequencies along the modes leading to decay are given on the surface. The direction of the initial gradient is represented by arrows on the top structure.The frequency along the C2h SP mode which correlates with the initial gradient is given in parentheses (see also ref. 22). Torsional coordinates are indicated by curly arrows on the right and left structures. S1/S0 CI has a real frequency at the S1 planar minimum (101 cm~1). The two has –ve small imaginary frequencies which indicate a —at but ridge-like region I. Two out of –ve of these frequencies (211 i and 183 i cm~1) are related to torsional motion of the terminal CH groups of the molecule. Another two (92 i and 259 i cm~1) describe 2 torsional motion about the two single bonds. Finally one (144 i cm~1) describes a skeletal deformation leading to a bent in-plane carbon framework. Analysis of the MEP co-ordinate along region II suggests that the 259 i mode is the one which best describes the reactive mode.Thus as a –rst approximation the other modes are of minor importance since their motion is either not related to the reactive mode or the corresponding frequencies (e.g. 144 i cm~1) become real as the relaxation proceeds along the totally symmetric stretching coordinate. In particular the mode related to the deformation towards modes involving CH torsional motions remain very small and imaginary. It has already 2 been reported22 that deformation along these two modes lead to slightly out-of-plane conformers which are only \1 kcal mol~1 more stable than SP. It has also been reported22 that close to S2/S1 CI the MEP coordinate is dominated by a stretching mode. However close to SP the MEP becomes dominated by a lower frequency CwH bending mode.Thus in general the gradient along the MEP is a combination of these two modes. At SP the values of these modes are 1803 and 414 cm~1 respectively. According to our frequency computations the S PES of all-trans-HT has a shape which is qualitatively diÜerent from that of 2-cis-C5H6NH2 `. The shape of region I is substantially —at if not slightly ridge-like. After relaxation along the double-bond 1 expansion single-bond contraction modes the surface remains very —at with a slight valley-like curvature. The relaxation dynamics along this energy surface is therefore expected to be diÜerent from that seen for the surface in Fig. 3. The —atness of the surface and the more valley like nature of region I should lead to a slower motion towards the CI.Furthermore while the CI in protonated SchiÜ bases lies below the S1 planar stationary point in all-trans-HT it lies at the same if not slightly higher energy. 63 M. Garavelli et al. 3.3 S energy surface of the CHD ring-opening 1 1 2 1 Both all-trans-HT and the protonated SchiÜ bases discussed above undergo photoinduced double-bond isomerization. S relaxation of these molecules occurs starting from their FC structures (even in the case of all-trans-HT due to an early S2/S1 conical intersection). In this Subsection we investigate the S relaxation process of CHD which results in a ring-opening reaction leading to s-cis-hexa-1,3,5-triene (cZc-HT). There are a few major diÜerences between CHD and the molecules discussed above.First CHD is not planar at the FC point and has C symmetry. Secondly in CHD the role of the non-totally symmetric torsional deformation is played by a mode which can be described as a bending of the molecular skeleton. Thirdly and most important S relaxation of CHD does not begin near the FC point. u 2 1 The spectroscopic state of CHD (i.e. the ionic 1B -like state) is not the lowest (relaxed) excited state. Investigation of the structure of the S energy surface19 indicates that the molecule does not lose its initial symmetry upon relaxation on this state. However in contrast with all-trans-HT the initial geometry undergoes important structural changes before entering the S energy surface at an S1/S2 conical intersection given in Fig.8(a) (open squares). It is clear from the structure of the 2S point. The structure of the MEP describing the relaxation along the S energy surface is 2/S1 intersection that relaxation on S1 1 C5wC6 r-bond. Here the structure of the S1 energy surface is investigated by computabegins at an almost open structure with a ca. 2.00 ” stretched tions of the fully unconstrained and C symmetry-constrained MEPs and the surface is characterized via analytical frequency computations along the constrained MEP. The initial part of the unconstrained MEP [Fig. 8(a)] is immediately directed (in the 2/S1 1 2 vicinity of the S point) along a non-totally symmetric deformation and then relaxes towards an asymmetric S SP. The type of molecular motion which characterises this initial relaxation process can be de–ned by comparing the S2/S1 CI and SP structures.Basically one has a further opening of the ring (C5wC6 changes from 2.0 to 2.3 ”) coupled with a change in the bending angles so as to move the CH group at C 2 1 1 g 1 6 Fig. 8 (a) Energy pro–les along the unconstrained MEP describing the S relaxation from the S2/S1 CI point to the S1/S0 CI decay point of CHD. (b) Energy pro–les along the C2-symmetry- S relaxation from the S2/S1 CI point to the 1/S0 CI decay S C2 2 constrained MEP describing the point of CHD. ()) S (2A -like) branch of the photochemical ring-opening path; (K) S initial (L) position of the conical intersection (1B -like) 1/S0 CI. branch of the path (data from ref.19) ; S The energies of all points have been scaled to match the PT2F energies of the initial and –nal u MEP points. The structures (geometrical parameters in ” and degrees) document the geometrical progression along the path. The MEP coordinate is in mass-weighted cartesians (au\u1@2 a0). 64 PESs for ultrafast photochemistry towards the CH group at C4 . Once this point is reached the system can evolve along a substantially barrierless path towards a conical intersection (S1/S0 CI). This motion involves a shortening of the C5wC6 and C4wC6 bonds up to a point where they are both ca. 2.2 ”. In order to investigate the structure of region I (i.e. the S region controlling initial C symmetry-constrained 1MEPC2 . The result of this relaxation) we have computed the computation is shown in Fig.8(b). Relaxation along this path ends at a previously 2 reported39 C stationary point [C SP in Fig. 8(b)] which has an energy only \1.0 kcal 2 mol~1 above the asymmetric SP structure reached along the unconstrained MEP. 2 2 C5wC6 and C4wC6 distances. These C SP to SP. Signi–cantly SP is ca. 1 au closer to S1/S0 CI) than C2 SP. surface given in Fig. 9. This structure is con–rmed by analytical frequency computations Comparison of the asymmetric and symmetric paths suggest the structure of the S1 along the MEPC2 . It can be seen that initially the surface appears as a ridge by a large imaginary (2700 i cm~1) frequency. The mode corresponding to the imaginary frequency (called ìbendingœ in Fig. 9) describes essentially the same type of distortion given by the initial part of the unconstrained MEP.Further insight can be obtained by comparing the structure of the symmetric stationary point C SP and asymmetric structure SP. The most relevant geometrical change is related to the are both shortened by passing from the decay channel (i.e. 2 Even if the ridge-like curvature decreases along MEPC2 it does not change to that of a valley. In fact at C SP one has a small imaginary frequency. One interesting feature 2 2 Fig. 9 Structure of the S energy surface of CHD. The initial part of the MEP [see framed regions in Fig. 8(a)] and MEP 1[see Fig 8(b)] are indicated by a full line and by a dashed line respec- C2 tively. The frequencies are computed at 1.0 and 1.3 au distance from the S2/S1 CI point (K) and at C SP (=).The frequencies along the modes leading to decay (or towards the unconstrained 2 MEP) are given on the surface. The direction of the initial gradient is represented by arrows on the top structure. The frequency along the C SP mode which correlate with the initial gradient is given in parentheses. 65 M. Garavelli et al. coordinate is the change in the type of dominant totally symmetric mode of the MEPC2 along the MEP. Initially i.e. close to S2/S1 CI the MEP coordinate is dominated (as in all-trans-HT and the protonated SchiÜ bases) by a stretching mode. However halfway along the path the bond lengths become constant and the dominating mode becomes the expansion of the spiral-like structure of the molecule (i.e.a breathing mode). Thus in general the gradient along MEPC2 is a combination of these two modes. At C SP these stretching and breathing modes have 2147 and 531 cm~1 frequencies respectively. Note 2 that these high-frequency (stretching) and low-frequency (breathing) modes are the analogues of the two modes describing the totally symmetric part of the MEP in all-trans- HT22 and loosely of the two modes describing that of 2-cis-C5H6NH2 `.20 4 Dynamical eÜects As mentioned in Section 1 in conditions when the photoexcited reactant initiates S1 relaxation with little initial kinetic energy along the reaction path the structure of the S1 energy surface should provide qualitative information on the excited-state dynamics and lifetime. In this section we argue that the double-bond isomerization of 2-cisof three diÜerent classes of ultrafast (i.e.barrierless) reactions characterized by a diÜerent C5H6NH2 ` and all-trans-HT and the ring opening of CHD may be seen as prototypes excited-state dynamics. In fact a diÜerent motion can be predicted on the basis of the three energy surfaces seen in Fig. 3(a) 7 and 9. This motion is illustrated in terms of classical trajectories in Scheme 3 (shaded areas correspond to the region spanned by the unconstrained MEP). 5H6NH2 `. This is The –rst energy surface in Scheme 3 corresponds to that of 2-cis-C characterized by a valley along the initial part of region I and by a gradual change of this valley into a ridge in region II. The structure of these regions rationalises the excited-state dynamics [see trajectory illustrated in Scheme 3(a)] which we now discuss in more detail.After initial acceleration through region I the motion of the system involves high-frequency (ca. 1450 cm~1) totally symmetric stretching oscillations. However each oscillation must pass through the highly anharmonic region II. Therefore in the presence of a small torsional perturbation the molecule will rapidly accelerate along the non-totally symmetric torsional mode. The steep slope connecting region II to the CI point will provide an increasing acceleration along the torsional coordinate. In other words the structure of this PES induces IVR from stretching to the torsional modes and then further acceleration along the torsional coordinate. The view above is supported by the result of semi-classical trajectory computations40 on 2-cis-C5H6NH2 `.These are shown in Fig. 10(a) and (b). It can be seen that three diÜerent trajectories (starting with diÜerent initial torsional deformations) do not show impulsive acceleration along the torsional coordinate [see plateau in Fig. 10(a)]. In contrast in Fig. 10(b) one notices that expansion of the central double bond begins immediately. However torsional motion begins after a certain time delay which is proportional to the initial torsional deformation given to the FC structure. energy surface of the a-methyl derivative of 2-cis-C5H6NH2 ` The structure of the S shows a steeper region I centred on a more shallow valley (see Fig. 3). This suggests that 1 the major eÜect of methyl substitution must be that of accelerating the molecule along the double-bond expansion coordinate.Nevertheless the IVR process from the initially excited stretching modes to the torsional mode may occur on similar timescales since the shapes of region II appear to be similar. 5H6NH2 `. The result of this computation is compared with that of This conclusion is supported by a semi-classical trajectory computation on a-methyl substituted 2-cis-C the unsubstituted molecule in Fig. 11. It can be seen that despite the fact that the substituted molecule has a slightly shorter excited-state lifetime the two quantities are similar. This seems in agreement with the fact that the frequencies along the torsional 66 PESs for ultrafast photochemistry Scheme 3 mode are similar and the systems are likely to be accelerated out of plane from similar force –elds.This does not occur in the case of the longer protonated SchiÜ base examined above (i.e. for the all-trans-C7H8NH2 `). In this case one expects a slower reaction with a longer excited-state lifetime. In fact while torsional motion about either the IVR from the initially populated stretching modes the extreme —atness of region II C2wC3 or C4wC5 double-bond isomerization channels must again be prompted by would not provide large acceleration along the torsional coordinates. The all-trans-HT S surface in Scheme 3(b) features a substantially —at shape all along region I. Therefore also in this system we do not expect a large initial acceler- 1 ation along non-totally symmetric modes.Initial acceleration would then result in the formation of an initial transient with vibrationally excited stretching modes. However in contrast to the surface in Scheme 3(a) there will be no acceleration along region II since this is completely —at. Population of the non-totally symmetric mode is expected to occur on a longer timescale. Therefore we predict the formation of a symmetric (C transient ì species œ which should last for at least a few stretching oscillations [see oscil- 2h) lation in the trajectory of Scheme 3(b)]. This species is expected to have a longer lifetime than 2-cis-C5H6NH2 `. The character of the motion of all-trans-HT has been investigated by trajectory computations on an MM-VB PES (see Section 2).The results of this simulation are given in Fig. 10(c) and (d) where they can be compared with the trajectories discussed above for 67 M. Garavelli et al. cis-C5H6NH2 `. (c) Torsion about the central a1 and single a2bonds of all-trans- 1 ]S0 decay occurs when both angles approach a 90° twist. The trajectory is started 2/S1 CI twisted structure (S2/S1 CItwisted ) with 2\20°. Note that in this a and a (a1B80° and a2B30°) of the decay at 600 fs and then a decay around 1500 fs (a1B80° a2B80°). (K) ( and L) Time steps. (d) Stretch of the central r1 and single r2 bonds of all- Fig. 10 Time evolution of the relevant geometrical parameters along the computed S trajectories cis-C5H6NH2 ` (data from ref. 40) and all-trans-HT. (a) for the double-bond isomerization of 1 cis-C Torsion about the central bond of 5H6NH2 `.The three curves correspond to diÜerent initial conditions with 5° 10° and 20°-twisted FC structures (FCtwisted The ). S1 ]S0 decay occurs when the torsional coordinate is nearly 90°-twisted. After the decay the value of the torsional angle continues to increase (see also Fig. 11). (K) Time steps for the 10° curve only. (b) Stretch of the central bond of HT. The S from a S 1\35° molecule the evolution towards the decay channel displays an oscillatory behaviour. There is a near miss and trans-HT in the time window comprising the decay process. 2-cis-C5H6NH2 `. These representative trajectories were begun with an initial torsional distortion about the C3wC4 (35°) and C4wC5 (20°) bonds.The resulting initial structure is ca. 12 kcal mol~1 less stable than the planar S2/S1 CI (or FC) geometry (i.e. ca. three times the energy rise occurring by distorting the 2-cis-C5H6NH2 ` by 20°).40 It can be seen that in both systems there is no initial acceleration along the torsional coordinate [cf. Fig. 10(a) and (c)]. However in 2-cis-C5H6NH2 ` one observes that once torsional motion is initiated by IVR (on a timescale of 15»30 fs depending on the value of the torsional angle at time zero) it continues directly up to a ca. 90° twisted structure corresponding to the CI. In other words S oscillations along the torsional modes are not observed. In sharp contrast in all-trans-HT one can see that despite the consider- 1 able initial distortion and 12 kcal mol~1 energy rise given to the S2/S1 CI structure several oscillations along the torsional coordinates are observed.This behaviour re—ects the very diÜerent shape of region II in these systems. In fact the slight stability (101 cm~1) rather than instability (264 i cm~1) in the region of the planar S structure and 1 68 C Fig. 11 Time evolution of the central bond torsional angle (a in Fig. 10) along the computed S1 trajectories for the double-bond isomerization of 5H6NH2 ` ()). The two curves refer to the initial conditions with 10° twisted FC structures. the —atness of the path to CI lead to oscillating behaviour and to an increase in S lifetime. (However since our MM-VB force –eld exaggerates the stability of the planar 1 S structure with respect to the CI the computed lifetime of 1.5 ps appears to be at least 1 three times longer than observed1h5 in these systems).conical intersection with little initial velocity (i.e. ideally from 0»0 excitation to S 2/S is1given in A possible trajectory along the CHD S surface and starting from the S 1 Scheme 3(c). The shape of the S energy surface along region I appears to be inverted 2) with respect to the 2-cis-C5H6NH2 ` surface [Scheme 3(a)]. In fact the initial part of region I has a ridge with a very large curvature. In these conditions one expects that 1 even a small perturbation along the non-totally symmetric mode will induce a large impulsive acceleration along region II. Therefore in this case the S1/S0 CI decay channel may be entered on a timescale which must be short with respect to all-trans-HT.However owing to the —atness of the CHD surface along region II no additional acceleration towards S The results (MEP MEPC2 and related frequencies) given above suggest that the 1/S0 CI will be delivered on the ìrelaxedœ excited-state molecule. previously proposed mechanism for the photochemical CHD ring opening,19,39 needs to be modi–ed. In fact we proposed on the basis of CI and SP computations that decay at S2/S1 CI resulted in the production of a C2 SP intermediate (cZc-HT* in ref. 39) vibrationally excited along totally symmetric (stretching and breathing) modes. Similarly to all-trans-HT decay to the ground state would have then occurred via IVR and ìslowœ C SP to the S motion along the —at region II from 1/S0 CI point.However the novel 2 ridge-shaped S surface given in Scheme 3(c) indicates an excited-state dynamics for the CHD ring opening which must be diÜerent from that of the other two molecules con- 1 sidered in this paper. In fact the molecule will probably never reach the bottom of region I where C SP is located. The ridge-like S surface is expected to push the mol- 2 1 ecule along the unconstrained MEP (see Fig. 9) since the very beginning of the relaxation process. IVR from totally symmetric to non-totally symmetric modes is not important in this system which should accelerate directly along the asymmetric mode prompting decay to S 5 Conclusions Above we have reported the structure of the barrierless S PESs which control ultrafast photochemical processes in three diÜerent conjugated compounds.Such a structure is 1 PESs for ultrafast photochemistry cis-C5H6NH2 ` (K) and a-methyl-cis- 0 . 69 M. Garavelli et al. characterized by the shape of two domains called region I and region II. Region I controls the initial excited-state relaxation of the system. In contrast region II is a low-lying region of the S PES and controls the evolution of the excited-state molecule towards fully efficient decay to the ground state. Region I develops along a totally symmetric 1 coordinate corresponding to a stretching deformation (at least in the region where the molecule initiates its S state motion). In contrast region II develops along a non-totally symmetric coordinate.For the protonated SchiÜ bases and all-trans-HT this coordinate 1 is dominated by torsional deformation about one or more bonds of the original reactant. However in the case of CHD this motion involves a sort of bending deformation of the carbon skeleton. Recent experimental investigations of the excited-state dynamics of PSB11 and other retinal protonated SchiÜ base (PSB) isomers all-trans-HT and CHD have probed the timescale for the depletion of the S transient in these systems. DiÜerent PSBs in solu- 1 tion show a biexponetial decay dynamics with ca. 2»4 and 3»12 ps lifetimes.11h14 On the other hand while there are no experimental data on the excited-state dynamics of the short protonated SchiÜ bases 2-cis-C5H6NH2 ` and a-methyl-2-cis-C5H6NH2 ` semiclassical dynamics simulations suggests a ca.60 fs lifetime and a monoexponential decay (see Fig. 11). On the basis of the present study we tentatively assign the diÜerent timescales to the structure of region II. In other words the —atness of this region (or the presence of a small barrier along it) and the presence of diÜerent decay channels (see Fig. 7) in all-trans-C7H8NH2 ` (a longer retinal protonated SchiÜ base model) would explain the two-order-of-magnitude slower dynamics and biexponential decay respectively. Alltrans-HT shows a ca. 250 and 500 fs S lifetime in the gas phase4 and in solution,5 respectively. This timescale is ca. one order of magnitude longer than computed for 1 2-cis-C5H6NH2 ` but ca. one order of magnitude shorter than that observed in PSBs.The —at region II of all-trans-HT resembles that of the longer model all-trans- C7H8NH2 `and this should be the reason for the slower dynamics (with respect to 2-cisshape. This region may thus prompt a certain amount of dispersion in a bunch of trajec- C5H6NH2 `). However region I in all-trans-HT features a —at (if not slightly ridge-like) tories leaving the initial structure. Finally a recent experimental investigation of CHD has suggested that the ring-opening reaction must occur in ca. 80 fs in the gas phase9 and in \1 ps in solution.10 These very short lifetimes seem to agree nicely with the hypothetical picture for the CHD S dynamics given above. In this case the ridge-like structure of region I provides an almost impulsive motion towards the decay channel.1 M.O. and M.A.R. are grateful to NATO for a travel grant (CRG 950748). This research has been supported in part by an EU TMR network grant (ERB 4061 PL95 1290 Quantum Chemistry for the Excited State). The dynamics computations were carried out on an IBM-SP2 funded jointly by IBM-UK and HEFCE (UK). The dynamics computations on 2-cis-C5H6NH2 ` were carried out in collaboration with Berny Schlegel. References Fuê T. Schikarski W. E. Schmid S. Trushin K. L. Kompa and P. Hering J. Chem. Phys. 1997 1 H. Petek A. J. Bell R. L. Christensen and K. Yoshihara J. Chem. Phys. 1992 96 2412. 2 C. C. Hayden and D. W. Chandler J. Phys. Chem. 1995 99 7897. 3 W. 106 2205. 4 D. R. Cyr and C. C. Hayden J. Chem. Phys. 1996 104 771.5 K. Ohta Y. Naitoh K. Saitow K. Tominaga N. Hirota and K. Yoshihara Chem. Phys. L ett. 1996 256 629. 6 P. J. Reid S. J. Doig S. D. Wickham and R. A. Mathies J. Am. Chem. Soc. 1993 115 4754. 7 M. O. Trulson G. D. Dollinger and R. A. Mathies J. Chem. Phys. 1989 90 4274. 8 P. J. Reid S. J. Doig and R. A. Mathies Chem. Phys. L ett. 1989 156 163. 9 S. A. Trushin W. Fuê T. Schikarski W. E. Schmid and K. L. Kompa J. Chem. Phys. 1997 106 9386. 70 PESs for ultrafast photochemistry 10 S. H. Pullen N. A. Anderson L. A. Walker II and R. J. Sension J. Chem. Phys. 1998 108 556. 11 H. Kandori Y. Katsuta M. Ito and H. Sasabe J. Am. Chem. Soc. 1995 117 2669. 12 S. L. Logunov L. Song and M. El-Sayed J. Phys. Chem. 1996 100 18586. 13 P. Hamm M. Zurek T. Roé schinger H.Patzelt D. Oesterhelt and W. Zinth Chem. Phys. L ett. 1996 263 613. 14 H. Kandori and H. Sasabe Chem. Phys. L ett. 1993 216 126. 15 F. Bernardi M. Olivucci and M. A. Robb Chem. Soc. Rev. 1996 25 321. 16 (a) J. Michl and M. Klessinger Excited States and Photochemistry of Organic Molecules VCH New York 1995; (b) J. Michl and V. Bonacic-Koutecky Electronic Aspects of Organic Photochemistry Wiley New York 1990. 17 M. Klessinger Angew. Chem. Int. Ed. Engl. 1995 34 549. 18 A. Gilbert and J. Baggott Essentials of Molecular Photochemistry Blackwell Oxford 1991. 19 P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Phys. Chem. 1996 100 19364. 20 M. Garavelli P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Am. Chem. Soc. 1997 119 6891. 21 M.Garavelli P. Celani M. Fato M. J. Bearpark B. R. Smith M. Olivucci and M. A. Robb J. Phys. Chem. 1997 101 2023. 22 M. Garavelli P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Am. Chem. Soc. 1997 119 11487. 23 M. Garavelli T. Vreven P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Am. Chem. Soc. 1998 120 1285. 24 P. Celani M. A. Robb M. Garavelli F. Bernardi and M. Olivucci Chem. Phys. L ett. 1995 234 1. 25 (a) The MC-SCF programme we used is implemented in Gaussian 94 Revision B.2 M. J. Frisch G. W. Trucks H. B. Schlegel P. M. W. Gill B. G. Johnson M. A. Robb J. R. Cheeseman T. Keith G. A. Petersson J. A. Montgomery K. Raghavachari M. A. Al-Laham V. G. Zakrzewski J. V. Ortiz J. B. Foresman C. Y. Peng P. Y. Ayala W. Chen M. W. Wong J. L. Andres E. S.Replogle R. Gomperts R. L. Martin D. J. Fox J. S. Binkley D. J. Defrees J. Baker J. P. Stewart M. Head-Gordon C. Gonzalez and J. A. Pople Gaussian Inc. Pittsburgh PA 1995. 26 (a) K. Andersson P-A. Malmqvist and B. O. Roos J. Chem. Phys. 1992 96 1218; (b) MOLCAS Version 3 K. Andersson M. R. A. Blomberg M. Fué lscher V. Kelloé R. Lindh P-A. Malmqvist J. Noga J. Olsen B. O. Roos A. J. Sadlej P. E. M. Siegbahn M. Urban P. O. Widmark University of Lund Sweden 1994. 27 W. H. Miller N. C. Handy and J. E. Adams J. Chem. Phys. 1980 72 99. 28 D. G. Truhlar and M. S. Gordon Science 1990 249 491. 29 F. Bernardi M. Olivucci and M. A. Robb J. Am. Chem. Soc. 1992 114 1606. 30 M. J. Bearpark M. A. Robb F. Bernardi and M. Olivucci Chem. Phys. L ett. 1994 217 513. 31 B. R. Smith M. J. Bearpark M. A. Robb F. Bernardi and M. Olivucci Chem. Phys. L ett. 1995 242 27. 32 T. Helgaker E. Uggerud and H. J. A. Jensen Chem. Phys. L ett. 1990 173 145. 33 W. Chen W. L. Hase and H. B. Schlegel Chem. Phys. L ett. 1994 228 436. 34 (a) A. Warshel and M. Karplus Chem. Phys. L ett. 1975 32 11; (b) The dynamics of PSB11 photoisomerization has been investigated at the semiempirical level of theory by Warshel et al. (see A. Warshel Nature (L ondon) 1976 260 679; A. Warshel. Proc. Natl. Acad. Sci. USA 1978 75 2558; A. Warshel Z. T. Chu and J-K. Hwang Chem Phys. 1991 158 303. 35 (a) R. K. Preston and J. C. Tully J. Chem. Phys. 1971 54 4297; (b) J. C. Tully and R. K. Preston J. Chem. Phys. 1971 55 562. 36 (a) S. Klein M. J. Bearpark M. A. Robb F. Bernali and M. Olivucci Chem. Phys. L ett. 1998 in press ; (b) W. Domcke and G. Stock Adv. Chem. Phys. 1997 100 1. 37 The importance of methyl substitution for the ultrafast dynamics of PSB11 in rhodopsin is under current investigation. See Q. Wang G. G. Kochendoerfer R. W. Schoenlein P. J. E. Verdegem J. Lugtenburg R. A. Mathies and C. V. Shank J. Phys. Chem. 1996 100 17388; R. W. Schoenlein L. A. Peteanu Q. Wang R. A. Mathies and C. V. Shank J. Phys. Chem. 1993 97 12087. 38 T. Vreven F. Bernardi M. Garavelli M. Olivucci M. A. Robb and H. B. Schlegel J. Am. Chem. Soc. 1997 119 12687. 39 P. Celani S. Ottani M. Olivucci F. Bernardi and M. A. Robb J. Am. Chem. Soc. 1994 116 10141. 40 M. Garavelli T. Vreven P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Am. Chem. Soc. 1998 120 1285. Paper 8/02270D; Received 23rd March 1998

ISSN:1359-6640    

DOI:10.1039/a802270d

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 71»89 Potential-energy surfaces and their dynamic implications Keiji Morokuma Qiang Cui and Zhiwei Liu Cherry L . Emerson Center for Scienti–c Computation and Department of Chemistry Emory University Atlanta GA 30322 USA Accurate density functional and ab initio calculations have been performed to study the potential-energy surfaces (PESs) and their implications for kinetics and dynamics of (1) the spin-forbidden reaction CH(2%) ]N2 ]HCN]N(4S) ; PES characteristics are calculated and used to evaluate the overall rate using non-adiabatic transition-state theory. (2) Gasphase ion»molecule reactions C2H2 `]NH3 ; PESs are calculated and the mechanism of efficient charge transfer and proton transfer competing with stable complex formation is discussed.C2H2 `]CH4; the modeenhancement eÜect has been elucidated in terms of the new transition state and by direct trajectory calculations. 1 Spin-forbidden reaction CH(2P) + N CH(X 2%)]N2 ]HCN]N(4S) The reaction 2 is a very important reaction in combustion chemistry and is believed to be the key step in the production of NO in hydrocarbon —ames.1 The pressure and temperature dependences of the reaction rate indicate that the reaction proceeds via formation of a long-lived complex and that diÜerent mechanisms dominate at low and high temperatures respectively.2 The PESs for this reaction have been studied by many including Manna and Yarkony,3 Martin and Taylor4 and Walch.5 It has been established from these studies that the mechanism of the reaction can be summarized as No barrier exists between the reactants and the dative minimum while a ca.0.7 eV barrier exists between the reactants and the C minimum. A large barrier exists between C the dative and the 2v structure and therefore isomerization between the two does not happen directly. A 2D model potential surface for the channel involving the C2v doublet 2v minimum has been constructed by Walsh and Seideman and used for calculations of the cumulative reaction probability N(E) and the thermal rate constant.6 It would be very useful to have a simple expression for the non-adiabatic reaction rate constant as the standard transition-state theory (TST) for the adiabatic reaction. Such an idea has been discussed by a few authors in a diÜerent context.7 Among those the work of Lorquet et al.7a is the closest to the extension we plan to make here.1.1 PESs There still remain some uncertainties on the barrier heights as well as the energetics of the crossing point. Some structures are found to be not directly connected and some 71 72 PESs and their dynamic implications intermediates are missing. Therefore we have –rst carried out high-level ab initio calculations of the structures and energies of the minima transition states and the minimum on the seam of crossing (MSX). The structures were optimized and vibrational frequencies were calculated at the B3LYP/6-311G(d,p) level and the energies were recalculated at the G2M(RCC) level.8 complex 2 2v as explored by Walch,5 goes over a C transition state d4-ts to a C interme- s The global potential-energy pro–le for the present system thus calculated is shown in Fig.1. The two dative complexes HCNN d1 in 2AA and d1-2 in 2A@ have been found of which d1 is the ground state. No encounter barrier exists from the reactants to either d1 or d1-2. The path leading to the other type of CHN complex the so-called C (2A d3 2) diate d4 (2A@) which is converted to a stable complex d3 (2A2) after a small barrier at 1 d3-d4-ts. As pointed out by Martin,4 a large barrier exists at the transition state d2-ts (2AA) for isomerization between d1 and d3. C2v region. HCN]N(4S) can be formed CH(4&~)]N via the two energetically similar The quartet state potential-energy pro–le can also be divided into two diÜerent regions the dative region and the C2v region.The C2v region includes the wellcharacterized minimum q3(4B1) and the transition state q3-ts (4AA) for dissociation to HCN]N(4S). They can be accessed at relatively low energy only through intersystem crossing from the doublet state surface in the directly from the quartet reactants of 2 trans and cis dative pathways via encounter transition states q1-ts-cis and q1-ts-trans through dative minima q2-cis and q2-trans and dissociation transition states q2-ts-cis and q2-ts-trans. (2A and quartet 4B 2) (1) MSX-dq3 is very close The optimized MSX between doublet to that of Yarkony.3 At the G2M(RCC) level without ZPE the energy of MSX-dq3 is nearly the same as the doublet reactants. The search for MSX for the dative channel between 2AA and 4AA converged to a structure very close to the trans-minimum on the quartet state surface q2-trans.Judging from the higher energy of q2-trans the high exit barrier at q2-ts-cis or trans and the smaller spin»orbit coupling element (calculated at the CASCF level with the one-electron Breit»Pauli Hamiltonian and empirically –t nuclear charge) between 2AA and 4AA the dative intersystem crossing channel is not likely to compete with the C2v intersystem crossing mechanism. In the rate constant calculations with non-adiabatic TST to be discussed below frequencies of vibrations orthogonal to the norm of the seam at the MSX have to be calculated. The vibrational frequencies at the MSX for the doublet and the quartet states were found to diÜer by up to as much as 20%.Much tighter convergence criteria9 did not reduce the diÜerence and the diÜerence in the projected vibrational frequencies of the two electronic states comes intrinsically from the de–nition of the norm of the seam which is expressed as the energy diÜerence gradient and causes some errors on the degeneracy condition starting from the second order. 1.2 Non-adiabatic TST As is well established the central issue in the rate constant calculations is the derivation of the cumulative reaction probability N(E) which within the framework of classical mechanics is de–ned as the following :10 (1) N(E)\2n+(2n+)~F Pdp Pdq d[E[H(p q)]F(p q)s(p q) where H(p q) is the classical Hamiltonian of the molecular system.The —ux operator F(p q) is de–ned in terms of a dividing surface f which in most cases is a function of the 73 CH(2%)]N2 ]HCN]N(4S). Energies are calculated with G2M(RCC) including Fig. 1 Global potential-energy pro–le for the reaction ZPC[B3LYP/6-311G(d,p)]. K. Morokuma et al. 74 PESs and their dynamic implications coordinate. To calculate N(E) within a TST framework one has to obtain the following quantities (i) the de–nition of the dividing surface ; (ii) the reaction coordinate; (iii) the Hamiltonian in terms of the reaction coordinate and those orthogonal to it and (iv) the characteristic function. We will de–ne these for a non-adiabatic extension of the TST. (i) Expression of the dividing surface. In the standard TST one uses the hypersurface that is perpendicular to the reaction coordinate and contains the saddle point qF\0.In the present non-adiabatic extension we choose the seam of crossing as the dividing surface and use a simple Taylor expansion of the two PESs around the MSX. Let us start by recalling the de–nition and a few geometrical properties of the MSX. Let V1(q) and V2(q) be the two PESs which behave like diabatic states in the usual sense owing to their diÜerent space]spin symmetries. The qs are the mass-weighted Cartesian coordinates. The seam surface can be de–ned with the simple equation (2) V1(q)[V2(q)\0 Expanding both (3) V1(q) [ and V1(q0 V ) 2 [ (q) V up 2(q to 0)] the ]* –rst q Æ [ order g1(q)[ we g2 may (q)] Kwrite \0 q0 Clearly the norm of the seam surface is nothing but the normalized energy diÜerence gradient K .q0 s� \ *g12(q) o*g12(q) o In the following s� is used to denote the normalized vector and s is used to denote the actual numerical value of the displacement along s� . The MSX is de–ned by eqn. (3) and thetion of the minimum on the seam of crossing (Iå [s� s� )gi(q)\0 8 (4) As discussed by many authors including us it is straightforward to set up a Newton» Rhapson optimization scheme to locate the MSX with those two constraints.11h13 Since the MSX is a true minimum on the seam of crossing it is valid to make a harmonic expansion around the MSX on the seam of crossing. Following Millerœs reaction path Hamiltonian,14 one can achieve this by diagonalizing a projected Hessian matrix (5) H@\(Iå [På )H(Iå [På ) where the projector På contains the normal vectors corresponding to the in–nitesimal total translation rotation and the energy diÜerence gradient vector s� .At the MSX it is well known that the gradient vectors of the two electronic states are either parallel or anti-parallel. s 15 Therefore the � vector is parallel or anti-parallel to the gradient of each PES and the projector in eqn. (5) becomes exactly the same as that in the case of the reaction-path Hamiltonian. Therefore one can calculate the projected vibrational frequencies at the MSX directly with any ab initio package that can handle reaction-path frequency calculations. Since the degeneracy condition is up to –rst order the error starts from the second order which implies that the projected vibrational frequencies for the two electronic states involved will not be exactly the same as shown in Section 1.1.From the normal mode analysis on the seam one obtains (3N[7) non-zero eigenvalues corresponding to the bound motion on the seam surface and seven zero eigenvalues corresponding to the total translation/rotation and the norm of the seam surface. One also obtains a new set of orthonormal coordinate MQN which are also orthogonal to s� . With this new set of coordinates the two diabatic potentials can then be expressed 75 K. Morokuma et al. as (6) Vi(q)4Vi(s Q)\Vi(s)]1 2 ; uk2(0)Qk2(0) k and the dividing surface can be expressed by the simple equation (7) s\(q[q0) Æ s� \0 (ii) The ìhoppingœ coordinate.s It is natural to choose � to be the reaction coordinate or more appropriately the hopping coordinate. We should emphasize that in most realistic reactions the hopping coordinate is not the coordinate that leads to the products. Therefore the N(E) one calculates at the MSX may only be a component of the total rate constant. One usually has to consider the reaction as a multi-step process and derive the rate constant for the entire reaction with the N(E)s calculated at several critical structures. The reaction of CH]N is a very typical example as we shall discuss in Section 1.3. 2 (8) Hi\ p 2 s2]Vi(s)]; Cp (iii) System Hamiltonian. The system Hamiltonian can then be expressed as 2 k2] 2 1 uk2Qk2D k (iv) The characteristic function.The characteristic function for a non-adiabatic process is clearly just a probability factor that the system makes a transition from one diabatic surface to the other (9) sr(J2Es Q)\Ptr(J2Es Q) In the standard TST all the trajectories with momenta pointing towards products are counted in the rate expression. In a non-adiabatic process the trajectory with momenta pointing towards the reactants can also make diabatic transitions and contribute to the rate constant. Therefore in the weak coupling limit no Heaviside function of the momentum appears in eqn. (9). With the de–nition of necessary ingredients above one can carry out the integration over s and N ps (E and )\ obtain 2](2n the +)~( following F~1)Pdp dQh[E[HF~1(pQ Q)]sr(J2Es Q) (10) P Q where the characteristic function depends in general on the energy in the s degree of freedom Es\E[HF~1 and also on Q.The factor 2 comes from the fact that the —ux of both directions can contribute to the rate constant. The evaluation of the phase-space integral in eqn. (10) can be rather complicated if one considers the transition probability as a function of Q coordinates. Great simpli–cation can be achieved if we consider the transition probability only as a function of energy in the hopping coordinate. Rewriting the Heaviside function as the integral of a d function c E N(E)\ we 2] can (2 write n+)~(F eqn. ~1) P(10) ~E as d : Es sr(Es) PdpQ dQd[E[Ec[Es[H@F~1(pQ Q)] P (11) 42]PE~Ec dE 0 v Ptr(E[Ec[Ev)oF~1(Ev) c 0 where E is the energy of the MSX and E denotes the energy in the ìboundœ modes on the seam.The physical meaning is very clear N(E) is nothing but a weighted sum of v density of states at the MSX. The quantum mechanical correspondence of eqn. (11) 76 PESs and their dynamic implications is given as (12) N(E)\2]; Ptr(E[EvKnL) KnL 1.3 Application of non-adiabatic TST to the reaction We would like to apply the non-adiabatic TST descibed above to the reaction rate of CH(2%)]N the reaction 2 ]HCN]N(4S) using the potential-energy properties calculated ab initio in Section 1.1. In the current study we have employed the simpli–ed reaction mechanism,6 as illustrated in Fig.2 without considering the dative channel. CH(2%)]N The entire process of 2 ]HCN]N(4S) is divided into three stages (i) overcoming the barrier d4-ts to form d3; (I) intersystem crossing from the doublet d3 to the quartet state q3 through the region around MSX-dq3; and (ii) overcoming the barrier q3-ts to form the quartet product HCN]N(4S). We treat each step independently and –nally combine them to obtain the rate constant from the whole process using the uni–ed statistical theory (UST) of Miller,16 noting that we have two rather deep and long-lived complexes d3 and q3. N(E) for the rate constant for the total spinforbidden reaction process is obtained as the following in terms of the N(E) at each critical structure (13) N0(E)\ (N N1N2N3NxNy 2Ny]N3Ny[N2N3)(N2Nx]N1Nx[N2N1) Ni(E)s.To calculate the N2(E) at MSX-dq3 we used the calculated Here as illustrated in Fig. 2 1 2 and 3 denote the –rst second and third transition states respectively and x and y denote the –rst and second complexes respectively. In the calculation according to eqn. (13) we have used harmonic direct counting to calculate all the projected vibrational frequencies. In order to calculate the transition probability another ingredient in the calculation of N2(E) as a function of energy in the hopping Fig. 2 Simpli–ed schematic potential-energy pro–le for the reaction CH(2%)]N2 ]HCN ]N(4S) used in the rate constant calculations K. Morokuma et al. 77 coordinate s� we –rst had to establish a 1D model. We carried out CASPT2/6-311G(d,p) along the NwCwN bending coordinate from d3 to q3 with other parameters –xed at those of MSX-dq3.We used the spin»orbit coupling elements calculated in Section 1.1. To carry out N21d(E) calculations quantum mechanically we arti–cially —attened out the potential-energy curves of the doublet (quartet) when the NwCwN angle was smaller(larger) than that in the d3(q3) to model the asymptote. Furthermore we also made the doublet arti–cially steeper after the NwCwN angle was close to that in q3. The goal was to calculate the transition probability which was only aÜected by a very small portion of the PESs near the MSX. Even with these arti–cial modi–cations the model is still far better than the linear potential»constant coupling model assumed in many transition formulae.After the 1D model was established the distorted wave approximation (DWA) was used to calculate the transition amplitude (the square root of the transition probability) (14) T0i4JP0i\2(k0 ki)~1@2 P= t(0) i (r@)Vi0(r@)t0(0)(r@) dr@ 0 N21d(E) is consistent with the 2D V where i0(r) is the diabatic coupling element. The parameters obtained in the ab initio calculations and used in the non-adiabatic thus calculated as a function of energy is presented in Fig. 3(a). Evidently the DWA TST calculations are shown in Table 1. The calculated transition probability N21d(E) exhibits ose transition probabilities which manifest as interference between the trajectories on the two diabatic surfaces. The absolute value of the transition probability is very small of the order of 10~4; the reaction is very ìdiabaticœ in the MSX region.To examine if the numerical value of N22d(E) obtained by Seideman,6b we convoluted N21d(E) with the asymmetric CwN 22d(E). stretch vibrational frequency at MSX-dq3 (943 cm~1 which is 1005 cm~1 for the quartet state). The derived result denoted as N22d(E) plotted in Fig. 3(b) is of the order of 10~3 and exhibits some moderate oscillations much less sharp than those of Seidemanœs N With the obtained N21d(E) we then calculated N(E) and k(T ) for the whole reaction. k2(T ) in Fig. 2(d). N(E) for J\0 is shown in Fig. 3(c). Even after convolution of the –ve vibrational degrees of freedom N(E) shows structures which are more visible when the 2AA frequencies at MSX-dq3 are employed in the calculations.In addition N(E) obtained with the 4AA frequencies is larger than that calculated with the 2AA frequencies as the numerical values of the frequencies are generally smaller for the former. The transition probability at MSX-dq3 is so small that the total N0(E) is reduced to N(E) at MSX-dq3. Since MSX-dq3 is the rate-determining structure we have simply employed J-shifting to derive the thermal rate constant. The resultant k(T ) in the temperature range of 1000» 3000 K shown in Fig. 3(d) can be –tted to an Arrhenius expression with a prefactor of 109.19 which seems to be too low compared with recent experimental measurements and an empirical RRKM study which has a prefactor of ca. 1011h12.Comparing our results with the previous empirical RRKM study of Rogers et al.17 obtained compared with experimental measurement unless several empirical frequencies [kfit(T ) in Fig. 3(d)] we note that they mentioned that a much smaller rate constant is are scaled down by a factor of 2 and a very large i value of 0.04 (larger than the L»Z probability 0.001»0.01 of Yarkony et al.) is used to describe the intersystem crossing probability. Indeed the scaled results contain three frequencies around 300 cm~1 much lower than the ab initio frequencies we have obtained. With their vibrational frequencies we found that our rate constant increased by more than an order of magnitude. Further assuming that the spin»orbit coupling element is two times larger than we have calculated the derived thermal rate constant becomes Obviously scaling the frequencies and coupling element is not a solution but rather it strongly suggests that some important issues may have been overlooked and require Table 1 Parameters used in the non-adiabatic TST calculation for the reaction of CH]N2 energeticsb/cm~1 frequenciesa/cm~1 structure CH2(2%) N2 d4-ts(2A) d3(2A2) 2804.3 2447.0 300.9 600.4 1036.2 1900.7 3008.2 718.1 838.7 926.3 1193.3 1636.4 3206.2 0.0 4932 [7310 1609 1600 MSX-dq3(2A2) 943.2 948.3 1370.7 1476.0 3025.8 MSX-dq3(4B1) 797.4 1005.0 1059.7 1358.6 3040.8 [700 6401 q3(4B1) q3-ts(4AA) 562.0 809.1 1103.4 1187.4 1225.2 2886.5 349.2 741.1 866.2 1929.5 3294.6 a Derived from the B3LYP/6-311G(d,p) results.b Derived from the UCCSD(T)/6-311G(d,p) results. c Reduced mass in the 1-D eÜective Hamiltonian in au. kID c Ia/u a0 2 4.2 30.0 52.2 19.9 14.6 41.4 29.1 17.1 63.5 16.6 13.2 63.5 16.6 13.2 98.0 13.1 11.5 68.2 10.0 8.7 16242.8 78 PESs and their dynamic implications 79 K. Morokuma et al. k2(T ) is computed with scaled vibrational frequencies of MSX-dq3 Fig. 3 (a) 1D transition probability obtained with the distorted wave approximation as a function N of energy. (b) 2D cumulative reaction probability 22d(E) obtained by energy shifting approx- N imation based on 21d(E) as a function of energy. The frequency for the bound degree of freedom is selected to be the asymmetric CwN stretch of the 2AA state at MSX-dq3. (c) N reaction calculated according to eqn.(13). The relatively smooth curve is obtained with the set of 0(E) for the total vibrational frequencies for the 4AA state at MSX-dq3 and the curve with more visible structures is production of quartet products. calculated with the 2AA frequencies at MSX-dq3. (d) Calculated thermal rate constant k1(T ) for the from ref. 17 and two-fold larger spin»orbit coupling constant. kfit(T ) is from ref. 17. further investigation. For instance although a 1D model for the intersystem crossing sounds very reasonable judging from the structure of d3 MSX-dq3 and q3 it is possible that another degree of freedom possibly the CwN stretch is also crucial to the spinforbidden transition. By this we mean that the transition probability s which in the current work is assumed to depend only on the energy in the hopping coordinate might actually vary signi–cantly along one or more degree of freedom orthogonal to the norm of the seam s.We note that at the q3 structure the doublet electronic state is not very high in energy and therefore the seam might cover a larger region of the PES at a certain energy than that based on simple harmonic expansion at MSX-dq3. Indeed as we mentioned above N2d(E) appears to increase too slowly as a function of the energy compared to the explicit 2D quantum mechanical calculation of Seideman.6b Therefore, 80 PESs and their dynamic implications rigorous quantum dynamic calculations with at least two degrees of freedom with accurate spin»orbit coupling elements in an extended region together with approximate treatment of other degrees of freedom are required to examine the situation in more depth.1.4 Conclusions The detailed reaction mechanism of the spin-forbidden reaction CH(2%)]N2 ]HCN ]N(4S) has been studied with high-level ab initio methods. Although a few new structures have been found on both the doublet and quartet electronic states the dominant C mechanism remains the intersystem crossing mechanism summarized by Walch.5 Vibrational frequencies orthogonal to the norm of the seam have been calculated at the 2v MSX and applied to calculate the rate with an extension of the TST for spin-forbidden reactions. A one-dimensional model was set up to consider the spin-forbidden transition probability. The eÜect of other degrees of freedom was then considered by energy convolution with the vibrational frequencies orthogonal to the seam of crossing at the MSX.The calculated cumulative reaction probability N(E) seems to be consistent with that obtained by Seideman.6b with the ABC-DVR-Fermi-Golden-rule approach and a 2D model. Nevertheless with such a TST expression and the 1D model for the intersystem crossing process the thermal rate constant k(T ) seems to be too low by two orders of magnitude compared with experimental measurements. Indeed the vibrational frequencies obtained in the current study are much larger than that from an empirical RRKM study where empirical vibrational frequencies at the MSX had to be scaled by a factor of two in order to derive a reasonable k(T ).Such a discrepancy strongly suggests that some important issues might have been overlooked. In particular the assumption that the spin-forbidden transition takes place with uniform probability on the seam may be a poor assumption in the particular case we are considering. (III) (IV) (V) (VI) 2 Dynamics of gas-phase ionñmolecule reactions 2.1 C2H2 ë + NH3 In the recent experiment by Anderson and co-workers,18 the only product channels observed over a wide range of collision energy are charge transfer (CT) and proton transfer (PT). (PT) C2H2 `]NH3 ]C2H]NH4 ` (CT) ]C2H2]NH3 ` (complex) ]C2H5N`] (H-abstraction) ]C2H3 `]NH2 Surprisingly no evidence for the formation of the stable C2H5N` intermediates has been observed.The H-abstraction channel which was dominant at high collision energy for the isoelectronic system of C2H2 `]CH4 ,19 was also not observed. It is not surprising for thermodynamic reasons that the CT and PT channels become open for the NH system than for the CH system. However the thermodynamic properties of the 3 complex channel and the H-abstraction channel are qualitatively similar for both 4 systems. Therefore it is really intriguing why these channels have been observed experimentally for C2H2 `]CH4 but not for C2H2 `]NH3 over a wide range of collision energy. All the observations suggest that PT is a direct channel with a proton-stripping mechanism. As for the CT channel two diÜerent mechanisms seem to exist. At low 81 K. Morokuma et al.energy CT takes place through a weakly bound complex with a lifetime longer than ca. 1 ps. At higher energy the charge seems to be transferred by a long-range electron hopping mechanism. 2H5N` complexes in the ground electronic state.20 In the present study Finally the eÜect of vibrational excitation in the reactant C2H2 ` on the reaction has also been examined as in the study of C2H2 `]CH4 where a large eÜect has been observed. The eÜect is much smaller in the present system albeit very mode-speci–c.19 In the PT channel the reaction is enhanced by C2H2 ` bending and inhibited by the CC stretch. In the CT channel both CC stretch and bending in C2H2 ` inhibit the reaction at high collision energy. At low collision energy the reaction is inhibited by the CC stretch but enhanced by the HCC bending in C2H2 `.All these facts suggest a reaction mechanism where the C2H2 ` vibration in—uences the probability of a favourable reactant geometry arrangement while the branching between product channels is determined later in the collision by factors not strongly dependent on the reactant vibration. However the origin of the mode-speci–city observed remains unclear. Very little is known about the PESs of the present system except for some structures of the stable C in order to –nd the relationship between the potential-energy characteristics and the reaction dynamics we have performed detailed and highly accurate calculations of the PESs. The overall potential-energy pro–le calculated at the G2M(RCC)//B3PW91/ 6-311G(d,p) level with sketches of some important structures is shown in Fig.4. PT channel. For both the reactants and products of the PT channel close-lying 2H2 `(2%)]NH3 is doubly degenerate with C2H2]NH3 ` state. For the products C2H(2&`) 4 ` is the lowest asymptote with the upper dissociation asymptote C2H(2%) 4 `only 0.50 eV higher in energy and also lower than the reactants of electronic states exist. For the reactants C the low lying non-degenerate ]NH ]NH C2H2 `]NH3. C If the reaction proceeds from 2H2 `]NH3 with perfect linear C2H(2&`)]NH4 ` both fall into the 2A1 sym- NwCwC framework which is the minimum energy path PT1 C symmetry is main- 2H2 `(2%)]NH3 3v C tained during the reaction.In 2H(2%)]NH4 ` fall 3v C both and C into the doubly degenerate 2E symmetry and are directly correlated. The PT1 channel in the 2E symmetry does not involve any entrance or exit barrier and proceeds through an intermediate complex PT-1 which resembles the products C2H(2%)]NH4 ` as shown in Fig. 4. Although the 2E state in C is subject to Jahn»Teller distortion in general the 3v eÜect is evidently very small in the case of PT-1. Although the calculation is carried out in C the A@ and AA states are nearly degenerate and all the vibrational frequencies are s real. This is not unexpected because the structure of PT-1 is very product-like. In the experiment the detection of the velocity distribution of NH4 ` (or actually its isoeven at low collision energy.The present channel supports the fast ìstrippingœ mechatopomer ND3H`) indicates that the PT channel follows a direct stripping mechanism nism proposed by Anderson and co-workers. The PT2 channel on the other hand proceeds from the ground state of the reactant C2H2]NH3 ` through a transition state PT-TS1 shown in Fig. 4 and no intermediate is involved. C2H2]NH3 ` and metry and are directly correlated. In the PT2 channel it is actually an H atom that has been transferred. The existence of the barrier in the PT2 channel is not surprising either since unlike PT processes most H-atom transfer reactions proceed with barriers. PT-1 lies 1.41 eV below In addition we have found an unexpected 2A@ TS structure in C PT-TS2 as shown C2H2 `]NH3 C and PT-TS1 lies 0.77 eV above 2H2]NH3 `.s in Fig. 4. IRC calculation indicates that PT-TS2 actually connects intermediate PT-1 and CC-1 a covalent NH CHCH` complex which will be discussed later. Therefore the 5 PT channel and the CC channel are now connected via PT-TS2 which is only 0.30 eV above PT-1. 3]C2H2 `. Energies are obtained from the G2M//B3PW91/6-311G(d,p) level includ- Fig. 4 Overall potential-energy pro–les for the reaction of NH ing ZPE. The B3PW91/6-311G(d,p) optimized geometries (in é A and degrees) of some key structures are also shown. 82 PESs and their dynamic implications 83 K. Morokuma et al. CC-3 CC-2 CC-1 (NH3 CHCH2)` C2H5N` 2.1.2 Covalent C complex (CC) formation channel. Several isomers of 2H5Në CHCH)` (NH (NHCHCH3)` and CC-4 (NH2CCH3)` and several isomerization TSs between them CC-TS-12 CC-TS-23 CC- 2 TS-24 and CC-TS-34 have been located.The results for the ground electronic-state structures agree well with the previous study,20 except for CC-1 (NH CHCH)` (which connects the Pt channel to the CC channel as discussed above) and CC-TS-12 which seem to have been overlooked previously. No entrance channel barrier has been found on the lowest adiabatic PES which makes it even more mysterious that no products from the covalent To rationalize the experimental fact that no CC channel products have been 2.1.4 Conclusions. In 3v 3 C2H5N` complexes have been found in the experimental work.18 observed one recognizes that the reactants NH3]C2H2 ` correlate adiabatically to the excited states of these covalent C2H5N` species.Although not very high in energy the formation of these complexes requires signi–cant alternation of the C2H2 ` geometry and electronic structure. Therefore the system is most likely to follow the PT channel instead of visiting the CC formation channel. However the experimental fact that no products from the CCs have been observed still remains to be a mystery. The fact that no H-abstraction products NH2]C2H3 ` have been observed can be understood similarly although no calculations have been carried out. 2.1.3 CT channel. For the CT process two pathways seem to exist according to the diÜerent product recoil velocity distribution.18 Although it is not totally clear from the current study what are these two paths one may make some speculations.First one may imagine diÜerent crossing structures depending on the angles of approach of the two fragments. As we have seen in Section 1.3 the two asymptotes NH3]C2H2 ` and NH3 `]C2H2 are rather close in energy at the NH3]C2H2 ` geometry. As the two fragments NH3(`) C and 2H2(`) approach with large angles the two 2A@ states interact strongly and the 2 2A@ adiabatic state becomes repulsive. Therefore there exists a good chance for the PES to cross at long separation. CT cannot take place at very far nuclear separation however owing to the weak interaction between the electronic states and therefore the small coupling element. Consequently to have a good Franck»Condon factor as well as a reasonably large coupling element an intermediate-range crossing structure is desired.On the other hand the situation is rather diÜerent when the two fragments approach with small angles close to linear. In this case the two 2A@ surfaces interact rather weakly owing to their diabatic characters and undergo weakly avoided crossings. As a result the CT channel for the linear con–guration case might yield products with quite diÜerent characteristics. In addition one may also suspect that the two CT paths come from diÜerent non-adiabatic processes namely A@]A@ CT and AA]A@ CT. To have more quantitative results one needs to optimize the MSX structures for 2A@/2A@ and also 2A@/2AA probably as a function of the relative angle of approach of the fragments. These calculations have not been carried out in the current study.C symmetry where the NwCwC framework is linear the 3]C2H2 ` lead to the PT products reactants NH 4 `]C2H(2%) without any NH barrier and only through a moderately bound complex which is very product-like. This path supports the fast stripping mechanism proposed by Anderson and co-workers. We have also located on the 1 2A@ state surface a transition state that connects the intermediates in the PT channel to the covalent species CC-1. Some trajectories may take this pathway. Several isomers of C2H5N` and the isomerization TSs between them have been located. To rationalize the experimental fact that no CC channel has been observed we argue that the reactants NH3]C2H2 ` correlate adiabatically to the excited states of these covalent C2H5N` species.Although not very high in energy the formation of these complexes requires signi–cant alteration of the C2H2 ` geometry and electronic structure. Therefore the system is most likely to follow the PT channel rather than 84 PESs and their dynamic implications visiting the CC formation channel. However the experimental fact that no products from the CCs have been observed still remains a mystery. The fact that no H-abstraction products NH2]C2H3 ` have been observed can be understood similarly although no calculations have been carried out in the current work. For the CT channel the 2AA state is repulsive in most regions except around the linear con–guration where the potential energy is attractive and readily leads to CT or PT.The shape of the 1 2A@ state con–rms the existence of a saddle point between the potential well at the linear con–guration and CC-1 which has been optimized as PT-TS2. The 2 2A@ state is mostly repulsive at all angles of approach. CT at diÜerent angles of approach or CT between electronic states of diÜerent symmetries (A@]A@ AA]A@) may produce –nal products with diÜerent characteristics and might account for the two pathways proposed by Anderson co-workers. 2H2 .21g 2.2 CH4+ C2H2 ë 2.2.1 Mode-enhancement eÜect. Controlling the outcome of reaction by ìmodeselective excitationœ concentrates on the local mode and emphasizes selective excitation of a particular motion which encourages reaction toward the desired channel.21 Many beautiful examples have been accumulated over the years in the photodissociation processes of vibrationally or rovibrationally selected small molecules including H2O,21ahd HNCO,21e,f and C Recent experiments19 provide some new examples in the case of more complex ion» molecule reactions that are vibrationally selective.Zare et al. studied the reaction of ammonium ion and ND and found that the umbrella mode of NH3 ` enhances CT and deuterium abstraction signi–cantly while the isoenergetic excitation of the breathing 3 mode does not induce any eÜect.19c In another experiment Anderson and coworkers19a, b have studied the eÜects of collision energy and mode-selective vibrational excitation on the reaction of C2H2 ` with CH4 and CD4 via a guided-ion beam scattering instrument.Two distinct reaction mechanisms are active in the energy range below 5 eV. At low energies a long-lived C3H6 ` complex forms efficiently and then decomposes primarily to ]C C3H5 `]H C and 3H4 `]H2 . C2H2 `]CH4 ]C3H6 `]C3H5 `]H (complex) 3H4 `]H2 (complex) (H-abstraction) (VII) (VIII) (IX) ]C2H3 `]CH3 Competing with reactions (7) and (8) is a hydrogen transfer reaction (9) producing 3 C2H3 `]CH3 with little atom scrambling. Channel (IX) is strongly enhanced by collision energy and becomes dominant above 0.4 eV. One interesting feature about this channel is that while CC stretching provides a weaker enhancement than collision energy two quanta of CwH bending modes (ca. 155 meV) enhance the reaction at least ca.ten-fold. Based on the isotope study with CD two possible reaction mechanisms for reaction (9) 4CH elimination from a long-lived they concluded that there exist C3H6 ` complex and direct H-atom abstraction through an oscillating intermediate where the latter is dominant by a factor of ca. 5»10 1. They also predicted an early barrier 150^50 meV. The enhancing eÜect of the CwH bending mode on the reaction is explained by the necessity of carbon atom rehybridization from sp to sp2 thus forming a bent TS during the bonding process. The amazing experimental results have certainly attracted the attention of theoreticians. A combined quantum and TS theory (TST) study has been carried out by Klippenstein to unravel the detailed reaction mechanism and the observed modeenhancement eÜect.22 Based on the structure and frequencies of several intermediates obtained at the level of MP2/6-31G(d) and G223 energetics TST calculation was found to yield qualitatively correct cross-sections for the direct channel (9) but not so satisfac-85 K.Morokuma et al. tory for the complex channels (VII) and (VIII). It was found that if the energetics of the TS involved in the complex channels was lowered by 4.5 kcal mol~1 the magnitude of the cross-section for the complex channel could be qualitatively reproduced. Most importantly no entrance or exit channel barrier was found for the H-abstraction channel (IX). However it was found that enhancement of the H-abstraction crosssection by the HwCwC bending excitation in C2H2 ` was qualitatively reproduced if the two quanta of HwCwC bending were assumed to be totally randomized.It might look as if all the issues have been solved by the work of Klippenstein. However the necessity of arti–cially lowering the energy of the TS involved in the complex channel looks somewhat questionable. Judging from the structure of the TS presented in ref. 22 we suspect that there may be a lower saddle point with totally diÜerent structure involved in the complex channel. In addition the system might be an ideal one to test the capacity of the direct trajectory method.24 In the present study we have performed detailed and highly accurate calculations of the PESs and compared these with the previous results. We also performed a few direct trajectory calculations.The overall potential-energy pro–le calculated at the G2M(RCC)//B3PW91/6-311G(d,p) level with sketches of some important structures is shown in Fig. 5. H-abstraction channel. For the H-abstraction channel as shown in Fig. 5 there exists … … …C2H2 ` no entrance barrier to reach the structure of abs-1 the so-called classical CH4 minimum or no exit barrier from abs-1 to the product the results are essentially the … … …C same as in ref. 22. However the so-called bridged CH 2H2 ` is a minimum at the 4 MP2/6-31G(d) level while it is a second-order saddle point abs-TS1 at the present B3PW91/6-311G(d,p) level connecting abs-1 and its pseudo-mirror image abs-1º. Optimization at the CCSD(T)/6-31G(d,p) level agrees with the present results suggesting strongly that B3PW91/6-311G(d,p) is closer to reality than MP2.To summarize the overall mechanism of the H-abstraction channel is the so-called ì direct œ abstraction channel actually proceeding through a moderately bound (ca. 15 kcal mol~1) complex without entrance or exit barrier 2H2 ` C but more like a 2H3 ` isomerization TS between its two conforma- 1 Covalent C CC channel. In ref. 22 the complex channel (7) and (8) leading to 3H6 ë C3H6 ` has been shown to proceed from the so-called classical CH4… … …C2H2 ` through a saddle point com-TS1-K which is also included in Fig. 5. However with the high energy for this TS the calculated cross-section is even qualitatively too small. Indeed the structure of com-TS1-K does not look like a transition state for CwH activation of CH by C tions perturbed by a CH fragment.We have actually located a new C transition state 4 3 com-TS1 shown in Fig. 5 which was missed in the previous study.22 The IRC25 calculations verify that this TS actually connects abs-1 and C3H6 `. At the G2M level the energy of com-TS1 is 0.38 eV (8.7 kcal mol~1) below the reactants and the barrier height measured from abs-1 is 0.28 eV (6.5 kcal mol~1). In ref. 22 a qualitatively correct cross-section was obtained by arti–cially lowering the barrier at com-TS1-K from 11.7 to 7.2 kcal mol~1. Therefore we are con–dent that with the present results reasonable cross-sections comparable to the experimentally measured value can be obtained. Direct trajectory calculations.Our ab initio calculations indicate that HwCwC bending is clearly strongly coupled to the reaction path and therefore additional energy in this mode may contribute eÜectively to the reaction rate while the CwC stretch is nearly inert during the whole reaction. In order to –nd some characteristics of the dynamical process we have ran three direct trajectories with the C constraint at the B3PW91/6-31G(d,p) level. s 2H2 ` and is In all the trajectories the impact parameter b is taken to be zero and the initial velocity is along the line that joins the centres of mass of CH and C 4 CH4]C2H2 `. Energies are obtained from the G2M//B3PW91/6-311G(d,p) level includ- Fig. 5 Overall potential-energy pro–les for the reaction of ing ZPE. The B3PW91/6-311G(d,p) (CCSD(T)/6-31G(d,p) in parentheses) optimized geometries (in é A and degrees) of some key structures are also shown.86 PESs and their dynamic implications 87 K. Morokuma et al. Fig. 6 Results from three direct trajectory calculations for the reaction of CH4]C2H2 `. The –rst column [(a) (d) and (g)] represents kinetic and potential energies in kcal mol~1. The second [(b) (e) and (h)] and the third column [(c) (f) and (i)] give essential bond distances (in Aé ) and bond angles (in degrees) respectively as indicated in the –gures. perpendicular to the CC triple bond in C2H2 `. All the trajectories start with the centre of mass separation of 4.0 Aé . In trajectory 1 the initial velocity in the centre of mass frame has been scaled so that the total initial kinetic energy is 5.0 kcal mol~1.In the trajectories 2 and 3 the initial velocities of the two H atoms in C2H2 ` have been modi–ed so that approximately two modes of asymmetric and symmetric HwCwC bending respectively are excited. The results are shown in Fig. 6. 88 PESs and their dynamic implications 2H2 ` dances for a while and then —ees. 4 Unfortunately all three trajectories with the initial conditions selected here are nonreactive. The CH molecule comes close to C Nevertheless we may make some observations on the dynamical processes based on these results. First by comparing the results of the third and the –rst trajectory we see that the symmetric HwCwC bending does not have much eÜect on the ìreactionœ process. The HwCwC bending is nearly adiabatic in the whole process.This may not be very surprising considering the ìsymmetryœ of this trajectory. We expect a larger eÜect for cases with non-zero impact parameters. Secondly we see that trajectory 2 with the initial asymmetric bending excitation reveals interesting features of the process. Although the trajectory starts with a nearly symmetric con–guration CH favours one carbon atom in C2H2 ` as it propagates and 4 forms the con–guration that resembles abs-1. Clearly the HwCwC asymmetric bending mode is far from being adiabatic and participates the ìreactionœ process actively. It is also noted that the CwH bond that needs to be broken for reaction has been signi–- cantly stretched compared to the other two trajectories and has been as long as 1.3 Aé .At the same time the CwH bond that needs to be formed has been as short as 1.15 Aé whereas in other trajectories the closest contact was only 1.45 Aé . In other words the second trajectory is nearly reactive. Clearly the initial excitation of the asymmetric bending in C2H2 ` makes it easier to form the highly asymmetric classical complex abs-1 which is a critical step in the H-abstraction channel. 4]C2H2 `. Compared with the pre- 2.2.2 Conclusions. High quality ab initio calculations have been carried out to study the mechanism of the ion»molecule reaction CH vious work of Klippenstein,22 very similar pro–le of the H-abstraction channel (9) was obtained despite some delicate diÜerences. No entrance or exit barrier was found and the reaction proceeds through a moderately bound (ca.15 kcal mol~1) intermediate complex. For the complex channel (7) and (8) a new transition state com-TS1 with a C1 structure has been located. The geometry and energetics of this structure are more consistent with experimental –ndings and it is expected that qualitatively a correct crosssection can be derived using the results of the current work. Direct trajectory calculation reveals that asymmetric HwCwC bending participates in the reaction actively. The authors are grateful to Prof. Joel M. Bowman and Prof. Steven J. Klippenstein for collaboration in the non-adiabatic TST project. This work was in part supported by Grants F49620-95-1-0182 and F49620-98-1-0063 from the Air Force Office of Scienti–c Research.References 1 J. A. Miller and C. T. Bowman Prog. Energy Combust. Sci. 1989 15 287. 2 For a recent summary of experimental work see J. W. Bozzelli M. H. U. Karim and A. M. Dean Proceedings of the 6th T oyota Conference on T urbulence and Molecular Processes in Combustion (Elsevier New York 1993). 3 (a) M. R. Manaa and D. R. Yarkony J. Chem. Phys. 1991 95 1808; (b) M. R. Manaa and D. R. Yarkony Chem. Phys. L ett. 1991 188 352. 4 J. M. L. Martin and P. R. Taylor Chem. Phys. L ett. 1993 209 143. 5 S. P. Walch Chem. Phys. L ett. 1993 208 214. 6 (a) T. Seideman and S. P. Walch J. Chem. Phys. 1994 101 3656; (b) T. Seideman J. Chem. Phys. 1994 101 3662. 7 (a) See e.g. J. C. Lorquet and B. Leyh-Nihant J. Phys. Chem. 1988 92 4778; (b) A. J. Marks and D.L. Thompson J. Chem. Phys. 1992 96 1911; (c) S. Hammes-SchiÜer and J. Tully J. Chem. Phys. 1995 103 8528; (d) E. J. Heller and R. C. Brown J. Chem. Phys. 1983 79 3336; (e) G. E. Zahr R. K. Preston and W. H. Miller J. Chem. Phys. 1975 62 1127. 8 A. M. Mebel K. Morokuma and M. C. Lin J. Chem. Phys. 1995 103 7414. 9 A. B. Baboul and H. B. Schlegel J. Chem. Phys. 1997 107 9413. 10 (a) J. C. Keck Adv. Chem. Phys. 1967 13 85; (b) W. H. Miller J. Chem. Phys. 1974 61 1823. 89 K. Morokuma et al. 11 For a review see e.g. D. R. Yarkony in Modern Electronic Structure T heory ed. D. R. Yarkony World Scienti–c Singapore 1995. 12 (a) N. Koga and K. Morokuma Chem. Phys. L ett. 1985 119 371; (b) Q. Cui Ph.D. Thesis Emory University 1997. 13 M. J. Bearpark M.A. Robb and H. B. Schlegel Chem. Phys. L ett. 1994 223 269. 14 W. H. Miller N. C. Handy and J. E. Adams J. Chem. Phys. 1980 72 99. 15 S. Kato R. L. JaÜe A. Komonicki and K. Morokuma 1983 78 4567. 16 W. H. Miller J. Chem. Phys. 1976 65 2216. 17 A. S. Rodgers and G. P. Smith Chem. Phys. L ett. 1996 253 313. 18 J. Qian H. Fu and S. L. Anderson J. Phys. Chem. 1997 101 6504. 19 (a) Y. Chiu H. Fu J. Huang and S. L. Anderson J. Chem. Phys. 102 1119; (b) Y. Chiu H. Fu J. Huang and S. L. Anderson J. Chem. Phys. 1994 101 5410; (c) R. D. Guettler G. C. Jones Jr. L. A. Posey and R. N. Zare Science 1994 266 259. 20 G. Bouchoux F. Penaud-Berruyer and M. T. Nguyen J. Am. Chem. Soc. 1993 115 9728. 21 See e.g. (a) A. Singa M. C. Hsiao and F. F. Crim J. Chem. Phys. 1990 92 6333; (b) A. Singa M. C. Hsiao and F. F. Crim J. Chem. Phys. 1991 94 4928; (c) M. J. Bronikowski W. R. Simpson B. Girard and R. N. Zare J. Chem. Phys. 1991 95 8647; (d) A. Sinha J. D. Thoemke and F. F. Crim J. Chem. Phys. 1992 96 372; (e) S. S. Brown R. B. Metz H. L. Berghout and F. F. Crim 1996 105 6293; ( f ) S. S. Brown R. B. Metz H. L. Berghout and F. F. Crim 1996 105 8103; (g) R. P. Schmid T. Arusi- Parpar R-J. Li I. Bar and S. Rosenwaks J. Chem. Phys. 1997 107 385 and references therein. 22 S. J. Klippenstein J. Chem. Phys. 1996 104 5437. 23 L. A. Curtiss K. Raghavachari G. W. Trucks and J. A. Pople J. Chem. Phys. 1991 94 7221. 24 See e.g. (a) R. Car and M. Parrinello Phys. Rev. L ett. 1985 55 2471; (b) B. Harktke and E. A. Carter J. Chem. Phys. 1992 97 6596; (c) V. Keshari and Y. Ishikawa Chem. Phys. L ett. 1994 218 406; (d) A. I. Krylov and R. B. Gerber J. Chem. Phys. 1997 106 6574; (e) M. S. Gordon G. C. Chaban and T. Kaketsugu J. Phys. Chem. 1996 100 11512. 25 (a) K. Fukui J. Phys. Chem. 1970 74 23; (b) K. Fukui S. Kato and H. Fujimoto J. Am. Chem. Soc. 1974 97 1; (c) K. Fukui Acc. Chem. Res. 1981 14 363. Paper 8/01186I; Received 10th February 1998

ISSN:1359-6640    

DOI:10.1039/a801186i

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 91»104 General Discussion Prof. Truhlar opened the discussion of the Spiers Memorial Lecture Prof. Miller has described an intriguing hybrid theory in which a quantal Boltzmann operator is combined with classical recrossing dynamics. It may be useful to distinguish various classes of recrossing dynamics and ask which of these classes would be treated qualitatively correctly by the hybrid theory. First we distinguish local recrossing from global recrossing. Local recrossing is recrossing that could be eliminated by local optimization of the location orientation and shape of the generalized transition state dividing surface while keeping the shape basically similar to a hyperplane in coordinate space. Global recrossing is that which persists even when the variational transition state is locally optimized under the constraint that its shape is not globally convoluted.Prof. Millerœs example of kinematic recrossing in the O]OH¢O2]H reaction is presumably an example of global recrossing. Similar examples were noted earlier in ion»molecule reactions by Hase and co-workers1 and in the F]H reaction by Garrett and myself.2 In both these cases the variational transition state is early where reaction-path curvature is 2 small and local recrossing is probably negligible. But some trajectories after passing the variational transition state continue on for some distance and then re—ect from a repulsive wall which reverses their momentum along the reaction path and causes them to proceed back to the variational transition state and recross.One can imagine that these trajectories may sometimes return to the variational transition state with less than the required zero point energy in transverse degrees of freedom. A similar situation may occur in ion»molecule reactions with tight transition states such as the Cl~]CH3Cl S reaction.3 After crossing the transition state (at time t\0) a trajectory may have its motion reversed in the ion-dipole well causing it to return to the variational transition N2 state. If it does so but with less than zero point energy it will be important to enforce the zero point constraint not just at t\0 but for longer times where semiclassical methods are typically less accurate. Another example of global recrossing involving a quantized energy constraint is the phenomenon of supernumerary transition states.4 An example would be where a global variational transition state (a global minimum in the integrated density of states) lets pass trajectories in transverse state n which then encounter a nonadiabatic region changing their state to n@ which in turn causes re—ection from a later local minimum in the integrated density of states or local maximum in the state-speci–c vibrationally adiabatic potential curve corresponding to state n@.Accurate quantum mechanical scattering calculations show that quantization is important at both locations.4 Would the new hybrid theory treat such eÜects correctly ? 1 K. N. Swamy and W L. Hase J. Chem. Phys. 1982 77 3011; S. L. Mondro S. Vande Linde and W.L. Hase J. Chem. Phys. 1986 84 3783; S. R. Vande Linde and W. L. Hase Comput. Phys. Commun. 1988 51 1734. 2 D. G. Truhlar and B. C. Garrett Faraday Discuss. Chem. Soc. 1987 84 464. 3 Y. J. Cho S. R. Vande Linde L. Zhu and W. L. Hase J. Chem. Phys. 1992 96 8275; H. Wang G. H. Perlherbe and W. L. Hase J. Am. Chem. Soc. 1995 116 9644; H. Wang and W. L. Hase Chem. Phys. 1996 212 247. 4 D. C. Chat–eld R. S. Friedman G. C. Lynch D. G. Truhlar and D. W. Schwenke J. Chem. Phys. 1993 98 342; D. C. Chat–eld R. S. Friedman S. L. Mielke G. C. Lynch T. C. Allison and D. G. Truhlar in Dynamics of Molecules and Chemical Reactions ed. R. E. Wyatt and J. Z. H. Zhang Marcel Dekker New York 1996 p. 323. Prof. Miller responded The linearization approximation to the semiclassical initial value representation (SC-IVR) described in Section IVb of my paper (and more completely in ref.58 and 59 therein) will in general provide a good description of quantum 91 92 General Discussion eÜects in the —ux correlation function for times of order +b but at longer times it essentially gives the results of classical mechanics. This is seen quite clearly in the applications presented in ref. 59b. Thus I do not think that it will describe the long time quantum eÜects that you ask about e.g. zero point energy constraints in recrossing trajectories. The full SC-IVR treatment should be able to do so and that is why we are putting so much eÜort in trying to –nd ways to facilitate such calculations (cf. Section IVd of my paper). Prof.Kuppermann asked You have given results of accurate calculations of microcanonical rate constants for total angular momentum J\0. The full rate constants involve sums over all J. Have you done accurate calculations of such sums? Cl]H Prof. Miller answered We have indeed carried out exact calculations for J[0 for some of the applications I discussed. Speci–cally for the 2 ]HCl]H reaction of ref. 42 we treated J[0 exactly and also via various approximations (J-shifting helicity conserving principle axis helicity conserving) to test them. Summing over all J[0 to obtain k(T ) is actually not so difficult because log kJ(T ) is an almost linear function of J(J]1). One thus does calculations for ca. 5»10 widely spaced values of J and then interpolates.Dr Dellago said In the Spiers Memorial Lecture Prof. Miller explained how to calculate reaction rate constants using semiclassical mechanics with an initial value representation and a linear approximation for the action. In this case one must calculate the Wigner transform of the Boltzmannized —ux operator. Is there an efficient way to calculate this distribution function for complex molecular systems? Prof. Miller responded This is a very perceptive question ! It is true that for a complex system the most difficult aspect of applying the linearized SC-IVR approach is the determination of the Wigner transform of the Boltzmannized —ux operator. For the applications to the system-harmonic bath problems described in my paper (and more fully in ref. 59a 59b) we neglected the system-bath coupling in the Boltzmann operator so that the Wigner transform could be carried out analytically.One way to proceed more generally is to determine the Boltzmann operator also semiclassically i.e. by a pure imaginary time SC-IVR. This approach is described more fully in ref. 59b where it was used successfully (and justi–ed our earlier neglect of system-bath coupling in the Boltzmann operator). Prof. Bowman asked How well do you think treating some degrees of freedom classically and some semiclassically might work? Prof. Miller replied I think that this may often work very well and be a more dynamically consistent alternative to ìmixed quantum»classical œ approaches»more dynamically consistent because all the degrees of freedom are treated by classical mechanics with the phase information being retained only for the degrees of freedom being treated semiclassically.This is precisely the methodology developed by Sun and myself in ref. 58 of my paper. Prof. Basilevsky said I would like to learn how important are the eÜects of quantum recrossings in a particular case of isomerization reactions. In this case we deal with a product well in which energy levels are quantized and the question is how important is this quantization when isomerization processes are con-93 General Discussion sidered ? In other words how reliable is statistical theory (completely disregarding these eÜects) as applied to isomerizations ? Prof. Miller responded In our calculations for a symmetric double well coupled to a harmonic bath (from ref.59a) recrossing eÜects were not signi–cant in the case of strong coupling [cf. Fig. 5(a)] because the particle is thermalized by the bath as soon as it leaves the transition state region. Stated another way one may say that the strong coupling to the bath broadens the levels of the 1D double well so that no level structure remains. In the weak coupling case however recrossing eÜects are very prominent [cf. Fig. 5(b)] because the particle is not thermalized during its –rst sojourn of the product well. In this case one may say that the level structure of the 1D double well persists. Prof. Schulten asked What questions can be answered by IVR theory when applied to polyatomic molecules or biomolecular systems? In the latter case errors in the computed energy potential surfaces render a quantitative improvement of reaction rates by use of IVR not the most relevant issue ; however qualitative quantum eÜects described properly by the theory might be of interest.Can you suggest what those eÜects might be e.g. temperature dependence of reaction rates ? Prof. Miller replied You are certainly correct that uncertainties about the potential energy surface(s) involved in biomolecular systems greatly limit the reliability of theoretical simulations of these phenomena (Wignerœs observation that I quoted in the Introduction of my paper is still very relevant today!). The reliability of classical mechanics for carrying out the molecular dynamics (MD) however is also uncertain particularly so for bond-breaking processes H atom motion and electronically nonadiabatic dynamics.For example the enormous zero point vibrational energy in the many CH bonds may give rise to artifacts in classical simulations if one includes these motions in biomolecular processes. The semiclassical IVR provides a way to incorporate quantum interference and tunneling eÜects into classical MD simulations and all the evidence to date suggests that it provides a very good description of all these quantum eÜects in molecular dynamics. The primary remaining task is to develop general and practical ways of implementing the SC-IVR approach for complex systems. Once this is available one will be able to see precisely where quantum eÜects do and do not matter in these processes.Prof. Truhlar added Prof. Schulten raised the interesting question of whether improved semiclassical theories of electronically nonadiabatic processes as envisioned in Prof. Millerœs lecture would be useful for providing insight to experimentalists seeking qualitative understanding of processes involving complex molecules. I believe that the answer is yes. In recent work on the unimolecular decay of collinear NaFH exciplexes1 and three-dimensional NaH exciplexes,2 we have found that existing semiclassical theories can be very inaccurate for the lifetime for nonadiabatic decay the dependence of 2 that lifetime on changes in the potential energy surface and the reactive/nonreactive branching probability. Predicting an incorrect time scale for nonadiabatic decay can lead to incorrect conclusions about the extent of vibrational energy transfer and randomization prior to decay and about other dynamical attributes that can be important for qualitative understanding and that can have profound eÜects on reactive branching ratios.Thus I agree with Prof. Millerœs assessment that further progress on developing improved semiclassical theories can be very valuable. 1 Y. Zeiri G. Katz R. KosloÜ M. S. Topaler D. G. Truhlar and J. C. Polanyi poster presentation at this Discussion. 94 General Discussion 2 M. D. Hack and D. G. Truhlar unpublished work. Prof. Truhlar opened the discussion of Prof. Schaeferœs paper This question is addressed to Prof. Schaefer and Dr Allen. In your paper you –nally estimate a classical barrier height of 975 cm~1 (2.8 kcal) to which you add 514 cm~1 (1.5 kcal) of zero point energy for a zero-point inclusive barrier of 1489 cm~1 (4.3 kcal).The latter is compared to the most recent experimentally based estimate of the eÜective barrier height which is 1281 cm~1 (3.7 kcal) indicating a possible discrepancy between theory and experiment of 208 cm~1 (0.6 kcal). However I would estimate that the zero point energy of transi- (u and u may drop another 50»100 cm~1 due to the increase of ca. tional modes 7»u9) 5 0.08 ” in the CwC bond length. This would reduce the disparity with experiment by a factor of about two. However one should actually –nd the variational state where the sum of the Born»Oppenheimer potential energy and the vibrational»rotational free energy is a maximum or»at or near 0 K»where the sum of the Born»Oppenheimer potential energy and the zero point energy is a maximum.This variational optimization will raise the zero-point inclusive barrier with respect to its value at the saddle point canceling part of that improvement. My question is have you explored the variation in local zero point energy along the reaction coordinate in the vicinity of the transition state ? Dr Allen responded In the past few months we have performed several computations to investigate the remaining disparity between theory and experiment on the barrier height for triplet ketene fragmentation. (1) Basis set extrapolation. Additional cc-pVQZ/TZ cc-pVQZ and cc-pV5Z UMP2 energies were obtained at 0.6 au intervals along the reaction path and used to determine OUMP2 basis set extrapolation limits.Therefore we are now able to augment Table 8 V of our paper with a true focal-point extrapolated curve E[R(s)] which entails ccpVQZ/ TZ UCCSD(T) reference points]OUMP2 basis set corrections]cc-pVDZ UCCSDT correlation corrections. The improved curve exhibits R*\2.262 ” V *\988 cm~1 and u*\317i cm~1. Compared to our best previous (VD) result the new transition state has migrated outward slightly (]0.005 ”) the barrier has diminished somewhat (57 cm~1) and the barrier frequency has hardly changed ([4i cm~1). Alternatively the estimated cc-pV5Z UCCSDT//TZ(2d1f,2p) CCSD classical barrier height of 975 cm~1 in Table 7 to which Prof. Truhlar refers is altered by only ]13 cm~1 in our most recent work as the upward shift upon CwC bond elongation is nearly cancelled by the downward basis-set extrapolation eÜect.(2) Higher-order correlation eÜects. Contributions from connected quadruple excitations in the electronic wavefunction have been evaluated by means of cc-pVDZ UBD(TQ) computations1 at the TZ(2d1f,2p) CCSD reference structures. The UCCSDT]UBD(TQ) correlation shift for the classical barrier is found to be ]44 cm~1. (3) Zero-point vibrational energy. Both quadratic force constants and gradients have now been evaluated via the TZ(2d1f,2p) CCSD method at 0.6 au intervals along the fragmentation path. Subsequent interpolation procedures have provided the TZ(2d1f,2p) CCSD quadratic force –eld at the transition state structure of the aforementioned V curve.Under the assumption that the TZ(2d1f,2p) CCSD quadratic force constants inE the chosen internal coordinate representation (Table 3) approximate those for the level of theory of the V curve and that the corresponding residual forces can be neglected,2 one arrives at the following harmonic frequency predictions for the migrated V tran- E (u1 u2 u3 u4)\(3352 3138 2169 1149) cm~1 for the intrafragment sition state E modes and (u5 u6 u8 u9)\(358 203 308 120) cm~1 for the transitional modes. Therefore by comparison with the TZ(2d1f,2p) CCSD//TZ(2d1f,2p) CCSD values of Table 4 we see that outward migration of the transition state does indeed engender a 65 95 General Discussion cm~1 drop in the transitional-mode harmonic zero-point vibrational energy (ZPVE); however bond-length contraction in the fragments is responsible for a[11 cm~1 compensation of this eÜect.In summary at our best (VE) predicted structure for the true transition state we now estimate a harmonic ZPVE contribution to the fragmentation barrier of ]462 cm~1. With respect to explicit applications of microcanonical variational transition-state theory to this problem there are some intricacies associated with the treatment of ZPVE for the transverse modes viz. the reaction coordinate used to de–ne the dividing surface must be precisely and astutely de–ned and the coordinates for the reduced-dimensionality transverse-mode vibrational problem must be speci–ed accordingly. Our new data for the variation of the TZ(2d1f,2p) CCSD harmonic force –eld along the reaction path now makes study of these issues possible.The net result from our new computations is that the vibrationally adiabatic association barrier for triplet ketene fragmentation is 988]44]462\1494 cm~1 which is still just over 200 cm~1 above the eÜective barrier from experiment. Among the three sources of error considered above we speculate that a proper accounting of anharmonicity in the zero-point vibrational energy is essential to closing the gap between theory and experiment. Three phenomena whose contributions to the barrier we have yet to quantify are core correlation relativistic eÜects and the Born»Oppenheimer diagonal correction (BODC); however there is reason to expect all of these terms to be small.1 K. Raghavachari J. A. Pople E. S. Replogle and M. Head-Gordon J. Phys. Chem. 1990 94 5579. 2 W. D. Allen and A. G. Csaç szaç r J. Chem. Phys. 1993 98 2983. Prof. Child asked What eÜect does the curvature of the reaction path have on the rate of tunnelling through the activation barrier ? Dr Allen replied The new and extensive TZ(2d1f,2p) CCSD data we have obtained just within the last few months will allow us to parametrize the complete variation of all quadratic force constants along our intrinsic reaction path for triplet ketene fragmentation. From this information the total curvature i(s) and curvature coupling functions B of the corresponding reaction path Hamiltonian1 will be evaluated. Thus while we have not yet quanti–ed the eÜect of curvature on the tunnelling rate reliable results k,3N(s) should soon be within reach.I will defer to Prof. Miller to speculate on the outcome of this eÜort and to comment on previous work by Gezelter and Miller2 which has considered the coupling of the reaction coordinate to a limited number of transverse vibrational modes. 1 See W. H. Miller in T he T heory of Chemical Reaction Dynamics ed. D. C. Clary Reidel Dordrecht 1986 pp. 27»45 as well as references therein. 2 J. D. Gezelter and W. H. Miller J. Chem. Phys. 1996 104 3546. Prof. Miller added Some of the calculations by J. D. Gezelter and myself1 involved fully rigorous quantum rate calculations with a global multi-dimensional potential energy surface (–t to earlier ab initio calculations by Prof.Schaeferœs group). Thus reaction path curvature and all other eÜects are included in these calculations and they (unfortunately) show that no step structure is obtained in the energy dependence of k(E) (unless the barrier is much wider than the ab initio calculations suggest). 1 J. D. Gezelter and W. H. Miller J. Chem. Phys. 1996 104 3546. Prof. Truhlar added Prof. Child raised the interesting question of the eÜect of reaction path curvature on the tunnelling calculation reported by Prof. Schaefer and Dr 96 General Discussion Allen. Actually including reaction path curvature would increase the tunnelling eÜect,1,2 which is the opposite of what they require to reconcile their calculations with experiment. 1 R. T. Skodje D. G. Truhlar and B.C. Garrett J. Chem. Phys. 1982 77 5955. 2 D. G. Truhar J. Chem. Soc. Faraday T rans. 1994 90 1740. Prof. Botschwina commented The accuracy of CCSD calculations with a not very —exible TZ (2d1f,2p) basis set under neglect of the basis set superposition error is not too well established for a weakly hydrogen-bonded complex like OC… … …HCH. I have therefore carried out some quick calculations at a higher level of sophistication. They make use of the ROHF-CCSD(T) method1 in the implementation of Knowles and coworkers2,3 and were carried out with the MOLPRO suite of programs.4 A basis set of 267 contracted Gaussian-type orbitals is employed which is more than twice as large as that used by King et al. in their study of the long-range complex. It consists of the (s p d f) parts of the aug-cc-pVQZ basis sets and the g functions from the cc-pVQZ set for carbon and oxygen5,6 while the hydrogen basis may be brie—y described as sp(avqz)]d(vqz).The changes in the equilibrium geometrical parameters of the monomers occurring upon complex formation are too small to be of any signi–cance and the monomers were therefore kept –xed in their ROHF-CCSD(T) equilibrium geometries [R (CO)\1.1316 ”; r (CH)\1.0774 ” and a (HCH)\133.70°]. The results are given e e here in the Table 1. The contribution of connected triple substitutions to the equilibrium e dissociation energy D is substantial and amounts to 38 cm~1 or 16%. On the whole the agreement with the CCSD results is quite good since the CCSD calculations very e likely bene–t from error compensation.2v For the nuclear con–guration with parallel orientation of the electric dipole (C moments symmetry) we calculate an optimum C………C separation of 3.85 ” and a relative energy of [112 cm~1. An estimate of the basis set superposition error by means of the familiar counterpoise procedure of Boys and Bernardi8 is ]18 cm~1. Table 1 Comparison of CCSDa and ROHF-CCSD(T)b calculations ROHF-CCSD(T)b 267 cGTO CCSDa 111 cGTO Re(C… … …C)/” a (OC… … …C)/degrees 3.995 171.4 7.0 3.922 172.9 6.6 [208 [242 ([227) e be(C… … …CH)/degrees Erel/cm~1 c a Ref. 7. b Valence electrons correlated. Counterpoise corrected value is given in parentheses. c Energy diÜerence with respect to CH (3X3 3B 2 1) ]CO(X 1&`) in their equilibrium geometries.1 J. D. Watts J. Gauss and R. J. Bartlett J. Chem. Phys. 1993 98 8718. 2 P. J. Knowles C. Hampel and H.-J. Werner J. Chem. Phys. 1993 99 5219. 3 M. J. O. Deegan and P. J. Knowles Chem. Phys. L ett. 1994 227 321. 4 MOLPRO is a package of ab initio programs written by H.-J. Werner and P. J. Knowles with contributions from J. Almloé f R. D. Amos M. J. O. Deegan S. T. Elbert C. Hampel W. Meyer K. A. Peterson R. Pitzer A. J. Stone P. R. Taylor R. Lindh M. E. Mura and T. Thorsteinsson. 5 T. H. Dunning Jr. J. Chem. Phys. 1998 90 1007. 97 General Discussion 6 K. A. Peterson R. A. Kendall and T. H. Dunning Jr. J. Chem. Phys. 1993 99 9790. 7 R. A. King W. D. Allen B. Ma and H. F. Schaefer III Faraday Discuss.1998 110 23. 8 S. F. Boys and F. Bernardi Mol. Phys. 1970 19 553. Dr Allen responded In our explicit mapping of the intrinsic reaction coordinate for C transition state did not seem to be connected in the triplet ketene fragmentation the s% forward direction to the long-range complex reported in Fig. 3 of our paper. Because this complex is thus not likely to be important in the photofragmentation dynamics we did not attempt to de–nitively characterize it. Nonetheless the structures and vibra- 3CH tional eigenstates of long-range 2]CO complexes are of inherent interest and we are pleased that Prof. Botschwina has nicely con–rmed the validity of our predictions by means of higher-level computations. In our experience basis-set superposition error (BSSE) in a method such as TZ(2d1f,2p) CCSD may aÜect already minuscule binding energies by a nonnegligible percentage and may change interfragment separations somewhat but predictions of the basic structure and fragment orientations in weakly bound complexes are not likely to be compromised.In this spirit we consider the agreement between our TZ(2d1f,2p) CCSD results and the 267 cGTO-basis CCSD(T) structure reported here by Prof. Botschwina to be excellent. Historically we have preferred to deal with most BSSE problems by making extra and sometimes exceptional eÜorts to extend the one-particle basis set toward completeness rather than to invoke counterpoise-type procedures which are often fraught with inconsistencies. In research on the water dimer with correlation-consistent series of basis sets Feller1 demonstrated that counterpoisecorrected dimerization energies converge monotonically from below the limiting D value whereas their uncorrected counterparts converge monotonically from above.Fore the cc-pVxZ series the counterpoise binding energies exhibited better convergence; however for the aug-cc-pVxZ sets the counterpoise terms only functioned to retard the already rapid convergence of the uncorrected D values. These lessons are certainly apt for the long-range complex on the triplet ketene surface. Finally it is worth mentioning e that various basis set contraction schemes and local electron-correlation methods are in current use which serve to minimize BSSE eÜects. 1 D. Feller J. Chem. Phys. 1992 96 6104.Mr Chan said I have a comment related to the previous one. The authors (King et al.,1 suggest that the structure of the long-range complex (Fig. 3 of their paper simpli–ed here) may not be due to classical electrostatics. I have a suggestion. If one draws lines through the symmetry axes of the molecules one sees a T shape. This has long been known to be the equilibrium structure when quadrupole»quadrupole interactions dominate. O C H C H In any case these molecules have small dipoles but larger quadrupoles. Bearing in mind the large separation of the molecules one would expect the electrostatic model to work and certainly the quadrupole»quadrupole interaction which goes as R~5 is likely to be more important here than the dispersion force which goes as R~6 in governing the structure.1 R. A. King W. D. Allen B. Ma and H. F. Schaefer III Faraday Discuss. 1998 110 xxx. 98 General Discussion 2]CO complex. By manipulating earlier Dr Allen responded We have followed up on this idea and have gained additional insight into the nature of the long-range 3CH expressions of Buckingham,1 we –nd that the quadrupole»quadrupole interaction energy to be added to eqn. (1) in our paper is WQQ\ 12R5 [35(nTQAn)(nTQBn)]2tr(QAQB)[20nTQAQBn] 1 Reducing this form to our speci–c case provides an additional term in eqn. (3) of our paper namely WQQ\ Qzz CO 12 Q R zz CH 5 2 17(1]eCO sin2 /)(1]eCH2 sin2 s)[4eCOeCH2 sin 2/ sin 2sD C2(eCO]2)(eCH2]2)]2(1]eCO cos2 /)(1]eCH2 cos2 s)] This latter expression is seen to be consistent with eqn.(56) of Buckingham1 for the special case of two dipolar linear molecules if each e parameter is set to its axially symmetric value of [3/2. At the TZ(2d1f,2p) CCSD equilibrium geometry of the long-range complex (Fig. 3) the quadrupole»quadrupole interaction energy is [98 cm~1 whereas our earlier analysis found that the dipole»dipole (DD) and dipole»quadrupole (DQ) contributions are [3 and [39 cm~1 respectively. Therefore the quadrupole»quadrupole (QQ) interaction is indeed the predominant electrostatic term although the multipole expansion W through DD]WDQ]WQQ only accounts for 67% of the TZ(2d1f,2p) CCSD binding energy. Accordingly we surmise that about one-third of the binding energy is due to non-classical dispersion forces under the important caveat that the dipole»octupole and higher-order electrostatic terms as well as the induction energy are negligible.While the W interaction energy by itself provides less than half of the binding energy of the complex it does seem to govern the orientation of the monomers in the QQ structure. We have generated detailed contour plots of WDD]WDQ]WQQ and have found that the quadrupole»quadrupole term indeed provides a substantial driving force for a T-shaped geometry. The addition of WQQ to eqn. (3) changes the (/\180° s\0°) dipole-alignment structure from a minimum at W \[79 cm~1 to a saddle point at W \[46 cm~1. Speci–cally this structure becomes a transition state connecting equivalent global minima appearing at (/\180^5.5° s\^79.6°) with W \[143 cm~1.Remarkably the actual TZ(2d1f,2p) CCSD structure has the orientational angles (/\171.4° s\73.9°) within 6° of those predicted by the electrostatic analysis. WDD]WDQ]WQQ function also exhibits equivalent pairs of secondary minima The with CwOwC frameworks in which the CO monomer is —ipped. These equilibrium structures occur at (/\^5.8° s\<81.0°) and lie 23 cm~1 above the primary minima. However upon execution of very tight TZ(2d1f,2p) CCSD geometry optimizations starting from such electrostatic initial con–gurations a rather diÜerent secondary structure ultimately appears one whose optimum geometric parameters are r(CwO)\1.1254 ” t(CwH)\1.0760^0.0001 a(HwCwH)\133.60° R(CwC)\4.0064 /\[62.34° and s\]132.31°. This alternative TZ(2d1f,2p) CCSD structure has ” minimum appears to be a better candidate for termination of the intrinsic reaction path De\148 cm~1 and lies 60 cm~1 above the primary minimum of Fig.3. This secondary for triplet ketene fragmentation because it would require no long-range turnover in the variation of d(CwCwO) with respect to the arc-length parameter (cf. Fig. 8 of our C fragmenta- paper). Nevertheless the span of the intrinsic reaction path between the s% tion transition state and this weakly bound minimum is so large that it may be difficult to de–nitively establish whether such a connectivity exists. ” 1 A. D. Buckingham Adv. Chem. Phys. 1967 12 107. General Discussion 99 Dr Lendvay communicated This comment has been prompted by the point raised by Prof.Botschwina in connection with the accuracy of the calculations of the longrange complex on the decomposition path of triplet ketene described in Prof. Schaeferœs introduction to his paper. As Prof. Botschwina also suggested if a ì relatively in—exibleœ basis set is used in the calculations on the long-range complex then a proper correction for the basis set superposition error (BSSE) is necessary. However I would like to call the attention of the community that the commonly used ìcounterpoiseœ (CP)1 technique of correcting BSSE may lead to contradictions when a complete potential energy surface (PES) is to be determined so it is not quite satisfactory from a strict conceptual point of view. The problem is that usually one performs a BSSE correction for the long-range complex only which produces a discontinuity on the calculated potential surface.To avoid this one should extend the CP corrections to the entire potential surface including the region of the equilibrium conformation of the starting molecule. This implies that the total energy of the latter will be diÜerent from that calculated by the standard techniques used for ìnormalœ molecules. Moreover if the same molecule participates in diÜerent reactions one has to attribute diÜerent values to its equilibrium energy depending on what reaction is actually considered which is obviously unacceptable. This conceptual difficulty is implicitly present in all PES calculations performed for reacting systems except of course if the limit when the complete basis set is closely approached and the BSSE is absent.Two theoretically consistent solutions seem to be possible to this problem. One can obtain unequivocal and continuous energies for any geometry of any system if it is decomposed into the smallest possible units i.e. atoms and the BSSE correction is calculated for them using a hierarchical CP scheme as described in ref. 2 and 3. The serious drawback of this approach is however that one needs to know the energy of all atoms and fragments in so many basis sets (e.g. in a four-atom system 125 calculations per geometry) that such calculations for polyatomic systems are essentially not feasible. The other alternative is that a method is developed in which the energy of the system is calculated a priori free of the BSSE.Such a technique may be worked out based on the ìchemical Hamiltonian approachœ (CHA).4 1 S. B. Boys and F. Bernardi Mol. Phys. 1970 19 553. 2 J. C. White and E. R. Davidson J. Chem. Phys. 1990 93 8029. 3 P. Valiron and I. Mayer Chem. Phys. L ett. 1997 275 46. 4 I. Mayer Int. J. Quantum Chem. 1983 23 341; 1998 in press. MSX. The most likely pathway for 1/T2 1/T1 S MSX that exists near the Franck»Condon region and the inaccessible S1 ]Tn (n\1 2) conversion is S1 ]S0 ]Tn. S The 1 ]S0 internal conversion is expected to be very effi- 1 T which is likely to be Prof. Morokuma communicated The present paper discusses very accurate calculations of the structure energy and harmonic frequencies at the transition state for dissociation of ketene on the lowest triplet state (T1-TS Although these are undoubtedly important in the photodissociation dynamics of the and T2-TS in the Fig.1 here). triplet ketene another probably more important factor is how the system comes down to the triplet state after it was originally excited to the S singlet state. Our calculations1 S1 ]Tn (n\1 2) intersystem conversion is indicate as shown in Fig. 1 that the direct 1 unlikely because of small spin»orbit coupling at the minimum on the seam of crossing (MSX) S high energy of the process cient as they cross extensively from the Franck»Condon region all the way beyond the or S 1-Cs-II) though S0/S1 MSX. S0 crosses the lowest energy triplet state (T minimum (S 2) ( at rather low energy near the triplet minimumT1 and or -Cs-II T2-Cs-I) S0/Tn 1 crossing persists all along the CwC dissociation pathway.As the CwC bond is stretched the energy of the crossing increases and the crossing geometry deviates substantially from the reaction path. Therefore the photodissociation dynamics of ketene will re—ect the initial distribution on the triplet state (T or T2) 1 100 General Discussion Fig. 1 Potential energy pro–le of the lower excited states of ketene non-statistical re—ecting the dynamics of intersystem crossing/internal conversion as well as the dynamics on the triplet potential surface. 1 Q. Cui and K. Morokuma J. Chem. Phys. 1997 107 4951. Dr Allen communicated in response We are thankful to Prof. Morokuma for additional summarizing remarks regarding his work on nonadiabatic interactions in the photodissociation of ketene.We both agree that elucidation of the interplay between S0/T1 2 intersystem crossing and subsequent fragmentation on the triplet surface is critical to explaining the observed steplike structure in k(E). To quote Cui and Morokuma1 ìThe exact eÜect of the intersystem crossing between the S and T (n\1 2) states on 0 the microcanonical rate constant of the triplet ketene is not very clear at the moment.œ n They go on to outline two limiting dynamical possibilities (1) Intersystem crossing is relatively fast compared to dissociation taking place at low-energy S0/T1 MSX structures near the triplet minimum geometry; in this case one would expect a statistical initial distribution on the triplet surface and any structure in k(E) to arise solely from transition-state vibrational thresholds.(2) Intersystem crossing occurs at high-energy S0/T1 2 MSX structures which deviate substantially from the intrinsic reaction paths for triplet ketene fragmentation; nonstatistical dynamics in the intersystem crossing process might then give rise to additional structure in k(E) or perhaps prevent tunnelling from washing out steps in the dissociation rate. Quantum dynamical studies are sorely needed to determine where the actual fragmentation process lies between these limits. In their paper Cui and Morokuma seem careful not to advocate one case over the other but if I interpret his remarks correctly Prof. Morokuma is now no longer noncomittal on this issue favoring the second scenario.We need to see the results of some concrete dynamical treatment to be persuaded fully in this direction. 1 Q. Cui and K. Morokuma J. Chem. Phys. 1997 107 4951. General Discussion 101 Prof. Schulten opened the discussion of Dr Olivucciœs paper You describe in your paper (Fig. 2 10 11) an interesting eÜect of an H]methyl substitution on the excited state potential energy surface of a polyene SchiÜ base. Can one explain this eÜect qualitatively ? The isomerizations studied in your paper involve the central double bond of polyene SchiÜœs bases with three double bonds. Did you investigate how the systems select the double bonds to be twisted in the case of the compounds investigated presently or in case of longer SchiÜœs base polyenes? Polyenes exhibit two closely spaced excited states a lower lying covalent optically ñ1 and retinalñ2.forbidden A state and an ionic optically allowed B state. Formation of a protonated g SchiÜœs base mixes these states and favours energetically the ionic state moving it below u the covalent state. However the latter state is favoured energetically when the polyene length is increased such that one would expect two close excited states for biological chromophores with six or seven double bonds i.e. retinal Did you always observe the widening of the S1»S2 energy gap upon initiation of photoisomerization reactions ? Dr Olivucci responded (a) The analysis of the energy pro–le along the computed S H]CH substitution in the1 reaction coordinate indicates that the major eÜect of the cis-pentadieniminium cation is a marked increase of the initial slope of the S relaxation 3 path [see Fig.2(a) and (b)]. The steeper slope in the methyl substituted system is 1 explained by the non-bonding repulsive interaction between the CH group in a and the 3 hydrogen in d. Such a non-bonding interaction may be decreased by (i) expansion of the central double bond and (ii) torsional motion about the same double bond. The computed cis-pentadieniminium cation isomerization path indicates that this chromophore remains planar during the initial S relaxation and that double bond expansion dominates the relaxation motion. Thus for the substituted molecule one has 1 a larger energy decrease due to the relief of the repulsive interaction.However due to the increase in length (from 1.35 to 1.53 ”) of the central CxC bond upon relaxation the repulsive interaction is decreased and the energy pro–les for the substituted and parent compounds become very similar at larger values of the reaction coordinate. This eÜect is also seen in the curvature of the energy surface with respect to the torsional mode. In fact as reported in Fig. 3 of our paper the curvature (i.e. the frequency) of the surface along the torsional coordinate in the region of the relaxed planar stationary point (SP) is substantially equal (within 10 wavenumbers) to that for the substituted and unsubstituted systems respectively. The possible role of the methyl substituent in accelerating the rate of the photoisomerization of the retinal chromophore in rhodopsin has been recently investigated by Mathies et al.1 In our short retinal model we do not see a large acceleration of the isomerization dynamics on S (see Fig.10). This is mostly due to the fact that there is no torsional component along the initial part of the path where the non-bonding inter- 1 action is important. However it can be easily recognised that in a situation where the potential energy surface is asymmetric or/and unstable with respect to the torsional mode (compare Fig. 2 here with Fig. 5 in our paper) the methyl substitution may be an 102 General Discussion Fig. 2 Hypothetical asymmetric structure of the S energy surface along the initial relaxation path 1 of a retinal protonated SchiÜ base model (e.g.the all-trans C7H8NH2 ` discussed in our paper or the realistic model 4-cis-c-methylnona-2,4,6,8-tetraeniminium cation of ref. 2 see also ref. 23 in our paper) induced by an asymmetric potential which couples the stretching and torsional modes (e.g. a protein cavity). The stream of arrows indicates the relaxation path starting from the Franck» Condon point (FC) and going towards the S1/S0 conical intersection CI. The full line represents a hypothetical classical trajectory released near FC. important factor in controlling the photoisomerization time scale. In such a situation the initial relaxation is expected to occur along strongly mixed torsional/bondexpansion coordinate. As discussed elsewhere2 such a potential energy surface may arise as an eÜect of the highly asymmetric rhodopsin cavity.(b) Recently we have investigated the S relaxation/isomerization paths for all double 1 bonds in the pentadieniminium3 (three double bonds) and eptatrieniminium (four double bonds) cations. The results show that the S isomerization of the two terminal groups cannot compete with the isomerization of the central double bonds. In fact we 1 –nd that in the pentadieniminium cation the 90° rotation of the terminal CH and NH2 groups leads to a higher lying S intermediate. In contrast the two central double bonds 2 in the eptatrieniminium cation may isomerize along independent but competitive paths 1 on the S energy surface (see Fig. 4 of our paper). In these molecules the two processes 1 are almost barrierless but the barrier is slightly (\1 kcal mol~1) higher for the isomerization of the double bond in position d,c.This seems to be in agreement with the fact that the photoisomerization of the protonated SchiÜœs base of all-trans retinal in solution produces a mixture of cis»trans stereoisomers and not a single product. (c) Presently the S and S energy pro–les (computed using the CASPT2 procedure) 2 along the S isomerization coordinate (computed using the CASSCF gradient) S and 1 1 2 S have been determined only for two systems the pentadieniminium3 (three double bonds) and eptatrieniminium (four double bonds) cations. There is no doubt that in 1 both the three double bond (see Fig. 1 of our paper) and four double bond (paper in preparation) systems the S2»S1 energy gap increases along the isomerization coordinate.103 General Discussion 1 (1B -like) state while the S state corresponds to the dark (2A 2 g-like) More speci–cally the gap increases from 20»30 (FC point) to 60»70 (S1/S0 crossing) kcal mol~1. It has also been possible to analyze the nature of the wavefunctions associated to S (i.e. the lower lying excited state) and S2 . The analysis shows that along the entire path the S state corresponds (along the isomerization coordinate) to the spectro- 1 scopic state. It is therefore apparent that the behaviour of protonated SchiÜœs bases is very diÜerent from u that of the corresponding polyenic hydrocarbon or unprotonated SchiÜœs base where the lowest lying state is the dark state.4,5 2»S1 gap going from three to four Since we do not observe a large decrease in the S conjugated double bonds and considering that the gap increases as a function of the torsional angle there is no evidence for a possible inversion in longer protonated SchiÜœs bases (computations on a –ve and six double bond system are in progress).Unless the protein cavity changes this trend our computational results suggest that the correct mechanistic model to be considered for the photoisomerization of these important chromophores involves two electronic states. This seems to be in contrast with the previously proposed three state model for the isomerization in bacteriorhodopsin6 (see also the paper by Prof. Schulten and co-workers7 presented at this meeting). 1 Q. Wang G.G. Kochendoerfer R. W. Schoenlein P. J. E. Verdegem J. Lugtenburg R. A. Mathies and C. V. J. Shank Phys. Chem. 1996 100 17388. 2 M. Garavelli F. Negri and M. Olivucci J. Am. Chem. Soc. submitted. 3 M. Garavelli P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Am. Chem. Soc. 1997 119 6891. 4 M. Garavelli F. Bernardi P. Celani M. A. Robb and M. Olivucci J. Photochem. Photobiol. A Chem. 1998 114 109. 5 M. Garavelli P. Celani F. Bernardi M. A. Robb and M. Olivucci J. Am. Chem. Soc. 1997 119 11487. 6 F. Gai K. C. Hasson J. Cooper McDonald and P. A. An–nrud Science 1998 279 1886. 7 M. Ben-Nun F. Molnar H. Lu J. C. Phillips T. J. Martïç nez and K. Schulten Faraday Discuss. 1998 110 447. Prof. Martïç nez communicated In your paper you refer to dynamical calculations of photoisomerization of a retinal analog using CASSCF wavefunctions.You have emphasized the importance of true intersections in photochemistry and it is known that these intersections diÜer qualitatively from avoided crossings in at least one aspect. The presence of the intersection and the requirement of wavefunction continuity introduces a geometric phase which requires special treatment in the dynamics and can lead to novel quantum interference eÜects. How have you dealt with this phase and what can you say about its importance for organic photochemical problems? Dr Olivucci communicated in response In our paper we focus on the time scale required to reach the conical intersection which is taken as an estimate of the excited state lifetime.This time scale is essentially determined by the nature of the adiabatic motion (redistribution of the vibrational energy) on the excited state energy surface prior to decay. Thus the use of semiclassical dynamics calculations (see ref. 1 and 2 below for a description of the methods used) is only required to ìclockœ the occurence of the nonadiabatic event. Prof. Martinezœs question regards a general and outstanding aspect of non-adiabatic dynamics computations which is concerned with the detailed description of the decay event and ultimately the prediction of ground state product distribution (i.e. quantum yields). In particular the questions concern the role played by the quantum mechanical properties of the conical intersection on the decay dynamics and their eÜect on the ground state relaxation process.The role of the geometric phase has been the subject of a contemporary and authoritative review by Frey and Davidson3 who also discuss many interesting mathematical properties of real surface crossings. We recommend this article for some of the background associated with this interesting problem. 104 General Discussion In our view there are two aspects of the problem (i) the use of the change in phase of the real electronic wavefunction as a test for a real surface crossing and (ii) the quantum adiabatic phase problem. We discuss each very brie—y. (i) Herzberg and Longuet-Higgins4 showed that if the wavefunction changes sign on traversing a closed loop around a point then there must be a conical intersection within the region of geometrical space.Please refer to ref. 1 where some instructive examples are given. Since all of our work has involved many nuclear degrees of freedom it is not easy to perform this test. However since we have accurate gradient driven methods for searching for conical intersections,5 we do not see that this test is very useful in studies of organic photochemistry. In the same way that a conical intersection in just a Jahn- Teller point in the absence of symmetry a true avoided crossing would be like a Renner- Teller degeneracy that occurs for degenerate states of linear molecules. In the latter case the gradients of both surfaces go to zero at the crossing point. Thus conical intersections and true avoided crossings can be easily distinguished by the gradients without the need to use the Herzberg and Longuet-Higgins topological test.(ii) Now let us turn to the question of the quantum adiabatic phase problem. Here the overall phase of the wavefunction needs to be conserved. Thus the nuclear wavefunction must also change sign around a closed loop to make the total wavefunction (a product of electronic and nuclear parts) single valued. However in all our work we have treated nuclear motion classically and electronic motion quantum mechanically. Thus we have not looked at the eÜects arising from this condition. The eÜects of quantum adiabatic phase are detected spectroscopically in Jahn-Teller systems (see the discussion in ref. 1 and references cited therein). However the study of such eÜects in organic photochemistry requires a quantum mechanical treatment of the nuclear motion. We believe that without a few signi–cant case studies on sizable organic systems it is difficult to predict if the quantum adiabatic phase is of practical relevance to organic photochemistry in general. 1 B. R. Smith M. J. Bearpark M. A. Robb F. Bernardi and M. Olivucci Chem. Phys. L ett. 1995 27. 2 S. Klein M. J. Bearpark M. A. Robb F. Bernardi and M. Olivucci Chem. Phys. L ett. 1998 in press. 3 R. F. Frey and E. R. Davidson Adv. Mol. Electron. Struct. T heory 1990 1 213. 4 G. H. Herzberg and J. C. Longuet-Higgins Discuss. Faraday Soc. 1963 35 77. 5 M. J. Bearpark M. A. Robb and H. B. Schlegel Chem. Phys. L ett. 1994 223 269.

ISSN:1359-6640    

DOI:10.1039/FD110091

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 105»118 The quantum transition state wavepacket method John C. Light and Dong Hui Zhang§ Department of Chemistry and T he James Franck Institute T he University of Chicago Chicago IL 60637 USA The accurate calculation of thermal rate constants for reactions in the gas phase often requires both accurate potential energy surfaces (PESs) and the use of quantum mechanics particularly in the case of light atom (H) transfers in reactions with activation energy barriers between reactants and products. The thermal rate constant k(T ) can be calculated directly or as a thermal average over the cumulative reaction probability N(E). Both k(T ) and N(E) can be calculated exactly and directly in terms of —ux formulations –rst presented by Miller et al.In this paper we review the recent reformulation of the calculation of N(E) in terms of the time evolution of transition state wavepackets (TSWPs) which then provides a very eÜective method for reactions with activation energy barriers. This method requires a single time propagation for each TSWP contributing to the desired thermal rate constant from which the required contributions to N(E) for all E can be obtained. We then apply this to the calculation of N(E) and k(T ) for the interesting four atom reaction H2(D2)]CN]HCN(DCN)]H(D). The system has a metastable well in the PES at the linear CNHH con–guration. The results and a discussion of the in—uence of this secondary TS well are presented. 1 Introduction The calculation of thermal rate constants of chemical reactions has long been an important goal of theoretical chemical physicists.For a large class of reactions the reaction rate constants are dominated by the energy required to pass over a barrier on the potential energy surface (PES) leading (approximately) to the Arrhenius form of the rate constants. Classical transition state theory (TST)1,2 and further developments of variational TSTs3h9 provide in many cases excellent approximations to thermal rate constants for heavier systems for which the PES is known with reasonable accuracy. However the dynamical approximations in such TST calculations make it desirable to have a quantum formulation that is both exact and practical. Such calculations will require accurate knowledge of the PES for the full system at least in the regions of the potential energy barriers separating reactants and products.In addition the dynamics must be simulated adequately; if light atom transfers are involved in the reactions the use of quantum dynamics which includes both discrete energy level eÜects and quantum tunneling is highly desirable. Reaction rate constants can be calculated exactly from thermal averages of exact quantum state-to-state reaction probabilities i.e. from the S-matrices obtained from full solutions to the Schroé dinger equation at each energy. For reactions with barriers and with relatively sparse reactant and product quantum states the full S-matrix can be calculated. Alternatively the time dependent Schroé dinger equation can be solved for § Present address Department of Computational Science National University of Singapore Singapore 119260.105 106 Quantum transition state wavepacket method each initial state to obtain the reaction probability as a function of energy from that state. Substantial improvements in the methods used in such initial state selected wavepacket approaches (ISSWP) now permit four atom systems to be solved exactly.10h15 However for reactions with a relatively dense distribution of reactant and product states at the energies of interest the number of energetically open states contributing to the rate constant will be very large. In these cases the full S-matrices or even the initial state selected reaction probabilities may be very difficult to calculate.In addition the full S-matrices contain much information on state-to-state probabilities that is averaged to obtain the rate constants. If there is a well de–ned transition state region with an activation energy barrier it will be much more efficient to attack the calculation of the rate constant directly or via the direct calculation of such averaged quantities as the cumulative reaction probability N(E). In our discussions below we focus on reactions with activation energies for which transition state type theories are most appropriate. The quantum methods discussed are exact for all types of reactions but their efficiency is greatest for reactions with barriers. Approaches to the direct quantum calculation of k(T ) and N(E) were formulated some time ago in very important papers by Miller and co-workers.16,17 In these formulations dividing surface(s) between reactants and products can be de–ned as in TST.However the rate constants or reaction probabilities are given as traces of quantum mechanical (—ux) operators. These were discussed by Miller at a Faraday Discussion meeting more than ten years ago,18 and have been developed and used in a number of applications since that time.19h27 The original formulations were quite general permitting the desired rate information to be obtained in a number of exact and formally equivalent ways. The choice between formally equivalent approaches was not obvious and there have been a variety of actual computational approaches developed with the basic aims of providing rigorous and efficient methods to determine the speci–c dynamical information of interest.Ideally there should be a small number of intuitive and efficient methods each tailored for the speci–c system information desired. Speci–c methods using the formal de–nitions of the thermal rate constants k(T ) have been developed to calculate it directly. Other approaches can determine the cumulative reaction probability N(E) from which k(T ) can be generated by a Boltzmann average. Yet other approaches can yield more detailed information about reaction from a speci–ed initial state or to a speci–ed –nal state by varying the position of the dividing surface(s). The –rst calculation of thermal rate constants by these methods for a ì real œ system i.e. a three-dimensional (3-D) calculation of the reaction of a triatomic system on a realistic surface was that of Park and Light23 for naturally the hydrogen exchange reaction.This was in a sense a ìbrute forceœ calculation in which the 3-D Hamiltonian was diagonalized on an L 2 basis. The trace was carried out on the basis of the thermal —ux operator determined by Lanczos reduction. The diagonalization permitted the rate constants to be calculated analytically at diÜerent temperatures as the time integral of the —ux»—ux correlation function. The fact that the —ux operator is of low rank22 was utilized to simplify the evaluation of the trace of the correlation function. The validity of this approach was veri–ed by Day and Truhlar28 and applied by these authors to the O]HD reactions.29 As a general method however the calculation had several —aws the diagonalization of Hå becomes prohibitive for larger systems and since no absorbing potential was used accurate results were limited to low temperatures where the eÜects of re—ections from the grid boundaries could be minimized.The obvious utility of absorbing boundary conditions (optical potentials) developed by Neuhauser and Baer28c in removing the eÜect of grid boundaries and permitting the convergence of the time integral of the correlation functions was demonstrated by Brown and Light.25 More recently a number of improved approaches have been developed. The ì transition state probability operatorœ approach of Manthe and co-workers29h31 for the 107 J. C. L ight and D.H. Zhang determination of N(E) is elegant. The eigenvalues of this operator are probabilities of reaction from each ì transition state œ and can be obtained by iterative procedures. This approach requires the use of optical potentials absorbing potentials are also utilized in all subsequent approaches. The advantages of the probability operator approach are that it is variational and relatively few eigenvalues are required [if the dividing transition state surface (TSS) is well placed at the top of an activation energy barrier]. The disadvantages are that the eigenvalues [essentially of the Green operator (H[E[ie)~1] are not easy to extract by iterative methods and that they must be determined again at each energy. At each energy the number of eigenvalues required is basically the number of transition states contributing to the reaction at that energy.Two other approaches to the direct calculation of rate constants,26,32,33 following the approach proposed by Park and Light,22 are based on the evaluation of the eigenvectors and eigenvalues of the thermal —ux operator via Lanczos reduction and propagation of these wavepackets. The use of optical potentials makes short time propagations adequate. However the propagations must be repeated at each temperature at which k(T ) is desired. These approaches have been applied to several impor- Cl]H tant reactions including 2 ]HCl]H.34 Manthe and co-workers have combined this approach with an approximate multi-con–guration time dependent Hartree (MCTDH) propagation that is applicable to larger systems.35,36 Our TSWP method37 diÜers from the above in that the cumulative reaction probabilities at all energies desired are evaluated from a single propagation of each TSWP forward and backward in time followed by the appropriate Fourier transforms.The N(E)s so obtained can then be thermally averaged to produce the thermal rate constants at all desired temperatures. This has the advantages that only TSWPs in the energy range of interest are required ; the Fourier transforms at each E must be accumulated only on a dividing (TS) surface ; and only one propagation (in]t and [t) is required per transition state. The TSWP approach has been successfully applied to calculate N(E) for J\038 and recently N(E) (summed over all J)39 for the prototype four atom reaction H2]OH]H2O]H which provided the –rst fairly accurate theoretical rate constant for a four atom reaction.In the following we review the formulation of the TSWP theory and discuss the application to a difficult reaction of interest the H2(D2)]CN reaction. Since the PES of this system has a transition state in the HwHwCwN con–guration and a metastable well in the CwNwHwH con–guration it is not a simple transition state problem. The location of the dividing surface (the transition state) and the convergence of the cumulative reaction probability with respect to the number of transition states included illustrate the difficulties encountered when reactions with ìoddœ PESs are considered. 2 Theory 2.1 TSWP approach The TSWP approach is similar to the regular time-dependent ISSWP approach to reactive scattering40,41 except for the initial wavepacket construction.In the ISSWP approach the initial wavepacket is usually a direct product of a gaussian wavepacket for the translational motion located in the reactant asymptotic region and a speci–c (N[1 dimensional) internal state for reactants. In the TSWP approach we –rst choose a dividing surface S separating the products from reactants preferably located to minimize the value of the density-of-states for the energy region considered. Then initial TSWPs are 1 constructed as the direct products of the (N[1 dimensional) Hamiltonian eigenstates on the surface (1) HS o/iT\ei o/iT 108 Quantum transition state wavepacket method and the —ux operator eigenstate o]T with positive eigenvalue j.(2) F o]T\j o]T Thus we have (3) o/i `T\o/iT o]T The cumulative reaction probability N(E) from the TSWP approach was derived simply from the formulation for N(E) given by Miller and co-workers,17 (4) 1 N(E)\2p2tr[d(E[H)F2d(E[H)F1] where F and F are the quantum —ux operators at dividing surfaces S and S between 1 2 reactants and products. The surfaces can be identical and for calculation of N(E) only 2 would normally both be the transition state surface. Here the —ux operator F is de–ned as (5) F\ 1 2k [d(q[q0)p� q]p� q d(q[q0)] where k is the reduced mass q is the coordinate normal to the dividing surface located at q\q0 which separates products from reactants and p� is the momentum operator q conjugate to the coordinate q.In evaluating the trace we take advantage of the fact that the —ux operator has only two non-zero eigenvalues for one-dimension,21,22,42 ^j with eigenfunctions that are complex conjugates of each other o]T\o[T*. We then evaluate the trace in eqn. (4) efficiently in a direct product basis of the —ux operator eigenstates which are highly localized near the dividing surface and the eigenstates of the (N[1 dimensional) Hamiltonian on the dividing surface the transition states. The microcanonical density operator d(E[H) will eliminate contributions to N(E) from transition states with internal energy much higher than the total energy E thus limiting the number of transition states that must be considered.(Owing to tunneling of course the internal states with energy somewhat greater than E can contribute to the cumulative reaction probability.) Thus by choosing a dividing surface with the lowest density of internal states we can minimize the number of transition states required to converge N(E). Since N(E) in eqn. (4) is represented in terms of d function operators the evaluation is most efficiently carried out utilizing the Fourier transform identity between the energy and time domains. After constructing the initial wavepackets we propagated them in time as in the ISSWP approach. otiT are calculated on the second divid- The components of the TSWPs at energy E ing surface (at x\x as 0) (6) oti(E)T\Jj P`=ei(E~H)tdt o/i `T. and with ~= Fo]T\j o]T o/i `T\o/iT o]T HS1 o/iT\ei o/iT (7) The cumulative reaction probability N(E) can be computed as 1 (8) Im[Sti oti@T] oq/q0 k i i N(E)\; Ni(E)\;Sti oF2 otiT\; i S2 (q\q0).Here q can be any coordinate as long as the ot where the i@T are the derivatives of otiT with respect to q the coordinate normal to the surface evaluated on the surface F2 q\q0 divides the products from reactants. —ux operator surface of There are several points to note concerning this compact formulation. First since the F exists only at the surface S t and its deriv- the Fourier transform of i 2 2 , 109 J. C. L ight and D. H. Zhang ative are required only on that surface not throughout space. If the surface for F is placed in an asymptotic region then the —ux can be projected onto the asymptotic 2 internal states of the reactants (or products) to obtain the cumulative reaction probabilities (and rate constants) from each initial state i.e.the information obtained from the ISSWP approach. For a single initial state the ISSWP is to be preferred since only one propagation is required. However when information for many initial states is desired and there is a barrier to reaction then the TSWP approach will converge with many fewer wavepacket propagations. Ni(E) above which are the contributions to A perhaps surprising point is that the N(E) from each TSWP are not probabilities i.e. Ni(E) can be negative at some energies and greater than one at others (usually by very small amounts). However N(E) itself is a cumulative probability always greater than zero.Finally since the TSWPs are determined by the position of the dividing surface S the convergence and behaviour of Ni(E) vary with the surface. Placement of S in the ì traditional œ transition state region seems 1 to yield the most rapid convergence with respect to the number of wavepackets required 1 and also seems to produce N values that are ìalmostœ probabilities. We shall see in the i application below that because some ì tion state œ wavepackets for the HHCN reaction start out in the HHNC con–guration the contributions of these wavepackets are zero at low energies but oscillate with E at higher energies and do not behave like probabilities. (9) 3 A 3 Applications to the H2(D2) + CN reaction In this section we show the results of the application of the TSWP approach to the H2(D2)]CN reaction.These two reactions have been the subject of active theoretical and experimental research43h51 because of their practical importance in combustion and atmospheric chemistry. In particular a new and much improved global PES for the reaction has recently been reported by ter Horst Schatz and Harding (denoted TSH3).46 Quasiclassical trajectory calculations,48 a reduced dimensionality quantum scattering calculation,47 as well as a recent full dimensional ISSWP calculation49 have been carried out for these reactions on the TSH3 PES. It is found that the thermal rate constant approximated by the rate constant for the ground initial state from the ISSWP approach49 is signi–cantly smaller than the experimental values in the low temperature region for the H2]CN reaction.Since the number of open rotational channels for this reaction is huge even at quite modest translational energy owing to the tiny rotational constant of CN it is obviously not feasible to use the ISSWP approach to obtain the rotationally averaged thermal rate constant. On the other hand reactions such as this which have a barrier and therefore a much lower density of states in the transition state region than the asymptotic region are ideal for the TSWP approach.37 We have used this approach to calculate the exact N(E) (J\0) on the TSH3 PES from which the rate constants were computed by using the energy shifting approximation. 3.1 The Hamiltonian and numerical parameters The Hamiltonian for the diatom»diatom system in mass-scaled Jacobi coordinates can be written as38,52 H\ 1 2k ; [L L s 2 i2]s ji2 i2B]V (s1 s2 s3 h1 h2 /) resented as i/1 where j j and are the rotational angular momenta for H2(D2) and CN which are then 2 3 coupled to form j23 .In the body-–xed frame the orbital angular momentum j1 is rep- (J[j23)2 where J is the total angular momentum. In eqn. (9) k is the 110 Quantum transition state wavepacket method reduced mass of the system (10) with Mi k\(M1M2M3)1@3 (i\1»3) being the reduced masses for the system H2(D2) and CN i.e. 1\m (mH]mH)(mC]mN) M H]mH]mC]mN 2\m mHmH M H]mH 3\m mCmN M C]mN . The mass-scaled coordinates s are de–ned as i (11) si2\ Mi k Ri2 where R (i\1»3) are the intermolecular distance between the centers of mass of H and i 2 CN and the bond lengths for H2(D2) and CN respectively.Now we de–ne two new ìreaction coordinateœ variables 2 q and q by translating and 1 2 rotating the s s and axes,37,53 1 (12) q \A cos s cos sin s sBAs1[s1 0 [sin s s2[s2 0B. 12 2 (13) H\ 1 s i(q1 q2 ji2 s1 0 s i 2 0)2] j3 2 3 2B]V 3 0 Aq B (s It can be seen from this equation that we –rst move the origins of the 1 s2) coordinates to (s1 0 s2 0 ) then we rotate these two axes by the angle s. The new coordinates q1 and q2 can now be de–ned as the ìreaction coordinateœ and the ìsymmetric stretch coordinateœ for the collinear HwHw(CN) transition state.The Hamiltonian in eqn. (9) can be written in term of s q (i\1 2) and as 2kA[; 2 1 L L q 2 i2[L L s 2 3 2 ];1 s q1\0 we can calculate the ì internal œ tranq1\ 0 in eqn. (13). After constructing the initial 1 q2 s3 h1 h2 ( /) coordinates we transfer them to the s1 s2 s3 h1 By choosing the dividing surface S at sition states for the other –ve degrees of freedom by solving for the eigenstates of the 1 5-D Hamiltonian obtained by setting wavepackets in (q h2 /) coordinates and propagate them as in the regular wavepacket approach. The calculation of the bound states in 5-D has been presented in detail elsewhere,54 and the propagation of the 6-D wavepacket in the diatom»diatom coordinates has also been shown;12 thus we will not present them again here.The parameters used in the current study are based on those employed in the initial state selected total reaction probability calculation carried out recently by Zhu et al.49 We used a total number of 55 sine functions (among them 21 for the interaction region) for the translational coordinate s in a range of [3.5 11.5]a0 . The number of vibrational basis functions used for the reagent CN is 2 or 3 depending on whether CN is vibra- 1 tionally excited. A total of 19 and 22 vibrational functions are employed for s in the 2.5]a for the reagents H and D2 respectively. For the rotational basis range of [0.3 2 j3max\41 2 2max\12 H2 j2max\16 D2 . j (for CN) s1 0 s2 0 for and we used for The values of and s that de–ne the transition state surface were carefully chosen to be 5.1 0.8 and 47° for H2]CN and 5.5 0.8 and 48° for D2]CN to minimize the density-of-states on the 111 J.C. L ight and D. H. Zhang dividing surface. We propagated the TSWPs for 9000 au for the H2]CN reaction and 10 500 au for the D2]CN reaction. In addition to the separation of even and odd parities which are related to the wave function symmetry with respect to torsion angle /\055 for the total angular momentum J\0 the even and odd rotation states of H2(D2) can also be separated. In the present study we only calculated the N(E) for the even rotation of H2 . Based on the fact that the transition state on the PES is quite rigid we approximated the N(E) for the odd rotation of H2(D2) by that for the even rotation.odd parity) to converge the N(E) for energies up to 0.5 eV with respect to the ground For the H2]CN reaction we propagated 60 wavepackets (40 for even parity 20 for rovibrational states of reagents. For the D2]CN reaction we propagated 45 wavepackets (30 for even parity 15 for odd parity) to converge the N(E) for energies up to 0.4 eV. 3.2 The cumulative reaction probability N(E) (J= 0) We –rst show in Fig. 1 the number of open states of even parity as a function of energy in the asymptotic region and on the dividing surface S for the H2]CN reaction. Very obviously and importantly the number of open states on S is much smaller than the 1 1 number open in the asymptotic region. For E\0.4 eV for example the ratio between these is about 16.Thus for this reaction the TSWP should require many fewer wavepacket propagations in order to calculate N(E) than the ISSWP approach. N of diÜerent transition states to N(E) (i\1»5 i(E) Ni(E) for the N Fig. 2 shows the contributions of 10 15 20 25 30 35 for even parity). As can be seen from the –gure the –rst few states rise very quickly from 0 to about 1 in about 0.1 eV energy interval then N slightly decrease as the energy increases further. This rise in i(E) is signi–cantly faster than that for the H2]OH reaction shown in Fig. 5 in ref. 38. This probably indicates that there is much less tunneling in the H2]CN reaction. Also we can see that the contributions to N(E) for the highly excited transition states are negligible for energy lower than 0.25 eV.The contributions to N(E) in that energy region almost all come from the –rst 4»5 transition states. For energy up to 0.5 eV the –rst 40 wavepackets have already given very well converged N(E). Among the Ni(E) shown in the –gure however we can see the ìnon-probabilityœ nature of i(E) ; some have values slightly 2 Fig. 1 Number of open states as a function of total energy on transition state dividing surface and in asymptotic region divided by 10 for theH (even rotation)]CN reaction in even parity 112 rotation)]CN reaction in even parity Fig. 2 Some Ni(E)s larger than 1 and some have very small negative contributions (a few percent) at some energies. To our surprise we found there are –ve out of the –rst 40 transition states that give rise to very strange oscillatory Ni(E) as shown in Fig.3. Further investigation revealed that these –ve transition states correspond to the HwHwNwC con–guration in the transition state region. These resonance-like structures in these Ni(E) are due to a deep well (close to 0.4 eV) along the reaction path in the HwHwNwC transition state region as shown in Fig. 4. Because the well is quite deep the TSWPs initially located inside the well are resonantly trapped in the well causing the oscillatory behavior in these Ni(E). Since it is believed that the well is very likely unphysical we just ignore these TSWPs. Thus the N(E) and later the rate constant presented in this paper are for the H2 ]CN]HwCwN]H reaction. 2 N Fig. 3 Oscillating i(E)s total energy for the H (even rotation)]CN reaction in even parity Quantum transition state wavepacket method (i\1»5 10 15 20 25 30 35 40) as a function of total energy for the H (even 2 (i\16 24 31 37 39) for the HwHwNwC TSWPs as a function of 113 J.C. L ight and D. H. Zhang Fig. 4 PES for collinear HwHwNwC con–guration with CN bond length optimized 2]OH reaction.38 Fig. 6(b) shows the huge diÜerence between 2 and ]CN H2]OH in the low-energy region. The N(E) for the H2]CN N for i\1»40 excluding those shown in Fig. 3 gives the N(E) The summation of i(E) for even parity shown in Fig. 5. The N(E) for odd parity shown in the –gure are obtained in the same way. The total N(E) is simply the summation of Ni(E) for even and odd parities. From the –gure we can see that N(E) is very slightly oscillatory not as smooth as that for the H N(E) for H reaction is much smaller than that for the H2]OH reaction for energy lower than 0.14 eV.The implies the tunneling eÜect in the H2]OH reaction is much more signi–cant than that in the H2]CN reaction. The N(E) for the D2]CN reaction for energies up to 0.4 eV is shown in Fig. 6 together with that for the H2]CN reaction. First we can see the N(E) curve for the D2]CN reaction also shows slight oscillatory behavior like that for the H2]CN reaction. For energy higher than 0.16 eV the N(E) for the D2]CN reaction is higher than that for the H2]CN reaction because of the larger density-of-states for D2]CN in the transition state region. For energies lower than about 0.16 eV where tunneling eÜects are dominant the N(E) for the D2]CN reaction becomes smaller than that for the H2]CN reaction because of the heavier mass of the D atoms.3.3 Thermal rate constants The thermal rate constant can be calculated from the cumulative reaction probability via a Boltzmann average (14) k(T )\[2pQr(T )]~1 P`=dE e~E@kBTNtot(E) ~= Qr(T ) is the reactant partition function (per unit volume) Ntot(E) is the total where cumulative reaction probability (15) Ntot(E)\; (2J]1)NJ(E) J 114 Quantum transition state wavepacket method Fig. 5 Cumulative reaction probability N(E) as a function of total energy for the H (even 2 rotation)]CN]HCN]H reaction compared with that for the H (even rotation)]OH reaction (a) linear scale ; (b) logarithmic scale with J being the total angular momentum.Since the calculation of the NJ/0(E) alone takes more than 1 month of CPU time on an SGI R10000 processor it is obviously not feasible to calculate the total cumulative reaction probability by using the same method with our current computer power. Thus we invoke the energy shifting approximation to N calculate the approximate rate constant from the exact J/0(E). Since the reaction has a linear transition state the approximate rate constant should be calculated as56,57 (16) energy shifting(E)B 2 Q n rot t Q Ct = dEe~E@kBTNJ/0(E) k P 2 2 r(T ) ~= where Qrot t is the rotational partition function for the linear H CN(D CN) complex at the (transition state) saddle point and can be easily calculated as 2 (17) Q Erot J \Brot t J(J]1) (2J]1)exp([Erot J /kBT ) with rot t \;J where Brot t is the rotational constant of the linear reaction complex at the saddle point ; it is 0.757 cm~1 for the H2CN complex and 0.495 cm~1 for the D2CN complex.Cî in eqn. 115 J. C. L ight and D. H. Zhang (even (even rotation)]CN reaction and the D2 Fig. 6 Comparison of N(E) for the H2 rotation)]CN reaction (a) linear scale ; (b) logarithmic scale (16) is a kind of bending partition function of a linear (doubly degenerate) transition state. For a linear four atom system there are two doubly degenerate bending states and Cî can be calculated as56 (18) Ct\1] 2 exp([+u1 t /kBT ) 1 [ [exp([+u1 t /kBT ) ] 1 2 exp( exp( [ [ +u +u 2 t / 2 t k /k B T BT ) ) .are 114.6 cm~1 and 562.3 cm~1 respectively. For For the H2CN complex the D2CN complex u1 t u and 1 t u2 t u2 t and are 103.5 cm~1 and 398.62 cm~1 respectively. The rate constants calculated using eqn. (16) are shown in Fig. 7 for temperatures between 200 and 750 K for the H2]CN reaction and for temperatures between 200 and 600 K for the D2]CN reaction. Also shown in Fig. 7 are the experimental rate constants for these two reactions.43,45 For the H2]CN reaction the agreement between the theory and experiment is only good in the high-temperature region. As the temperature decreases the experimental value becomes increasingly larger than the theoretical one. At T \209 K the experimental value is larger than the theoretical one by a factor of 2.4.For the D2]CN reaction the theoretical rate constants agree quite well with all the available experimental data. However because there is no experimental value measured at T \209 K it is very hard to say if the theory can agree with experi-116 Quantum transition state wavepacket method 2(D2)]CN reaction Fig. 7 Comparison of the present theoretical rate constant with the experimental measurements for the H ment at that low temperature. We also should bear in mind that we invoked the energy shifting approximation to calculate these rate constants from N(E) (J\0). The accuracy of this approximation for these two reactions still must be assessed. Meanwhile if we assume that the energy shifting approximation is quite good for these two reactions then the comparison between theoretical and experimental results indicates that the tunneling eÜect is not signi–cant enough on the TSH3 PES.We also compare the present rate constant with other theoretical results on the same PES in Fig. 8. In the high-temperature region all the calculations agree with each other quite well. Over the whole temperature region the rate constants for the ground initial state agree very well with our results. In the low-temperature region the present value is slightly higher than that for the ground initial state (at T \200 K the present value is higher than that for the ground initial state by 16%). In the high-temperature region the rate constant for the ground initial state is slightly higher than the present value (at T \700 K the rate constant for the ground initial state is higher than the present value H2]CN reaction Fig.8 Comparison of the present theoretical rate constant with other theoretical results for the 117 J. C. L ight and D. H. Zhang by 10%). It is worth pointing out that the approximation of the thermal rate constant by that for the ground initial state neglects the reagent rotational eÜect on rate constants whereas it treats the high J eÜect by the CS approximation. On the other hand the eÜects from initial reagent rotation are treated accurately in the current study while the high J eÜect is treated quite crudely by the energy shifting approximation. Realizing the essential diÜerence in the approximations made in these two calculations it is quite interesting to see such good agreement between them.It is found in a recent study of the H2]OH reaction39 that in the high-temperature region the rate constant for the ground initial state is slightly higher than the accurate value whereas in the lowtemperature region the rate constant for the ground initial state is slightly lower than the accurate result. If this is also true for the H2]CN reaction this would mean that the rate constant from the energy shifting approximation is quite accurate. Finally we can see that in the low-temperature region the RB (rotating bond) 4-D results47 are slightly higher than the current results whereas the rate constants obtained from quasiclassical trajectory calculations which agree very well with the experimental results are signi–cantly larger than the current values.The role of the metastable well in the HHNC con–guration on the TSH3 PES has not been examined as the well might be an artifact. Hence the current results exclude transition states located in this well. However if the Ni(E) from these transition states were included the eÜect on the overall rate constants would be small. The sum of Ni(E) from these states is less than 0.1 for energies below 0.40 eV and remains between about [0.2 and]0.6 for energies between 0.40 eV and 0.5 eV. At energies above 0.4 eV the N(E) from the ìtrueœ transition states is larger than 15. Thus the contribution from the metastable states real or not is not signi–cant. It also should be noted that the reaction probability from these HHNC con–gurations might result in formation of the HNC isomer.The current dividing surface lumps the two isomers as ìproductœ since it does not depend on the polar angle of CN with respect to the intermolecular axis. It will be interesting to investigate this possibility in the future. 4 Conclusions We have evaluated the cumulative reaction probability for the reaction of H2(D2) with CN for J\0 by the TSWP method. We –nd that the method is highly advantageous for this system because of the relatively high barrier and low density-of-states in the transition state region. The cumulative reaction probability is a smooth function of energy except for contributions from (possibly spurious) transition states located in the metastable HHNC minimum.The thermal rate constants are approximated from N(E) for J\0 by the energy shifting approximation. Surprisingly good agreement is obtained between this calculation and two earlier calculations the –rst based on a 4-D RB approximation, 47 and the second an exact quantum initial state selected calculation for the ground state system (with initially non-rotating H2 and CN). However all three quantum calculations are not in good agreement with experiment or with quasiclassical calculations at low temperatures. This suggests (but certainly doesnœt prove) that the PES might not permit enough tunneling and/or that the zero point energies at the transition state are too large. The agreement with experiment at higher temperatures for both and H2]CN D2]CN is excellent.This research was supported in part by a grant from the Department of Energy DEF602-87ER 13679 and by Academic research grant RP970632 National University of Singapore. References 1 E. Wigner J. Chem. Phys. 1937 5 720. 2 S. Glasston K. J. Laidler and H. Eyring T he T heory of Rate Processes McGraw-Hill New York 1941. 118 Quantum transition state wavepacket method 3 J. C. Keck J. Chem. Phys. 1960 32 1035. 4 J. C. Keck Adv. Chem. Phys. 1967 13 85. 5 P. Pechukas and F. J. McLaÜerty J. Chem. Phys. 1973 58 1622. 6 P. Pechukas Modern T heoretical Chemistry ed. W. H. Miller Plenum New York Part B vol. 2 1976. 7 B. C. Garrett and D. G. Truhlar J. Chem. Phys. 1979 70 1593. 8 D. G. Truhlar and B. C. Garrett Annu. Rev.Phys. Chem. 1984 35 159. 9 D. G. Truhlar B. C. Garrett and S. J. Klippenstein J. Phys. Chem. 1996 100 12771. 10 D. H. Zhang and J. Z. H. Zhang J. Chem. Phys. 1993 99 5615. 11 D. Neuhauser J. Chem. Phys. 1994 100 9272. 12 D. H. Zhang and J. Z. H. Zhang J. Chem. Phys. 1994 101 1146. 13 D. H. Zhang and J. C. Light J. Chem. Phys. 1996 104 4544. 14 D. H. Zhang and J. C. Light J. Chem. Phys. 1996 105 1291. 15 W. Zhu J. Dai J. Z. H. Zhang and D. H. Zhang J. Chem. Phys. 1996 105 4881. 16 W. H. Miller J. Chem. Phys. 1974 61 1823. 17 W. H. Miller S. D. Schwartz and J. W. Tromp J. Chem. Phys. 1983 79 4889. 18 J. W. Tromp and W. H. Miller Faraday Discuss. Chem. Soc. 1987 84 441. 19 K. Yamashita and W. H. Miller J. Chem. Phys. 1985 82 5475. 20 J. W. Tromp and W.H. Miller J. Phys. Chem. 1986 90 3482. 21 T. P. Park and J. C. Light J. Chem. Phys. 1986 85 5870. 22 T. P. Park and J. C. Light J. Chem. Phys. 1988 88 4897. 23 T. P. Park and J. C. Light J. Chem. Phys. 1989 91 974. 24 T. P. Park and J. C. Light J. Chem. Phys. 1991 94 2946. 25 D. Brown and J. C. Light J. Chem. Phys. 1992 97 5465. 26 W. H. Thompson and W. H. Miller J. Chem. Phys. 1995 102 7409. 27 W. H. Thompson and W. H. Miller J. Chem. Phys. 1994 101 8620. 28 (a) P. Day and D. Truhlar J. Chem. Phys. 1991 94 2045; (b) P. Day and D. Truhlar J. Chem. Phys. 1991 95 5097; (c) D. Neuhauser and M. Baer J. Chem. Phys. 1989 90 4351. 29 U. Manthe and W. H. Miller J. Chem. Phys. 1993 99 3411. 30 U. Manthe T. Seideman and W. H. Miller J. Chem. Phys. 1993 99 10078.31 U. Manthe T. Seideman and W. H. Miller J. Chem. Phys. 1994 101 4759. 32 U. Manthe J. Chem. Phys. 1995 102 9204. 33 W. H. Thompson and W. H. Miller J. Chem. Phys. 1997 106 142. 34 H. Wang W. H. Thompson and W. H. Miller J. Chem. Phys. 1997 107 7194. 35 U. Manthe and F. Matzkies Chem. Phys. L ett. 1996 252 7. 36 (a) F. Matzkies and U. Manthe J. Chem. Phys. 1997 106 2646; (b) U. Mauthe and F. Matzkies Chem. Phys. L ett. 1998 282 442; (c) F. Matzkies and U. Manthe J. Chem. Phys. 1998 108 4828. 37 D. H. Zhang and J. C. Light J. Chem. Phys. 1996 104 6184. 38 D. H. Zhang and J. C. Light J. Chem. Phys. 1997 106 551. 39 D. H. Zhang J. C. Light and S. Y. Lee J. Chem. Phys. 1998 109 79. 40 B. Jackson Annu. Rev. Phys. Chem. 1995 46 251. 41 D. H. Zhang and J.Z. H. Zhang in Dynamics of molecules and chemical reactions ed. R. E. Wyatt and J. Z. H. Zhang Marcel Dekker New York 1996. 42 T. Seideman and W. H. Miller J. Chem. Phys. 1991 95 1768. 43 I. R. Sims and I. W. M. Smith Chem. Phys. L ett. 1988 149 564. 44 Q. Sun and J. M. Bowman J. Chem. Phys. 1990 92 5201. 45 Q. Sun D. L. Yan N. S. Wang J. M. Bowman and M. C. Lin J. Chem. Phys. 1990 93 4730. 46 M. A. ter Horst G. c. Schatz and L. B. Harding J. Chem. Phys. 1996 105 2309. 47 T. Takayanagi and G. C. Schatz J. Chem. Phys. 1997 106 3227. 48 J. H. Wang K. Liu G. C. Schatz and M. ter Horst J. Chem. Phys. 1997 107 7869. 49 W. Zhu J. Z. H. Zhang Y. C. Zhang Y. B. Zhang L. X. Zhang and S. L. Zhang J. Chem. Phys. 1998 108 3509. 50 L.-H. Lai J.-H. Wang D. C. Che and K. Liu J. Chem. Phys. 1996 105 3332. 51 J. M. Bowman and G. C. Schatz Annu. Rev. Phys. Chem. 1995 46 169. 52 D. C. Clary J. Chem. Phys. 1991 95 7298. 53 J. M. Bowman J. Zuniga and A. Wierzbicki J. Chem. Phys. 1989 90 2708. 54 D. H. Zhang Q. Wu J. Z. H. Zhang M. Dirke and Z. Bacó icç J. Chem. Phys. 1995 102 2315. 55 D. H. Zhang and J. Z. H. Zhang J. Chem. Phys. 1994 100 2697. 56 Q. Sun J. M. Bowman G. C. Schatz J. R. Sharp and J. N. L. Connor J. Chem. Phys. 1990 92 1677. 57 J. M. Bowman J. Phys. Chem. 1991 95 4960. Paper 8/01188E; Received 10th February 1998

ISSN:1359-6640    

DOI:10.1039/a801188e

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 119»138 Quantum mechanical angular distributions for the F+ HD reaction Jesus F. Castillo and David E. Manolopoulos* Physical and T heoretical Chemistry L aboratory South Parks Road Oxford UK OX1 3QZ Quantum-mechanical integral and diÜerential cross-sections have been calculated for the title reaction at the two collision energies (Ecoll\1.35 and 1.98 kcal mol~1) studied in the 1985 molecular beam experiment of Lee and co-workers using the new ab initio potential-energy surface of Stark and Werner. The DF]H product channel is found to behave essentially classically the present quantum-mechanical angular distributions for this channel are in good agreement with both the earlier quasi-classical trajectory results of Aoiz and co-workers on the same potential-energy surface and the results of the molecular beam experiment.However the HF]D product channel in which the light H atom is transferred between two heavier atoms is inherently more quantum-mechanical our computed angular distributions for this channel diÜer signi–cantly from the quasiclassical trajectory results and agree better with the results of the experiment (especially at the higher of the two experimental collision energies). The main quantum-mechanical eÜect that is identi–ed in the calculations is a reactive scattering resonance that gives rise to a pronounced forwardscattering peak in the calculated F]HD(v\0 j\0)]HF(v@\3)]D diÜerential cross-section. The in—uence of this resonance on the reaction dynamics is discussed in some detail together with the implications of our results at the lower of the two collision energies for an improvement to the Stark»Werner potential-energy surface.119 1 Introduction The F]H reaction and its isotopic variants have played a central role at various stages during the development of gas-phase reaction dynamics and they are currently 2 very much in the limelight.1h28 The key experimental advance that stimulated this revival of interest was the FH2~ anion photoelectron spectroscopy experiment performed by Neumark and co-workers,1,2 which provided us with a wonderfully detailed picture of the dynamics of the F]H reaction in the transition-state region.3 The corresponding theoretical advance was the development of a highly correlated ab initio 2 potential-energy surface (PES) for the F]H reaction by Stark and Werner (SW),4 2 which brought theory and experiment together for this reaction for the –rst time.3,5,10,11 These developments are summarised in a recent review article which discusses the dynamics calculations that have been performed on the SW surface and highlights some interesting directions for future research.20 Since this review was written studies of the reaction have continued to —ourish,21h28 with at least two new experiments having been applied to the reaction21,24 and several more F]H dynamics calculations having been 2 published.25h28 120 Quantum-mechanical angular distributions for F]HD largest skewing angles and therefore lie at opposite ends of the Fig.1 Collinear potential-energy contours and kinematic skewing angles for the four deuteriumsubstituted isotopomers of the F]H reaction.Note that the F]HD reaction is especially 2 interesting because its HF]D and DF]H product channels have respectively the smallest and F]H ì resonance hierarchyœ.31 2 The motivation for the present study is an issue that has been central to discussions of the F]H reaction ever since 1975 when the pioneering collinear quantum scattering calculations of Schatz et al. showed the –rst evidence of reactive scattering reso- 2 nances in the reaction.29 These resonances were invoked in particular by Lee and co-workers in order to explain the observed state-selective forward scattering of the HF(v@\3) product in their celebrated F]H molecular beam experiment in 1985.30 However we now know that the results of this molecular beam experiment can be 2 reproduced quite satisfactorily in a calculation on the SW surface that shows no indication of any resonance contribution to computed angular distributions.10 Instead the observed state-selective forward scattering of the HF(v@\3) product in the F]H reaction can be attributed simply to quantum-mechanical tunnelling through the reaction 2 barrier at the high initial orbital angular momenta that lead to forward scattering with the product quantum state-speci–city arising from the thermoneutrality of F]H2(v\ 0)]HF(v@\3)]H.10 Whether or not the resonances in the F]H reaction can be observed experimentally in some other way is still however an open question that is 2 attracting new research.16 Now although resonances are not necessary to explain the observed F]H2 ]HF ]H angular distributions there is some reason to suppose that they might be necessary for F]HD]HF]D which is expected to show both a higher degree of tunnelling and a more pronounced quantum-mechanical resonance structure than any other iso-121 J.F. Castillo and D. E. Manolopoulos topic variant of the reaction on account of its smaller kinematic skewing angle and the fact that it is the only isotopomer in which a light H atom is exchanged between two heavier atoms29,31 (see Fig. 1). Further support for this idea is provided by the recent quasi-classical trajectory (QCT) study of the F]HD reaction by Aoiz and co-workers,5 the results of which agree very well with the experimental angular distributions for F]HD]DF]H but are in striking disagreement with experiment for F]HD]HF]D.The implication is clearly a very large quantum-mechanical eÜect for this particular isotopic variant of the reaction larger by far than the eÜect observed for F]H2 and the purpose of the present paper is to examine this eÜect in more detail. In order to do this we have performed the –rst fully three-dimensional quantum reactive scattering calculations for the F]HD reaction at the two average collision energies (Ecoll\1.35 and 1.98 kcal mol~1) studied in the 1985 molecular beam experiment, 31 using the now standard SW PES.4 The details of these calculations and the convergence tests we have performed to ensure their accuracy are presented in Section 2 and the resulting integral and diÜerential cross-sections are presented in Section 3.These diÜerential cross-sections are interpreted in Section 4 via an analysis of the computed opacity functions and cumulative reaction probabilities and the implications of this analysis for the experimental observation of resonances in the reaction are discussed in Section 5. Our conclusions are presented in Section 6. 2 Method Computational details The present quantum reactive scattering calculations were performed using an exact coupled-channel hyperspherical coordinate method similar to the one described by Schatz,32 with canonical orthogonalisation of the coupled-channel basis set to avoid overcompleteness problems in the exchange region as described by Parker and Pack.33 For higher total angular momenta (J[0) we used a parity-adapted body-frame angular momentum basis set as described by Zhang,34 retaining all basis functions with helicity quantum numbers kOmin(J j kmax) in each of the three chemical arrangements (F]HD H]DF and D]HF).The same method has been used previously to calculate quantum-mechanical integral and diÜerential cross-sections for the F]H2 reaction10 and product rotational polarisations for H]D2 ,35 but the present F]HD calculations are the largest that have been performed with this method to date. In order to simulate the F]HD molecular beam experiment,31 it is necessary to perform calculations for a large number of total angular momenta triatomic parity blocks and total energies.However given the expense of the calculations we have con- –ned our attention in this preliminary study to the ground (v\0 j\0) rovibrational state of the reactant HD molecule. This simpli–cation reduces both the number of total energies and the number of parity blocks that have to be considered by a factor of two and it is justi–ed by the fact that the thermal beam used by Lee and co-workers consisted of 90% HD(v\0 j\0) and 10% HD(v\0 j\1).31 (The fact that the QCT calculations of Aoiz and co-workers5 were found to predict very diÜerent reactivities for F]HD(v\0 j\0) and F]HD(v\0 j\1) suggests that the 10% HD(v\0 j\1) contribution may be quite signi–cant however and this should therefore be borne in mind when comparing our results with experiment.) The quantum reactive scattering calculations that we have performed on the F]HD(v\0 j\0) reaction are summarised in Table 1 where the quoted total energies E were obtained from the average E experimental collision energies tot by adding the calculated HD zero-point energy on the SW PES (0.2325 eV).coll Having performed these calculations quantum-mechanical integral and diÜerential SJ cross-sections were extracted from the computed S-matrix elements (where a@ a{v{j{k{,avjk 122 Quantum-mechanical angular distributions for F]HD Table 1 Summary of the F]HD(v\0 j\0) quantum reactive scattering calculations that have been performed on the SW PES triatomic parity angular momentum E eVtot/ Ecoll/ kcal mol~1 ([1)J only J\0»25 0.2911 0.3184 1.35 1.98 labels either the DF(v@ j@,k@)]H or the HF(v@ j@,k@)]D product arrangement and a the F]HD(v j k) reactant arrangement) using the standard helicity representation equations36 (1) dpa{v{j{k{Havjk dX K2 J (h)\K 1 ; (2J]1)dk{,k J (n[h)SJa {v{j{k{,avjk 2ikavjk and J pa{v{j{k{Havjk\Pdpa{v d { j{ X k {Havjk (h)dX\k n ; (2J]1) o SJa {v{j{k{,avjk o2 (2) avjk 2 and then appropriately summed and averaged to compare with the experimental results.As discussed by Zhang and Miller,36 and again by us in our earlier study of the F]H2 reaction,10 the use of n[h rather than h in eqn. (1) makes the equation consistent with the experimental de–nition of forward scattering with h\0° corresponding to the –nal velocity vector of the DF or HF product molecule lying parallel (rather than antiparallel) to the initial velocity vector of the F atom in the centre-of-mass frame.30,31 Convergence tests Since these are the –rst fully three-dimensional calculations on the F]HD reaction to have been performed and they have been performed on a well documented PES it is possible that they will serve as a useful benchmark for future reactive scattering calculations.We have therefore gone to some trouble to ensure that our results are wellconverged and we shall now brie—y describe the convergence tests we have performed to demonstrate this. Three key convergence parameters in our coupled-channel hyperspherical coordinate Emax jmax and k program are denoted max .37 The –rst two of these parameters de–ne the coupled-channel basis set which contains all F]HD DF]H and HF]D channels with diatomic energy levels less than or equal to E and rotational quantum numbers jmax .Since these parameters play the same role for all values of the less than or equal to max total angular momentum quantum number J it suffices to examine their convergence for J\0. Tables 2 and 3 show the convergence of selected J\0 reaction probabilities for the F]HD(v\0 j\0) reaction as a function of E and j max (Ecoll\1.98 kcal mol~1). It is clear higher of the two molecular beam collision energies max respectively at the from these tables that the dominant reaction probabilities are all well converged with production runs. Emax\1.7 eV and jmax\15 which are the values of these parameters we used in our The –nal convergence parameter kmax is only relevant for higher total angular momenta J[0 where it serves as an upper limit on the helicity quantum number k.Examining the convergence of the calculation with respect to this parameter is therefore more expensive since it requires calculations for higher values of J. The representative values of J we considered in our convergence tests were J\8 12 and 16 and the 123 J. F. Castillo and D. E. Manolopoulos Table 2 Convergence of selected F]HD(v\0 j\0) reaction probabilities as a function of (with J\0 Etot\0.3184 eV and jmax\17) Emax HF(v@ all j@)]D DF(v@ all j@)]H v@\3 v@\2 v@\1 v@\4 v@\3 v@\2 v@\1 Emax/eV 0.009 0.012 0.012 0.243 0.247 0.247 0.119 0.117 0.117 0.140 0.147 0.147 0.276 0.272 0.272 0.139 0.133 0.133 0.023 0.022 0.021 1.5 1.7 1.9 convergence of the computed reaction probabilities with respect to kmax at each of these values of J is shown in Table 4.It can be seen that the dominant reaction probabilities are all again well converged by the time kmax\4 which is the value of this parameter we used in our production runs. max\1.7 max\15 E max\4 j eV k and Combining all three production parameters results in a coupled-channel basis set containing a total of 302 channels with parity ([1)J for J\0 583 channels for J\1 843 channels for J\2 1082 channels for J\3 and 1300 channels for JP4. Since the eÜort required to integrate close-coupled equations scales as the cube of the number of channels our production calculations were rather expensive requiring ca.26 h of computer time for each partial wave JP4 on a Silicon Graphics R10000 processor (with both scattering energies included). We will therefore be interested to see whether or not our results can be reproduced in the future using a cheaper and/or more approximate version of quantum reactive scattering theory. Table 3 Convergence of selected F]HD(v\0 j\0) reaction probabilities as a function of (with J\0 Etot\0.3184 eV and Emax\1.7 eV) jmax DF(v@ all j@)]H HF(v@ all j@)]D v@\3 v@\2 v@\1 v@\4 v@\3 v@\2 v@\1 jmax 0.012 0.012 0.012 0.249 0.246 0.247 0.117 0.117 0.117 0.148 0.147 0.147 0.270 0.272 0.272 0.134 0.134 0.133 0.022 0.022 0.022 13 15 17 eV and function of Table 4 Convergence of selected F]HD(v\0 j\0) reaction probabilities as a eV jmax\15) Emax\1.7 kmax HF(v@ all j@)]D (with Etot\0.3184 DF(v@ all j@)]H v@\3 v@\2 v@\1 v@\4 v@\3 v@\2 v@\1 J kmax 8 0.012 0.012 0.012 0.252 0.252 0.252 0.073 0.073 0.073 0.154 0.155 0.155 0.243 0.247 0.247 0.090 0.092 0.091 0.013 0.012 0.012 3 45 12 0.001 0.001 0.001 0.160 0.161 0.162 0.027 0.027 0.027 0.054 0.055 0.056 0.116 0.128 0.129 0.041 0.042 0.041 0.006 0.005 0.004 3 45 16 0.000 0.000 0.000 0.111 0.111 0.111 0.005 0.005 0.005 0.002 0.002 0.002 0.009 0.010 0.011 0.004 0.004 0.004 0.001 0.000 0.000 3 45 124 a 90% HD(v\0 j\0)]10% HD(v\0 j\1).Table 6 Comparison of quantum-mechanical and quasi-classical trajectory integral cross-sections pR(v@) (in and vibrational branching ratios parentheses) for the F]HD]HF(v@)]D reaction at E ”2) » 0.02(0.03) » 0.06(0.05) (0.02) 3 Results Integral cross-sections Our computed quantum-mechanical integral cross-sections for the F]HD(v\0 j\0)]DF(v@)]H and F]HD(v\0 j\0)]HF(v@)]D reactions are shown in Tables 5 and 6 where they are compared with both the results of the molecular beam experiment31 and the recent QCT results of Aoiz and co-workers.5 In the case of the molecular beam experiment the comparison is con–ned to vibrational branching ratios (shown in brackets) for the F]HD]DF]H reaction at E experiment did not measure absolute cross-sections and the laboratory to centre-of-mass coll\1.98 kcal mol~1.The frame transformation needed to obtain product vibrational branching ratios was only performed for the DF product channel at this higher collision energy.31 However the comparison with the QCT calculations is more complete. Several important conclusions can be drawn from these tables. The –rst and most striking is that the quantum-mechanical vibrational branching ratios are in signi–cantly better agreement with experiment than the QCT branching ratios in the one case where a direct comparison is possible (see Table 5).This is to be expected for an exact quantum reactive scattering calculation on an accurate PES and it shows in particular that quantum-mechanical eÜects are not entirely insigni–cant for the F]HD(v\0 j\0)]DF(v@)]H reaction. Indeed the remaining discrepancies between the quantummechanical and experimental DF(v@)]H vibrational branching ratios in Table 5 are less than 15% which is small enough to be attributed to the lack of HD(v\0 j\1) in the calculations and the 0.1 kcal mol~1 spread of experimental collision energies.31 The second important observation that should be made from Tables 5 and 6 is that the computed quantum-mechanical cross-sections are consistently larger than the QCT cross-sections due to tunnelling. This eÜect is more pronounced for the HF]D product channel than for the DF]H product channel and it is more pronounced in both channels at the lower of the two experimental collision energies.As a result of the tun- Table 5 Comparison of quantum-mechanical quasi-classical trajectory and experimental integral cross-sections pR(v@) (in and vibrational branching ratios (in parentheses) for the F]HD]DF(v@)]H reaction at E mol~1 Ecoll 1.35 1.98 Ecoll 1.35 1.98 Quantum-mechanical angular distributions for F]HD pR(v@)/pR(v@\3) coll\1.35 and 1.98 kcal method all v@ v@\4 v@\3 v@\2 v@\1 0.92 1.34 0.42(1.11) 0.47(0.72) 0.38(1.00) 0.65(1.00) 0.12(0.32) 0.20(0.31) QCT( j\0) QM( j\0) 2.36 2.40 0.81(0.79) 0.65(0.54) (0.48) 1.02(1.00) 1.21(1.00) (1.00) 0.53(0.52) 0.46(0.38) (0.40) QCT( j\0) QM( j\0) Expt.(a) pR(v@)/pR(v@\3) (in coll\1.35 and 1.98 kcal mol~1 method all v@ v@\3 v@\2 v@\1 0.52 1.40 0.21(0.68) » 0.31(1.00) 1.24(1.00) » 0.15(0.12) QCT( j\0) QM( j\0) ”2) v@\0 » 0.01(0.01) » 0.02(0.01) 1.08 2.20 0.24(0.35) 0.09(0.05) 0.69(1.00) 1.71(1.00) 0.15(0.22) 0.37(0.12) QCT( j\0) QM( j\0) 125 J. F. Castillo and D. E. Manolopoulos (a) 1.35 and (b) 1.98 kcal mol~1 Fig. 2 Quantum-mechanical diÜerential cross-sections for the F]HD(v\0 j\0)]DF(v@)]H reaction at Ecoll nelling contribution the quantum-mechanical cross-sections for the production of HF and DF are very similar at both collision energies whereas the QCT cross-sections are ca.two times larger for DF than for HF. The quantum-mechanical results in Tables 5 and 6 are therefore more consistent with F]HD thermal rate constant measurements, 38 which –nd the rate of H atom transfer to be ca. 1.5 times faster than the rate of D atom transfer at 298 K. The third and –nal conclusion that can be drawn from Table 6 concerns the production of HF(v@\3) from the F]HD(v\0 j\0) reaction at Ecoll\1.35 kcal mol~1. The HF(v@\3 j@\0) product channel has a threshold of 0.3035 eV on the SW surface and is therefore closed at Etot\0.2911 eV which explains why the quantum-mechanical cross-section for this channel is zero. The fact that the QCT cross-section is non-zero at this energy is therefore an artifact of the –nal state ìboxingœ used in the QCT method which was performed in ref.5 using a Dunham expansion of HF rovibrational energy levels based on the experimental data in Huber and Herzberg.39 The fact that the SW surface incorrectly predicts the HF(v@\3 j@\0) channel to be closed at Ecoll\1.35 kcal mol~1 when the experimental Dunham expansion predicts it to be open is clearly a 126 Quantum-mechanical angular distributions for F]HD de–ciency in the surface and we shall see several other rami–cations of this de–ciency in what follows. DiÜerential cross sections Our computed quantum-mechanical diÜerential cross-sections for the F]HD(v\0 j\0)]DF(v@)]H and F]HD(v\0 j\0)]HF(v@)]D reactions are shown in Fig. 2 and 3. These diÜerential cross-sections were calculated from eqn.(2) by summing over the product rotational ( j@) and helicity (k@) quantum numbers and we have veri–ed in each case that they integrate correctly to give the integral cross-sections in Tables 5 and 6. Although the diÜerence between the DF]H and HF]D product channels was not especially apparent in these integral cross-sections except perhaps for the increased tunnelling into the HF]D product channel the diÜerence between Fig. 2 and 3 is quite remarkable. The F]HD]DF]H diÜerential cross-sections in Fig. 2 are all strongly backward peaked as seen in both the earlier QCT calculations and the molecular beam experiment.31 However the F]HD]HF]D diÜerential cross-sections in Fig. 3 are Fig. 3 As Fig. 2 but for the F]HD(v\0 j\0)]HF(v@)]D reaction 127 J.F. Castillo and D. E. Manolopoulos molecular beam experiment31 DF(v@)]H reaction at E Fig. 4 Quantum-mechanical LAB frame angular distributions for the F]HD(v\0 j\0)] coll (a) 1.35 and (b) 1.98 kcal mol~1 compared with the results of the coll\1.98 kcal mol~1 for reasons explained in the more structured and bear little if any resemblance to the corresponding QCT results.5 In particular there is clear evidence in Fig. 3(b) of a state-selective HF(v@\3) forward scattering peak at the higher of the two experimental collision energies. This peak was entirely absent from the QCT calculation and is therefore a purely quantum-mechanical eÜect and we shall spend much of the remainder of this paper examining this eÜect in more detail. Before we do this however it is interesting to compare the present quantummechanical diÜerential cross-sections directly with the results of the F]HD molecular beam experiment.Unfortunately this is only possible in the centre-of-mass (CM) frame for the DF]H product channel at E experimental paper.31 However a recent study of F]H reaction angular distributions has shown that in order to obtain an unbiased comparison between theory and experi- 2 ment it is preferable to transform the theoretical results into the laboratory (LAB) frame.27 This CM to LAB transformation of our quantum-mechanical diÜerential crosssections has been performed for us by Aoiz and co-workers40 using the methods detailed 128 Quantum-mechanical angular distributions for F]HD in ref. 27 and these authors have kindly permitted us to present the resulting laboratory frame angular distributions here.Laboratory angular distributions The F]HD molecular beam experiment of Lee and co-workers consisted of a —uorine atom beam intersecting an HD beam at right angles with time-of-—ight detection of the HF and DF product molecules as a function of the polar laboratory scattering angle hLAB measured from the direction of the —uorine atom beam.31 Our transformed40 quantum-mechanical diÜerential cross-sections are shown as a function of this laboratory scattering angle in Fig. 4 and 5. The total experimental signals for each reaction are also shown in these –gures for comparison with the overall normalisation of the theoretical results having been adjusted within each panel to give the same integrated signal as in the experiment.It can be seen from Fig. 4 that the agreement between the computed and measured LAB frame angular distributions for the F]HD]DF]H reaction is essentially perfect at both collision energies. In fact the agreement is quite remarkable given the Fig. 5 As Fig. 4 but for the F]HD(v\0 j\0)]HF(v@)]D reaction 129 J. F. Castillo and D. E. Manolopoulos 10% contribution of HD(v\0 j\1) in the experimental HD beam and the 0.1 kcal mol~1 spread of experimental collision energies neither of which was taken into account in the calculations. The DF]H angular distributions in Fig. 4 do not really provide a very stringent test of the calculation however since equally good agreement was seen previously in a comparison of the same experimental results with the QCT crosssections of Aoiz and co-workers,5 which diÜer appreciably from the present crosssections at both collision energies (see Table 5).By contrast the agreement between the computed and measured LAB frame angular distributions for the F]HD]HF]D reaction is at best qualitative [at the higher of the two collision energies in Fig. 5(b)] and at worst non-existent [at the lower of the two collision energies in Fig. 5(a)]. The complete lack of agreement between theory and experiment at Ecoll\1.35 kcal mol~1 can be traced to the fact that the HF(v@\3) channel is closed on the SW surface at this collision energy since this product channel is known to be responsible for the dominant peak in the experimental angular distribution near hLAB\20°.31 Fig.5(a) therefore provides a direct experimental demonstration of the de–ciency in the SW surface discussed above and suggests in particular that the surface should be modi–ed in the future to incorporate more accurate diatomic potential-energy curves. coll\1.98 kcal mol~1 in Fig. 5(b) is also The qualitative agreement between the computed and experimental angular distributions for the F]HD]HF]D reaction at E worse than one might have expected on the basis of our earlier results for F]H2 ] HF]H.10,27 However it is conceivable that the agreement could be improved in this case by including a 10% contribution from HD(v\0 j\1) in the calculations and by considering a 0.1 kcal mol~1 spread of collision energies.In any event the quantummechanical angular distributions in Fig. 5(b) agree signi–cantly better with the experiment than the earlier QCT angular distributions of Aoiz and co-workers which completely missed the dominant HF(v@\3) peak near hLAB\15°. This peak arises in the present calculations from the state-selective forward scattering of HF(v@\3) in the CM frame diÜerential cross-sections in Fig. 3(b). 4 Analysis Opacity functions One of the main reasons for performing reactive scattering calculations is that they can provide far more detailed information about the dynamics of chemical reactions than it is possible to extract from experiments. The dependence of the computed reaction probability on the impact parameter is straightforward to extract from a scattering calculation for example but it is still a holy grail of reaction dynamics to be able to measure this dependence experimentally.The following analysis of F]HD opacity functions was inspired in particular by the pioneering reduced-dimensionality work of Lee and Bowman.41 In the context of the F]HD(v\0 j\0) reaction for which the initial orbital angular momentum quantum number l coincides with the total angular momentum quantum number J the impact parameter b is given by b\cJJ(J ]1) with c\ +/J2kFhHDEcoll . The quantum-mechanical opacity functions for this reaction therefore contain the same information as plots of the computed reaction probabilities as a function of J which are shown for the F]HD(v\0 j\0)]DF(v@)]H and F]HD(v\0 j\0)]HF(v@)]D reactions in Fig.6 and 7. As in the case of the diÜerential cross-sections in Fig. 2 and 3 these opacity functions are clearly very diÜerent for the DF]H and HF]D product channels and moreover they provide a clear indication as to why the diÜerential cross-sections in the two channels behave so diÜerently. The opacity functions for the F]HD]DF]H reaction in Fig. 6 are similar to 130 Quantum-mechanical angular distributions for F]HD coll (a) 1.35 and (b) 1.98 kcal mol~1 Fig. 6 Quantum-mechanical opacity functions (reaction probabilities as a function of J) for the F]HD(v\0 j\0)]DF(v@)]H reaction at E the QCT opacity functions reported by Aoiz and co-workers,5 the only important diÜerences being the dip in the reaction probability at J\3 in Fig.6(a) and the fact that the present quantum-mechanical reaction probabilities extend to slightly higher values of J than the QCT results (which were found to be zero for all JP9 at 1.35 kcal mol~1 and for all JP14 at 1.98 kcal mol~1 in Aoiz and co-workersœ calculations5). The dip at J\3 in the reaction probability at the lower of the two collision energies is simply a remnant of what is happening in the other (HF]D) product channel as will become apparent when we come on to consider this channel in Fig. 7. The fact that the present quantum-mechanical reaction probabilities continue to higher values of J than the QCT F]H results is similar to the situation encountered previously for 2 ]HF]H,10 and is simply a manifestation of tunnelling through the combined centrifugal and potentialenergy barrier to reaction.This tunnelling is however less pronounced for the F]HD]DF]H reaction than it was for F]H its larger kinematic skewing angle (see Fig. 1). The quantum-mechanical opacity func- 2 ]HF]H as to be expected from tions in Fig. 6 are therefore still dominated by comparatively low values of J (or equivalently b) which explains why the F]HD]DF]H diÜerential cross-sections in Fig. 1 are backward peaked.42 131 J. F. Castillo and D. E. Manolopoulos Fig. 7 As Fig. 6 but for the F]HD(v\0 j\0)]HF(v@)]D reaction In contrast our computed opacity functions for the F]HD]HF]D reaction in Fig. 7 are strikingly diÜerent from the corresponding QCT results,5 which show no evidence of the two dramatic quantum-mechanical reaction probability peaks A and B.The –rst of these peaks occurs in the tunnelling region (JP9) at the lower of the two experimental collision energies whereas the second occurs at the same classically allowed value of the angular momentum (J\3) as the dip in the reaction probability to the DF]H channel in Fig. 6. Both peaks occur in the tunnelling region (JP14) at the higher collision energy where peak B is especially interesting because it accounts for virtually all of the computed reaction cross-section into HF(v@\3). Overall as a result of the two peaks the quantum-mechanical opacity functions for the F]HD]HF]D reaction in Fig. 7 have a signi–cant contribution from high values of J (or equivalently b) which explains why there is so much forward scattering in the diÜerential crosssections for this reaction in Fig.3. Cumulative reaction probabilities In order to understand the origin of peaks A and B in Fig. 7 we have also calculated cumulative reaction probabilities for the F]HD]DF]H and F]HD]HF]D 132 Quantum-mechanical angular distributions for F]HD reactions at the total angular momentum quantum numbers J\0 and 9. These cumulative reaction probabilities Na{Ha J (E) are de–ned in terms of the reactive S-matrix elements of eqn. (1) and (2) by43 SJ (E) a{v{j{k{,avjk (3) Na{Ha J (E)\ ; ; o SJa {v{j{k{,avjk(E) o2 v{j{k{ vjk and they are shown for the two reactions in Fig. 8. The diÜerence between the two product channels is once again apparent from these reaction probabilities which vary smoothly with energy in the DF]H channel but show considerably more structure for HF]D.In particular one can identify two distinct cumulative reaction probability peaks (A and B) in the low-energy threshold region of the F]HD]HF]D reaction both of which occur below the lowest molecular beam collision energy for total angular momentum J\0. As the angular momentum increases both peaks shift to higher energy and by the time J\9 the higher-energy peak (B) is between the two molecular beam collision energies. Since this J-shifting is reminiscent of the situation in Fig. 7 in which the two F]HD]HF]D opacity The arrows at Fig. 8 Quantum-mechanical cumulative reaction probabilities for the F]HD]DF]H and F]HD]HF]D reactions at total angular momentum quantum numbers J (a) 0 and (b) 9.Etot\0.2911 and 0.3184 eV on the energy axis mark the positions of the two molecular beam collision energies on the SW PES. 133 J. F. Castillo and D. E. Manolopoulos function peaks A and B move to higher J with increasing collision energy one is naturally led to imagine that the physical origin of the peaks in the two –gures might be the same. That this is indeed the case is demonstrated in Fig. 9 which plots the total energy E(J) at which each peak occurs as a function of J(J]1). The open circles in this –gure correspond to the cumulative reaction probability peaks in Fig. 8 and the solid circles to the opacity function peaks in Fig. 7. (The error bars on the opacity function data re—ect the fact that quantum-mechanical reaction probabilities are only available at a given energy for integer values of J whereas the cumulative reaction probability peaks in Fig.8 were obtained to graphical accuracy at J\0 and J\9 by using an energy grid with a spacing of 0.5 meV.) Clearly both sets of data can be –tted to the same straight lines E(J)\E(0)]BJ(J]1) giving eÜective rotational constants of 0.222 meV and 0.133 meV for peaks A and B respectively. (Note however that there is also some indication in Fig. 9 of centrifugal distortion reducing the energy of peak B by the time J\15 which is to be expected from the physical interpretation of this peak discussed below.) 5 Discussion The rotational constants in Fig. 9 provide a clear indication as to the physical origin of peaks A and B the rotational constant of peak A is very similar to the average rotational constant B\+2/2kFhHDR2 \ FhHD 0.233 meV of the F]HD transition state region whereas the rotational constant of peak B is closer to the value of 0.126 meV that one would calculate for the linear equilibrium geometry of the D»HF van der Waals well on the SW surface (see Fig.10). This strongly suggests that peak A is associated with the dynamics of the reaction in the transition state region and that peak B is due to a quasi-bound resonance state localised in the D»HF van der Waals well region and there is a signi–cant body of related evidence to support these assignments.10,16 In discussing this evidence we shall concentrate almost exclusively on peak B which is responsible via Fig.7(b) for the experimentally observed state-selective forward scattering of HF(v@\3) in Fig. 5(b). The detailed interpretation of peak A turns out to be more subtle and is therefore best left to another paper in which there is more space to Fig. 9 E(J) vs. J(J]1) for peaks A and B in Fig. 7 and 8. The open circles correspond to the cumulative reaction probability peaks in Fig. 8 and the –lled circles to the opacity function peaks (J[1 in Fig. 7 (which have error bars ranging from 2)(J]12) ( to J]12)(J]32)). The straight line for peak B is shown as solid below and dashed above the threshold for producing HF(v@\3)]D which occurs at Etot\0.3035 eV on the SW PES. This line –ts the equation EB(J)\EB(0)]BBJ(J ]1) with E (0)\0.289 eV and B B E (0)\0.254 eV and B B\0.133 meV.The corresponding parameters for peak A are A\0.222 meV. A 134 Quantum-mechanical angular distributions for F]HD well in the HF]D product valley4 Fig. 10 Two stationary points on the SW PES (A) the non-linear saddle point of the transition state for the F]HD]HF]D reaction ; (B) the linear equilibrium geometry of a van der Waals discuss it in detail.44 In particular it is not at all clear from the evidence we have presented thus far whether peak A is due to a scattering resonance trapped in the transition state region or simply to a direct scattering state with a large amplitude in this region. The distinctively Lorentzian appearance of the peak in Fig. 8 seems to suggest that it is a resonance,45 and indeed similar Lorentzian peaks have been seen previously in the threshold region of the F]HD]HF]D reaction in both collinear29 and three-dimensional46 quantum scattering calculations.However the corresponding peak in the threshold region of F]H reaction on the SW surface has been shown conclu- 2 sively to be due to direct scattering without any resonance contribution.10,16 The only way to settle the question this conclusively for the F]HD]HF]D reaction is to examine the appropriate reactive scattering wavefunction,16 and such a study is currently in progress.44 The situation for peak B is considerably clearer however since the evidence for assigning this peak to a resonance is overwhelming. Perhaps the most compelling evidence comes from the eÜect of isotopic substitution a similar peak has been seen pre- F]H viously in calculations on the 2 ]HF]H reaction,10,16 where it occurs at the slightly higher energy of 0.293 eV (rather than 0.289 eV) for total angular momentum J\0.This F]H version of the peak has been studied in considerable detail both 2 within a simple vibrationally adiabatic approximation10 and via a fully coupled timedependent wavepacket calculation16 of the resonance wavefunction. Both of these calculations lead to the conclusion that the peak is due to a Feshbach resonance (for total angular momentum J\0) associated with the opening of the HF(v@\3) product manifold. 10,16 Moreover both calculations give the same set of approximate quantum numbers for the resonance wavefunction v@(HwF stretch)\3 j@(HF hindered rotation)\0 and t@(HwHF van der Waals stretch)\0.The present peak B in the F]HD reaction is entirely consistent with this assignment. Its eÜective rotational constant in Fig. 9 agrees with the predicted rotational constant of the D»HF van der Waals well in Fig. 10 and the fact that the peak occurs at slightly lower energy for F]HD than for F]H is to be expected from the lower zero point energy in the D»HF (vs. H»HF) van der Waals stretching mode. Peak B is there- 2 fore almost certainly due to a Feshbach resonance (for total angular momentum J\0) with the same approximate quantum numbers as in the F]H2 case v@(HwF stretch)\3 j@(HF hindered rotation)\0 and t@(DwHF van der Waals stretch)\0. One further piece of evidence we shall give in support of this assignment is based on the behaviour of the peak at higher J.It can be seen from Fig. 9 that peak B occurs below the threshold for the production of HF(v@\3)]D until J\10 by which time the centrifugal contribution to the resonance energy is large enough to make 135 J. F. Castillo and D. E. Manolopoulos HF(v@\3)]D an accessible decay product. Below J\10 the only way in which the resonance can decay into reaction products is therefore by vibrational predissociation into HF(v@O2)]D and all the indications from Fig. 7(a) (in which peak B occurs at J\3) are that it decays predominantly into HF(v@\2). Above J\10 however the resonance can decay into HF(v@\3)]D by tunnelling through the centrifugal (or ìorbitingœ) barrier and it can be seen from Fig. 7(b) (in which peak B occurs at J\15) that this is in fact the dominant high-J decay channel.This picture of a transition from a Feshbach resonance to a shape resonance with increasing J was originally suggested for the F]H reaction by Weaver and 2 Neumark,1 and it is illustrated here schematically in Fig. 11. The key feature as it relates to our assignment of peak B is that assuming rovibrational adiabaticity in the decay process one would predict from the approximate resonance quantum numbers Fig. 11 Schematic interpretation of peak B as a Feshbach resonance decaying into HF(v@\2)]D for J\10 and a shape resonance decaying into HF(v@\3 j@\0)]D for JP10. The quantum numbers of the Feshbach resonance are v@(HwF stretch)\3 j@(HF hindered rotation)\0 and t@(DwHF van der Waals stretch)\0.Panel (a) shows the situation for total angular momentum J\0 where the resonance occurs below the threshold for the production of HF(v\3)]D and can therefore only decay into HF(v@O2)]D. Panel (b) shows the situation for J\15 where the centrifugal potential has pushed the resonance above the threshold for the production of HF(v@\3)]D. 136 Quantum-mechanical angular distributions for F]HD oP (cos h) o2 resonance contribution 15 Fig. 12 Comparison of the computed HF(v@\3)]D diÜerential cross-section in Fig. 3(b) with its dominant o d0,0 15 (n[h) o2\oP15(cos h) o2. The extent to which this proportionality oP (cos h) o2 v@(HwF stretch)\3 and j@(HF hindered rotation)\0 that the dominant shape resonance decay product at high J would be HF(v@\3 j@\0)]D.This is indeed found to be the dominant HF(v@\3)]D product channel in our calculations accounting for some 88% of the total reaction cross-section into HF(v@\3)]D at Ecoll\1.98 kcal mol~1. We are now –nally also in a position to comment on the appearance of the computed HF(v@\3)]D diÜerential cross-section in Fig. 3(b). Since this diÜerential crosssection is due exclusively to peak B which can be seen to cross Ecoll\1.98 kcal mol~1 at J\15 in Fig. 7(b) and since the quasi-bound state associated with peak B decays predominantly into HF(v@\3 j@\0)]D at this collision energy it follows from eqn. (2) that the HF(v@\3)]D diÜerential cross-section will be dominated by a single term proportional to is actually realised in the calculation is shown in Fig.12 the dominant 15 contribution clearly captures the essence of the full HF(v@\3)]D diÜerential crosssection but the other contributions from the minor HF(v@\3 j@) product channels are nevertheless required to make the full cross-section more forward than backward peaked (as seen in the molecular beam experiment31). 6 Conclusions We have presented fully three-dimensional F]HD quantum reactive scattering calculations on the new ab initio PES of Stark and Werner.4 The results of these calculations have been analysed in some detail with regard to their implications for both the accuracy of the PES and the experimental observation of resonances in the reaction. Our main conclusion about the PES is that it should be improved in the future to incorporate more accurate diatomic potential-energy curves since the diatomic curves in the present surface are at the root of the disagreement between theory and experiment in Fig.5(a). Our conclusion about resonances in so far as it is possible to draw such a conclusion from the qualitative agreement between theory and experiment in Fig. 5(b) is that the state-selective forward scattering of the HF(v@\3) product in the F]HD molecular beam experiment of Lee and co-workers31 is indeed due to a quantum reac-137 J. F. Castillo and D. E. Manolopoulos tive scattering resonance associated with the opening of the HF(v@\3) product manifold. It is a pleasure to thank F. J. Aoiz L. Banares and B. Martinez-Haya for transforming our diÜerential cross-sections into the laboratory frame and D.M. Neumark for a stimulating discussion. This work was supported by the EPSRC. References 1 A. Weaver and D. M. Neumark Faraday Discuss. Chem. Soc. 1991 91 5. 2 S. E. Bradforth D. W. Arnold D. M. Neumark and D. E. Manolopoulos J. Chem. Phys. 1993 99 6345. 3 D. E. Manolopoulos K. Stark H-J. Werner D. W. Arnold S. E. Bradforth and D. M. Neumark Science 1993 262 1852. 4 K. Stark and H-J. Werner J. Chem. Phys. 1996 104 6515. 5 F. J. Aoiz L. Banares V. J. Herrero V. S. Rabanos K. Stark and H-J. Werner Chem. Phys. L ett. 1994 223 215; J. Phys. Chem. 1994 98 10665; J. Chem. Phys. 1995 102 9248. 6 M. Gilibert and M. Baer J. Phys. Chem. 1994 98 12822; 1995 99 15748. 7 M. Faubel L. Rusin S. Schlemmer F. Sondermann U. Tappe and J. P. Toennies J.Chem. Phys. 1994 101 2106. 8 M. Faubel B. Martinez-Haya L. Y. Rusin U. Tappe and J. P. Toennies Chem. Phys. L ett. 1995 232 197. 9 G. D. Billing L. Y. Rusin and M. B. Sevryuk J. Chem. Phys. 1995 103 2482. 10 J. F. Castillo D. E. Manolopoulos K. Stark and H-J. Werner J. Chem. Phys. 1996 104 6531. 11 M. Baer M. Faubel B. Martinez-Haya L. Y. Rusin U. Tappe J. P. Toennies K. Stark and H-J. Werner J. Chem. Phys. 1996 104 2743. 12 J. Chang and N. J. Brown Chem. Phys. L ett. 1996 254 147. 13 F. J. Aoiz L. Banares V. Herrero K. Stark and H-J. Werner Chem. Phys. L ett. 1996 254 341. 14 M. Faubel B. Martinez-Haya L. Y. Rusin U. Tappe J. P. Toennies F. J. Aoiz and L. Banares Chem. Phys. 1996 207 227. 15 F. J. Aoiz L. Banares M. Faubel B. Martinez-Haya L.Y. Rusin U. Tappe and J. P. Toennies Chem. Phys. 1996 207 245. 16 C. L. Russell and D. E. Manolopoulos Chem. Phys. L ett. 1996 256 465. 17 E. Roseman S. Hochmankowal A. Persky and M. Baer Chem. Phys. L ett. 1996 257 421. 18 G. C. Schatz J. Chem. Phys. 1997 106 2277. 19 J. Lindner J. K. Lundberg C. M. Lovejoy and S. R. Leone J. Chem. Phys. 1997 106 2265. 20 D. E. Manolopoulos J. Chem. Soc. Faraday T rans. 1997 93 673. 21 G. Dharmasena T. R. Phillips K. N. Shokirev G. A. Parker and M. Keil J. Chem. Phys. 1997 106 22 M. Faubel B. Martinez-Haya L. Y. Rusin U. Tappe and J. P. Toennies J. Phys. Chem. A 1997 101 9950. 6415. 23 G. Dharmasena K. Copeland J. H. Young R. A. Lasell T. R. Phillips G. A. Parker and M. Keil J. Phys. Chem. A 1997 101 6429. 24 W.B. Chapman B. W. Blackmon and D. J. Nesbitt J. Chem. Phys. 1997 107 8193. 25 R. T. Skodje R. Sadeghi and J. L. Krause J. Chem. Soc. Faraday T rans. 1997 93 765. 26 A. J. R. da Silva H. Y. Cheng D. A. Gibson K. L. Sorge Z. H. Liu and E. A Carter Spectrochim. Acta A 1997 53 1285. 27 F. J. Aoiz L. Banares B. Martinez-Haya J. F. Castillo D. E. Manolopoulos K. Stark and H-J. Werner J. Phys. Chem. A 1997 101 6414. 28 B. Hartke and H-J. Werner Chem. Phys. L ett. 1997 280 430. 29 G. C. Schatz J. M. Bowman and A. Kuppermann J. Chem. Phys. 1975 63 674; 1975 63 685. 30 D. M. Neumark A. M. Wodtke G. N. Robinson C. C. Hayden and Y. T. Lee J. Chem. Phys. 1985 82 3045. 31 D. M. Neumark A. M. Wodtke G. N. Robinson C. C. Hayden R. Schobatake R. K. Sparks T. P. Schaefer and Y. T. Lee J. Chem. Phys. 1985 82 3067. 32 G. C. Schatz Chem. Phys. L ett. 1988 150 92. 33 G. A. Parker and R. T Pack J. Chem. Phys. 1993 98 6883. 34 J. Z. H. Zhang J. Chem. Phys. 1991 94 6047. 35 M. P. de Miranda D. C. Clary J. F. Castillo and D. E. Manolopoulos J. Chem. Phys. 1998 (in press). 36 J. Z. H. Zhang and W. H. Miller J. Chem. Phys. 1989 91 1528. 37 J. F. Castillo and D. E. Manolopoulos unpublished work. 38 A. Persky J. Chem. Phys. 1973 59 5578. Quantum-mechanical angular distributions for F]HD 138 39 K. P. Huber and G. Herzberg Molecular Spectra and Molecular Structure IV . Constants of Diatomic Molecules Van Nostrand New York 1979. 40 F. J. Aoiz L. Banares and B. Martinez-Haya unpublished work 1998. 41 K. T. Lee and J. M. Bowman J. Phys. Chem. 1982 86 2289. 42 See e.g. R. D. Levine and R. B. Bernstein Molecular Reaction Dynamics and Chemical Reactivity Oxford University Press Oxford 1987. 43 W. H. Miller J. Chem. Phys. 1975 62 1899. 44 T. Gonzalez and D. E. Manolopoulos comment at this Discussion meeting. 45 See e.g. M. S. Child Molecular Collision T heory Academic Press London 1974. 46 D. E. Manolopoulos M. DœMello R. E. Wyatt and R. B. Walker Chem. Phys. L ett. 1990 169 482. Paper 8/01227J; Received 12th February 1998

ISSN:1359-6640    

DOI:10.1039/a801227j

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 139»157 Quantum scattering studies of spinñorbit eÜects in the Cl(2P)+ HCl«ClH+ Cl(2P) reaction 1@2 1@2 1@2 1@2 § Visiting Scientist. Permanent address Department of Chemistry Northwestern University Evanston IL 60208-3113 USA. î Present address Clinical Trial Service Unit Nuffield Department of Clinical Medicine University of Oxford Harkness Building RadcliÜe In–rmary Woodstock Road Oxford UK OX2 6HE. George C. Schatz,a*§ Patrick McCabebî and J. N. L. Connorb* a T heoretical Chemistry Group Chemistry Division Argonne National L aboratory Argonne IL 60439 USA b Department of Chemistry University of Manchester Manchester UK M13 9PL 1@2 3@2 1@2 3@2 We present quantum scattering calculations for the Cl]HCl]ClH]Cl reaction in which we include the three electronic states that correlate asymptotically to the ground state of Cl(2P)]HCl(X 1&`).The potential surfaces and couplings are taken from the recent work of Maierle et al. J. Chem. Soc. Faraday T rans. 1997 93 709. They are based on extensive ab initio calculations for geometries in the vicinity of the lowest energy saddle-point and on an electrostatic expansion (plus empirical dispersion and repulsion) for long range geometries including the van der Waals wells. Spin»orbit coupling has been included using a spin»orbit coupling parameter j that is assumed to be independent of nuclear geometry and Coriolis interactions are incorporated accurately. The scattering calculations use a hyperspherical coordinate coupled channel method in full dimensionality.A J-shifting approximation is employed to convert cumulative reaction probabilities for total angular momentum quantum number J\1/2 into state selected and thermal rate coefficients. Two issues have been studied (i) the in—uence of the magnitude of j on the –ne-structure resolved cumulative probabilities and rate coefficients (we consider j values that vary from 0 to ^100% of the true Cl value) and (ii) the transition state resonance spectrum and its variation with j and with other parameters in the calculations. A surprising result is the existence of a range of j where the cumulative probability for the 2P 2P state of Cl is larger than that for the 2P state even though is disfavoured by statistical factors and only reacts via nonadiabatic coupling.This result which is not connected with resonance formation may arise from coherent mixing of the Xj\1/2 components of the 2P and 2P states in the van der Waals regions. The 2P state dominates for values of j between the statistical and adiabatic limits when mixing converts 2P into a state that is for linear geometries predominantly 2& near the barrier. We –nd two signi–cant resonances for total energies below 0.7 eV. They are associated with two quanta of asymmetric stretch excitation of the transition state and with zero or one quanta of bend excitation. These resonances are most prominent (i.e. narrowest) in the adiabatic limit of large o j o . For o j oB0 the resonances are largely washed out due to strong mixing between attractive –ne-structure states that support the resonances and repulsive ones that produce decay.139 140 Quantum scattering studies of spin»orbit eÜects Cl(2P)]HCl]ClH]Cl(2P) I Introduction One of the most important recent developments in the quantum theory of gas phase chemical reactions is that it has become possible to carry out converged threedimensional scattering calculations for reactions where two or more potential energy surfaces are coupled together during reaction.1h6 Such reactions are extremely common in gas phase kinetics when one or both of the reagent species is a radical and in fact all reactions that have previously been studied using single surface dynamics methods (e.g. H]H2 F]H2 O]H2 Cl]H2 Cl]HCl) involve multiple potential surfaces in some sense.Most of these reactions have several potential surfaces which are asymptotically degenerate (i.e. degenerate in the reagents and/or products). However these surfaces split when the reagents approach giving rise to surfaces with diÜerent barriers. In this situation it is usual to ignore surfaces other than the one with the lowest barrier height ; however the validity of this approximation is generally not known. A common situation that arises for these reactions is that the surface with the deepest van der Waals wells is not the one with the lowest barrier height. This means there is always a crossing between diÜerent potential curves as one moves along the reaction path which provides an opportunity for signi–cant nonadiabatic coupling.Complicating this picture is the presence of spin»orbit coupling which partially lifts the asymptotic degeneracy providing additional mechanisms for nonadiabatic coupling. There are several levels of sophistication when performing nonadiabatic quantum dynamics calculations on reactions with asymptotically degenerate potential surfaces. The most rigorous approach uses basis functions in which the electronic orbital and spin angular momenta of the separated reagents and products are explicitly included along with coupling to the orbital and rotational angular momenta of the nuclei. It is this approach that we have adopted (with a few approximations) in our recent work.3,6 It leads straightforwardly to the inclusion of Coriolis coupling between the basis states and to the incorporation of electrostatic nonadiabatic and spin»orbit coupling.A simpler approach is to ignore the electronic angular momenta and thereby regard the electronic degrees of freedom as internal variables without vector properties. This is a commonly used level of treatment ;2,4 it still allows for the inclusion of nonadiabatic and some aspects of spin»orbit coupling but the electronic part of the Coriolis coupling is not included. Further simpli–cations involve the use of rotational sudden approximations or linear models where the nuclear rotational coupling is missing and important aspects of the nonadiabatic coupling are lost. This level of treatment was common in quantum scattering calculations done in the late 1970s (reviewed in ref.1). The high level of sophistication in our treatment of electronic/nuclear coupling allows us to study nonadiabatic reactions in ways that have not previously been considered. In the present paper we present the results of quantum scattering calculations that explore two new issues namely (i) the in—uence of the magnitude of the spin»orbit coupling parameter j on the –ne-structure resolved and cumulative reaction probabilities and rate coefficients ; and (ii) the in—uence of j and the nonadiabatic coupling on the transition state resonance spectrum. To study these issues we have performed calculations for the reaction including the three potential surfaces 1 2A@ 22A@ and 1 2AA which correlate to the ground states of the reagents and products.Cl]HCl is a simple hydrogen transfer reaction that serves as a canonical model both for heavy»light»heavy atom reactions and for the reactions of halogen atoms with closed shell molecules. We have chosen this reaction in part because we and others have studied its single surface dynamics thoroughly in the past using quantum scattering methods,7,8 and because it is one of the few reactions for which there are global diabatic potential surfaces and couplings that are 141 G. C. Schatz et al. valid both near the lowest energy saddle-point and in the long range van der Waals regions. Cl]HCl is also of interest because transition state photodetachment measurements9 have been performed and the results are suggestive of resonance formation. Our dynamics calculations are based on a coupled-channel quantum scattering method in hyperspherical coordinates introduced in ref.3. This follows the scattering formalism of Rebentrost and Lester,10 which explicitly includes all nuclear and electronic angular momenta and their vector couplings. The electronic state expansion uses a diabatic basis so that geometric phase eÜects associated with the conical intersection are automatically included. The diabatic potentials and couplings that we use are from ref. 6. These were developed using a new approach in which the long range potentials (van der Waals regions) are represented by an electrostatic expansion11 while the short range potentials (barrier regions) are derived from ab initio electronic structure calculations.The electrostatic expansion is developed in a diabatic representation and the coupling terms in this representation are assumed to be valid even at short range thereby providing the couplings needed to construct diabatic surfaces at short range using the adiabatic data that come from the ab initio calculations. These diabatic surfaces do not include spin»orbit coupling but it is not difficult to add this with the assumption that j is independent of nuclear geometry. Justi–cation for this assumption is provided by comparison with recent relativistic ab initio calculations which show a shift in the barrier height due to spin»orbit eÜects,12 that is similar to our simple treatment. In the present study we have performed calculations for values of the spin»orbit wavenumber parameter ranging from 0 to ^100% of the true Cl atom value ([588 cm~1) so as to determine how the dynamics change with j.We already know3,6 that the lowest barrier height (that for the surface we denote the 2& diabat) increases as j becomes more negative because the spin»orbit Hamiltonian (for negative j) preferentially stabilizes the asymptotic 2P3@2 state relative to the 2& barrier where there is a partial quenching of the spin»orbit eÜect. A point that will be emphasized in the present 2P paper is the way in which spin»orbit in—uences the reactivity of 2P relative to 1@2 as this provides a direct indication of the importance of nonadiabatic dynamics which 3@2 has long been of interest in experimental studies of halogen atom reactions.13 Whilst varying j is an arti–cial procedure we will show that it helps us to separate the spin» orbit contribution from other sources of nonadiabaticity thus clarifying the role of spin»orbit in determining the branching between product –ne-structure states.Also we note that many other atoms have spin»orbit wavenumber parameters that are in the range we consider [F([265 cm~1) O([80 cm~1) C(13 cm~1) Na(11.5 cm~1) and K(38.5 cm~1)] so the present calculations can be used to initiate studies of other reactions. Another point of interest is the ClHCl transition state resonance spectrum. Past work using single surfaces7,8,14 has demonstrated the existence of one important transition state resonance feature which corresponds to transition state quantum numbers (0 0 2) where (v1 v2 v3) stands for the (symmetric stretch bend antisymmetric stretch) of the ClHCl intermediate.The corresponding (0 0 0) state is not sufficiently stable to exhibit well de–ned (narrow) features which means that only states with asymmetric stretch excitation are stabilized sufficiently to support narrow resonances. However what we do not know is whether the presence of multiple coupled potential surfaces will stabilize or destabilize the transition state resonances. We now summarize the rest of this paper. In Section 2 we brie—y give details of the global diabatic surfaces and couplings that were developed in ref. 6. Section 3 describes the quantum scattering method and the numerical parameters for the coupled channel calculations.The dynamics calculations are presented and discussed in Section 4 whilst Section 5 contains our conclusions. 142 Quantum scattering studies of spin»orbit eÜects 2 Diabatic potential surfaces and couplings Fig. 1 presents a schematic drawing of the potential curves along the reaction path (for linear geometries) for ClHCl with (a) showing what occurs for j\0 and (b) including spin»orbit eÜects. The reaction gives rise to three doublet adiabatic potentials namely 1 2A@ 2 2A@ and 1 2AA (in C symmetry) or 2& and 2% (for linear geometries) all of which s correlate to the ground state of Cl(2P)]HCl in the reagents and products. Fig. 1(a) shows that the 2& curve has the lower barrier for hydrogen atom transfer. This corresponds to the singly occupied p-orbital of the reagent Cl atom pointing directly towards the H atom of HCl.In contrast the singly occupied p-orbital for the 2% curve is perpendicular to the ClHCl axis so the barrier is much higher (0.7 eV versus 0.4 eV for the scaled surfaces of ref. 6). This orientation of the p-orbital also gives rise to a more attractive long range potential due to stronger electrostatic interactions between the doubly occupied p-orbital pointing towards HCl and the positively polarized hydrogen atom in HCl. As a result the 2& and 2% curves cross giving rise to a conical intersection between 1A@ and 2A@ (hereafter we drop the spin multiplicity label). correlates with 2P Fig. 1(b) shows how spin»orbit coupling changes the curves. For the Cl atom in the reagents or products the 2P1@2»2P3@2 splitting is 882 cm~1 which is about 25% of the 2& barrier height and about twice the van der Waals well depth.The asymptotic energy Cl(2P of the state is lowered by spin»orbit interaction and since spin»orbit has little 3@2) eÜect on the 2& potential near the barrier top the overall barrier for the 2& curve is higher (by approximately 33% of the atomic splitting) than it would be in the absence of 2& 2% spin»orbit. Note that the curves labelled 2P and correlate with whilst 1@2 3@2 3@2 2%1@2 1@2 . Because of this one might expect the reaction probability Fig. 1 Schematic pro–les of the potential surfaces for linear ClHCl along the reaction path joining reagents and products. (a) Non-relativistic pro–les (b) relativistic pro–les that include the spin» orbit interaction.143 G. C. Schatz et al. associated with Cl(2P1@2) to be much smaller than that for Cl(2P3@2). However these adiabatic correlations can in some situations be misleading as we will see in Section 4. The global diabatic & and % potentials and couplings that we use are described in rHCl\5.0 a0 rHCl{\2.4 a0 . The saddle-point occurs at rHCl\ detail in ref. 6 so here we give just a few key features. The diabats are as mentioned in the Introduction based on ab initio calculations for geometries near the lowest barrier and on an electrostatic expansion at long range with the switch between the two r occurring near HCl\4.3 a0 independent of rHCl{ (note however that the diabats are invariant to interchanging the two Cl atoms).This region is close to the bottom of the barrier to reaction with the van der Waals wells at larger distances. The lowest energy of the conical intersection occurs somewhat inside the switch region at rHCl\3.4 a0 rHCl{\2.6 a0 for linear geometries. The van der Waals minima are located at larger distances approximately rHCl{\2.879 a0 on the & diabat and at rHCl\rHCl{\2.953 a0 on the % diabat. Note that the saddle-point geometry is bent on the & diabat (an internal bond angle of 152°) but is linear on the % diabat. The electronic Hamiltonian is represented in a diabatic basis that is de–ned using a set of p-orbitals on the reacting Cl atom. The explicit form for the Hamiltonian matrix is p p p q n z x y t t V00]2 5 V20 1 5 61@2 V21 0 pz t t H 0 p (1) el\t t 1t t V00[1 5 V20]1 5 61@2 V22 5 61@2 V21t tx t t t t 0 0 V00[1 5 V20[1 5 61@2 V22 py s p where the V are coefficients of a Legendre expansion of the electrostatic potential in renormalized spherical harmonics which are functions of the polar angles ha /a that lm V locate the orientation of the singly occupied p-orbital.V and V00\(V&]2V%)/3 and 20 V20\5(V&[V%)/3. The are obtained from the 00 & and % diabats using the formulae V interaction terms V and are taken from the electrostatic potentials given in ref. 11. 21 22 They vanish for linear geometries and drop oÜ as R~4 where R is the Jacobi distance from the HCl centre of mass to Cl.3 Quantum reactive scattering calculations 3.1 Method The quantum scattering method we use is the same as that in ref. 6 which is very similar to the technique described by one of us3 in an earlier study of Cl]HCl using multiple potential surfaces. Here we describe the method brie—y so that notation can be introduced to indicate the calculations we have done and for our discussion of the results. We use the notation of Rebentrost and Lester10 wherein the electronic orbital angular momentum vector of the atom A is denoted L the electron spin angular momentum is denoted S the nuclear rotational angular momentum of the diatomic BC is denoted N and the nuclear orbital angular momentum of A relative to BC is denoted l. The electronic total angular momentum is denoted j and the electronic plus nuclear total angular momentum is denoted J so that j\L]S and J\j]N]l.The corresponding angular momentum quantum numbers are denoted L S N l j J. In the present application L \1 and S\1/2 in the pure precession limit so that j\1/2 or 3/2. 144 Quantum scattering studies of spin»orbit eÜects The allowed values for the remaining quantum numbers are N\0 1 2 . . . l\0 1 2 . . . and J\1/2 3/2 5/2 . . . (or sometimes J\3/2 5/2 7/2 . . .). Body-–xed projection quantum numbers associated with N j and J are XN Xj and X respectively. Note that X\XN]Xj and the body-–xed z-axis is chosen to be along the Jacobi vector R from the center of mass of the diatom to the atom. o jXjT are related In terms of these quantum numbers the body-–xed electronic states to the spin and orbital parts of the electronic wavefunctions by (2) o jXjT\ ; o L KT o SRTSL KSRo j)jT (3) (4) and (5) H [1 to H to Table 2j j\^1/2.3.2 Coupled channel expansion K R jT and the following equations as they have –xed values. We use where Sl1m1l2m2 o l3m3T is a Clebsch»Gordan coefficient. The labels L \1 S\1/2 have been omitted from o jX the states in eqn. (2) to represent the electronic Hamiltonian and as a starting point for the coupled channel expansion. If we now assume that R is mass-scaled,15 and de–ne r to be the mass-scaled diatom internuclear vector then the Hamiltonian is given by H\P2/2k]l2/2kR2]p2/2k]N2/2kr2]Hel]Hso where k is the scaled reduced mass,15 P and p are the radial momenta associated with the distances R and r respectively H is the nonrelativistic electronic Hamiltonian and H el is the spin»orbit Hamiltonian.The Hamiltonian in eqn. (3) neglects mass- so polarization terms in the electronic Hamiltonian,10 which are not likely to be important for the low energy processes we are considering. We next replace l by J[j[N in the centrifugal term in eqn. (3) which gives l2/2kR2\(J2]j2]N2)/2kR2[(2J … j]2J … N[2N … j)/2kR2 The cross terms in eqn. (4) produce three types of Coriolis coupling orbital»electronic orbital»rotational and electronic»rotational. We evaluate all these terms accurately in the coupled-channel expansion given below. The electronic Hamiltonian H in eqn. (3) is de–ned using eqn. (1). When expressed el o jX in terms of the basis functions jT of eqn.(2) we obtain the results in Table 1 (which include a spin»orbit energy adjustment discussed below). Note that in this representa- V V only appear in oÜ-diagonal terms but tion V appears both in the diago- 21 22 nal and oÜ-diagonal terms. 20 The spin»orbit Hamiltonian Hso is taken in the usual form so\jL … S \1 2 j( j2[L2[S2) 2P1@2»2P3@2 states is 0.109 eV which jT basis set the matrix elements of eqn. (5) are Eso\12 j[j( j]1)[L (L ]1)[S(S]1)] along the diagonal of where the spin»orbit parameter j is assumed to be constant and independent of the internuclear distances. The atomic splitting of the means that j is [0.073 eV. In the o jX easily evaluated giving the matrix and zero for all oÜ-diagonal matrix elements.It is convenient to add E 2P so so that the state has zero energy. This makes the contributions of 3@2 so 1 non-zero only for the diagonal terms with j\1/2 X We begin by coupling the electronic states in eqn. (2) with angular eigenstates associated with the rotational and orbital motion of the nuclei. To do this we couple the vectors j and N to form a resultant vector F where F\j]N. Note that the z-projection quantum number of F along R is X. The resulting electron»nuclear wavefunction associ-j@ Xj{ 3/2 3/2 3/2 1/2 3/2 [1/2 3/2 [3/2 1/2 1/2 1/2 [1/2 Only the upper portion is shown because the matrix is symmetrical. ated with F and X is given by BC molecule is oNX where NT is a rotational state ket. We can now write down the coupled channel expansion for the wavefunction associated with each partial wave J and space-–xed z-projection quantum number M v{N{j{F{X{ where D is a rotation matrix that depends on the polar angles h / associated with R U is an eigenfunction of the BC rovibrational Hamiltonian and g is an R-dependent expansion coefficient which is determined numerically by solving a set of coupled Schroé dinger equations.In the present case the Schroé dinger equation for the isolated where l(r) is the diatomic internuclear potential and v\0 1 2 . . . . We next substitute eqn. (7) into the Schroé dinger equation and use the Hamiltonian in eqn. (3) to obtain the coupled channel equations where g is the matrix of expansion coefficients and U is a matrix that can be written in the form XN Xj WvNjFX JM \ ; DM J X{(/ h 0)Uv{N{(r) oN@j@F@X@Tgv JvNjF {N{j{FX{X{(R) 2k] N(N 2k ] r2 1)+2 (U)tt{\[2 + k 2 (E[evN)dtt{](Uel)tt{](Uso)tt{](Uco)tt{ 145 G.C. Schatz et al. Table 1 Electronic matrix elements j Xj 1/2 1/2 3/2 1/2 3/2 3/2 1/2 [1/2 3/2 [3/2 3/2 [1/2 0 [ J2 V 5 [ J2 5 2 5 V22 1 5 V21 22 V21 V00[ 1 5 V20 0 V [ J3 V 5 [ J2 V 5 [ J2 V 5 21 20 22 00] 1 5 V20 J2 J2 5 [ J3 5 5 V00] 1 5 V20 V21 V21 V20 V00[ 1 5 V20 [ 2 5 V22 1 5 V21 0 V00[ 3 2 j V00[ 3 2 j (6) o NjFXT\ ; oNXNT o jXjTSNXN jXj oFXT (7) (8) ]l(r)DUvN(r)\evNUvN(r) Cp2 (9) dR2\Ug d2g (10) 146 Quantum scattering studies of spin»orbit eÜects Here we use the collective index ì t œ to denote the set of quantum numbers (vNjFX).In eqn. (10) the –rst term contains the total energy E the second term includes the electro- H static potential couplings induced by the diÜerence between and l(r) the third describes spin»orbit eÜects and the fourth describes Coriolis eÜects. Speci–c expressions el for the second third and fourth terms are (a) for the electrostatic terms (11) tt{\2 + k 2 SDM J X o SUvN o SNjFXoHel[l(r) oN@j@F@X@T oUv{N{T oDM J X{T (Uel) (b) for the spin»orbit terms (12) tt{\ kj +2 [ j( j]1)[L (L ]1)[S(S]1)]dtt{ (13) (Uco) where mB(J X)\[J(J]1)[X(X^1)]1@2. Note that in deriving eqn.(13) the Coriolis Hamiltonian in eqn. (3) has been simpli–ed by use of F and the angular momentum eigenfunctions in eqn. (6). Parity decoupling can be introduced into the coupled channel equations by generalizing the theory for single surface scattering.15 In particular parity adapted rotation matrices are used in eqn. (7). This leads to a factorization of the problem into two smaller problems each with an equal number of channels (for half-odd integer J values). In the present calculations we have approximated the parity decoupling by using basis functions of the form (1/J2)( o jXjT^o j[XjT). When the matrix in Table 1 is reexpressed in terms of these functions it is necessary to neglect terms involving V (a relatively small term) in order to decouple into two equivalent parity sub-blocks.One of 22 the resulting sub-blocks is identical to the elements in Table 1 that refer only to positive X values and the other to negative X values. Each sub-block gives the same cumulative j j probabilities so only one has been explicitly considered in our calculations. 3.3 Electrostatic coupling matrix To relate the matrix elements in eqn. (11) to those in Table 1 we substitute eqn. (6) into o jX eqn. (11) thereby converting the matrix elements to the basis jT. Assuming that the primed and unprimed variables in the new eqn. (11) refer to the same arrangement channel index a then the Wigner rotation matrices are orthogonal and eqn. (11) reduces to (Uel) (Uso) (c) for the Coriolis terms tt{\ J(J]1)]F R (F 2 ]1)[2X2 dtt{ [ 2 R2 [m~(J X@)m~(F X@)dX X{~1]m`(J X@)m`(F X@)dX X{`1]dFF{ dNN{ djj{ dvv{ tt{\2 + k 2 ; ; SNXNjXj oFXTSN@XN{ j@Xj{ oF@X@T (14) XN Xj XN{Xj{ ]SUvN o SNXN o SjXj oHel[l(r) o j@Xj{T oN@XN{T oUv{N{T If the primed and unprimed variables refer to diÜerent arrangement channels then the overlap of rotation matrices yields d-matrices that should be inserted into the middle of eqn.(14) as described in ref. 3. 147 G. C. Schatz et al. 3.4 Reactive scattering and reaction probabilities The coupled channel equations given by eqn. (9) are appropriate for the description of nonreactive atom»diatom scattering. They must be modi–ed for reactive collisions because the use of isolated BC rovibrational eigenstates for expanding the wavefunction is inappropriate for product AB]C states.To treat reactive problems we introduce Delvesœ hyperspherical coordinates,16 following the theory given previously for single surface reactions by one of us17 and by Koizumi and Schatz.18 The generalization to multisurface problems is identical to that described in ref. 3 so we omit the details. The –nal outcome of the calculations is the scattering matrix S which is labelled by initial and –nal values of the quantum numbers v N j F X and by the arrangement channel index a. The partial-wave cumulative reaction probability Pcum J which is needed to calculate rate coefficients is given by (15) Pcum J (E)\; PJa ta{t{(E) tt{ where the sums are over all open states at energy E and the arrangement channel indices a and a@ are chosen to be appropriate for reaction.The partial-wave state-to-state reaction probabilities are related to the S matrix elements by (16) PJa ta{t{(E)\o SJa ta{t{(E) o2 We also de–ne cumulative probabilities Pcum J (E; j,j@) that are labeled by the initial and –nal values of the electronic quantum number ( j and j@) by restricting the sum in eqn. (15) appropriately. 3.5 Thermal rate coefficients The multisurface thermal rate coefficient k(T ) at temperature T is related to the cumulative probability by the standard formula 1 (17) cum(E)exp([E/kBT ) dE k(T )\hQ P=P reag nu (T )Qreag el (T ) 0 Qreag nu (T ) is the reagent nuclear partition function per unit volume (describing Qreag el (T ) is the corresponding state of where translational vibrational and rotational motions) and reagent electronic partition function.We measure energies relative to the 2P3@2 Cl and recall that j\3/2 has a degeneracy of 4 while j\1/2 has a degeneracy of 2 then j\3/2)\4 and that are labelled by the initial and –nal values of the electronic quantum numbers ( j and Qreag el (T )\4]2 exp(3j/2kBT ). We also de–ne state selected rate coefficients k(T ; j j@) j@) using the cumulative probabilities Pcum J (E; j j@) from the previous section and replacing Qreag el (T ) by the electronic partition function Qreag el (T ; j) appropriate for the initial Qreag el (T ; reag el (T ; Q quantum number j in eqn. (17) i.e. j\1/2)\ 2 exp(3j/2kBT ). Pcum(E) in eqn.(17) is a weighted sum over partial waves The cumulative probability of Pcum J (E) (18) Pcum(E)\; (2J]1)Pcum J (E) J J[1 by means of the J-shifting approximation,19 In the present application we have used the J\1 reaction probability to determine the reaction probabilities for 2 2 (19) Pcum J (E)BPcum J/1@2(E[Erot J ]Erot J/1@2) J\1 2 3 2 5 2 . . . 148 Quantum scattering studies of spin»orbit eÜects Erot J is the rotational-electronic energy of the ClHClt complex for angular momen- where tum quantum number J. We approximated Erot J by Be t J(J]1) where Be t\1.4]10~5 eV is the rotor constant associated with the saddle-point on the & diabat (this value is actually that for the BCMR potential of ref.20). Other details of the rate coefficient calculations including the inclusion of bend excited states (having oXN o[0) and the accuracy of J-shifting (typically 20%) have been discussed previously.6,19 3.6 Basis sets and numerical parameters Most of our scattering computations were done using a basis of 244 states consisting of rotational states N\0»13 for vibrational state v\0 and N\0»7 for v\1 for each of the two Cl]HCl arrangement channels. We also present results for a 172 state basis set (Nmax\9 for v\0 max\5 ( for v\1) and for a 292 state basis set Nmax\13 for from V (which was 00 N v\0 and Nmax\11 for v\1). These three basis sets will be used to estimate the degree of convergence of the results. In addition we have examined the eÜect of using two diÜerent reference potentials to generate the underlying hyperspherical basis functions.All of the results that we present below use the average electronic potential V eqn. (1) as the reference potential but we also did calculations using employed in our previous work6 with smaller basis sets). The scattering results are gen- & (Pcum J is within 20% for nonresonant scattering for E\0.60 V -based results for studying the resonances at higher E. erally in good agreement eV); however one diÜerence is that the V -based results show additional resonances & close to the reaction threshold (i.e. for E well below the energy range where the resonances to be discussed later occur). These additional resonances only appear when a large number of rotational states are included in the v\1 basis (NmaxP7) and only V N when V is included in the Hamiltonian.For smaller values the 21 &-based results do not exhibit additional resonances and are then similar to the V max results. An analysis of the dependence of the hyperspherical adiabatic states on hyperradius suggests that the 00 additional resonances are due to spurious eigenvalues of the Hamiltonian matrix that arise from overcompleteness of the basis set. We therefore conclude that these additional resonances obtained from the V -based calculations are not physically signi–cant. As a & result we only use below the 00 The complete set of 2P electronic states appropriate for J\1/2 was used in all m calculations although it was only necessary to consider one of the two identical parity components.The cumulative probabilities are multiplied by two when calculating rate coefficients in order to include the other parity component. The masses used in the computations are H\1.008 u and mCl\34.969 u. 4 Results and discussion 4.1 Cumulative reaction probabilities P Fig. 2 presents the cumulative reaction probability cum J (E) for J\1/2 as a function of E from our multisurface calculations comparing results from the 292 and 244 basis sets. P Also plotted are the –ne-structure state selected cumulative probabilities cum J (E; j j@) for J\1/2 with the quantum numbers j and j@ chosen to be j\j@\3/2 j\j@\1/2 and j\3/2 j@\1/2. Note that microscopic reversibility requires the j\1/2 j@\3/2 cumulative probability to equal that for j\3/2 j@\1/2.Pcum J (E) and j\j@\3/2 cumulative probabilities show that the eÜective reaction The threshold is near E\0.37 eV. At higher E there is a gradual rise in the probabilities along with broader peaks near E\0.58 eV and 0.61 eV. These peaks can be assigned to resonances associated with the transition state region of the & diabat as discussed in Section 4.3. The cumulative probabilities for 244 and 292 states agree within a few 149 G. C. Schatz et al. Fig. 2 Cumulative reaction probability Pcum J (E) (thick solid curve) for J\12 and state-selected cumulative reaction probabilities Pcum J (E; j j@) for J\12 versus total energy E with ( j j@)\(1/2 1/2) (dashed) (3/2 1/2) (dotted) (3/2 3/2) (solid) and a linear scaling factor of s\1.0.(a) Results for 244 states (b) results for 292 states. percent for E\0.55 eV and are still in good agreement for E\0.65 eV. This indicates good convergence with respect to basis set size. The –ne-structure-resolved cumulative probabilities for j\j@\1/2 and j\3/2 j@\1/2 are always much smaller than the j\j@\3/2 one and the resonance structure is less noticeable. The small values of the j\j@\1/2 and j\3/2 j@\1/2 cumulative probabilities relative to j\j@\3/2 is the expected behaviour if the electronic states evolve adiabatically between the reagents and products since the reactive —ux connecting j\j@\3/2 can cross the lowest barrier by a purely adiabatic route. 4.2 Variation of cumulative reaction probabilities with spinñorbit parameter j\sjCl and jCl is the correct spin»orbit parameter for Cl([0.073 eV).The state is 1@2 2P 2P to have zero energy). The 3@2 3@2 Fig. 3 presents results analogous to those in Fig. 2 for four diÜerent values of the spin» orbit parameter j. From here on we express j in terms of a linear scaling parameter s where values chosen for s are [0.5 0.0 0.1 and 1.0 where for negative s the 2P lower in energy than (however we still de–ne results in Fig. 3 show two important trends. First as s increases the cumulative probabilities including both direct and resonance features shift linearly to higher E with beingB[1 the energy shift 2j. This is the previously studied3,6 energy shift of the barrier height of the & diabat relative to the 2P state as described in Section 2.Secondly 3@2 for EB0.40 eV the 1/2]1/2 cumulative probability is larger than 3/2]1/2 and 3/2]3/2 for s\[0.5 then it is smaller for s\0.0 it is larger than the others for 150 Quantum scattering studies of spin»orbit eÜects Fig. 3 Same as Fig. 2 for 244 states except that (a) s\[0.5 (b) s\0.0 (c) s\0.1 (d) s\1.0 s\0.1 and –nally it drops to nearly zero for s\1.0. The 3/2]3/2 cumulative probability does approximately the reverse of 1/2]1/2 being larger for s\1.0 and s\0.0 and smaller for s\[0.5 and 0.1. 3@2 The unusual behaviour in Fig. 3 is further illustrated in Fig. 4 where we plot each cumulative probability as a function of s (with [1OsO1) for E\0.40 eV. The results in Fig. 4 were obtained from 172 state calculations but are not signi–cantly diÜerent from the 244 state results.Fig. 4 shows that the 1/2]1/2 cumulative probability is largest for negative s values with a dip at s\[0.05 then it has a peak between s\0.05 and s\0.1 before dropping to near zero at s\1.0. The 3/2]3/2 cumulative probability starts out small for s\[1.0 then exhibits a sharp peak at s\[0.05 with P a modest dip on either side of the peak and then approaches cum J (E) near s\1.0. In P contrast to the peaks and dips in the state-resolved probabilities cum J (E) decreases smoothly and monotonically with s ; this is expected because of the decrease in barrier height of the & diabat relative to 2P with increasing j. The behaviour in Fig. 3 and 4 can be understood from the way in which the –ne structure states evolve from the asymptotic region to the barrier top of the & diabat.To simplify the analysis let us consider the Hamiltonian matrix in Table 1 for linear geome- V tries where V and are identically zero. In this case the only source of coupling 21 between states with diÜerent ( j 22 Xj) is V20 . Now V20 is proportional to the ìdiÜerence potentialœ V&[V% and thus vanishes asymptotically. This means that the o jXjT states are true eigenstates asymptotically ; in particular there are three states that we need to consider (for one parity sub-block) namely j\3/2 Xj\3/2 and j\3/2 Xj\1/2 and j\1/2 Xj\1/2. (Note that the parity basis functions involve symmetric or antisymmetric linear combination of functions with positive and negative values of X as j 151 G.C. Schatz et al. Fig. 4 Pcum J (E) and Pcum J (E; j j@) for 172 states with J\12 E\0.40 eV versus the linear scaling parameter s explained previously so the sign of X is not meaningful in this list.) Of these three j states we note from the structure of Table 1 that only two states are coupled by V20 namely the two with Xj\1/2. This means that the Xj\3/2 state is uncoupled to the other states except for Coriolis coupling which is weak for J\1/2 . The Xj\3/2 state correlates with %3@2 and it thus has the attractive van der Waals wells at large distances and the high reaction barrier at short range shown in Fig. 1. This state should therefore have low reactivity. j\1/2 correlate with & and % at short range. The asymptotic states having X 1@2 1@2 2P 2P & Whether this evolution from %1@2 respectively is adiabatic and to and 3@2 1@2 1@2 or not mainly depends on the magnitude of the dimensionless parameter f de–ned by J2 o V20 o (20) f\ spin»orbit energy separation \ coupling term in Table 1 5 3 2 o j o rHCl at its equilibrium value of 2.40 a0) whilst Fig.5(b) displays the three In Fig. 5(a) we show f as a function of the (unscaled) Jacobi coordinate R (for linear geometry with adiabatic potential functions for s\1.0 which have Xj\1/2 and 3/2. Fig. 5(b) is similar to Fig. 1 at least for geometries outside the conical intersection i.e. R[5.9 a0 . It is evident from Fig. 5(a) that f\1 for all R beyond the conical intersection. This suggests that adiabatic behaviour should prevail as indeed is observed.If s is decreased to 0.1 then the values along the ordinate of Fig. 5 are multiplied by 10 and f[1 for R values in the van der Waals region suggesting signi–cant nonadiabatic interaction occurs there. Still smaller values of s lead to even larger values of f and in the limit s\0 the states are completely mixed i.e. we obtain statistical behaviour in this limit with the 2 1 ratio of j\3/2 1/2 degeneracies giving ratios of 4 2 1 for the 3/2]3/2 3/2]1/2 1/2]1/2 cumulative probabilities. 152 Quantum scattering studies of spin»orbit eÜects 0 Fig. 5 (a) Adiabaticity parameter f versus Jacobi coordinate R for rHCl\2.40 a0 and linear ClHCl. (b) Adiabats for s\1.0 associated with the electronic Hamiltonian for Xj\1/2 [2&1@2 2% (solid curve) and j\3/2 [2%3@2 (dashed)] as a function of the Jacobi (dotted)] and for X 1@2 coordinate R for r\2.40 a and linear ClHCl.The adiabats have been labelled in the same way as Fig. 1. To test these suggestions concerning the correlation between the asymptotic states and the states closer in we have used our hyperspherical scattering program to examine the scattering wavefunction for a variety of asymptotic states for both linear and nonlinear geometries. When the asymptotic state is 2P1@2 we –nd the scattering wavefunction shows substantial mixing between j\3/2 and j\1/2 (the precise details depend on the value of j) when the inner repulsive wall of the van der Waals well has been reached. This is consistent with the behaviour in Fig.5. Our analysis using eqn. (20) of the importance of nonadiabatic coupling between and %1@2 does not depend on the sign of s. However it is important to note that for 1@2 j\1/2 state has % character in the van der Waals region closer in whereas for s[0 the lower energy Xj\1/2 state is & both at long &1@2 s\0 the lower energy X and &1@2 and short range. This means that the possibility of nonadiabatic dynamics is more 1@2 important for s\0 than for s[0. Fig. 4 con–rms this in the sense that the 1/2]1/2 cumulative probability contributes less (in per cent) to Pcum J (E) for s\[1.0 than does 3/2]3/2 at s\1.0. Perhaps the most surprising results in Fig. 3 and 4 are for s just slightly above or slightly below zero as here we see behaviour that is neither adiabatic nor statistical and which does not interpolate monotonically between the two limits.This behaviour is diÜerent from what was observed in ref. 3 where surfaces without a conical intersection and van der Waals wells gave cumulative probabilities that evolved monotonically from the statistical to the adiabatic limit with 3/2]3/2 always being the dominant probability (for s[0). This suggests that the results in Fig. 3 and 4 for sB0 might arise from quantum coherence in the interaction between the two Xj\1/2 states. What we mean 153 G. C. Schatz et al. by this is that there is a range of s values where f is large enough to cause substantial ([50%) population transfer between the two states during the approach phase of the collision thereby favouring the nonadiabatic process.Larger values of o s o give adiabatic behaviour whilst smaller values leads to continuous cycling of —ux between the two states leading to statistical behavior. This type of behaviour was not observed in ref. 3 because there were no van der Waals wells so the magnitude of the diÜerence potential o V20 o was much smaller and the adiabatic limit remained dominant down to smaller o s o values. Population transfer could still occur for small o s o but the coupling arises from the diÜerence potential in the more repulsive region near the barrier and the transition between adiabatic and statistical limits occurs much more rapidly. 4.3 Transition state resonances We now consider the resonance structures in Fig. 2 and 3.We begin by recalling from the Introduction that past (single surface) studies of ClHCl using the BCMR surface20 have observed one signi–cant resonance feature over the energy range considered,8 namely a peak near E\0.642 eV with a full width at half maximum of about 0.004 eV. This resonance has been assigned the saddle-point quantum numbers (l1 l2 l3)\(0 0 2). A similar resonance is seen in exact quantum calculations for linear ClHCl as well as resonances at higher energies that correspond to higher excitations of the l mode; the l3\0 are not sufficiently stable to show narrow struc- lower energy resonances e.g. for 3 ture.20 Note also that there are no progressions in either symmetric stretch or bend quantum numbers associated with the (0 0 2) resonance on the BCMR surface.7 The results in Fig.2(b) for s\1.0 are similar to what we have just described for the BCMR surface except that the resonance occurs at E\0.58 eV rather than at 0.642 eV and there is an excited state at 0.61 eV. To understand these results we have performed additional 244 state calculations (not shown) in which the interaction potential V is set to zero everywhere but no other parameters are changed. We –nd that the resulting 21 cumulative probabilities are very similar to those in Fig. 2(b). This suggests that the E\0.58 eV resonance is not associated with the coupled multisurface nature of the dynamics and is therefore the analogue of the (0 0 2) resonance seen in BCMR calculations. It occurs at a lower energy than for BCMR in spite of the somewhat higher barrier height (0.393 eV for the & diabat versus 0.371 eV for BCMR) because the resonance is probably delocalized over a broader region of space which includes the van der Waals wells.Further support for this idea can be gained by noticing that only the j\j@\3/2 cumulative probability is strongly perturbed by this resonance for s\1.0. For s\[0.5 it is the j\j@\1/2 cumulative probability that is strongly perturbed which is consistent with the adiabatic correlations that occur when the 2P state is of lower energy as discussed near the end of Section 4.2. 1@2 If this assignment of the resonance is correct then the higher energy resonance can be assigned as a bend excited state of the & diabat transition state based on the behavior of cumulative probabilities where the value of the initial and –nal values of X are N speci–ed.These cumulative probabilities show that for s\1.0 the resonance at E\0.58 eV is associated with XN\XN{\0 as one would expect for the ground bend state while that at E\0.61 eV is exclusively XN\XN{\^1 which corresponds to one quantum of vibrational angular momentum for the –rst excited bend state. Note that N\^1 are allowed in our calculation whereas they XN\^1 states are included in the J\1/2 basis set along with XN\0 and XN\^2 so the excited bend states with X would not be present in single surface calculations where J\0. The energy shift between the ground and –rst excited bend states is thus B0.03 eV which is substantially smaller than the energy shift using the harmonic saddle-point bending energy of 0.09 eV on the & diabat but it is likely that the resonance is considerably delocalized away from the saddle-point region to geometries in the van der Waals wells where the 154 Quantum scattering studies of spin»orbit eÜects eÜective bend energy will be much smaller.For s\[0.5 a similar analysis of the dependence of the cumulative probabilities on X gives similar results namely that the N lowest energy resonance mostly perturbs the XN\0 probabilities and the next one per- N\^1. However the separation between turbs X XN\0 and oXN o\1 is not as clean as for s\1.0. This suggests stronger nonadiabaticity for negative s which is consistent with comparisons of Fig. 3(a) and 3(d) presented in Section 4.2.Another clue comes in comparing the graphs in Fig. 3 (a)»(d). The position of the lowest resonance energy varies with s as if determined by the properties of the & diabatic 2P barrier relative to Eres J (s)\Eres J (0)[12 sjCl\Eres J (0)](0.0363 eV)s. This 3@2 namely relation says that the s\1.0 resonance energy should be 0.055 eV higher than the s\[0.5 one which is what we –nd. Notice also in Fig. 3 that the resonances are quite distinct for s\[0.5 and 1.0 but are barely discernible (i.e. broader) for s\0.0 and 0.1. This broadening of the resonances for small o s o is most likely due to the strong coupling between diÜerent –ne-structure components that occurs in this limit. This coupling mixes together the states that support the resonance (those correlating with the & diabat sumably what produces the broader resonances.near the transition state) with those that produce resonance decay (%1@2) which is pre- 4.4 Rate coefficients Fig. 6 presents an Arrhenius plot for the thermal rate coefficient k(T ) which is derived from Pcum J (E) for J\12 using eqn. (17)»(19). Also shown are normalized rate coefficients (de–ned below) for the three ( j j@) combinations considered in Fig. 2»4. The rate coeffi- Fig. 6 Arrhenius plot of the thermal rate coefficient log k(T ) versus 1/T (thick solid line). Also plotted are the normalized –ne-structure rate coefficients log knorm(T ; j j@) for j\3/2]j@\3/2 (solid) j\3/2]j@\1/2 (dotted) and j\1/2]j@\1/2 (dashed). Note that knorm(T ; j\3/2 j@\1/2)4knorm(T ; j\1/2 j@\3/2).All results are from calculations using 244 states. 155 G. C. Schatz et al. cients are calculated from the 244 state cumulative probabilities shown in Fig. 3 and are for the scale factors s\[0.5 0.0 0.1 and 1.0. Over the temperature range 300»1200 K the s\1.0 rate coefficients agree with those from the 292 state calculations to within 20% so we have not plotted the 292 state results separately. In ref. 6 we compared very similar 244 state rate coefficients for s\1.0 with results from earlier single surface calculations with quasiclassical trajectory calculations and with experimental data so we omit a discussion here. The normalized state-selected rate coefficients are de–ned by (21) Q knorm(T ; j j@)\ Qreag el (T ; j) reag el (T ) k(T ; j j@) where from Section 3.5 (22) P=Pcum(E; j j@)exp([E/kBT ) dE k(T ; j j@)\hQ 1 reag nu (T )Qreag el (T ; j) 0 It follows from the de–nition in eqn.(21) that (23) k(T )\; knorm(T ; j j@) jj{ The relation in eqn. (23) has the advantage that it simpli–es Arrhenius plots since it is easier to see how diÜerent spin states contribute to the total rate coefficient k(T ). Furthermore the normalized rate coefficients for j\3/2]j@\1/2 and j\1/2]j@\3/2 are equal so we only need plot one of them. The k(T ) values in Fig. 6 show typical Arrhenius behaviour with a dependence on s state by which for positive s is easily described using the simple energy shift of the 2P the spin»orbit interaction i.e. 3@2 (24) 2 s o jCl o /kBT B k(T s[0)Bk(T s\0)expA[1 For negative s we need to take into account the fact that 2P is the lower energy state and the barrier relative to this state increases as s decreases.In this case the energy shift 1@2 is twice what it is when 2P is the ground state so eqn. (24) is replaced by 3@2 k(T s\0)Bk(T s\0)exp(s o jCl o /kBT ) (25) The normalized rate coefficients in Fig. 6 show considerable variation with s as expected from our analysis of Fig. 3 and 4 with the 3/2]3/2 result being dominant for s\1.0. All three rate coefficients are comparable in magnitude for s\0.1. The s\0.0 result shows the expected 4 2 1 ratio (see Section 4.2) and 1/2]1/2 is dominant for s\[0.5 (for T \600 K). For s\1.0 at 300 K the normalized 1/2]1/2 rate coefficient is only 0.7% of k(T ) 3/2]1/2 is 3.5% of k(T ) and 3/2]3/2 is 92.3% of k(T ).The corresponding percentages for s\0.1 are 34.1% 22.4% and 21.0% whereas for s\[0.5 they are 41.6% 17.8% and 22.9%. 5 Conclusions Cl(2P This paper has explored two interrelated aspects of the j)]HCl reaction and more generally of open shell atoms colliding with closed shell diatomic molecules. First we have studied the evolution of the asymptotic –ne-structure states towards the barrier region and the way in which this in—uences the –ne-structure resolved reactivity. Our results in Fig. 5 indicate that there are many diÜerent kinds of behaviour which are primarily determined by the magnitude of the van der Waals well depth (which determines the ìdiÜerence potentialœ V that couples the states at large distances) compared 20 156 Quantum scattering studies of spin»orbit eÜects with the magnitude of the spin»orbit coupling constant (which determines the splitting of the asymptotic states).When the spin»orbit constant is negative and the splitting is larger than the van der Waals well depth as happens for ì real œ chlorine atom reactions the dynamics are predominantly adiabatic which means that the reactivity of the 2P state is much smaller (few per cent) than 2P3@2 . Adiabatic dynamics is also recovered for 1@2 large positive values of the spin»orbit constant but in this limit it is the 2P state that has the larger reactivity at least close to the reaction threshold (see Fig. 4). If the 1@2 spin»orbit constant is zero (or much smaller than the van der Waals well depth) then the –ne structure states are strongly mixed and the –ne structure propensities are controlled by simple statistics.This means that 2P higher reactivity. There is an intermediate regime between the statistical and adiabatic 3@2 with its greater degeneracy has limits where nonadiabatic coupling produces ìinvertedœ propensities where for negative spin»orbit parameters the 2P state has the higher reactivity whilst for positive parameters the 2P3@2 state has the higher reactivity. This behaviour is found for both cumula- 1@2 tive probabilities and thermal rate coefficients. Cl(2Pj)]HCl reaction that we studied are transition state The second aspect of the resonances. We found that using coupled potential surfaces does not change the resonance spectrum much compared with the single surface result.In particular we –nd that the two lowest energy resonances correspond to the asymmetric stretch excited state (0 0 2) and the state (0 1 2) having one quantum of bend excitation. 1@2 The connection between the present results and experiment is an important task for 2P future studies. The small reactivity that we –nd when the true chlorine spin»orbit parameter is used has often been observed in chlorine atom reactions ;13 it will be interesting to study reactions that exhibit the more dramatic eÜects that we have predicted. Our results indicate that these eÜects should occur in atoms whose spin»orbit parameters have smaller magnitudes such as are found in the –rst row of the Periodic Table.The experimental observation of transition state resonances of any type has proven to be a major experimental challenge. However ClHCl is a candidate for such observation through photodetachment spectroscopy.9 Recently an improved formalism for calculating photodetachment spectra in ClHCl~ has been presented,21 so an important task for future theoretical work will be the implementation of this theory (e.g. using the wavefunctions generated with the scattering code described in this paper). Before this is done however it will be necessary to check the accuracy of the potential energy surfaces and couplings that we have used. This should be possible soon as Besley and Knowles22 have recently performed ab initio calculations for this reaction that provide both surfaces and couplings of higher quality than have been available previously.This research was supported by the US Department of Energy Office of Basic Energy Sciences Division of Chemical Sciences under Contract No. W-31-109-ENG-38 and by the UK Engineering and Physical Sciences Research Council. We thank Dr A. J. Dobbyn for many important comments. References and M. Baer Chem. Phys. L ett. 1997 265 629; R. Baer D. M. Charutz R. KosloÜ and M. Baer J. Chem. Phys. 1996 105 9141; I. Last M. Gilibert and M. Baer J. Chem. Phys. 1997 107 1451. 00 In Section IV for j read o j o . In the caption to Fig. 3 for ref. 24 read ref. 25. In ref. 1 H. Nakamura Int. Rev. Phys. Chem. 1991 10 123; M. Baer T heory of Chemical Reaction Dynamics ed.M. Baer CRC Press Boca Raton FL 1985 vol. II ch. 4. 2 M. Gilibert and M. Baer J. Phys. Chem. 1994 98 12822; ibid. 1995 99 15748; D. M. Charutz R. Baer 3 G. C. Schatz J. Phys. Chem. 1995 99 7522. In eqn. (2) for ]2N … j read [2N … j. In Table 1 for v v read V00 V20 . 20 25 for 91 5496 read 93 251. 4 S. L. Mielke G. J. Tawa D. G. Truhlar and D. W. Schwenke Chem. Phys. L ett. 1995 234 57; S. L. Mielke D. G. Truhlar and D. W. Schwenke J. Phys. Chem. 1995 99 16210; M. S. Topaler M. D. Hack T. C. Allison Y.-P. Liu S. L. Mielke D. W. Schwenke and D. G. Truhlar J. Chem. Phys. 1997, 157 G. C. Schatz et al. Pcum J (E; 1/2 1/2) ; in Figs. 4(c) and 4(d) for ClCHI read ClHCl. 106 8699; T. C. Allison S. L. Mielke D. W. Schwenke and D. G. Truhlar J. Chem. Soc. Faraday T rans.1997 93 825. 5 J. Qi and J. M. Bowman J. Chem. Phys. 1996 104 7545; A. J. C. Varandas and H. G. Yu Chem. Phys. L ett. 1996 259 336; T. J. Martïç nez Chem. Phys. L ett. 1997 272 139. 6 C. S. Maierle G. C. Schatz M. S. Gordon P. McCabe and J. N. L. Connor J. Chem. Soc. Faraday T rans. 1997 93 709. On p. 712 for c\152° read h\152°; in eqn. (2) for ]2N … j read [2N … j ; below eqn. (11) for substitution of eqn. (1) read substitution of eqn. (10) ; in Fig. 6 for Pcum J (E; 11/2 1/2) read 7 G. C. Schatz D. Sokolovski and J. N. L. Connor in Advances in Molecular V ibrations and Collision Dynamics ed. J. M. Bowman JAI Press Greenwich CT 1994 vol. 2B pp. 1»26. 8 J. N. L. Connor P. McCabe D. Sokolovski and G. C. Schatz Chem. Phys. L ett. 1993 206 119; G.C. Schatz D. Sokolovski and J. N. L. Connor Can. J. Chem. 1994 72 903; D. Sokolovski J. N. L. Connor and G. C. Schatz Chem. Phys. L ett. 1995 238 127; J. Chem. Phys. 1995 103 5979; Chem. Phys. 1996 207 461; N. Rougeau and C. Kubach Chem. Phys. L ett. 1994 228 207; A. B. McCoy Mol. Phys. 1995 85 965; M. J. Cohen A. Willetts and N. C. Handy J. Chem. Phys. 1993 99 5885; A. Lagana` A. Aguilar X. Gimenez and J. M. Lucas in Advances in Molecular V ibrations and Collision Dynamics ed. J. M. Bowman JAI Press Greenwich CT 1994 vol. 2A pp. 183»201. 9 R. B. Metz T. Kitsopoulos A. Weaver and D. M. Neumark J. Chem. Phys. 1988 88 1463; R. B. Metz A. Weaver S. E. Bradforth T. N. Kitsopoulos and D. M. Neumark J. Phys. Chem. 1990 94 1377. 10 F. Rebentrost and W. A. Lester Jr.J. Chem. Phys. 1975 63 3737; 1976 64 3879; 1976 64 4223; 1977 67 3367; F. Rebentrost in T heoretical Chemistry Advances and Perspectives T heory of Scattering Papers in Honor of Henry Eyring ed. D. Henderson Academic New York 1981 vol. 6B pp. 1»77. 11 M.-L. Dubernet and J. M. Hutson J. Phys. Chem. 1994 98 5844. 12 L. Visscher and K. G. Dyall Chem. Phys. L ett. 1995 239 181. 13 D. Husain and R. J. Donovan Adv. Photochem. 1971 8 1; R. J. Donovan and D. Husain Chem. Rev. 1970 70 489; P. J. Dagdigian and M. L. Campbell Chem. Rev. 1987 87 1; P. J. Dagdigian in Selectivity in Chemical Reactions Proceedings of the NATO Advanced Research Workshop Bownesson-Windermere UK 7»11 September 1987 ed. J. C. Whitehead Kluwer Dordrecht 1988 pp. 147» 177; A. Gonzaç lez-Uren8 a and R. Vetter J. Chem. Soc. Faraday T rans. 1995 91 389; M. Alagia N. Balucani P. Casavecchia D. Stranges and G. G. Volpi J. Chem. Soc. Faraday T rans. 1995 91 575. 14 G. C. Schatz J. Phys. Chem. 1990 94 6157. 15 G. C. Schatz and A. Kuppermann J. Chem. Phys. 1976 65 4642; G. C. Schatz L. M. Hubbard P. S. Dardi and W. H. Miller J. Chem. Phys. 1984 81 231. 16 L. M. Delves Nucl. Phys. 1959 9 391; 1960 20 275. 17 G. C. Schatz Chem. Phys. L ett. 1988 150 92. 18 H. Koizumi and G. C. Schatz in Advances in Molecular V ibrations and Collision Dynamics ed. J. M. Bowman and M. A. Ratner JAI Press Greenwich CT 1991 vol. 1A pp. 139»164. 19 Q. Sun J. M. Bowman G. C. Schatz J. R. Sharp and J. N. L. Connor J. Chem. Phys. 1990 92 1677; M. C. Colton and G. C. Schatz Int. J. Chem. Kinet. 1986 18 961. 20 D. K. Bondi J. N. L. Connor J. Manz and J. Roé melt Mol. Phys. 1983 50 467. 21 G. G. Balint-Kurti and G. C. Schatz J. Chem. Soc. Faraday T rans. 1997 93 755. 22 N. A. Besley and P. J. Knowles 1998 personal communication. Paper 8/01825A; Received 5th March 1998

ISSN:1359-6640    

DOI:10.1039/a801825a

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 159»167 Reactantñproduct decoupling approach to state-to-state dynamics calculation for bimolecular reaction and unimolecular fragmentation Tong Peng Wei Zhu Dunyou Wang and John Z. H. Zhang* Department of Chemistry New Y ork University New Y ork NY 10003 USA The main purpose of the study is to explore a number of computational methods to be used in the RPD approach to state-to-state quantum dynamics calculation of polyatomic reactions. Speci–cally a mixed timedependent (TD) and energy-dependent (ED) approach to solve the RPD equations is investigated. In the mixed TD and ED approach the reactant wavefunction is computed by the method of wavepacket propagation while the product wavefunction is calculated by energy-dependent methods.As a result the reactant»product coordinate transformation only needs to be carried out for the number of energies at which the state-to-state S-matrix elements are needed which is advantageous if state-to-state information at only a few energies are needed. Similar implementation of the mixed RPD approach is given for calculation of product state distribution in molecular photofragmentation dynamics. 159 1 Introduction The TD wavepacket approach currently provides a practical means for studying polyatomic reaction dynamics beyond triatomic reactions.1 The TD method is especially suited for computing total reaction probabilities (i.e. probabilities summed over –nal states of the product arrangement). In the simplest approach to computing total reaction probabilities one can employ the Jacobi coordinates corresponding to the reactant arrangement to carry out the time propagation for the scattering wavefunction and employ an absorbing boundary condition just beyond the transition-state region.Thus one can obtain total reaction probabilities by using a single set of (reactant) Jacobi coordinates to propagate the wavepacket in a relatively small region of coordinate space excluding asymptotic regions of the product arrangements. Such an approach has proven to be very robust in computational studies of tetraatomic reactions such as H2]OH HO]CO etc. as discussed in ref. 1. However if state-to-state dynamics information is needed the use of a single set of Jacobi coordinates is numerically inefficient because one has to extend the grid all the way from the reactant asymptotic region to the product asymptotic region.For example if the product arrangement involves long-range inelastic interactions using the reactant Jacobi coordinates for the entire con–guration space is very inefficient. Although the use of the product Jacobi coordinates could help ease the problem of long-range inelastic interaction in the product arrangement it will be inefficient if the reactant arrangement also has long-range inelastic scattering. In any event the use of a single set of Jacobi coordinates (corresponding to either reactant or product) to calculate state-to-state quantum dynamics throughout the entire con–guration space is numerically inefficient. This is an inherent difficulty associated with reactive scattering because the reactant and product arrangements are properly described by diÜerent sets of Jacobi coordinates.

ISSN:1359-6640    

DOI:10.1039/a800808f

出版商:RSC    

年代:1998

数据来源: RSC

摘要:

Faraday Discuss. 1998 110 169»183 Gabriel G. Balint-Kurti,a,* Ana I. Gonzalez,a,§ Evelyn M. Gold–eldb and Stephen K. Grayc a School of Chemistry T he University of Bristol Bristol UK BS8 1T S b Department of Chemistry W ayne State University Detroit MI 48202 USA c Chemistry Division Argonne National L aboratory Argonne IL 60439 USA A recently developed wavepacket method is applied to study the quantum dynamics of the reactions O(1D)]H2 ]OH]H and O(1D) ]HD]OH(OD)]D(H). The ab initio based global 1A@ potential energy surface of Ho et al. is employed. The results of three-dimensional total angular momentum J\0 scattering calculations are presented and discussed with emphasis on qualitative features of the dynamics. 2 Quantum reactive scattering of 1 Introduction The reaction O(1D)]H2(1&g `)]OH(2%)]H(2S) has been the focus of considerable research eÜorts because of its importance in atmospheric and combustion chemistry and also because of interest in its basic dynamics.The dynamics of this highly exoergic reaction is interesting because short-lived water complexes are believed to play an important role. There have been a large number of experimental and theoretical studies of this system and of its isotopic variants. The process has been and is currently being investigated by several leading experimental groups.1h9 Theoretical studies include numerous classical trajectory calculations6,7,10h14 as well as some quantum mechanical studies.15h18 A number of fascinating dynamical issues have been raised by such work including the role of direct abstraction vs.insertion in the reaction mechanism the importance of electronically non-adiabatic eÜects the nature of the water complexes and the validity of statistical descriptions of the reaction dynamics. In this paper we hope to contribute to the understanding of some of these issues by carrying out several three-dimensional (total angular momentum J\0) wavepacket studies utilizing a novel approach recently developed by two of us.19 The electronic structure of the system is complicated by the presence of –ve electronic states20 that correlate with the reactants. The lowest singlet electronic state the 1A@ state also correlates with ground electronic state water and is believed to be the most important one for the reaction dynamics.A high quality global potential energy surface for this state has recently been developed by Ho et al.12 We should note however that the experimental work of Hsu and co-workers4,5 has pointed to direct abstraction on an excited state as playing a role in the dynamics. The trajectory work of § Permanent address Departamento de Quimica. C-9 Facultad de Ciencias Universidad Autonoma de Madrid 28039 Madrid Spain. 169 O(1D) + H and O(1D) + HD (I) 170 Quantum reactive scattering of O(1D)]H /HD 2 Fitzcharles and Schatz10 has also pointed to the importance of the 1AA and 2A@ electronic states in the dynamics. Our studies here however will focus on adiabatic dynamics on the 1A@ state. Section 2 below sketches the methodological and computational details.Section 3 presents and discusses our results and a summary is given in Section 4. (2) (3) i+ Lt(t@) Lt@ \f (H)t(t@) (4) s) f (H)\[ + cos~1(H q 2 Methods 2.1 Quantum dynamics Most of the detailed theory behind our approach is discussed in ref. 19. Here we brie—y review the key aspects of this theory and expand on some new features of relevance to the present applications. The approach is particularly advantageous for difficult problems (such as the present one) that require relatively large grid and basis set sizes. This is because only the real part of the wavepacket is employed thus signi–cantly reducing both computer memory and computation times. We should emphasize that our approach shares many common features with the important work of Kouri and coworkers, 21,22 Mandelshtam and Taylor,23,24 Kroes and Neuhauser,25 and Chen and Guo.26,27 Let t be a numerical approximation of a wavepacket i.e.a –nite vector of complex numbers representing the values of the wavepacket at discrete grid points and/or the coefficients associated with a basis set expansion. Let H denote the associated Hamiltonian matrix. Furthermore let q denote the real part of t q\ReMtN. The central computational equation of our approach as in the work of Mandelshtam and Taylor,23,24 is (1) qk`1\A([Aqk~1]2Hs qk) where k\1 . . . N and A is some appropriate operator that damps the wavepacket amplitude as it approaches the grid edges and Hs\asH]bs a and b chosen so that the minimum and maximum eigenvalues of the scaled s H s s lie within the interval ([1 1).Given q and q eqn. (1) can be used to 0 1 qk . (Similar considerations are involved in Tal-Ezer and with Hamiltonian generate all subsequent iterates KosloÜœs expansion of the time-dependent propagator.28) q has been chosen appropriately it is possible19 to interpret q as being the real If 1 part of the solution of a modi–ed time-dependent Schroé dinger equation. In this case k qk\ReMt(t@\kq)N where t(t@) is the solution of the equation where with q being some arbitrary time step. The time step q should (and does) cancel out in any correct formulae for any observable. We depart slightly from the notation in ref. 19 in that the evolution variable for the modi–ed Schroé dinger equation is denoted by t@ and not t.We will show here that it is possible to connect the dynamics of eqn. (3) with ordinary wavepacket dynamics [see discussion of eqn. (9) below] and we reserve t to denote the more physically meaningful time consistent with those dynamics. Given q0\ 171 G. G. Balint-Kurti et al. Re q Mt(t@\0)N and p0\ImMt(t@\0)N the appropriate formula for is19 1 (5) (p q1\Hs q0[J1[Hs2 p0 Our previous work,19 as well as other related ì real wavepacketœ work,25 employed a real initial condition 0\0) to de–ne the initial wavepacket t(0). In the present calculations we employ a complex initial condition and evaluate the eÜect of the square root operator in eqn. (5) with an appropriate Chebyshev series expansion. The complex initial condition corresponds to an incoming Gaussian wavepacket.The advantage of employing this type of initial condition which is commonly used in ordinary wavepacket propagations is that very little of the wavepacket is going ìthe wrong wayœ. Because of this potential errors arising from the application of the absorbing potential or the damping operator near the edges of the grid in the reactant part of the coordinate space are minimized. Generally 100»200 Chebyshev terms lead to an accurate evaluation of the square root term in eqn. (5). Successive terms in the Chebyshev series are obtained using the same recursion formula as employed for the rest of the dynamics eqn. (1). The eÜort required to evaluate the square root operator in eqn. (5) is insigni–cant in comparison with the thousands of iterations required for determining the subsequent dynamics.a 2 t(t@\0)\g(R The initial condition used in our calculations is thus a)/I(ra) where Ra or HD center-of-mass distance and r denotes all other (internal) represents a speci–c diatomic initial vibration»rotation state and in all denotes the O to H coordinates. / the calculations reported here we employ just the ground vibration»rotation state. I (Reactant coordinates are denoted with a subscript a to distinguish them from product coordinates.) The function g(R taken to be a) describing the initial relative motion of the reactants is (6) g(Ra)\p~1@4a~1@2 expA[(Ra 2 [a 2 R0)2 exp([ik0(Ra[R0)) B and its corresponding Fourier decomposition is (7) g(Ra)\P`= dk g6 (k)exp(ikRa) ~= where a g6 (k)\ 1 = dR exp[[ikR 2p a]g(Ra) P (8) \S 0 a 2p3@2 exp([ikR0)expA[ a2(k[ 2 k0)2B =.(With appropriate care it is also possible to centre the initial With k positive eqn. (6) corresponds to an incoming Gaussian wavepacket in the 0 reactant channel. Ref. 19 gives the relevant expression for S matrix elements [see eqn. (22) of ref. 19] and the only diÜerence here is that owing to the incoming initial condition above the g6 term in the expression is now given by eqn. (8). The S matrix expression was derived using ideas similar to those in wavepacket studies of photodissociation.29 It involves analysing the wavepacket at a particular asymptotic value of a relevant scattering coordinate.In our previous study,19 we focused on reactive S matrix elements and so analysis was carried out at R\R= where R denotes the atom»diatom separation in either of the two possible product channels in an A]BC system and R is some appropriate large value. Actually if non-reactive probabilities are desired then the same formulae apply but the analysis line is then placed in the = reactant channel i.e. Ra\R= . In such calculations we centre the initial wavepacket at R values smaller than R a 172 Quantum reactive scattering of O(1D)]H /HD 2 R values than wavepacket at larger R=.25) Speci–cally for each iteration k one forms the projections onto all possible asymptotic diatomic states at the analysis line. A simple a Fourier sum over all the iterations k\0 . .. N of these projections then yields the desired S matrix element at any given energy E for which the wavepacket has reasonable amplitude.19 It should be noted that a t(t@) arising from eqn. (3) is not the same as a wavepacket s(t) satisfying the usual Schroé dinger equation (9) i+ Ls(t) Lt \Hs(t) Thus the iterations concerning the real part of t eqn. (1) strictly speaking do not correspond precisely to the evolution of the real part of a usual wavepacket satisfying eqn. (9). However the stationary states of H are also the stationary states of f (H) H/E\E/E implies f (H)/E\f (E)/E . Thus if t(t@\0)\s(t\0) i.e. if the initial conditions are the same at the respective zeros of time both t(t@) and s(t) are sums (or integrals in the continuous case) over the same stationary state wavefunctions / and E the contribution of any given stationary state to each of the wavepackets is also the same.What diÜers is only the time-dependent phase associated with each stationary state. At a qualitative level one expects t(t@) and s(t) to exhibit the same ìmechanismœ in terms of —ow into various parts of space as they evolve. [Similarly one expects q(t@) and ReMs(t)N to exhibit qualitatively similar behaviour.] One can go even further and draw an approximate but direct connection between t(t@) and s(t). An approximation to f (H) can be obtained via a Taylor series expansion. Consider the function f (E) where the operator H is replaced by the real number E. Carry out a –rst order Taylor series expansion of f (E) about the mean wavepacket energy E1 f (E)Bf (E1 )]f @(E1 )(E[E1 ).Replacing E with H in this expansion results in a reasonable approximation to f (H). It is reasonable because generally speaking the spectral range of H is very large and the wavepacket contains signi–cant probability only for a relatively small range of energies about E1 . When inserted into eqn. (3) the resulting approximate modi–ed Schroé dinger equation is of course linear in H. Assuming the same initial condition is involved in both the modi–ed and usual Schroé dinger equations then there is a direct relation between t(t@) and s(t). Restricting t@ to multiples of kq where k is the iteration number the relation is found to be (10) s(t\bk)B eick tk where (11) b\ as + J1[E1 s\as E1 s2 with E1 being the mean wavepacket energy in scaled energy units [see eqn.(2)] E1 s ]bs . since it cancels out of any expectation value. Eqn. (10) is a useful relation because physic E is a real-valued function of as bs and 1 and is not relevant to our arguments cal time scales t can be related to the iteration numbers k i.e. to a reasonable approximation k iterations of eqn. (1) correspond to a physical time of t\bk (12) All our calculations are for total angular momentum J\0. In our earlier work,19 we carried out propagations directly in the product channel with coordinates R r and c where for a particular reaction A]BC]AB]C R represents the distance from C of the center-of-mass of AB r is the AB internuclear distance and c is the angle between the vectors formed from R and r.For this work we have also implemented a version of our code that propagates in the reactant channel. This code has proved to be very useful 173 G. G. Balint-Kurti et al. in computing total reaction probabilities but is not suitable for computing product quantum state distributions. We report results using both codes below. (An economical procedure for computing product quantum state distributions might be to propagate initially in the reactant channel utilizing a grid adapted to that channel and to the interaction region. Following this reactant channel propagation the grid could be changed at some point to one better suited to propagating the wavepacket in the product channel and the propagation should be continued on the new grid.30) A further diÜerence between the calculations reported here and the application dis- (D]H cussed in our earlier work 2 ]DH]H),19 is that while we retain a mixed gridbasis representation for the wavepacket based on grids in the Jacobi coordinates R and r and Legendre basis functions P (cos c) to describe the diatomic rotational states the j eÜect of the potential matrix acting on the wavepacket is accomplished by transforming from the basis to an angular grid representation multiplication by the appropriate potential energy values on the R r and cos c grid and then transformation back to the basis function representation.The relevant transformation matrices with Gauss» Legendre quadrature points used to de–ne the grid points in cos c are simply related to the corresponding quadrature weights and the Legendre functions.31 (This is a particular kind of discrete variable and –nite basis representation transformation.32) This saves additional memory since no potential matrix needs to be kept.(Of course an alternative would be to carry out the computation entirely within a pure grid representation. Owing to the need to impose an energy cut-oÜ on the value of the potential and on that of the centrifugal repulsion term this procedure would still require the same overall computational eÜort.) 2.2 Calculation details One reason the (1A@) O(1D)]H system is very challenging to describe with accurate 2 quantum dynamics methods is the presence of a deep 7.2 eV (692 kJ mol~1) potential well corresponding to water.Table 1 lists some relevant properties of the potential energy surface of Ho et al.12 that are used in our calculations. We have estimated that the deep H2O well supports 586 (J\0) bound states of even symmetry with respect to exchange of H atoms. In the case of HOD since there is no exchange symmetry and one of the atomic masses is heavier there are probably more than a 1000 bound vibrational states. Grid spacings must be of the order of the smallest de Broglie wavelengths of physical relevance and in the region of a deep potential well very high kinetic energies are possible leading to the possibility of very small de Broglie wavelengths. Thus unlike many scattering calculations where converged dynamical results can be obtained with (1A@)O(1D)]H potential energy surface Ho et 2 Table 1 Some properties of the al.12 employed in the present calculationsa Potential values at appropriate equilibrium geometries O(1D)]H2 V \0 eV H2O V \[7.18 eV OH]H V \[1.86 eV Total energies including zero-point energies O(1D)]H2 E\0.27 eV H2O E\[6.60 eV OH]H E\[1.63 eV O(1D)]HD E\0.23 eV HOD E\[6.70 eV OD]H E\[1.69 eV Lowest OH vibrational state energies EvOH/eV v\0 1 .. . [1.63 [1.19 [0.77 [0.37 0.01 0.36 0.70 Lowest OD vibrational state energies EvOD/eV v\0 1 . . . [1.69 [1.37 [1.06 [0.75 [0.46 [0.18 0.09 0.35 0.60 0.84 a The zero of energy is taken to be separated reactants O(1D)]H2 H2 at its with equilibrium distance ra\1.41 a0 . 1 a0\0.0529177 nm 1 eV\96.49 kJ mol~1. 174 Quantum reactive scattering of O(1D)]H /HD 2 Table 2 Grid/basis set and initial condition details O]DH]OH]D product coordinates O]H reactant 2 coordinates 0»13 199 0.8»12.5 i) range/a0 0»15.5 239 0.5»10.5 143 143 90 100 50 4 3 90 0.005 8.0 0.25 0.25 0.015 9.5 0.4 0.3 scattering coordinate (R number of grid points in R i) range/a0 diatomic coordinate (r number of grid points in ri number of angular grid points number of angular basis functions absorption length/a absorption strength/au 0 centre of initial wavepacket (R0)/a0 Gaussian width factor (a) initial kinetic energy/eV grid spacings of 0.1 a or larger we –nd that grid spacings less than 0.1 a are required.0 0 (See also the previous wavepacket work17,18 on the Schinke»Lester33 potential surface for water.) In addition to the nature of the problem demanding relatively dense grids we have also found it necessary that the relevant scattering coordinate grids extend out to rather large values which is due in part to the possibility of long-range interactions.Another feature that makes this system difficult to study particularly if state-to-state information is desired is the large exoergicity of 1.9 eV. This implies that a large number of product (OH or OD) vibration»rotation states can be open and a large range of de Broglie wavelengths is possible for these outgoing product states making absorption at O]H the grid edges difficult to carry out adequately. In the 2(v\j\0)]OH(v@ j@) ]H case for example with a collision energy of 0.2 eV (20 kJ mol~1) and thus a total energy EB0.5 eV when zero-point energy is included Table 1 shows that the –rst six OH vibrational states and of course many associated rotational states are open.In the sub-sections below we report results for the total reaction probability for O]H with zero total angular (J\0) and preliminary results for O]DH reacting to give OH in particular product quantum states. The wavepacket calculations are 2 analysed to yield information about the mechanism of the reaction process and some conclusions are drawn concerning the product quantum state distributions and the relative probabilities of producing OH or OD in O]HD collisions. The important technical details that determine the wavepacket calculation are the grid size number of radial and angular grid points the details of the absorbing operator A [see eqn.(1)] the position and details of the initial wavepacket and the kinetic energy imparted to the wavepacket. The incoming Gaussian initial condition eqn. (6) is employed. As in our previous work,19 absorptions of the form A(x)\exp[[cabs(x [xabs)2] where x corresponds to either of the Jacobi radial variables are applied to the wavepacket for x[x out with a great variety of parameters. Some of the relevant details consistent with two abs . A large number of numerical experiments have been carried of our most ambitious calculations which were used to estimate reaction probabilities are listed in Table 2. 2(v\j\0)]OH]H 3 Results and discussion 3.1 O(1D) + H2«OH+ H Fig.1 shows the total reaction probability for the O(1D)]H reaction. This –gure was obtained from a calculation carried out in reactant Jacobi 175 G. G. Balint-Kurti et al. Fig. 1 Total reaction probability vs. energy for the reaction. The zero of energy is taken to be the bottom of the O]H entrance channel. O(1D)]H2(v\j\0)]OH]H (J\0) 2 (R c coordinates a ra cos a). [The total non-reactive probability Pnr(E) was computed by placing the analysis line in the reactant channel as indicated in Section 2.1 and then the M1[P total reaction probability was inferred as nr(E)N. Numerous checks with other grid/basis set combinations have indicated that our result is adequately converged.] We have found that it is much easier to obtain converged total reaction probabilities in these coordinates than in the product Jacobi coordinates.There are three reasons for this. First since one is not calculating detailed state-to-state reaction probabilities into the product channel the r grid need not be too large. The second is that the initial wavepacket is of course accurately described using a much smaller number of angular a basis functions than would be the case in product Jacobi coordinates. The third is that the resonance states of the OH system do not appear to have large amplitude motions 2 in the OwH Jacobi scattering coordinate. This therefore permits the use of shorter grids in this coordinate. 2 The –rst important thing to notice about Fig. 1 is that the reaction probability is generally quite high.(Of course this total reaction probability is the sum of the two in this case equivalent product channel possibilities.) One should also note that this reaction has no collision energy threshold i.e. the reaction probability rises rapidly to nearly unity as total energy increases just slightly beyond the zero-point energy (0.27 eV see Table 1) of the reactants. Such large reaction probabilities were also found in the calculations of Peng et al.17 and Dai18 in relation to this reaction on the Schinke»Lester surface,33 and have been attributed to the large reaction exoergicity and relatively direct dynamics. However there is a lot of structure in the total reaction probability shown in Fig. 1. This structure might originate from two aspects of the dynamics.The –rst of these is that a major component of the reaction will be shown to occur via a short-lived complex in which the oxygen atom inserts itself into the centre between the two H atoms (see below). This short-lived complex is probably responsible for the gross features of the reaction probability. The second aspect responsible for structure is that 176 Quantum reactive scattering of O(1D)]H /HD 2 OwH angle c Fig. 2 Contour plots of the wavepacket for the O(1D)]H2(v\j\0)]OH]H (J\0) reaction. The plots in the left-hand column show contours of the wavepacket density plotted against the radical reactant Jacobi coordinates Ra(OwH2) and ra(HwH) integrated over the angular volume element. The right-hand column shows contour plots of the 2 a weight by the volume element sin(ca).there are probably many (possibly overlapping) water resonances which contribute to the –ne details of the reaction probability in any given energy range. (The short-lived complex could be viewed as a particular superposition of such resonances.) The threshold region in particular appears to have some closely spaced sharp resonance structures. Examination of the product quantum state distributions for the O]HD case (see 177 G. G. Balint-Kurti et al. below) shows that a portion of the products depart with little relative kinetic energy and a lot of their available energy trapped in internal motion of the products. There may be some long lived Feschbach resonance states of this type where so much of the energy is trapped in product OH vibrational»rotational motion that there is insufficient energy available for the system to decompose.Fig. 2 shows some contour plots of a wavepacket. (The particular wavepacket displayed was initiated slightly closer to the interaction region and had a slightly more modest grid and basis set parameters than those of Table 2 but led to the same reaction probability as in Fig. 1 to within 10%.) The four plots in the left hand column show contours of the wavepacket density plotted against the radial reactant Jacobi coordinates Ra(OwH2) and ra(HwH) integrated over the angular volume element. The right hand column shows contour plots of the wavepacket against the reactant Jacobi scat- R tering coordinate a(OwH2) O and the wH2 angle ca weighted by the volume element sin(ca) the contribution to the full density from the real part of the wavepacket since that is all integrated over the HwH vibrational coordinate.(By ìdensityœ we always mean that we compute. This density is sufficient for inferring mechanistic issues since the contribution from the imaginary part of the wavepacket although perhaps exhibiting some diÜerent phase oscillations is always qualitatively similar.) We have used the approximate relationship between the iteration number k and the physical time t eqn. (12) to assign times to the diÜerent panels in Fig. 2. (For this calculation b\3.46]10~2 fs. Note b depends on the scaling factor a and the mean wavepacket energy and is thus peculiar to a given calculation.) The left hand column shows the s initial wavepacket (k\0).It then approaches the reaction region (k\1000 or tB35 fs) and most of it goes out towards the products (k\2000 tB70 fs). The lowest frame in the left hand column shows the remnant of the wavepacket coming back out in the entrance channel. The third frame down on the right hand side is by far the most interesting. It shows that during the crucial period when the reaction is occurring the angular —ux is concentrated around ca\90° and Ra\0. This indicates that the reaction proceeds predominantly through an insertion mechanism. R r and as a function of Fig. 3 shows the averaged values of the radial coordinates a iteration number for the same wavepacket used to generate Fig. 2. (As with the densities a these average values are based on just the real part of the wavepacket.) The average a Fig.3 Averaged values of the radical coordinates R r and as a function of iteration number a 178 Quantum reactive scattering of O(1D)]H /HD 2 R (the OwH distance) decreases as the wavepacket approaches the strong 2 a value of interaction region of the deep well which coincides with a noticeable increase of the a average value of r (the HwH separation). The rough point where R has its –rst a a minimum and r has a maximum is approximately at k\1700 iterations or tB60 fs. The fact that r has increased so much (more than 1 a0) at this point is consistent with insertion being important. Unlike direct reactions (such as H]H2 which we have a examined in a similar manner) however Fig.3 shows that the average scattering coordinate does not simply increase after its –rst minimum but exhibits a maximum (and ra exhibits a minimum) at about 4000 iterations (tB140 fs). This behaviour is consistent with a single large amplitude oscillation of a water complex i.e. by tB140 fs on average the oxygen atom has passed once through the centre of the HwH bond has then rebounded. The subsequent behaviour of both average values is difficult to quantify owing to the necessity of absorption in our calculations ; i.e. as iterations continue the bulk of the wavepacket reacts and is absorbed thus obscuring the meaning of the average values. (We have carried out test calculations with much larger grids and no absorption to verify that the behaviour in Fig.3 is real and not an artefact of absorption.) The main mechanism of the reaction therefore appears to proceed by an insertion of O into the HwH bond. This is followed by a single large amplitude vibrational motion which might involve the oxygen atom coming back through the centre of the bond after which the bulk of the wavepacket proceeds to react. We should however emphasize that although ìmostœ of the wavepacket has reacted by the later iterations indicated in Fig. 3 which correspond to physical times on the order of 300 fs we –nd that there is a small fraction (2»5%) of the wavepacket that reacts on a much slower timescale. Typically to resolve the key bumps and wiggles in a reaction probability curve such as in Fig. 1 particularly at the lower collision energies requires us to carry out 2»3 times more iterations than displayed in Fig.3 which corresponds to physical times of the order of a picosecond. 3.2 O(1D) + HD«OD(OH) + H(D) Calculations involving HD as reactant are signi–cantly more challenging than those involving H since symmetry cannot be used to reduce the number of reactant rotational functions and since there are many more closely spaced HDO bound and quasi- 2 bound states than for H2O. Our examination of the HD case is on-going and the results presented here are preliminary because they are not yet fully converged. (Certain key average properties however are reasonably converged.) Detailed inspection and comparison between calculations using progressively larger and denser grids shows that the results for O(1D)]HD]OH]D are much better converged than those for O(1D)]HD]OD]H.The reason for this seems to be associated with the role of complexes exhibiting large amplitude motions in exit channel dynamics of the reaction. We speculate that the amplitudes of the motions of these complexes are smaller in the OHwD coordinate than in the ODwH channel. Whatever the reason our comparison calculations con–rm that all the qualitative features of our O(1D)]HD]OH]D calculations are converged. The detailed positions of the peaks in the reaction probability energy will however change as we achieve complete convergence. Fig. 4 presents the total reaction probabilities for both O(1D) ]HD(v\j\0)]OD]H and O(1D)]HD(v\j\0)]OH]D reactions computed by propagating the wavepacket in product Jacobi coordinates.As already noted the –ner details of the reaction probability vs. energy plots are not completely converged but the overall picture given by our current results is certainly correct. The sums of the reaction probabilities rise somewhat above unity [to a maximum of (1.1)] near the threshold and the details of the oscillations in the –gure can be expected to change as 179 G. G. Balint-Kurti et al. Fig. 4 Total reaction probabilities for both O(1D)]HD(v\j\0)]OD]H (J\0) and O(1D)]HD(v\j\0)]OH]D (J\0) reactions we converge the calculations. This –gure shows that the reaction O(1D)]HD produces OD in preference to OH in a ratio of approximately 1.7 1 averaged over the energy range 0.23»1.25 eV. This preference for production of OD over OH agrees with the classical trajectory calculations of Schatz et al.10 and arises from the fact that the H atom due to its smaller mass makes oscillations of greater amplitude away from the heavy oxygen atom than does the D atom.Fig. 5 shows the reaction probability for the production of OH(v@) in the reaction O(1D)]HD(v\j\0)]OH(v@)]D averaged over the energy interval 0.23»1.25 eV. The average was taken by –rst calculating the total reaction probability for production of each product vibrational state as a function of the energy and then averaging over the energy interval giving all energies at which the probability was non-zero being an equal weighting. The –gure demonstrates that the most probable product OH quantum state is v@\3.Note that product vibrational states v@\0»3 are all open at the threshold energy when the reactants have zero relative kinetic energy (see Table 1) and that v@\4 is only just closed at this energy. We have also examined the vibrational populations averaged over a narrower energy range (^0.1 eV) about E\0.49 eV which corresponds to a collision energy of 0.22 eV (5 kcal mol~1 or 21 kJ mol~1). The results do not signi–cantly change with v@\3 still being most probable. This result diÜers from quasiclassical trajectory results12 on the same potential surface that do not show such an inverted vibrational distribution. However the trajectory results are averages over all impact parameters and our J\0 quantum results are more consistent with just the zero impact parameter limit.Fig. 6 shows the OH product rotational state distribution averaged over the same energy interval for diÜerent OH product vibrational states. The –gure demonstrates that 180 Quantum reactive scattering of O(1D)]H /HD 2 Fig. 5 Reaction probability for the production of OH (v@) in the reaction O(1D) ]HD(v\j\0)]OH(v@)]D (J\0) averaged over the energy interval 0.23»1.25 eV ]D (J\0) Fig. 6 OH product rotational state distribution averaged over the energy interval 0.23»1.25 eV for diÜerent OH product vibrational states from the reaction O(1D)]HD(v\j\0)] OH(v@ j@) 181 G. G. Balint-Kurti et al. the most probable product rotational state decreases with increasing product vibrational state quantum number. It also shows that in every case a very substantial portion of the available energy is tied up in internal excitation of the products.In Fig. 7 we show the development of the wavepacket in product Jacobi coordinates with time at two –xed product Jacobi angles for the reaction O(1D)]HD]OD]H . (The plots have been made from calculations using somewhat smaller grids than those reported in Table 2 but are typical of the reaction dynamics.) The three panels on the left of the diagram show the wavepacket represented as contour plots in the radial coordinates R and r for ìsmallœ Jacobi angles (2.28°). Such angles correspond to both HwDwO and DwHwO near linear con–gurations. The fact that a single Jacobi angle is associated with two diÜerent con–gurations accounts for the two areas of maximum density in the plots.The plots in the right hand panels are for a product Jacobi angle of 177.72° and correspond to near collinear DwOwH geometries. The –rst panel on the right hand side is missing because it requires a little time before the oxygen can insert between the two hydrogen atoms. The central two panels correspond to tB15 fs [see Fig. 7 Development of the wavepacket with time at two –xed product Jacobi angles for the reaction O(1D)]HD(v\j\0)]OD]H. The three panels on the left show the wavepacket represented as contour plots in the radical coordinates R and r for ìsmallœ Jacobi angles (2.28°) corresponding to both HwDwO and DwHwO near linear con–gurations. The plots in the right-hand panels are for a product Jacobi angle of 177.72° and correspond to near collinear DwOwH geometries.182 Quantum reactive scattering of O(1D)]H /HD 2 eqn. (12)]. The left hand panel shows reduction in r (the HwOD separation plotted along the y axis). The central right hand panel (upper of the two) shows the build up of particle —ux in HwOwD geometries corresponding to insertion of O into the DwH bond. The lower panels show the densities at longer times tB200 fs. The most signi–- cant panel is the top panel on the right hand side of the –gure. This panel shows the insertion of the oxygen atom into the center of the bond. The two bottom panels which both correspond to long times at which only a minor fraction of the original particle —ux remains on the grid show that a small fraction of the —ux remains trapped in both HwHwO (or DwHwO) and HwOwD type con–gurations for a longer time period.This —ux con–rms the presence of some long-lived complexes. 4 Conclusion We have presented and discussed reactive scattering results for the O(1D)]H2(v\j\ 0)]OH]H O(1D)]HD(v\j\0)]OH]D and O(1D)]HD(v\j\0)] OD]H reactions computed using a newly developed wavepacket technique.19 There is a wealth of detailed structure in the reaction probability vs. energy plots. This structure must in some way be related to resonance dynamical behaviour although it seems unlikely that isolated resonances will exist except possibly at energies very near the zero collision energy threshold. Our analysis of the averaged behaviour of the two Jacobi radial coordinates strongly suggests that the bulk of the reaction occurs via a short-lived complex that lives for one vibrational oscillation.The complex corresponds to the oxygen inserting into the HwH bond passing through it and rebounding back once before going on to react. The O(1D)]HD system has been studied in product Jacobi coordinates. Our calculations for these systems are not fully converged but the reaction to form OH products is much better converged than that to form OD products. Our calculations show that in the energy range 0.23 to 1.25 eV OD is formed in preference to OH by an average ratio of about 1.7 1 . This conclusion seems robust and is unlikely to change when fully converged results become available in the near future. Examination of the product quantum state distributions for the O(1D)]HD(v\j\0)]OH]D reaction shows that the OH(v@\3) state is preferentially produced over the energy range up to total energy of 0.125 eV (reactant kinetic energy of 0.102 eV).The product rotational state distributions show that the products are highly rotationally excited and that the average rotational excitation decreases with increasing vibrational excitation. A portion of the products have very high internal energies and correspondingly low relative kinetic energies. These products might well result from the break-up of longer lived complexes. Many questions remain as to the origin of the rich structure in the total reaction probability. We also presented plots showing the development of the wavepacket in both reactant and product Jacobi coordinates (Fig.2 and 7). These also con–rm that the reaction proceeds mainly by insertion of the O into the middle of the HwH or HwD bond. The plots also show that the amplitude of the wavepacket in HwOwD geometries persists to long times and indicates the presence of a small fraction of longer lived complexes. We are very grateful to Dr Wei Zhu for helpful discussions and sharing with us some of his results on the O]H reaction. We thank Dr R. J. Allan for assistance and advice with the computer program during the initial stages of the work and Dr C. Wilson for 2 assistance with computing problems. We also bene–ted from numerous discussions with Professor George C. Schatz. S.K.G. is grateful to the EPSRC for a Visiting Fellowship to the University of Bristol where this project was initiated.S.K.G. was also supported by the Office of Basic Energy Sciences Division of Chemical Sciences US Department of Energy under Contract No. W-31-109-ENG-38. G.G.B.K. thanks the UK Computa-183 G. G. 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ISSN:1359-6640    

DOI:10.1039/a801712c

出版商:RSC    

年代:1998

数据来源: RSC