Department of Chemistry, The University of Chicago, Chicago, IL 60637

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1 Appendix: Overview of our methodology in "Regularity Non-recrossing Paths in Transition States of Chemical Reactions" Tamiki Komatsuzaki R. Stephen Berry Department of Chemistry, The University of Chicago, Chicago, IL (February 8, 999) The crux of our method, Lie canonical perturbation theory, abbreviated as LCPT, is a classical version of Van Vleck perturbation theory. These theories a priori assume existence of M invariants of motion for the M-dimensional Hamiltonian systems in question. It is well known that w such methods, the new, transformed Hamiltonian diverges if the system encounters (near-)resonances, becomes meaningless. However, we clarify in this work that LCPT is still powerful enough to reveal an invariant manifold even if almost action variables fail to maintain constants of motion. Rather, the method provides us w a new phase-space dividing surface free from recrossing up to moderately high energy. We briey describe our method to construct a model Hamiltonian from any arbitrary autonomous Hamiltonian to which LCPT applies. We use an ecient technique, so-called algebraic quantization, to carry out the LCPT wout any special mathematical manipulations achieve a reformulated (classical) transition state theory by using the hyperbolic phase space orbit in the sea of chaotic motions of the other stable modes. At the end, we describe shortly three distinct levels of local dynamics in the transition state in terms of regularity of the dynamics, which have not been addressed previously. APPENDI A: LIE CANONICAL PERTURBATION THEORY (LCPT) Canonical perturbation theories ( ) transform (p q) to a new (p q) coordinate system so as to make the new Hamiltonian H(p q) as close to integrable as possible. Lie canonical perturbation theory (LCPT) ({7) is the most sophisticated theory among them, applicable to implementing higher-order perturbations to treating systems w many degrees of freedom. The LCPT is based on Lie transforms, that is, the exponential of a Lie operator induces a canonical transformation: let L w be the Lie operator associated w a generating function w, L w fw g (A) where fg denotes the Poisson bracket. Then the transformation of an autonomous Hamiltonian H to a new Hamiltonian H, H(p q) ;! H(p q) = exp(lw )H(p q) (A) is canonical. We let H be an M-dimensional Hamiltonian expable in (=strength of the perturbation) where the zeroth-order Hamiltonian H 0 is assumed to be integrable, e.g., a system of M harmonic oscillators. Such a zero-order system is a function of action variables J only, does not depend on the conjugate angle variables, so H(p q) = n=0 n H n (p q) (A3) = H 0 (J)+ n= n H n (J ) (A4) =! k J k + n= n H n (J ) (A5) where! k is the fundamental frequency of the kth mode of H 0. Assuming the new Hamiltonian H the generating function w are also expable in, we substitute all H, H, w to Eq. A determine the new Hamiltonian, at each order in, to be as a simple a form as possible by eliminating its dependencies on the new angle variables.( 3 7) IftheH is obtained free from the angle (at the order of the perturbative calculation performed): H(p q) = H( J )= H( J)= n Hn ( J) (A6) n=0 the new action angle variables for the kth mode are expressed as d J k dt = _ Jk = H( k =0 (A7) J k = constant (k = 3:::M) (A8) _ k H( J k! k ( J)=constant (A9) k =! k ( J)t + k (A0) where k is the arbitrary initial phase factor of the kth mode. From there, the equations of motion w respect to the new coordinates q momenta p are obtained from the Hamiltonian equations of motion obeying H d q k (p q) dt +! kq k (p q) =0 (A) p k (p q) =! k dq k (p q)! k dt (A)

2 where! k (=! k ( J)=! k (p q)) is independent of time t because the J are constant through all t [Eq. (A7)]. The equations (A) (A) tell us that even though the motions look quite complicated in the old coordinate system, they could be reformulated as simple decoupled periodic orbits in the phase space. If one tries to regularize a general Hamiltonian globally, one almost surely encounters the problem that the (near-)commensurable conditions that an integer linear combination of fundamental frequencies vanishes identically at a given order n, n k! k O( n ) (A3) (n k is arbitrary integer), makes the corresponding new Hamiltonian diverge destroys invariants of motion. ( ) It is quite likely in many M-dimensional systems that the near-commensurable conditions densely distribute in the phase space, i.e., the occupation ratio of the M-dimensional tori in the phase space is negligibly small, hence Eqs. (A)-(A) become meaningless. Here, however, we consider only approximately regular behavior in local regions, specically saddles, where we nd that such diculties of fully-developed chaos for the full set of degrees of freedom only at quite high energies. We rst showed that, by monitoring the new action of the kth mode J k (p q) along molecular dynamics (MD) trajectories obeying equations of motion of the original Hamiltonian H(p q), one can detect whether the q k mode tends to maintain an invariant of motion. If the J k (p q) its associated! k (p q) exhibit nearconstants of motion through a certain range in time locality, it implies that p k q k are approximately decoupled from the other modes, represent the local dynamics analytically. Those quantities associated w a reactive mode exhibit such near-constants of motion up to a moderately high energy in the region of a saddle. Furthermore the dividing hypersurface, dened by the condition that the reactive coordinate in the transformed coordinates is zero, is almost free from recrossing problems. APPENDI B: THE REGIONAL HAMILTONIAN Despite its versatility, LCPT has not been applied to many-degrees of freedom (DOF) realistic atomic/molecular systems. One of the two main diculties is how one could hle the analytical derivative integral calculations that appear successively in the LCPT procedure. The other is the near-impossibility of obtaining even moderately simple analytical expressions to describe the accurate (e.g., ab initio) potential energy surfaces in full. As shown foravariety of atomic/molecular clusters, the dynamical properties of multidimensional Hamiltonian systems are strongly nonuniform at low moderate energies, depending on the local topography of their potential energy surfaces. (8 9) We rst exp the full 3N-DOF potential energy surface about a chosen stationary point, i.e., minimum, saddle, or higher rank saddle. By taking the zeroth order Hamiltonian as a harmonic oscillator system, which might include some negatively curved modes, i.e., reactive modes, we establish the higher-order perturbation terms to consist of nonlinear, anharmonic couplings which wemay express in arbitrary combinations of coordinates. Thus, where H = H 0 + H 0 = n= n H n j n= n H n (B) (p j +! j q j ) (B) = jkl + C jkl q j q k q l jklm C jklm q j q k q l q m + ::: (B3) (B4) Here, q j p j are the jth normal coordinate its conjugate momentum, respectively! j C jkl, C jklm,... are, respectively, the frequency of the jth mode, the coupling coecient among q j, q k, q l that among q j, q k, q l, q m so forth. The frequency associated w an unstable reactive modef those of the other stable modes B are pure-imaginary real, respectively. Atany stationary point there are six zerofrequency modes corresponding to the total translational innitesimal rotational motions, the normal coordinates of the innitesimal rotational motions appear in the perturbation terms H n (q) (n>0). The contribution of the total translational motion is simply separated. We make no more mention of this. If one deals w a system whose total angular momentum is zero, one could eliminate the contributions of the total rotational motions from H n (q) (n >0) by operating w a suitable projection operator (0 ) at the stationary point it corresponds to putting to zero each normal coordinate corresponding conjugate momentum representing the innitesimal total rotational motion. For the sake of simplicitywe focus on a (3N-6)-DOF Hamiltonian system w total linear angular momenta of zero, so that the kinetic potential energies are purely vibrational. For such a zeroth-order Hamiltonian! k 6=0for all k (= 3:::3N ; 6( M)), the associated actionangle variables of the stable modes B (! B <: real) the unstable mode F (! F = : imaginary) are expressed as J B = I p B dq B = p B +! B qb! B (B5)

3 B = tan ; pb! B q B J F = Zbarrier Im p F dq F = i p F j! F j ;j! F jq F F = i tanh ; pf j! F jq F (B6) (B7) (B8)! F ;j! F ji: (B9) Here the action associated w the reaction mode F, having rst been postulated in the semiclassical transition state theory, ( {3 ) is purely imaginary is connected w the barrier penetration integral in the semi-classical theory. It is easily veried (4 ) that any such set of variables J is canonical, including those associated w the unbound mode F. APPENDI C: THE ALGEBRAIC QUANTIZATION METHOD For practical LCPT calculations of the above Hamiltonians, a quite ecient method, called \algebraic quantization," has been developed. (6 7) This method rst transforms (p q) in Eqs. B-B4 to (a a) by the customary means: a k = p (p k + i! k q k ) a k = p (p k ; i! k q k )(C) which are expressed in terms of the old action variable, the associated frequency J k,! k, time obeying Hamiltonian H 0 as a k () =p! k J k e ik = p! k J k e i(!k+k) a k () = p! k J k e ;ik = p! k J k e ;i(!k+k) : (C) (C3) The cumbersome analytical calculations of Eqs. B4-B8 that appear in the LCPT calculations are then replaced by simultaneous algebraic, symbolic operations, thanks to the simple Poisson bracket rules for the (a a), fa j a kg = fa j a k g =0 fa j a kg = i! k jk (C4) where fg denote Poisson bracket Kronecker delta, respectively. One simply substitutes (C)-(C3) into (B), to set up the simultaneous algebraic equations, which readily yield the desired dynamical quantities w the help of (C4). (7 ) Finally,we obtain new transformed physical quantities A, i.e., the new Hamiltonian H, new action Jk, frequency! k, momentum p k, coordinate q k of the kth mode, in terms of the original p q as A = A(p q) = n=0 n An (p q): (C5) In the present work, we analyze A up to a (nite) i order (i=0,,): A = A (p q) = i n=0 n An (p q) (C6) where no (near-) commensurable conditions were encountered at these orders during our LCPT procedure. The details of the computational recipe are given in (7 ). To indicate the complexity of the transform, we show the expressions for p (p q) q (p q) up to rst order at saddle I of Ar 6 (Here mode is reactive the others are non-reactive). The contributions of the original p q in p (p q)q (p q) are not necessarily large almost all modes contribute to p q (p q) for i. p 0th = p (C7) p st = p +0:03353p q +0:09766q 7 p 8 + 0:08446q p 0 +0:0797q 4 p ; 0:075768p q 0 ; 0:07068p 4 q +0:06759q p ; 0:06966p q + 0:05790q 5 p 0 +0:056576q 6 p ; 0:056558p 7 q 8 + 0:0595q 0 p ; 0:048788p 6 q ; 0:044909p 5 q 0 ; 0:044499q 5 p +0:039476q p +0:038653p 5 q ; 0:038584q p ; 0:036376p 0 q ; 0:035646p q + 0:03543p q ; 0:03806p 6 q 6 +0:0383p 3 q 3 ; 0:03358q p 5 +0:06383p 7 q 7 +0:05753p q 5 + 0:0577p q ; 0:04068p q +0:09803p 8 q 8 + 0:07695q 4 p 6 ; 0:077p 9 q 9 +0:0660q p 0 ; 0:0435p q 0 +0:0048p 5 q 5 ; 0:008864p 4 q 6 + 0:005637q p 5 ; 0:00387p q 5 ; 0:00379p q + 0:003657p 4 q 4 ; 0:00603p 0 q 0 +0:0070q 3 p 9 ; 0:0043p 3 q 9 (C8) q 0th = q (C9) q st = q +0:599q 7 q 8 ; 0:559687q + 0:409p +0:338508q 7 +0:38906q 8 ; 0:3909q 6 +0:30549q 0q ; 0:9435q 9 + 0:8370q 3 +0:70034q 5 q 0 +0:688p 7 p 8 + 0:583q +0:46p +0:3669q 4 q ; 0:30785p 6 +0:8438q 6 q +0:685p 3 + 0:0580p 7 +0:009q 5 ; 0:0988q 5 q ; 0:095949p ; 0:090680q +0:078946p 8 + 0:075768q q 0 +0:0768q 4 q 6 ; 0:068455p 9 + 0:06966q q +0:059076p 0 p +0:05408q q ; 0:053049p p ; 0:05677q +0:048960p 5 p 0 3

4 ; 0:046850q 0 +0:0457p 5 +0:03648p 4p (D) + 0:03507p 4 p 6 +0:03457p p 0 +0:03050p 6 p where h(x) (x), respectively, denote the Heaviside ; 0:05753q q 5 ; 0:0577q q +0:0594q q 0 function Dirac's delta function of x. The canonical + 0:04569q 4 ; 0:03307p 5 p ; 0:0347p p 5 form is also formulated straightforwardly. + 0:004p p +0:0570p p ; 0:059p If no approximate invariant of motion exists in the saddle region, + 0:04577p 4 +0:0340q k q 5 ; 0:00375p GTST (E) deviates from the (classically) exact reaction rate constant k(e). Therefore one can in :009008p p 0 +0:007034p p 5 +0:00367q 3 q 9 troduce a new transmission coecient c, the deviation + 0:00p 3 p 9 : of the k GTST (E) from the k(e): (C0) The new momenta coordinates p k (p q) q k (p q) have the following forms respectively, p k (p q) = j q k (p q) = j c j p n; q m d j p n q m (C) (C) where c j d j denote the coecient of the jth term, n, m( 0) are arbitrary integers, q m = q m qm qm3 3 qm mm (P M j= m j = m) etc. The new p k (p q) q k (p q) maintain time reversibility. The units of p k q k are = m =, respectively. As we increase the order of LCPT to be evaluated, the total number of the terms rapidly increases. For example, 450 for p nd q nd. APPENDI D: REFORMULATION OF TRANSITION STATE THEORY In the cases that saddle crossing dynamics has approximate invariants of motion associated locally w the reactive mode F in a short time interval but long enough to determine the nal state of the saddle crossings, the q F can be identied as a \good" reaction coordinate. This is because there is no means or force returning the system to the new dividing surface S(q F =0)even though the system may recross the original naive surface S(q F = 0). The reformulated microcanonical (classical) transition state theory(tst) rate constant k GTST is obtained (7 ) as a thermal average of the one-way uxes j + (= _q F (p q)h(_q F (p q))) across S(q F =0)over microcanonical ensembles constructed over a range of energies E. k GTST (E) = hj + i E = h_q F (p q) [q F (p q)] h [_q F (p q)]i E Z Z = dq dp ::: dq N dp N N [E ; H(p q)] _q F (p q) [q F (p q)] h [_q F (p q)] k = c k GTST : (D) We may use c to estimate the barrier recrossing eect, as a measure of the extent to which the quasi-invariants of motion associated w the reactive mode, i.e., the local action its local frequency, cease to be approximate invariants. Their nonconstancy reects the degree of fully-developed chaos in which noinvariant of motion exists, if the vibrational energy relaxation is fast enough to let us assume quasi-equilibration in the reactant's potential well. In order to focus on how the recrossings overagiven dividing surface contribute to c,we estimate the timedependent quantities MD c (t S(q F = 0)) in terms of our MD trajectories dened by MD c (t S(q F h = 0)) = hj(t =0)h q _ F (p q) hj + (t =0)i E i i E (D3) where j(t = 0), j + (t = 0), respectively, denote the initial total, initial positive uxes crossing the order LCPT dividing surface S(q F (p q) = 0). The origin of time t is taken to be zero when the system trajectory rst crosses the given dividing surface. This equation should also tell us how the vanishing of the approximate invariants of motion of the reactive mode reects on the c. APPENDI E: DISTINCT CLASSES OF DYNAMICS IN THE TRANSITION STATE We found, w the analysis aorded by the minimalrecrossing trajectories provided by the LCPT analysis, that there are at least three distinct energy regions above the saddle point energy that can be classied in terms of the regularity of saddle-crossing dynamics. Let us now articulate the distinctions among them. Quasiregular region All or almost all the degrees of freedom of the system locally maintain approximate constants of motion in the region of the transition state. The saddle crossing dynamics from well to well is fully deterministic, obeying M-analytical solutions (see Eqs. A- A) for systems of M degrees of freedom. The dynamical correlation between incoming outgoing trajectories from to the transition state region is quite strong, the dimensionality of saddle crossings is essentially one, corresponding to the reactive modeq F in the (p q) 4

5 space. Barrier recrossing motions observed over a naive dividing surface dened in the conguration space are all rotated away to no-return single crossing motions across a phase space dividing surface S(q F (p q) = 0). If the vibrational energy relaxes fast enough to let us assume quasi-equilibration in the wells, the initial conditions (p(0) q(0)) of the system as it enters the transition state from eer of the stable states can be simply sampled from microcanonical ensembles. One may then evaluate the (classical) exact rate constant, free from the recrossing problem. The staircase energy dependence observed by Lovejoy etal.(5 ) for highly vibrationally excited ketene indicates that the transverse vibrational modes might indeed be approximately invariants of motion. (6 )We classify such a range of energy, in which the rate coecient shows staircase structure, as corresponding to this quasiregular region. Intermediate, semi-chaotic region Due both to significant (near-)resonances to strong anharmonic modemode couplings emerging at these intermediate energies, almost all the approximate invariants of motion disappear, consequently inducing a topological change in dynamics from quasiregular to chaotic in the regions of saddle crossings. However at least one approximate invariant of motion survives during the saddle crossings, associated w the reactive coordinate q F (p q). This is due to the fact that an arbitrary combination of modes cannot satisfy the resonance conditions of Eq. A3 if one mode has an imaginary frequency, the reactive mode in this case, is included in the combination. The other frequencies associated w bath modes fall on the real axis, orthogonal to the imaginary axis in the complex!-plane. That is, y nk! k j! F j >O( n ) (E) for arbitrary integers n k w n F 6= 0, where y denotes the combination including the reactive mode. This was rst pointed out by Hernez Miller. (3 ) In this region the dynamical correlation between incoming outgoing trajectories to from the transition state becomes weak (but non-zero!), the saddle crossings' dimensionalityis' M ;, excluding the one dimension of q F, in this region. If the associated imaginary frequency! F (p q) is approximately constant during a saddle crossing as the action J F (p q) is, the reaction coordinate q F decouples from the Z-subspace composed of the other bath-dof, in which the system dynamics is manifestly chaotic. The q F dynamics is then represented analytically during saddle crossings, a dividing surface S(q F (p q) = 0) can still be extracted free from the barrier recrossings, even for saddle crossings chaotic in the bath modes. This class does not exist near potential minima, but is inherent associated w the transition state. Stochastic (=fully-developed chaotic) region The system becomes subject to considerable nonlinearities of the potential energy surface at much higher energies, the convergence radius becomes negligibly small for the LCPT near the xed (saddle) point for the invariant of motion associated w the reactive coordinate q F. In this energy region, no approximate invariant of motion can be expected to exist, even in the passage over the saddle between wells. The saddle-crossing dynamics is entirely stochastic, w dimensionality essentially equal to the number of degrees of freedom of the system. Here it is probably not be possible to extract a dividing surface free from barrier recrossings. At these high energies above the lowest, presumably (but not necessarily) rst-rank saddle, the system trajectories may pass over higher-rank saddles of the potential energy surface. These provides us w a new, untouched problem, i.e., what is the role of resonance in the imaginary!-plane for the bifurcation? (This even arises in the degenerate bending modes for a linear transition state of a triatomic molecule.) W this, we encounter one of the related open subjects in statistical theories of many-dof systems. : A. J. Lichtenberg M. A. Lieberman, Regular Chaotic Dynamics nd Ed. (Springer-Verlag, New York, 99). : G. Hori, Pub. Astro. Soc. Japan 8, 87 (966) ibid. 9, 9 (967). 3: A. Deprit, Celest. Mech.,, (969). 4: A. J. Dragt J. M. Finn, J. Math. Phys. 7, 5 (976) ibid. 0, 649 (979). 5: J. R. Cary, Phys. Rep. 79,30 (98). 6: L. E. Fried G. S. Ezra, J. Phys. Chem. 9, 344 (988). 7: T. Komatsuzaki M. Nagaoka, J. Chem. Phys. 05, 0838 (996). 8: C. Amitrano R. S. Berry, Phys. Rev. E 47, 358 (993). 9: R. J. Hinde R. S. Berry, J. Chem. Phys. 99, 94 (993). 0: M. Page J. W. McIver. Jr. ibid. 88, 9 (988). : W. H. Miller, Faraday Discussions Chem. Soc. 6,40 (977). : T. Seideman W. H. Miller, J. Chem. Phys. 95, 768 (99). 3: R. Hernez W. H. Miller, Chem. Phys. Lett.4, 9 (993). 4: H. Goldstein, Classical Mechanics, (Addison-Wesley, Massachusetts, 950). 5: E. R. Lovejoy, S. K. Kim C. B. Moore, Science 56, 54 (99). 6: R. A. Marcus, ibid. 56, 53 (99). 5