Few-Body Systems 0, 6 (5) Few- Body Systems cfl y Springer-Verlag 5 PrintedinAustria Dynamical Hierarchy in Transition States of Reactions C.B. Li ; Λ, A. Shojiguchi ; ΛΛ,M.Toda ; ΛΛΛ,T. Komatsuzaki ;3; y Nonlinear Science Laoratory, Department of Earth and Planetary Sciences, Faculty of Science, Koe University, JST/CREST, Nada, Koe 657-850 Japan DepartmentofPhysics, Faculty of Science, Nara Women's University, Nara 630-8506, Japan 3 Department of Theoretical Studies, Institute for Molecular Science, Myodaiji, Okazaki 444-8585, Japan Astract. We present a partial normalization procedure of Lie canonical perturation theory to elucidate the phase space geometry of the transition state in the multidimensional phase space for a wide range of energy aove the threshold. State selectivity and dynamical correlation along the evolution of reactions will also e discussed. Introduction All reactions take place through the `edge of staility,' which chemists have long envisioned as transition state (TS). The concept of TS has een much more widely utilized than just for the prolems near instaility in chemistry, e.g., atomic physics, cluster physics, celestial mechanics, and iology []. In 930s, the TS is defined y Wigner as a no-return dividing hypersurface in the phase space through which reacting species passes only once on the way from the reactant to the product. In other terms, the no-recrossing condition requires that the `one dimensional' motion normal to the TS (reaction mode) is separale from the motions tangent to the TS (ath modes). While the existence and the definaility of such a no-return TS were well estalished in two degrees of freedom (dof) Hamiltonian systems, those dynamical prolems in more than two dof has een an open prolem until very recently [, 3, 4, 5, 6]. For a N-dof Hamiltonian system, we can expand it near a saddle with index one (an equilirium point with one negative Hessian eigenvalue) as H = E 0 + NX i= p i +! i q i Λ E-mail address: cli@koe-u.ac.jp ΛΛ E-mail address: shojiguchi@ki-rin.phys.nara-wu.ac.jp ΛΛΛ E-mail address: toda@ki-rin.phys.nara-wu.ac.jp y E-mail address: tamiki@koe-u.ac.jp + X n=3 H n ; ()
Dynamical Hierarchy in Transition States of Reactions where E 0 is the energy of the saddle point and H n consists of terms of n-th degree in (p ; ;p N ;q ; ;q N ). (p ;q ) are the momentum and coordinate of the reactive normal mode, and (p ; q )(= (p ; ;p N ;q ; ;q N )) are those of the ath normal modes, respectively. The unpertured frequency of the reactive mode! is pure imaginary and those of the ath modes! i (i ) are real. In an energy regime slightly aove the saddle point energy (i.e., P quasi-regular N regime []) where! i (i ) do not satisfy the resonance condition n i= i! i ο = 0 (for any integers fn i g unless all n i 's are zero), H can e fully normalized into the Birkhoff normal form H 0 (a classical analog of Dunham Hamiltonian in quantum mechanics) y a canonical transformation (p; q)! (p 0 ; q 0 ) in terms of the Lie canonical perturation theory (LCPT). H 0 (p 0 ; q 0 )=H 0 (J 0 )=E 0 + NX i=! i J 0 i + X i;j a ij J 0 ij 0 j + X i;j;k a ijk J 0 ij 0 jj 0 k + ; () where J 0 = i(p0 =j! j j! jq 0 )= and J 0 i =(p0 i =! i +! q 0 i )= are,respectively, the `action' of the transformed reactive normal form coordinate (p 0 ;q0 ) and those of the ath normal form coordinates (p 0 ; q0 ). a ij;a ijk ;::: are coupling constants among J 0.Notethat the fully normalized Hamiltonian H 0 does not involve any `angle' variale conjugated to J 0. Thus, all J 0 are invariants of motion (i.e., H 0 is integrale) and the crossing dynamics can e solved analytically in full. As the energy of the system increases, nonlinear resonances among the ath modes ecome significant and the overlapping of these resonances gives rise to the transition to gloal chaos. Nevertheless, in the region of index one saddles, any resonance condition can never e satisfied among the pure imaginary frequency (associated with the reactive mode) and the real frequencies (with the ath modes) [0]. This leads to the existence of the following partial normal form H for a wide range of energy aove the threshold (i.e., semi-chaotic regime [, 5]); H (J ; p ; q )=E 0 +! J + NX i= p i +! i q i + f (J ; p ; q )+f (p ; q ); (3) where J = i(p =j! j j! jq )= and the functions f and f contain anharmonic terms in power of the partial normal form variales (p ; q ). In particular, f is defined so that f = 0 when J = 0. Note that Eq. (3) does not involve the `angle' variale conjugated to J, resulting in that J is an invariant of motion. However, in f and f, resonance terms can exist among the ath modes (p ; q) (i.e., H is non-integrale). Since solely the reactive dof is normalized, only (p ;q ) should e called normal form coordinate while the ath modes (p ; q ) remains to e normal mode" coordinates. Note, however, that (p ; q) are functions of the original normal mode momenta and coordinates (p; q) and so they are different from the original set. Most of the theoretical studies [4, 6, 7, 8] have focused only on the quasiregular regime. In this paper, y using a partial LCPT to normalize solely the reactive mode, we discuss how nonlinear resonances among the ath modes, which appear as energy increases aove the threshold, affect the invariance of the normal form action of the reactive mode and the state selectivity of reactions.
C.B. Li et al. 3 The partial normal form in the region of index one saddles First, let us riefly look into the equation of motion of the partial normal form H in Eq. (3): dp = + @f!! q =!! q ; dt = @H @q dq dt = @H @p = dp dt = @H @q dq dt = @H @p +! @f @J @J =! q @f @q + @g = p + @f @p @p p =! @g ; p! ; @q ; =; ;N where! (J ; p; q )=@H =@J =! + @f=@j is the normalized frequency of the reactive dof. Note here that while! 0 in the full normal form H0 is an invariant of motion,! in H varies in time through its dependence on the ath modes (p ). Nevertheless, the equation of motion of (p ) is formally the same as ; q ;q that of (p 0 ;q0) in H0 [8], and preserves the hyperolic structure on the (q ;p) plane (ecause J = constant). For instance, the condition p = q = 0 (yielding dp =dt = dq =dt =0)definesa(N 3)-dimension normally hyperolic invariant manifold (NHIM) on the (N ) dimension equienergy surface in the phase space R N. Normal hyperolicity means that the stretching and contraction rates of the motion normal to the manifold (i.e., the `reactive' dof) dominate those of the motions tangent to the manifold (i.e., the ath dof). Fig. shows the geometrical structure in the multi-dimensional phase space in the region of an index one saddle of the partial normal form Hamiltonian. Note that Eq. (4) tells us that the velocity dq =dt does not change its sign does not change during the crossing. This implies that the = 0 (except ackward reactions). Therefore, the local no-return property ofs in H is guaranteed if! does not change its sign while the system crosses the saddle. The change of sign if the sign of! system never turns ack toahypersurface S defined y q of! requires the passage of! reakdown of normal hyperolicity, that is, ( + =! @f=@j (4) =0,which corresponds to an instantaneous" (J ; p; q)) J = =0 0: It was found [9] that, as the energy increases near to the reakdown of normal hyperolicity, the system recrosses the hypersurface S much earlier than finding this condition. However, one of the most important feature of this condition is the dependency on (p ; q ), implying that some portions of the NHIM" start to reak the normal hyperolicity, not gloally at once ut locally, as energy increases. We apply the partial LCPT to the following 3-dimensional Hamiltonian, which is regarded as a prototype of isomerization reactions, H = (p + p + p 3)+a q + a q 4 + (! q +! q ) + X i=0 e ff i(q q i ) (fi i q q 3 + fl i q (q + q 3)): (5)
4 Dynamical Hierarchy in Transition States of Reactions -ij" >0 p" -ij" =0 (W + ) p" N -ij" <0 -ij" <0 q"... q'' N -ij" >0 -ij" =0 (W - ) Figure. The geometrical structure of the multidimensional phase space in the region of index one saddle (Eq. 4) and NHIM M, its stale/unstale invariant manifolds W /W + and the `local' no-return TS S. M and W =W +, respectively, correspond to the origin, and the stale/unstale directions represented y p = q =0,andp = j! jq on the ) plane with the same `internal' structure composed of the ath dof represented (q ;p y H (J = 0; p ; q ) = E (the dash ox in the figure). S for forward/ackward reactions are denoted y the two old lines along q = 0 on the (q ;p ) plane (0 <p p ;max =p ;min» p < 0) with the same internal structure of the ath dof represented y H (J = p =(! ); p ; q ) = E. As in the case of the full normal form, one can classify reactive/nonreactive trajectories in terms of the sign of ij [8].» The parameters were chosen as follows: a = 35=75;a ==875;! =;! 3 = 0:809;q 0 = ;q = q = 4:4;ff 0 = =6;ff = ff = ;fi 0 = 8;fi = fi =;fl 0 =0:75;fl = fl =. The ratio etween! and! 3, approximately the golden mean, was chosen as eing avoided from linear resonance. The imaginary frequency associated with q at the saddle is estimated as! ' 0:94i. The nonzero value of q 0 aims at avoiding specific symmetry of the potential energy function in q. The original Hamiltonian Eq. (5) is transformed to the full normal form H 0 and the partial normal form H up to the fifteenth order. In Fig., we display the full and partial normal form actions of the reactive dof J 0 (p; q) and J (p; q) evaluated y the full and partial LCPTs along three distinct classes of trajectories (namely, on a torus, in a stochastic layer and in a gloal chaotic sea) on the NHIM oeying the original Hamiltonian H(p; q) at E = +0: aove the threshold, at which the gloal chaotic regime appears. It is found that while the full normal form action J 0 (p; q) only fairly persists as invariant, the partial normal form action J (p; q) strongly persists as invariant, especially at the gloal chaotic regime (see the difference of the scale of the vertical axis and compare J with J 0 at t!±large in Fig. ). For the full normal form H 0 at slightly aove the threshold, the motions of the ath dof are integrale with constants of actions J 0 corresponding to good quantum numers in the TS. It is expected that this yields state selectivity to give rise to reaction channels linking to specific product states. However, for the partial normal form H eing nonintegrale (except only J )atthe higher
C.B. Li et al. 5 (a) -ij' (-ij" ) () -ij' (-ij" ) (c) -ij' (-ij" ) 0.0 0.0-6 -4-4 6 t -6 0.8 0.4-4 - t 0.5 0. 0.05-6 -4-4 6 t Figure. J(p; 0 q) (dashed line) and J (p; q) (solid line) at E =+0: aove the threshold (a) on a torus, () in a stochastic layer and (c) in a gloal chaotic sea on the NHIM. The insets are the Poincaré surface of section defined y q 3 = 0 and p 3 > 0. The initial conditions were prepared y q (p; q) =p (p; q) =0attimet = 0 and the trajectories were propagated ackward and forward in time y the original Hamiltonian H (eventually at t!±large the system leaves from the (approximate) NHIM ecause of the finiteness of the order of LCPTs). Note that at E ' +0:5 the normal hyperolicity of the NHIM starts to e ruined due to the nonlinear resonances etween the reactive mode q and some of the hyperolic orits densely distriuted on the NHIM, whose imaginary frequencies are (near-)commensurale with! [9]. energy regime, one may not expect the state selectivity. Fig. 3 exemplifies this: at an energy regime slightly aove the threshold where most of tori remain on the NHIM, the system still `rememers' the initial distriution of the normal mode action J in the reactant and results in the `structured' J -distriution in the product state, where the different values of J at the initial state correspond to the well-defined distinct regions of J at the final state. As the energy increases more, the system having different J at the initial state can access any region of J at the final state, yielding a `non-structured' J -distriution. Note however that, as far as the original Hamiltonian can e transformed into the partial normal form (where normal hyperolicity holds), the dynamical correlation along the reaction coordinate q 3 Outlook and Conclusions (p; q) still persist with the invariant of action J (p; q). The hierarchical structure of the phase space was presented in the region of index one saddle y introducing a partial LCPT that normalizes solely the reactive dof. Quite recently, the quantum counterpart of this generic feature for Hamiltonian systems has also een explored [, ]. Acknowledgement. Parts of this work were supported y JSPS, Grant-in-Aid for Research on Priority Area `Control of Molecules in Intense Laser Fields' of the Ministry of Education, Science, Sports and Culture of Japan, Japan Science and Technology and st century COE (Center Of Excellence) of `Origin and Evolution of Planetary Systems (Koe Univ.)', MEXT. References. Toda, M. et al. (eds.): Adv. Chem. Phys. 30 (5) and references therein.
6 Dynamical Hierarchy in Transition States of Reactions P (a) 0. P () 0.4 0 0-0. -0. -0.4 0 0. -0.4-0.4 0 0.4 0.0 0.04 J 0. 0. J Figure 3. The product distriution of the normal mode action J at E = +0:05 (a) and +0: () aove the threshold. The initial conditions are prepared as follows: we turned off the coupling among the modes in the reactant y putting ff = in Eq. (5) in order to prepare awell-defined initial condition in the reactant state. At q = 4:4, we fixed p to confirm the crossings P to the product. Then, we set an initial distriution of the action J such that 3 P ini(j ) = i= ffi(j Ji) with three initial values of J indicated y three arrows (J3 is automatically determined with the fixed J and J at a given E) with the uniform sampling for the angle variale (and 3). Then, the ensemle of trajectories of the initial conditions were evolved using the original Hamiltonian untiltheyreach the product at q =4:4. The inset are the Poincaré surface of section defined y q 3 = 0 and p 3 > 0 on the NHIM (See the figure caption of Fig. in detail).. Komatsuzaki, T., Berry, R. S.: J. Chem. Phys. 0, 960 (999); Phys. Chem. Chem. Phys., 387 (999); Proc. Natl. Acad. Sci. USA 78, 7666 (); J. Chem. Phys. 5, 405 (); J. Chem. Phys. 6, 86 (); J. Phys. Chem. A 06, 0945 (); Adv. Chem. Phys. 3, 79 (); Adv. Chem. Phys. 30A, 43 (5) 3. Toda, M.: Phys. Rev. Lett. 74, 670 (995); Phys. Lett. A 7, 3 (997); Adv. Chem. Phys. 3, 53 (); Adv. Chem. Phys. 30A, 337 (5) 4. Koon, W.S. et al.: Chaos 0, 47 (0) 5. Wiggins, S. et al.: Phys. Rev. Lett. 86, 5478 () 6. Uzer, T. et al.: Nonlinearity 5, 957 () 7. Waalkens, H. et al.: J. Chem. Phys., 607 (4) 8. Li, C.B. et al.: J. Chem. Phys. 3, 8430 (5) 9. Li, C.B. et al.: (unpulished) 0. Moser, J.: Comm. Pure Appl. Math., 57 (958). Creagh, S.C.: Nonlinearity 7, 6 (4); Nonlinearity 8, 089 (5). Cargo, M. et al.: J. Phys. A 38, 977(5)