Phase space barriers and dividing surfaces in the absence of critical points of the potential energy
aa r X i v : . [ phy s i c s . c h e m - ph ] N ov Phase space barriers and dividing surfaces in the absence ofcritical points of the potential energy
Gregory S. Ezra ∗ Department of Chemistry and Chemical BiologyBaker LaboratoryCornell UniversityIthaca, NY 14853USA
Stephen Wiggins † School of MathematicsUniversity of BristolBristol BS8 1TWUnited Kingdom (Dated: October 29, 2018)
Abstract
We consider the existence of invariant manifolds in phase space governing reaction dynamics insituations where there are no saddle points on the potential energy surface in the relevant regionsof configuration space. We point out that such situations occur in a number of important classesof chemical reactions, and we illustrate this concretely by considering a model for transition stateswitching in an ion-molecule association reaction due to Chesnavich (J. Chem. Phys. , 2615(1986)). For this model we show that, in the region of configuration space relevant to the reaction,there are no saddle points on the potential energy surface, but that in phase space there is anormally hyperbolic invariant manifold (NHIM) bounding a dividing surface having the propertythat the reactive flux through this dividing surface is a minimum. We then describe two methodsfor finding NHIMs and their associated phase space structures in systems with more than twodegrees-of-freedom. These methods do not rely on the existence of saddle points, or any otherparticular feature, of the potential energy surface. . INTRODUCTION Critical points of the potential energy surface have played, and continue to play, a sig-nificant role in how one thinks about transformations of physical systems . The term‘transformation’ may refer to chemical reactions such as isomerizations or the analogueof phase transitions for finite size systems . A comprehensive description of this so-called ‘energy landscape paradigm’ is given in ref. 2. The energy landscape approach is anattempt to understand dynamics in the context of the geometrical features of the potentialenergy surface, i.e., a configuration space approach. However, the arena for dynamics isphase space , and numerous studies of nonlinear dynamical systems have taught us thatthe rich variety of dynamical behavior possible in nonlinear systems cannot be inferred fromgeometrical properties of the potential energy surface alone. (An instructive example is thefact that the well-studied and nonintegrable H´enon-Heiles potential can be obtained by se-ries expansion of the completely integrable Toda system .) Nevertheless, the configurationspace based landscape paradigm is physically very compelling, and there has been a greatdeal of work over the past ten years describing phase space signatures of index one saddles of the potential energy surface that are relevant to reaction dynamics (see, for example, refs22–24). More recently, index two and higher index saddles have been studied.The work on index one saddles has shown that, in phase space , the role of the saddle point is played by an invariant manifold of saddle stability type, a so-called normally hyperbolicinvariant manifold or NHIM . The NHIM proves to be the anchor for the constructionof dividing surfaces that have the properties of no (local) recrossing of trajectories andminimal (directional) flux . There is an even richer variety of phase space structures andinvariant manifolds associated with index two saddles of the potential energy surface, andtheir implications for reaction dynamics are currently under investigation . Fundamentaltheorems assure the existence of these phase space structures and invariant manifolds fora range of energy above that of the saddle . However, the precise extent of this range, aswell as the nature and consequences of any bifurcations of the phase space structures andinvariant manifolds that might occur as energy is increased, is not known and is a topic ofcurrent investigation .While work relating phase space structures and invariant manifolds to saddle points onthe potential energy surface has provided new insights and techniques for studying reaction2ynamics , it certainly does not exhaust all of the rich possibilities of dynamical phe-nomena associated with reactions. In fact, recent work has called into question the utilityof concepts such as the reaction path and/or transition state . Of particular interest forthe present work is the recognition that there are important classes of chemical reaction,such as ion-molecule reactions and association reactions in barrierless systems, for which thetransition state is not necessarily directly associated with the presence of a saddle point onthe potential energy surface (or even the amended potential, which includes centrifugal con-tributions to the energy ). The phenomenon of transition state switching in ion-moleculereactions provides a good example of the dynamical complexity possible in such systems.The lack of an appropriate critical point on the potential energy surface with which toassociate a dividing surface separating reactants from products in such systems does not however mean that there are no relevant geometric structures and invariant manifolds in phase space . In this paper we discuss the existence of NHIMs, along with their stableand unstable manifolds and associated dividing surfaces, in regions of phase space that donot correspond to saddle points of the potential energy surface. After presenting a simpleexample motivated by Chesnavich’s model for transition state switching in an ion-moleculeassociation reaction , we describe a theoretical framework for describing and computingsuch NHIMs. Like the methods associated with index one and two saddles, the method wedevelop for realizing the existence of NHIMs is based on normal form theory; however, ratherthan normal form theory for saddle-type equilibrium points of Hamilton’s equations (whichare the phase space manifestation of index one and two saddles of the potential energysurface), we use normal form theory for certain hyperbolic invariant tori. The hyperbolicinvariant tori (and their stable and unstable manifolds) alone are not adequate, in termsof their dimension, for constructing NHIMs that have codimension one stable and unstablemanifolds (in a fixed energy surface). However, by analogy with the use of index one saddlesto infer the existence of NHIMs (together with their stable and unstable manifolds, and otherdividing surfaces having appropriate dimensions), these particular hyperbolic invariant torican likewise be used to infer the existence of phase space structures that are appropriate fordescribing reaction dynamics in situations where there is no critical point of the potentialenergy surface in the relevant region of configuration space.Section II discusses our simplified version of Chesnavich’s model for transition stateswitching . For this 2 DoF system, we exhibit a NHIM (in this case, an unstable peri-3dic orbit) that is the rigorous dynamical manifestation of the mininimal flux surface ofvariational transition state theory . In Section III we describe a (time-dependent) nor-mal form based approach for finding such NHIMs in phase space. In particular, we presenttwo variations of the method. In Section III A we consider systems where (to leading order)the system can be separated into a two degree-of-freedom (DoF) subsystem and a collectionof decoupled “bath modes”. We assume that there exists a hyperbolic periodic orbit in the2 DoF subsystem, and show that this can be used to construct a hyperbolic torus for thefull system. We then show that this hyperbolic torus implies the existence of a NHIM, withstable and unstable manifolds that are codimension one in the energy surface. Appropriatedividing surfaces can then be constructed using the NHIM and the normal form Hamilto-nian. In Section III B we describe a method which requires knowledge of the appropriatehyperbolic invariant torus from the start. The advantage of the first method is that it ismore intuitive and can exploit the considerable number of methods for locating hyperbolicperiodic orbits in 2 DoF Hamiltonian systems. Method two is more general, but at presentthere are few techniques available for locating hyperbolic invariant tori of the appropriatedimension in general N DoF Hamiltonian systems that are not perturbations of integrablesystems. Section IV concludes. 4
I. A MOTIVATING EXAMPLE: VARIATIONAL TRANSITION STATE FOR AMODEL BARRIERLESS REACTIONA. Introduction
The conventional approach to variational transition state theory (VTST) for barrierlessreaction proceeds by minimizing the reactive flux with respect to variation of some reactioncoordinate chosen a priori . The value of the reaction coordinate so determined is there-fore the location of a flux bottleneck, which is identified with the transition state for theparticular association reaction. An invariant phase space characterization of such variation-ally determined dividing surfaces is highly desirable; for N = 2 DoF, such transition stateswill presumably be associated with unstable periodic orbit dividing surfaces (PODS) ,or, more generally, with NHIMs ( N ≥ . B. Model Hamiltonian
We consider a highly simplified model for a barrierless association reaction (cf. ref. 43).The system has 2 DoF: a radial coordinate r , identified as the reaction coordinate, and acoordinate s describing vibrations transverse to the reaction coordinate. The radial poten-tial has the character of a long-range attractive ion-neutral interaction, while the potentialtransverse to the reaction coordinate is harmonic. The system is nonseparable by virtue ofa dependence of the harmonic oscillation frequency on the coordinate r : H = p r p s − αr + 12 ω ( r ) s . (1)We take ω ( r ) to have the form ω ( r ) = ω e − βr (2)so that, for β >
0, the transverse vibration stiffens as r decreases.A contour plot of the potential v ( r, s ) = − αr + 12 ω ( r ) s (3)for parameter values α = 1, β = 1, ω = 8 is shown in Fig. 1.5 . Locating the bottleneck For the 2 DoF Hamiltonian (1), we can compute the action of the transverse vibrationalmode as a function of the coordinate r at fixed energy E : I ( r ; E ) = h E + αr i ω ( r ) = h E + αr i e βr ω . (4)As r decreases, there are two competing tendencies: • Decreasing r increases the amount of energy in the oscillator degree of freedom, therebytending to increase the action. • Decreasing r increases the frequency ω ( r ), tending to decrease the action of the tran-verse mode.The competition between these two trends can therefore result in the existence of a minimum in the action as a function of r : see Figure 2.The minimum of the action as a function of r corresponds to an extremum of the sum ofstates (phase space area) or flux as a function of r , and hence is interpreted as a bottleneck.In the variational transition state approach, the transition state for association is thenlocated at the value r = r ∗ corresponding to minimum flux. This bottleneck corresponds toan inner or “tight” transition state . D. Intrinsic characterization of variational TS: PODS
The formulation of VTST outlined above for the model association reaction is unsat-isfactory in that the minimum flux bottleneck so determined has no intrinsic dynamicalsignificance. It is natural to seek a dynamical, phase space based characterization of thevariational TS. For 2 DoF systems, transition states are identified as PODS . The invari-ant object defining the TS is a hyperbolic (unstable) periodic orbit; the 1D periodic orbitforms the boundary of a 2D dividing surface on the 3D energy shell in phase space, which isthe transition state. The minimal flux (local no-recrossing) property of the TS follows fromthe principle of stationary action .A search using the model potential (3) reveals the presence of a PODS in the vicinityof r ≃ r ∗ (the periodic orbit at E = 1 is shown in Fig. 1) . This PODS is the rigorousdynamical realization of the variational TS in this simple case.6 II. LOCATING NHIMS WHEN THERE ARE NO (RELEVANT) SADDLES ONTHE POTENTIAL ENERGY SURFACE
In this section we describe two methods for locating NHIMs of the type discussed inthe previous section. These methods are inherently phase space approaches, based on theexistence of a hyperbolic invariant torus solution of Hamilton’s equations. Normal formtheory for hyperbolic invariant tori can be used to provide “good coordinates” for computingexplicit formulae for a NHIM, its stable and unstable manifolds, and dividing surfaces in thephase space vicinity of the hyperbolic invariant torus on which we base our method, in muchthe same way that it is used to compute similar objects associated with index one saddlesof the potential energy surface . A large literature for normal form theory associatedwith invariant tori of Hamilton’s equations has been developed over the past twenty years,see ref. 53 for an overview, and ref. 54 for a survey of the issues associated with bifurcationof tori in Hamiltonian systems. For our present purposes we use the results contained inref. 55, which explicitly discusses the relevant normal form and also clarifies the issue of“hyperbolicity” of tori in Hamiltonian systems (concerning which there had previously beensome confusion in the literature).
A. Method 1: a relevant 2 DoF subsystem can be identified at leading order
Consider a Hamiltonian of the following form: H = p r p s V ( r, s ) + 12 n − X i =1 ω i (cid:0) u i + v i (cid:1) + f ( r, s, u , . . . , u n − , p r , p s , v , . . . , v n − ) (5)where f ( r, s, u , . . . , u n − , p r , p s , v , . . . , v n − ) is at least order 3, denoted O ( r, s, u , . . . , u n − , p r , p s , v , . . . , v n − ). In general this term serves to couple all ofthe variables, but we have written the Hamiltonian in such a way that we can identify aclearly defined 2 DoF subsystem, on which we make the following assumption: Assumption:
The 2 DoF subsystem defined by the Hamiltonian: H = p r p s V ( r, s ) , (6)has a hyperbolic periodic orbit, denoted P = ( r ( t ) , s ( t ) , p r ( t ) , p s ( t )).To construct a NHIM for (5) we proceed as follows.7 tep 1: Transform the 2 DoF subsystem to normal form in a neighborhood ofthe periodic orbit. Following ref. 55, we can find an invertible transformation T (as smooth as the Hamil-tonian) defined in a neighborhood of the periodic orbit T : ( r, s, p r , p s ) T ( r, s, p r , p s ) ≡ ( I, θ, x, y ) (7a) T − : ( I, θ, x, y ) T − ( I, θ, x, y ) ≡ ( r, s, p r , p s ) (7b)such that the 2 DoF Hamiltonian takes the form: K = ωI + λxy + O ( I ) + O ( I, x, y ) , (8)where we can take ω, λ >
0. Of course, nontrivial calculations are required in going from(6) to (8). In particular, after the hyperbolic periodic orbit is located, a time-dependenttranslation to “center” the coordinate system on the periodic orbit must be carried out; theresulting Hamiltonian is then Taylor expanded about the origin (i.e., the periodic orbit),a Floquet-type transformation constructed to make (to leading order) the dynamics in thenormal direction to the periodic orbit constant (i.e., λ is constant in (8)), then, finally,Hamiltonian normal form theory is applied to the result. Details of the methodology forcarrying out this procedure for specific examples are described in refs 56,57. For the purposesof demonstrating the existence of a NHIM, we only need to know that such transformationscan in principle be carried out. Step 2: Use the normal form transformation for the 2 DoF subsystem to rewrite (5) . We use the normal form transformation of the 2 DoF subsystem to express (5) as follows:¯ H = ωI + λxy + 12 n − X i =1 ω i (cid:0) u i + v i (cid:1) + F ( u , . . . , u n − , v , . . . , v n − , I, θ, x, y ) + O ( I ) + O ( I, x, y ) (9)where F ( u , . . . , u n − , v , . . . , v n − , I, θ, x, y ) = f ( r ( I, θ, x, y ) , s ( I, θ, x, y ) , u , . . . , u n − , p r ( I, θ, x, y ) , p s ( I, θ, x, y ) , v , . . . , v n − ) . (10) Step 3: Use Hamiltonian (9) to conclude the existence of a NHIM. I = 12 ( w + z ) , θ = tan − (cid:16) zw (cid:17) . (11)We then rewrite (9) as H = λxy + ω w + z ) + 12 n − X i =1 ω i (cid:0) u i + v i (cid:1) + F ( u , . . . , u n − , v , . . . , v n − , I ( w, z ) , θ ( w, z ) , x, y ) + O ( I ( w, z )) + O ( I ( w, z ) , x, y ) . (12)By construction, H (0) = 0. Neglecting higher order terms in (12), we obtain: H trunc = λxy + ω w + z ) + 12 n − X i =1 ω i (cid:0) u i + v i (cid:1) . (13)If we set x = y = 0, then on the energy surface H trunc = h > ω w + z ) + 12 n − X i =1 ω i (cid:0) u i + v i (cid:1) = h > , (14)is a normally hyperbolic invariant 2 n − n -dimensional space with coordinates( u , . . . , u n − , v , . . . , v n − , w, z, x, y ), having 2 n − n − h = 0 (cf. ref. 30).At this point we are in a position where normal form theory can be used on (12) toconstruct a new set of coordinates (the normal form coordinates) in which (12) assumesa particularly simple form that results in explicit formulae for the NHIM, its stable andunstable manifolds, and dividing surfaces between regions of the phase space correspondingto reactants and products. The normal form algorithm also provides the transformation fromthe original physical coordinates and its inverse, and this allows us to map these surfacesback into the original physical coordinates, as described in ref. 24. B. Method 2: A hyperbolic torus of dimension n − can be located in an n DoFsystem.
The advantage of method 1 is that it can make use of extensive prior work on locat-ing periodic orbits, and determining their stability, in 2 DoF systems. In method 2 the9tarting point is knowledge of the existence of a hyperbolic torus of dimension n − n -dimensional phase space, with the frequencies on the torus satisfying a diophantinecondition . In this case, it follows from ref. 55 that an invertible transformation of coordi-nates, valid in a neighborhood of the torus, can be found where the system has the followingform: H = ω I + . . . + ω n − I n − + λxy + O ( I , . . . , I n − ) + O ( I , . . . , I n − , x, y ) . (15)Now let I i = 12 ( u i + v i ) , θ i = tan − (cid:18) v i u i (cid:19) , i = 1 , . . . , n − . (16)In terms of these coordinates the Hamiltonian (17) has the form:¯ H = 12 n − X i =1 ω i ( u i + v i ) + λxy + O ( u + v , . . . , u n − + v n − ) + O ( u + v , . . . , u n − + v n − , x, y )(17)We now proceed exactly as for method 1. Neglecting the terms in (17) of order 3 andhigher gives: ¯ H trunc = 12 n − X i =1 ω i ( u i + v i ) + λxy. (18)If we set x = y = 0, then on the energy surface ¯ H trunc = h > n − X i =1 ω i ( u i + v i ) = h > , (19)is a normally hyperbolic invariant 2 n − n -dimensional space with coordinates( u , . . . , u n − , v , . . . , v n − , x, y ) having 2 n − n − h = 0.Method 2 relies on first finding an appropriate hyperbolic invariant torus, and in generalthe existence of this object will need to be verified numerically. There has been a great dealof activity developing such numerical methods in recent years. See, for example, refs 59–66.Finally, it is also worth noting the difference between the h = 0 limit for index 1 saddleson the potential energy surface compared with the torus case in phase space. In the formercase the NHIM shrinks down to a point on the potential energy surface (i.e. configurationspace), while in the latter case it shrinks down to a torus in phase space.10 V. SUMMARY AND CONCLUSIONS
In this paper we have exhibited a normally hyperbolic invariant manifold (NHIM , in thiscase a PODS , or unstable periodic orbit) defining a flux bottleneck in a simple modelof an ion-molecule reaction, for which there is no associated critical point of the potentialenergy surface. We have also developed a theoretical framework for showing the existenceof such NHIMs. Two methods were described that are in principle suitable for computingsuch phase space objects in the multidimensional case.For index one saddles a software package has been developed that allows one to com-pute the normal form associated with the corresponding saddle-center- . . . -center stabilitytype equilibrium point to high order for multi-dimensional systems, with control over theaccuracy . Accuracy is assessed by a battery of tests, and specifying an accuracy mayaffect the order of the normal form that can be computed as well as the dimensionality ofthe system that can be treated. In general, these issues must be analyzed on a case-by-casebasis. Nevertheless, the normal form, and most importantly the transformation from theoriginal physical coordinates and its inverse, allow us to realize the NHIM, its stable andunstable manifolds, and dividing surfaces between regions of the phase space correspondingto reactants and products. Moreover, flux through the dividing surfaces can be computedas an integral over the NHIM, and the normal form coordinates provide a natural way ofselecting distributions of initial conditions of trajectories on the dividing surfaces to com-pute gap times . It should be possible to develop similar software for computing normalforms associated with hyperbolic tori of the type discussed above, and one expects that thenormal form will, similarly, allow one to realize phase space structures relevant to reactiondynamics as well as to compute fluxes and sample distributions of initial conditions.This program has yet to be carried out, but the essential computational elements of theapproach can be found in refs 56,57. Such capabilities would be very useful for the study ofreaction dynamics in multimode systems exhibiting transition state switching, for exampleref. 42. 11 cknowledgments SW acknowledges the support of the Office of Naval Research Grant No. N00014-01-1-0769. GSE and SW both acknowledge the stimulating environment of the NSF sponsoredInstitute for Mathematics and its Applications (IMA) at the University of Minnesota, wherethis work was begun. ∗ [email protected] † [email protected] P. G. Mezey,
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