Page 1

1 A New Look at the

Transition State: Wigner’s

Dynamical Perspective

Revisited

CHARLES JAFF´E

Department of Chemistry, West Virginia University, Morgantown, WV

26506-6045, USA

SHINNOSUKE KAWAI

Department of Chemistry, Graduate School of Science, Kyoto University,

Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan

JES´US PALACI´AN and PATRICIA YANGUAS

Departamento de Matem´ atica e Inform´ atica, Universidad P´ ublica de Nava-

rra, 31006 - Pamplona, Spain

T. UZER

Center for Nonlinear Sciences, School of Physics, Georgia Institute of Tech-

nology, Atlanta, GA 30332-0430, USA

i

Page 2

Contents

1 A New Look at the Transition State: Wigner’s Dynamical

Perspective Revisited

Charles Jaff´ e

i

Shinnosuke Kawai

Jes´ us Palaci´ an and Patricia Yanguas

T. Uzer

1.1

1.2

1.3

Introduction: Wigner’s “Three Threes”

Brief History of Transition State Theory

Saddles in Energy Landscapes

1.3.1 Phase Space Versus Coordinate Space

1.3.2 Stability

1.3.3Energy Landscapes in Phase Space

Phase-Space Structure Around a Simple (Rank-One) Saddle

1.4.1Justification of the n-Degree-of-Freedom Hamiltonian

1.4.2Finding the “Apt” Coordinates Around a Saddle Using

a Normal Form

1.4.3Normally Hyperbolic Invariant Manifolds (NHIMs) and

their Stable and Unstable Manifolds

1.4.4 The Transition State

1.4.5Searching for the Transition State and Other Phase-

Space Structures

1.4.6Flux Through the Transition State

Normalization by Lie Transformations

1.5.1Lie Transformations

1.5.2 Dynamics Near the Transition State Using the Normal

Form Coordinates

The Isomerization of HCN

1.6.1 The Model System

1.6.2The Hamiltonian

1

2

8

9

10

11

12

12

1.4

13

15

19

21

22

23

23

1.5

26

28

28

30

1.6

ii

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CONTENTS

iii

1.6.3

1.6.4

1.6.5

1.6.6

1.6.7

Points of Stationary Flow

Preparation for Transformation to the Normal Form

Transformation to the Normal Form

Visualization of the HCN Dynamics

The Quantization of the Non-reactive Degrees of Free-

dom

Summary and Outlook

32

34

35

36

41

421.7

References45

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1

1.1INTRODUCTION: WIGNER’S “THREE THREES”

The proper departure point for a discussion of the Transition State are the

proceedings of the lively 67th General Discussion of the Faraday Society [1]. In

his admirable summary of this Discussion, given during the Spiers Memorial

Lecture of the 110th Faraday Discussion of 1998 [2], W. H. Miller recounts how

two distinct points of view emerged on rates of chemical reaction from the dis-

cussions of the 67th meeting: Eyring’s thermodynamic picture and Wigner’s

dynamical perspective, which, in the decades between the 1930’s and 1970’s

was buried by the enormous numbers of applications of the thermodynamic

picture [3]. In his perceptive paper [4], Wigner gave a clear outline of the sub-

ject in terms of his “Three Threes” [2]. First there were the three steps in the

theory of kinetics: (1) Constructing the potential energy surface, (2) calcu-

lating the rates of elementary reactions, and (3) combining many elementary

reactions into a complex reaction mechanism. Next came the three groups

of elementary reactions, and finally, were the three assumptions of Transition

State Theory (TST): (1) No electronically non-adiabatic transitions (2) va-

lidity of classical mechanics for the nuclear motion, and (3) the existence of

a dividing surface, separating the reactants and products, that no classical

trajectory passes through more than once. Wigner noted that the failure of

the last assumption will lead, in general, to values of the reaction rate that

are too large. Wigner’s formulation quickly leads to the recognition that the

Transition State (TS) is actually a general property of all dynamical systems,

provided that they evolve from “reactants” to “products”. The TS, therefore,

is not confined to chemical reaction dynamics [2], but it also controls rates

in a multitude of interesting systems, including, e.g., the rearrangements of

clusters [5], the ionization of atoms [6,7], conductance due to ballistic elec-

tron transport through microjunctions [8], diffusion jumps in solids [9], and

statistical rates of asteroid capture [10]. With the reemergence of Wigner’s

dynamical approach to TST [4,11,12] the search for these no-recrossing sur-

faces has been pursued vigorously, leading researchers to dynamical systems

theory [13,14] through the intermediate stages of variational TST [15] and

PODS [16, 17]. Despite this effort, the no-recrossing rule has been “more

honored in its breach than its observance.” The formalism presented here ad-

dresses this issue by constructing the dynamically correct higher-dimensional

geometrical structures in phase space that regulate transport between quali-

tatively different states (“reactants” and “products”). The salient features of

this new formulation are that

• It is a phase space rather than configuration space theory, so it can

treat Hamiltonian systems containing unconserved angular momenta

like Coriolis interactions which prevent the Hamiltonian from being writ-

ten as a sum of the kinetic and potential energies [6,18]. The resulting

hypersurfaces are dynamical in that they involve momenta as well as

coordinates.

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2

• It is designed for multidimensional systems. This is a qualitative dif-

ference from the existing formulations which attempt to extrapolate to

three or more degrees of freedom geometrical methods that work for

systems with two degrees of freedom (2dof).

• The Transition State it produces is locally a surface of no return.

• It identifies impenetrable barriers [19] which, by acting as phase-space

“scissors” [20], cut phase space into hypervolumes of initial conditions

that are destined to react and those which cannot.

• It is a “top-down” approach in providing explicit recipes for all these

geometrical structures, and is presumably equivalent to the “bottom-up”

approach of working through their manifestations in terms of ensembles

of trajectories [5].

Details of our approach, which is reviewed in Sections 4 and 5, can be found

in the original article [21]. Two of the most striking features of our solution,

namely the recrossing-free TS and the identification and construction of the

multidimensional separatrices, are illustrated in Figs. 1.1,1.2, and 1.3.

This account is organized as follows: After briefly reviewing the history

of TST, we present a pedagogical discussion of the geometry of the TS. This

is followed by the rigorous mathematical theory of the geometry of higher-

dimensional saddles and their associated separatrices in the context of the

theory of Normally Hyperbolic Invariant Manifolds (NHIMs) [22]. We begin

by showing that near an equilibrium point consisting of a reactive direction

and several non-reactive (’bath’) directions (technically of the saddle ⊗ cen-

ter ⊗···⊗ center stability type) the Hamiltonian can be transformed to a

normal form [13] from which the Normally Hyperbolic Invariant Manifold, its

stable and unstable manifolds, and the TS are straightforward to obtain an-

alytically. Moreover, if the center (or bath) vibrations are non-resonant, the

normal form, truncated at any finite order, is integrable. The theory leads

to an algorithm for identifying the TS (and other geometrical structures) an-

alytically. Finally, we apply this formalism to the isomerization of HCN. In

this process we identify the center manifold as the activated complex and

demonstrate how it can be quantized to obtain the quantized thresholds first

discussed by Chatfield et al. [23] and subsequently observed [24,25].

1.2 BRIEF HISTORY OF TRANSITION STATE THEORY

The idea of the existence of a boundary between “reactants” and “products”

can be traced to the scientific memoirs of Marcelin published in 1915 [26].

It was not until 1931 that this idea began to percolate into the thinking of

the chemistry community. In that year Eyring and Polanyi published their

seminal paper on the calculation of the absolute reaction rate for the collinear

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BRIEF HISTORY OF TRANSITION STATE THEORY

3

Fig. 1.1

crossed electric and magnetic fields [18,21]. The line passing through the saddle point

shows the projection of the conventional transition state which is a vertical plane.

Of course the trajectory can cross this plane many times. We cannot display our

true Transition State which is a complicated four-dimensional object, but note that

it is intersected only once, namely at the dot. The right-hand panels show the same

trajectory in the three Normal Form coordinates (see Section 1.6.6).

(in color) The trajectory of an electron ionizing under the influence of

H + H2 reaction [27]. It was in this paper, which must be viewed as the

origin of the modern theory of chemical reactions, that the concept of a TS

separating reactants from products is first quantified. They defined it in terms

of the morphology of the potential energy surface.

The impact of this idea was immense. Six years later, the 67thGeneral

Discussion of the Faraday Society addressed this subject under the guise of

Reaction Kinetics [1]. This Discussion, which has been alluded to above,

set the stage for the further development of these ideas and the general ac-

ceptance of them by the chemistry community at large. By this point two

distinct approaches to TST had developed. The first one, primarily due to

Eyring [3,28], was based in thermodynamics. Here the idea was to develop the

quantities of interest from a thermodynamic perspective and then to evaluate

the thermodynamic quantities in terms of simple molecular models. The sec-

ond approach, advocated by Wigner [4,11,12,29], was to calculate quantities

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4

Fig. 1.2

of the previous figure. The projection of the conventional Transition State in configu-

ration space is the vertical axis. The trajectory crosses it many times. Our new TS is

intersected only once, namely at the dot in the upper right-hand side of the ellipse.

(in color) Two-dimensional configuration-space projection of the trajectory

of interest directly from the dynamics. When properly implemented, both

approaches are expected to be equivalent.

In the decades following this Discussion the majority of the progress was

made in the development and application of TST followed the thermodynamic

path [30,31]. To a large extent this was due to the nature of the difficulties

encountered when following the dynamical path. The first of these was the fact

that the technology needed to numerically investigate the dynamics simply did

not exist. Furthermore, the theory of dynamical systems, despite the efforts

of Poincar´ e [32], was still in a primitive state. In the early 1970’s, when

interest in the classical mechanics was rekindled by the quest for quantum

chaos [33, 34], attention was again focused upon the dynamical version of

TST. The variational of TST together with the identification of the “periodic

orbit dividing surfaces” or PODS (both singular and plural) [16,17,35–37]

were just the initial steps in this reawaking of interest.

In the central idea in the variational treatment of TST [15,38,39] is to

consider all possible dividing surfaces that partition coordinate space into two

separate regions, one associated with reactants and the other with products.

One then considers the flux across each of these surfaces and chooses the

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BRIEF HISTORY OF TRANSITION STATE THEORY

5

Fig. 1.3

as one, approach the ionization saddle [6] from the top right. After some complex

dynamics at the saddle point, one reacts (by crossing the Transition State at the dot

in the upper left-hand corner of the bundle) and goes off to the top left, and the non-

reactive one returns to the bottom right. We engineered this outcome by selecting

their initial conditions on opposite sides of the impenetrable phase-space barrier (one

of the “scissors” [20]) between ionizing and non-ionizing hypervolumes.

(in color) Two electron trajectories, so close in phase space that they appear

surface with the minimum flux as the TS. The logic is as follows: We are

interested in the rate at which states cross the TS. If, during the course of a

reaction, the trajectory of each initial reactant state only crosses the TS once,

then the rate of reaction can be determined by simply counting the number

of crossings of the transition state, that is, the flux across the TS. However, if

there are recrossings of the TS, then simply counting the number of crossings

will over-estimate the true rate.

Pechukas argued that there exists a set of periodic orbits whose projections

into coordinate space, which he calls PODS, are solutions to the variational

problem [35,36,40]. The projections of these orbits touch the equipotentials

of the potential energy at two different points. At each of these points the

trajectory is reflected and retraces its path in coordinate space. Pechukas

recognized that the principle of least action for such a periodic orbit im-

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6

plied that it was a solution to the variational TS problem. Pechukas’ work

is particularly significant, despite shortcomings due to the formulation of the

variational principle in coordinate space, since it refocused attention on the

dynamical aspect of TST. The shortcomings of Pechukas’ approach are re-

flections of problems with the formulation of variational principle. The most

fundamental of these is that the TS is defined in coordinate space and not in

the state space, that is, phase space. While this does not pose significant dif-

ficulties for systems having just two degrees of freedom, it has prevented the

extension of Pechukas’ results to higher-dimensional systems. Recently, Jaff´ e

et al. [6,18] have shown how to apply Pechukas’ approach to the ionization of

hydrogen atom in crossed static electric and magnetic fields. Due to the pres-

ence of the magnetic field, the Hamiltonian of this system cannot be divided

in the kinetic and potential energy and, consequently it would appear that

Pechukas’ formalism, which requires the existence of a potential energy, will

fail. However, Jaff´ e et al. [6,18] show that in this case Pechukas’ approach

works provided the dynamics are projected into a different (two-dimensional)

plane.

In the 1970’s with the introduction of more sophisticated computer technol-

ogy, one of the primary difficulties facing the scientist wishing to investigate

the dynamics of realistic model systems vanished. It now became possible not

only to investigate numerically the dynamics of reactive systems but it also

become possible to implement a number of important tools such as Poincar´ e’s

Surface of Section. This set the stage for the next development, that of the

advancement of the theory of dynamical systems. Many examples can be

cited. Of particular interest is the work of MacKay, Meiss and Percival [41]

concerning the existence of cantori acting as space barriers in the vicinity of

the last surviving torus. Davis and co-workers applied these ideas to chemical

reactions [42–44]. Similarly, Jaff´ e and Tiyapan [45–47] demonstrated that in

the formation of complexes, the homoclinic tangle associated with the peri-

odic orbit defining the transition state imposes an invariant fractal structure

upon the energy shell and that the asymptotic scaling laws of this fractal are

related to the long time behavior. Unfortunately, these, and many other ef-

forts, rely heavily on techniques only applicable to systems with two degrees

of freedom and consequently are not readily extendable to systems with more

degrees of freedom.

Despite these difficulties significant progress was made. In particular, it was

observed that something remarkable happens to the dynamics near a saddle.

No matter how complicated the motion is leading up to the saddle, at the

saddle it becomes simpler only to become more complicated once more as the

system leaves the saddle region. The reason for this striking simplification has

been explained by Miller [48]: Simply put, at a saddle, the potential consists

of an inverted parabola in the reaction coordinate and ordinary parabolas in

the bath directions. Technically speaking, the frequency is imaginary in the

reactive direction and real in the bath directions, and since there cannot be a

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BRIEF HISTORY OF TRANSITION STATE THEORY

7

resonance between imaginary and real frequencies, whatever couplings there

may be, cannot be effective, thereby isolating the reactive direction from

the bath modes, and in simple cases, making the dynamics in the reaction

direction integrable [48–51].

The dynamics near the TS have been the subject of many theoretical ( [52]-

[66]) and experimental [24,25] investigations. The experiments of Lovejoy et

al. [24,25] see the TS via the photofragment excitation spectra for unimolec-

ular dissociation of excited ketene. They have shown that, in the vicinity

of the barrier, the reaction rate is controlled by the flux through quantized

thresholds. The observability of quantized thresholds in the TS was first dis-

cussed by Chatfield et al. [23]. Marcus pointed out that this indicates that

the transverse vibrational quantum numbers might indeed be approximate

constants of the motion in the saddle region [60].

During the same period, Berry and coworkers were exploring the non-

uniformity of the dynamical properties of Hamiltonian systems representing

atomic clusters with up to 13 atoms. In particular, they explored how regular

and chaotic behavior may vary locally with the topography of the potential en-

ergy surfaces (PES) [53,54,57,58,61–63,65,66]. By analyzing local Lyapunov

functions and Kolmogorov entropies, they showed that when systems have

just enough energy to go through the TS, the system’s trajectories become

collimated and regularized through the TS regions, developing approximate

local invariants of the motion different from those in the potential well. This

happens even though the dynamics in the potential well is fully chaotic under

these conditions. They also showed that at higher energies above the thresh-

old, intermode mixing wipes out these approximate invariants of the motion

even in the region of the TS [5].

Recently, Komatsuzaki and coworkers have investigated regularity observed

in the vicinity of the TS in many-body systems [5,67–74]. They used Lie

transformations [75, 76] together with microcanonical molecular dynamics

simulations of the region near a potential energy saddle point. They con-

struct a non-linear transformation that “rotates away” the recrossings and

irregular behavior. Using the intramolecular proton transfer reaction in mal-

onaldehyde [67, 68] and the isomerization of a simple cluster of six argon

atoms [69–73] they showed that this separation of the dynamical modes per-

sists up to energies well above the onset of chaos in the TS. In other words,

they observed that the action associated with the reaction coordinate re-

mains an approximate invariant of the motion throughout the region of the

TS. Moreover, they demonstrated that it is possible to choose a multidimen-

sional phase-space dividing surface satisfying the dynamical requirement of

TST [69]. They “visualized” the dividing surface in phase space by construct-

ing the projections onto smaller subspaces, revealing how the shape of the

reaction bottleneck depends on the energy of the system and the passage ve-

locity through the TS, and how the complexity of the recrossings emerges

over the saddle in the configuration space [70,71]. Using this visualization

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8

they further showed that changing the energy of the dynamics results in the

dividing surfaces “migrating” just as the PODS do.

We have taken the opposite, top-down approach by asking what the struc-

tures based on a simple rank-one saddle look like for a general Hamiltonian.

Indeed, in the early 1990’s the Wiggins group [14,22] investigated the geom-

etry associated with certain types of stationary points in phase space. They

proved the existence of Normally Hyperbolic Invariant Manifolds (NHIMs) in

the vicinity of these stationary points. With these results and with the tech-

nology Lie-Deprit transforms [75,76], we have been able to reformulate the

definition of the TS in terms of the geometrical objects that lie in the vicinity

of the phase-space barrier. A consequence of the phase space geometry in the

vicinity of the stationary point is that the Hamiltonian can be transformed

into a known normal form. It is this normal form that is responsible for

the separation between the dynamics in the reactive mode from those in the

internal modes. This is reflected in the equations of motion when they trans-

formed into the Normal Form variables. There are classical trajectories that

remain in the vicinity of the stationary point for all time. These trajectories

lie on a (2n − 2)-dimensional invariant manifold called the center manifold.

This manifold is of fundamental importance in the phase-space formulation of

TST. Its importance stems from the fact that it corresponds to the activated

complex.

Moreover, the intersection of the center manifold with an energy shell yields

a NHIM. The NHIM, which is a (2n − 3)-dimensional hypersphere, is the

higher-dimensional analog of Pechukas’ PODS. As this manifold is normally

hyperbolic it will possess stable and unstable manifolds. These manifolds are

the (2n − 2)-dimensional analogs of the separatrices. The NHIM is the edge

of the TS which is a (2n − 2)-dimensional hemisphere.

In the next two sections we will review the mathematical foundations of

the phase space formalism. This is followed in Section 1.5 by a discussion of

how the Hamiltonian is transformed into normal form and finally in the last

section, in order to illustrate these concepts, we quantize semiclassically the

TS for the isomerization of HCN.

1.3SADDLES IN ENERGY LANDSCAPES

Our intuition in the study of reaction dynamics is largely based upon our

understanding of the “geography” of simple two-dimensional potential energy

surfaces. While this basis has proven very useful, it also has produced a

false sense of security. The very simplicity of the geometry a two-dimensional

potential energy surface can lead one into difficulties that only arise in higher-

dimensional systems. Similar problems are encountered in 2dof systems. The

ionization of Rydberg atoms in the presence of external electromagnetic fields

provides an excellent example [6,18]. The complication that occurs is that

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SADDLES IN ENERGY LANDSCAPES

9

in the presence of the electromagnetic fields the Hamiltonian can no longer

be partitioned into the kinetic and potential energies. Consequently, it is

not possible to define the TS in terms of the morphology potential energy

surfaces, but rather, must be defined in terms of the geometry of the total

energy surface.

The difficulty is that we have trained ourselves to think of potential energy

surfaces defined on n-dimensional coordinate spaces. Instead we should be

thinking of the total energy defined on the 2n-dimensional phase space. The

method of analysis for the total energy is essentially the same as is used

for the potential energy surface. The first step is to identify the stationary

points (extrema) and their linear stabilities. Once these points are found

and characterized, the machinery of geometrical mechanics can be applied

to obtain the invariant manifolds and their stable and unstable manifolds

associated with the stationary points.

phase space can be partitioned into reactive and non-reactive regions and

such physically interesting properties as the rate of reaction or branching

ratios can be obtained from the geometry.

Finding and characterizing the stationary states of systems with more than

two degrees of freedom is an unsolved problem. Isolated stationary points are

the best known of these manifolds. In systems with two degrees of freedom, in

addition to isolated stationary points, it is also possible to find a closed loop of

stationary points [77]. These are associated with parabolic resonances. More

complicated manifolds will exist in systems having more than two degrees of

freedom. In the present discussion we will focus on the consequences of the

existence of isolated points of stationary flow in phase space.

Using these geometrical constructs

1.3.1Phase Space Versus Coordinate Space

In the coordinate-space treatment of TST certain assumptions must be made

concerning the nature of the Hamiltonian of the system. First, it must be

assumed that it can be partitioned into the sum of two terms, the kinetic and

the potential energy. Further, one must also assume that the kinetic energy

is positive definite and is quadratic in the momenta. With these assumptions

then the point of stationary flow in phase space and the saddle point of the

potential energy coincide. That is, the conditions placed on the kinetic energy

guarantee that the momenta characterizing the point of stationary flow are

equal to zero, consequently, the coordinate space configuration of this point

must be such that the various forces are balanced. In other words, that it is

extremum of the potential energy.

A major difference between phase and coordinate space is that phase space

is a state space, that is, each point corresponds to a unique state of the sys-

tem, whereas, a point in coordinate space determines only the physical con-

figuration of the system. Mathematical consequences of this seemingly small

difference are remarkable. First, the metric in phase space correspond to a

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volume. The physical interpretation of this volume as the number of states

follows from the fact that phase space is a state space. On the other hand, in

the coordinate space treatment, the metric provides us with a way of measur-

ing the “distance” between two physical configurations. This difference in the

metric results in a significant difference in the nature of the questions that can

be addressed. In the phase-space treatment we construct probabilities that

certain events occur by considering ratios of phase-space volumes. The rates

at which these events occur are readily obtained as fluxes across boundaries

between phase-space volumes. And in coordinate space one focuses primarily

on the rate at which the coordinate space configuration changes.

The coordinate space approach to TST encounters a number of significant

problems. The most serious of these concerns the definition of the TS. A

“rigorous” definition of the TS only exists for systems possessing two degrees

of freedom. Efforts at extending this definition to systems with three or

more degrees of freedom have not been successful. A similar difficulty occurs

when the distinction between the kinetic and potential energy is lost. Jaff´ e

et al. [6,18] have shown that this particular difficulty can be avoided by the

judicious choice of a different coordinate space.

must be defined in phase space and that if this object is projected into the

appropriate coordinate space, then the coordinate space formalism can be

applied. From this the origin of these difficulties becomes apparent: The

coordinate- space objects are shadows (projections) of the phase-space objects.

Seen from this point of view it is not surprising that difficulties are encountered

when trying to extend the coordinate space formalism to systems with more

than two degrees of freedom.

They argue that the TS

1.3.2Stability

The analysis of the stability of isolated stationary points is different in the

phase-space treatment from that in the coordinate space treatment. In the

coordinate space treatment the slope of the potential energy surface gives

the forces exerted on the system. Stationary points occur at extrema of the

potential energy. Their stability is determined by the eigenvalues of the matrix

of second derivatives evaluated at the extremum. Assuming the system has n

degrees of freedom, it will possess n eigenvalues. The stability of the extremum

is determined by the number of eigenvalues greater than or less than zero. If

all of the eigenvalues are positive, then the extremum is a minimum and it is

stable against all perturbations. In contrast, when all of the eigenvalues are

negative, then the extremum corresponds to a maximum and all perturbations

are unstable. Clearly, the extremum can be characterized by the number

of positive and negative eigenvalues it possesses. If it possesses m negative

eigenvalues it is said to be a rank-m saddle, that is, that it is unstable in m

degrees of freedom and stable in the remaining n − m degrees of freedom.

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SADDLES IN ENERGY LANDSCAPES

11

In the analogous analysis in the phase-space treatment one examines the

stability matrix

∂ ˙ q

∂q

∂ ˙ q

∂p

∂ ˙ p

∂q

∂ ˙ p

∂p

=

∂2H

∂q∂p

∂2H

∂2p

−∂2H

∂2q

−∂2H

∂q∂p

evaluated at the point of stationary flow. The eigenvalues of this matrix occur

in n pairs of numbers being either real or imaginary. A pair of imaginary

eigenvalues corresponds to a stable degree of freedom (elliptical) and a pair

of real eigenvalues to an unstable degree of freedom (hyperbolic). When both

treatments are both valid, the eigenvalues of the coordinate treatment are

equal to minus the square of eigenvalues of the corresponding phase-space

treatment (assuming mass weighted coordinates).

The case when the eigenvalues are equal to zero is special must be treated

separately. One is tempted to assume that such cases are rare. In fact these

cases occur when the manifolds of stationary flow are not isolated points. The

simplest of these cases give rise to parabolic resonances [77], however, they

are beyond the scope of this review. These resonances have been observed in

some of the simplest reactive systems [78].

1.3.3Energy Landscapes in Phase Space

In the coordinate space formulation one investigates the geometry of the po-

tential energy surface, which is defined on the n-dimensional coordinate space.

On this surface the minima are called potential wells. In turn, each of these

wells are separated from each other by a rank-one saddle. The transport from

one potential well to the next must pass over the saddle that separates them.

The rate of a reaction is formulated in terms of the flux across the saddle.

In systems with more than two degrees of freedom higher rank saddles occur.

These occur when the boundaries of more than three or more potential wells

coincide.

In the phase-space treatment the situation is very similar. However, rather

than study the morphology of the potential energy surface we most focus

on the total energy surface. The geometry of this surface, which is defined

on phase space instead of coordinate space, can also be characterized by its

stationary points and their stability. In this treatment, the rank-one saddles

play a fundamental result. They are, in essence, the traffic barriers in phase

space. For example, if two states approach such a point and one passes on

one side and the other passes on the other side, then one will be reactive and

the other non-reactive. Once the stationary points are identified, then the

boundaries between the reactive and non-reactive states can be constructed

and the dynamical structure of phase space has been determined. As in the

in the case of potential energy surfaces, saddles with rank greater than one

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12

occur, especially in systems with high symmetry between outcomes, as in the

dissociation of ozone.

In the next section we will review the mathematical foundations of the

phase space formalism.

1.4 PHASE-SPACE STRUCTURE AROUND A SIMPLE

(RANK-ONE) SADDLE

In this section we will develop the phase-space structure for a broad class of n-

degree-of-freedom Hamiltonian systems that are appropriate for the study of

reaction dynamics through a rank-one saddle. For this class of systems we will

show that on the energy surface there is always a higher-dimensional version

of a “saddle” (a normally hyperbolic invariant manifold (NHIM) [22]) with

codimension-one (i.e., with dimensionality one less than the energy surface)

stable and unstable manifolds. Within a region bounded by the stable and

unstable manifolds of the NHIM we can construct the TS which is a dynamical

surface of no return for the trajectories. Our approach is algorithmic in nature

in the sense that we provide a series of steps that can be carried out to locate

the NHIM, its stable and unstable manifolds, the TS, as well as describe all

possible trajectories near it.

1.4.1Justification of the n-Degree-of-Freedom Hamiltonian

Consider a Hamiltonian of the following form:

H

=

n−1

?

+f2(q1,...,qn−1,p1,...,pn−1),

(q1,...,qn,p1,...,pn) ∈ IR2n.

Here I ≡ pnqnand f1, f2are at least of third order, i.e., they are responsible

for the non-linear terms , and f1(q1,...,qn−1,p1,...,pn−1,0) = 0. In the

language of reaction dynamics, the coordinates (qn,pn) are the reaction coor-

dinates and the remaining coordinates are referred to as the bath coordinates.

The corresponding Hamiltonian vector field is given by:

i=1

ωi

2

?p2

i+ q2

i

?+ λqnpn+ f1(q1,...,qn−1,p1,...,pn−1,I)

(1.1)