Reaction dynamics through kinetic transition states
RReaction dynamics through kinetic transition states ¨Unver C¸ ift¸ci , and Holger Waalkens Department of Mathematics, Namık Kemal University, 59030 Tekirda˘g, Turkey Johann Bernoulli Institute for Mathematics and Computer Science,University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands (Dated: October 29, 2018)The transformation of a system from one state to another is often mediated by a bottleneck in thesystem’s phase space. In chemistry these bottlenecks are known as transition states through whichthe system has to pass in order to evolve from reactants to products. The chemical reactions areusually associated with configurational changes where the reactants and products states correspond,e.g., to two different isomers or the undissociated and dissociated state of a molecule or cluster. Inthis letter we report on a new type of bottleneck which mediates kinetic rather than configura-tional changes. The phase space structures associated with such kinetic transition states and theirdynamical implications are discussed for the rotational vibrational motion of a triatomic molecule.An outline of more general related phase space structures with important dynamical implications isgiven.
PACS numbers: 45.05.+x, 03.65.Nk, 03.65.Sq, 45.40.-j
Introduction.–
Reaction type dynamics is characterizedby the property that a system spends long times in onephase space region (the region of the ‘reactants’) and oc-casionally finds its way through a bottleneck to anotherphase space region (the region of the ‘products’). Themain examples are chemical reactions where the bottle-necks are induced by saddle points of the potential en-ergy surface which arise from a Born-Oppenheimer ap-proximation and determine the interactions between theconstituent atoms or molecules involved in the reaction.The reactions are then characterized by configurationalchanges like, e.g., in an isomerization or disscociationreaction. In this case the bottleneck is referred to a tran-sition state . The main idea of transition state theory which is the most frequently used approach to computereaction rates is then to define a surface in the tran-sition state region and compute the rate from the fluxthrough this so called dividing surface . For getting theexact rate this way, it is crucial to define the dividingsurface in such a way that it is crossed exactly once byreactive trajectories (trajectories that evolve from reac-tants to products) and not crossed at all by nonreactivetrajectories (i.e. trajectories which stay in the reactantsor products region). In the 1970’s it has been shown byPechukas, Pollak and others that for systems with twodegrees of freedom, such a dividing surface can be con-structed from an unstable periodic orbit which gives riseto the so called periodic orbit dividing surface (PODS)[1, 2]. It took several decades to understand how thisidea can be generalized to systems with three or moredegrees of freedom [3]. The object which replaces theperiodic orbit is a so called normally hyperbolic invariantmanifold (NHIM) [4]. The NHIM does not only allow forthe construction of a dividing surface with the desiredcrossing properties but it also has (like the unstable peri-odic orbit in the two-degree-of-freedom case) stable andunstable manifolds which have sufficient dimensionality to form separatrices that channel the reactive trajectoriesfrom reactants to products and separate them from thenonreactive ones [5]. In this paper we show that for sys-tems which are not of type ‘kinetic-plus-potential’, thereare saddle type equilibrium points which either insteadof the bottlenecks associated with configuration changesinduce bottlenecks of a kinetic nature or, more gener-ally, do not even induce bottlenecks but still give rise toNHIMs whose stable and unstable manifolds govern thedynamics near such equilibrium points.
The standard case.–
For a system of type ‘kinetic-plus-potential’ (so called natural mechanical systems [6]), sad-dle points of the potential lead to equilibrium points ofHamilton’s equations near which the Hamiltonian can bebrought through a suitable choice of canonical coordi-nates (so called normal form coordinates [5, 7]) into theform H = E + λ p − q ) + n (cid:88) k =1 ω k p k + q k ) + h.o.t. , (1)where f = n + 1 is the number of degrees of freedomand E is the energy of the saddle of the potential.The quadratic part of the Hamiltonian (1) consists ofa parabolic barrier in the first degree of freedom whosesteepness is characterized by λ > n harmonic oscillators with frequencies ω k > k = 1 , . . . , n .Let us ignore the higher order terms for a moment andrewrite the energy equation H = E in the form λ p + n (cid:88) k =1 ω k p k + q k ) = E − E + λ q . (2)Then one sees that for E > E (i.e. for energies above theenergy of the saddle), each fixed value of the reaction co-ordinate q defines a (2 f − S f − . a r X i v : . [ n li n . C D ] M a r The energy surface thus has the topology of a ‘sphericalcylinder’, S f − × R . This cylinder has a wide-narrow-wide geometry where the spheres S f − are ‘smallest’when q = 0 (see [8] for a more precise statement). In factthe (2 f − q = 0on the energy surface H = E > E defines a dividing sur-face which separates the energy surface into a reactantsregion q < q >
0. All forwardreactive trajectories cross it with p >
0. All backwardreactive trajectories cross it with p <
0. The condition p = 0 defines a (2 f − p > p < normally hyperbolic invariant manifold (NHIM) [4].It can be identified with the transition state: a kind ofunstable ‘super molecule’ sitting between reactants andproducts [7, 9]. The NHIM has stable and unstable man-ifolds S f − × R . They thus have one dimension lessthan the energy surface H = E . They form the phasespace conduits for reaction [10]. The NHIM is of cen-tral importance because it dominates the dynamics inthe neighborhood of the saddle. An important aspect ofthe theory of NHIMs is that they persist if the higherorder terms in (1) are taken into account and the energyis not too far above E [4]. The phase space structuresabove persist accordingly. Kinetic transition states.–
Let us now more generallyconsider the case of an equilibrium point of a Hamiltoniansystem which has one pair of real eigenvalues ± λ and n complex conjugate pairs of imaginary eigenvalues ± i ω k , k = 1 , . . . , n . Choosing normal form coordinates nearsuch a so called saddle-center- · · · -center equilibrium theHamiltonian assumes the form (1). In fact this generalcase already formed the starting point in [3, 5] and laterworks. However all these studies were restricted to thecase where the frequencies ω k are positive. In this paper we study what happens if we give up this assumption.To this end let us for simplicity consider a system with f = 2 degrees of freedom with Hamiltonian function H = E + λ p − q ) + ω p + q ) (3)with λ > ω <
0. In his case let us rewrite theenergy equation H = E in the form λ q − ω p + q ) = λ p + E − E . (4)This is the same as (2) however with the role of q and p exchanged and E − E replaced by E − E . Accordinglyfor E < E , the right hand side is positive for any p andthe energy surface has again the structure of a sphericalcylinder with a wide-narrow-wide geometry. The crucialdifference to the standard case of positive frequencies inEq. (2) is that the reaction coordinate is now p instead of q . The bottleneck is thus associated with kinetic rather m m m s φ s FIG. 1:
Definition of Jacobi coordinates for a triatomic molecule. than configurational changes. As the following exampleshows such kinetic bottlenecks do indeed exist in manyimportant applications.
Example: rotational vibrational motion of triatomicmolecules.–
The Hamiltonian of a triatomic molecule isgiven by [11, 12] H = 12 (cid:104) ρ + ρ cos φρ ρ sin φ J + 2 cos φρ sin φ J J + 1 ρ J + 1 ρ + ρ J + p + p + ρ + ρ ρ ρ ( p φ − ρ ρ + ρ J ) (cid:105) + V ( ρ , ρ , φ ) . (5)Here ( ρ , ρ , φ ) are Jacobi coordinates defined as ρ = (cid:107) s (cid:107) , ρ = (cid:107) s (cid:107) , s · s = ρ ρ cos φ , where s and s are the mass-weighted Jacobi vectors(see Fig. 1) s = √ µ ( x − x ) , s = √ µ ( x − m x + m x m + m ) , (6)computed from the position vectors x k of the atoms and the reduced masses µ = m m m + m and µ = m ( m + m ) m + m + m . (7)The momenta p , p and p φ in (5) are conjugate to ρ , ρ and φ , respectively, and J = ( J , J , J ) is the bodyangular momentum. The magnitude J of J is conservedunder the dynamics generated by the Hamiltonian (5).Let us at first consider a rigid molecule (i.e. the val-ues of ρ , ρ and φ are fixed). The body fixed frame FIG. 2:
Angular momentum sphere with contours of the Hamilto-nian in (8). can then be chosen such that the moment of inertial ten-sor becomes diagonal with the principal moments inertia M < M < M ordered by magnitude on the diagonal.The Hamiltonian (5) then reduces to H = 12 (cid:0) J M + J M + J M (cid:1) . (8)As the magnitude J of J is conserved the angular mo-mentum sphere J + J + J = J can be viewed asthe phase space of the rigid molecule [12]. The solutioncurves of this one-degree-of-freedom system are obtainedfrom the level sets of the Hamiltonian H on the angu-lar momentum sphere (see Fig. 2). The Hamiltonian H has local minima at ( J , J , J ) = (0 , , ± J ) of energy J / (2 M ), local maxima at ( J , J , J ) = ( ± J, ,
0) ofenergy J / (2 M ) and saddles at ( J , J , J ) = (0 , ± J, J / (2 M ). These correspond to the center equi-libria of stable rotations about the first and third prin-cipal axes and the saddle equilibria of unstable rotationsabout the second principal axis, respectively. A possi-ble choice of canonical coordinates ( q, p ) on the angularmomentum sphere is [12] J = (cid:112) J − p cos q, J = (cid:112) J − p sin q, J = p . (9)Since ( J , J , J ) = ( J, ,
0) resp. ( q, p ) = (0 ,
0) is amaximum of the Hamiltonian the normal form of theHamiltonian at this equilibrium is H = J / (2 M ) + ω ( p + q ) / negative frequency ω = − (cid:112) M M ( M − M )( M − M ) /M M M .Let us now consider a (flexible) triatomic molecule.For simplicity, we freeze ρ and ρ and consider thetwo-degree-of-freedom system consisting of pure bend-ing coupled with overall rotations. The potential V isthen a function of φ only and we choose it to be of theform shown in Fig. 3. This potential has two minima at φ = 0 and φ = π which correspond to two different lin-ear isomers which are separated by a barrier at φ = π/ φ -0.25-0.2-0.15-0.1-0.050 V FIG. 3:
Potential V as a function of the Jacobi coordinate φ . This type of potential occurs, e.g., in the HCN/CNH iso-merization problem [10]. We consider the equilibriumwhich for a given magnitude J of the angular momen-tum arises from the barrier at φ = π/ ω <
0. In fact alsoin the presence of coupling between the bending motionand the rotation this remains to be the case (at least fora moderate coupling strength). To illustrate the dynam-ical implication of this equilibrium we use the canonicalcoordinates ( q, p ) defined in (9) on the angular momen-tum sphere J + J + J = J and construct a Poincar´esurface of section with section condition q = 0 mod 2 π ,˙ q >
0. Using the canonical pair ( φ, p φ ) as coordinates onthe surface of section we obtain Fig. 4.We see that, as expected, there appears to be a bar-rier associated with the momentum reaction coordinate p φ . Near φ = π/ p φ > p φ < above the energy of the saddle whereas forenergies below the energy of the saddle, transitions arepossible. The rotational/vibrational motion of the tri-atomic molecule with the transition channel near φ = π/ φ near 0 and π ,respectively, coupled with rotations. For motions withthe channel near φ = π/ p φ can switchsign near φ = π/ local bottleneck for transitions between p φ > p φ < Globally such transitions always occur in thepresent example when a trajectory passes close to thecollinear configuration where φ is close to 0 or π . Thisexplains how a single trajectory can contribute points tothe lower and the upper half in Fig. 4(a) even though the a) φ -0.1-0.0500.050.1 p φ b) φ -0.1-0.0500.050.1 p φ FIG. 4:
Poincar´e surface of section for the Hamiltonian (5) withparameters J = 0 . ρ = 1 and ρ = 2 (see the text). Each pictureis generated from a single trajectory of energy 0.0205625 in (a) and0.0197526 in (b), respectively. local channel near φ = π/ More generally: mixed positive and negativefrequencies.–
Near a saddle-center- · · · -center equi-librium the Hamiltonian can always be brought into theform (1). The NHIM at energy E is then obtained fromthe intersection of the center manifold of the equilibriumgiven by q = p = 0 and the energy surface H = E , i.e. n (cid:88) k =1 ω k p k + q k ) + h.o.t. = E − E (10)All studies on the geometric theory of reactions stud-ied so far [13] concern the case of positive frequencies ω k . This corresponds to the Hamiltonian restricted tothe center manifold having a minimum. If all frequen-cies are negative then the Hamiltonian restricted to thecenter manifold has a maximum. In this case the NHIMis again a (2 f − p rather than q is the reaction coordinate as discussed in the presentexample for f = 2. Also the case of mixed signs of the ω k occurs in many applications. It, e.g., also occurs in ourexample of the rotational-vibrational motion of triatomicmolecule if we take the other vibrational degrees of free-dom into account which we for simplicity considered tobe frozen. In the case of mixed signs of the ω k the NHIMis not a sphere but a non-compact manifold. Although no bottleneck in the energy surface is induced in thesecases the NHIM has important dynamical implicationsas it has stable and unstable manifolds which are of onedimension less than the dimension of the energy surfaceand hence form impenetrable barriers. This bears somesimilarities to the case of non-compact NHIM’s that haverecently been considered for rank 2 and higher rank sad-dles [14, 15]. The case of mixed positive and negativefrequencies will form an interesting direction for futureresearch. [1] P. Pechukas and F. J. McLafferty. On transition-statetheory and the classical mechanics of collinear collisions. J. Chem. Phys. , 58:1622–1625, 1973.[2] P. Pechukas and E. Pollak. Transition states, trappedtrajectories, and classical bound states embedded in thecontinuum.
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