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Direct Construction of a Dividing Surface of Minimal Flux for Multi-Degree-of-Freedom Systems: The Equivalence of Conventional and Variational Transition State Theory

The fundamental assumption of conventional transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants must cross the surface only once and then proceed to products. Recently it has been shown how to construct a dividing surface in phase space for Hamiltonian systems with an arbitrary (but finite) number of degrees of freedom having the property that trajectories only cross once locally. In this letter we provide an argument showing that the flux across this dividing surface is a minimum with respect to certain types of variations of the dividing surface. In this sense, conventional transition state theory is shown to be equivalent to variational transition state theory.


Introduction
Transition state theory provides a fundamental framework for computing chemical reaction rates. The original ideas are due to Wigner, Polanyi, and Eyring, yet much work on different aspects of the subject continue to this day (see the recent reviews of [3] and [4]). In recent years transition state theory has been shown to be much more widely applicable than just 1 for problems in chemical reactions. It has been used in atomic physics [5], studies of the rearrangements of clusters [6], solid state and semi-conductor physics [7,8], cosmology [9], and celestial mechanics [10].
The fundamental assumption of transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants must cross the surface only once and then proceed to products. The reaction rate is proportional to the flux through the dividing surface. The construction of such a surface in specific systems, especially those with more than two degrees of freedom (DOF), has posed extreme difficulties. The phenomenon of recrossing of a dividing surface gives rise to a larger value for the flux, which naturally inspires the idea of varying the dividing surface so that the flux is minimised, this has lead to the subject of variational transition state theory ( [11], see also the review of [12]).
For systems with 2 DOF described by a Hamiltonian H of the simple type kinetic-plus-potential the problem of constructing a dividing surface (transition state) with trajectories only crossing once and having minimum flux was solved during the 70's by McLafferty, Pechukas and Pollak [13,14,15,16]. They considered the line in configuration space given by the projection of an unstable periodic orbit (the Lyapunov periodic orbit associated with a saddle point), see figure 1(a). This line which is referred to as periodic orbit dividing surface (PODS) separates the reactant region (x < 0) from the product region (x > 0) by connecting two pieces of an equipotential. This solves the problem of (local) recrossings as there is no trajectory evolving from reactants to products (or vice versa) whose projection to configuration space is tangent to this line. After a reacting trajectory has crossed the PODS it has to leave the neighbourhood of the PODS before it can possibly recross it.
To comprehend a PODS as a phase space object we rewrite the energy equation as For (x, y) on the projection of the periodic orbit in figure 1(a) the equation in (1) defines a family of circles in (p x , p y ) which shrink to points at the turning points of the periodic orbit, i.e. the PODS is a 2-sphere. The periodic orbit which has p x = 0 can be considered as the equator of the PODS. It divides the PODS into two hemispheres which have p x > 0 and p x < 0, respectively. The flux from x < 0 ("reactants") to x > 0 ("products") through the PODS in figure 1(a) is commonly written as  Figure 1: (a) Equipotentials and PODS (bold line) near a saddle. (b) Saddle plane (q 1 , p 1 ) with projections of the (2n − 1)-dimensional energy surface H = E for E > 0 (all but the hatched region), the stable and unstable spherical cylinder (the p 1 -axis and the q 1 -axis), the NHIM (the origin), and the (2n − 2)-spheres p 1 − q 1 = 0 (dividing surface). The light and dark grey regions are the projections of the energy surface volumes enclosed by the forward and backward reactive spherical cylinder W f (E) and W b (E), respectively. p 1 − q 1 = ±c, c > 0, delimit the region of validity of the normal form.
The step function Θ(p x ) effectively restricts the integral to one hemisphere of the PODS. Using Stokes' theorem it is not difficult to see that the flux in (2) is given by the action of the periodic orbit (p.o.), i.e.
Unfortunately, the methods of McLafferty, Pechukas and Pollak are specific to 2 DOF, and as well to Hamiltonian's of a certain form. Recently it has been shown [1,2] how to construct a dividing surface in phase space for Hamiltonian systems with an arbitrary (but finite) number of DOF having the property that trajectories only cross once locally. A computational algorithm for realising this construction was also given. However, it was not shown that this transition state was a surface of minimum flux. In this letter we give a geometrical argument demonstrating that this is indeed the case, and in the process show that for the phase space transition states constructed according to the techniques in [1,2], conventional transition state theory is equivalent to variational transition state theory. In order to do this we must first recall the elements of the many DOF theory in [1,2].

Dividing surface for multi-dimensional systems
In the general case of n DOF we consider an equilibrium point of saddlecenter-...-center type, i.e. the linearised vector field about the equilibrium point has one pair of real eigenvalues ±λ and n − 1 imaginary eigenvalues ±iω k , k = 2, . . . , n, where without restriction of generality λ, ω k > 0. Equilibria of this type are characteristic for systems with a reaction type dynamics, and they occur in all of the examples given in the introduction. The transport is controlled by various high-dimensional manifolds which can be realised and computed through a procedure based on Poincaré-Birkhoff normalisation. In fact, under general conditions (see [2] for the details), near a saddle-center-...-center one can construct a sequence of local, non-linear, symplectic transformations of the phase space coordinates that transform the Hamiltonian into up to any desired order. Here (q 1 , . . . , q n , p 1 , . . . , p n ) are canonical phase space coordinates, I = p 1 q 1 and f 1 , f 2 are at least of third order, i.e. they are responsible for the nonlinear term in the Hamiltonian vector field. Moreover f 1 has the property that it vanishes for I = 0. (q 1 , p 1 ) play the role of reaction coordinates. (q 2 , . . . , q n , p 2 , . . . , p n ) are the bath coordinates. The importance of saddle-center-...-center equilibria for reaction type dynamics can be inferred from the topology of energy surfaces H = E. First consider the quadratic Hamiltonian given by the first part in (3) and write the energy equation as For E < 0 the left hand side is positive for -spheres shrink to points. The energy surface thus appears as two disjoint spherical cones which correspond to "reactants" and "products", respectively. Increasing the energy to E > 0, the left hand side of (4) is strictly positive. The formerly disjoint components merge and the energy surface becomes a spherical cylinder S 2n−2 × R. Restricting to a sufficiently small neighbourhood by confining p 1 − q 1 to an interval I = [−c, c] with c > 0 sufficiently small and with E sufficiently close to zero, the topological consideration remain true if the non-linear terms f 1 and f 2 are taken into account. Moreover, for a high but finite order normal form, the error from neglecting the non-normalised "tail" of the Hamiltonian can be made as small as desired by choosing the interval [−c, c] sufficiently small and E sufficiently close to zero. The importance of the normal form arises from the fact that it gives explicit expressions for all the manifolds which control the dynamics from reactants to products. For a fixed energy E > 0 these manifolds are . It can be considered as a "big saddle". In fact, it is a socalled normally hyperbolic invariant manifold (NHIM) where normal hyperbolicity means that the expansion and contraction rates transverse to the manifold dominate those tangent to the manifold.
• The forward and backward reactive spherical cylinders W f (E) and The saddle sphere has stable and unstable manifolds W s (E) and W u (E) which are iso-energetic, i.e. contained in the energy surface, and which are explicitly given by q 1 = 0 and p 1 = 0, respectively. These invariant manifolds have the topology of spherical cylinders S 2n−3 × I. Since they are of codimension 1 in the energy surface, i.e. they are of one dimension less than the energy surface, they act as impenetrable barriers. W s (E) and W u (E) each appear as two branches. We call the branch of W s (E) which has p 1 > 0 the forward branch W s f (E) and the branch which has p 1 < 0 the backward branch W s b (E). Likewise the forward branch W u f (E) of W u (E) has q 1 > 0 and the backward branch W u b (E) has q 1 < 0. We call the union of the forward branches, , the forward reactive spherical cylinder. Similarly, we call the union of the backward branches, , the backward reactive spherical cylinder. The significance of these spherical cylinders arises from the fact that they enclose volumes of the energy surface which contain all forward and all backward reactive trajectories, respectively. All non-reactive trajectories are contained in the complement of these two volumes.
We define the dividing surface or transition state as follows.
• Transition state S 2n−2 ts (E): On the energy surface H = E the equation p 1 = q 1 defines a (2n − 2)-sphere which we denote by S 2n−2 ts (E). It is of codimension 1 in the energy surface. It divides the energy surface into two components: the reactant region p 1 − q 1 > 0 and the product region p 1 − q 1 < 0. The saddle sphere S 2n−3 NHIM (E) can be considered as the "equator" of the transition state. It divides S 2n−2 ts (E) into the two hemispheres: the forward hemisphere D 2n−2 ts, f (E), which has p 1 = q 1 > 0, and the backward hemisphere D 2n−2 ts, b (E), which has p 1 = q 1 < 0. D 2n−2 ts, f (E) and D 2n−2 ts, b (E) are topological (2n−2)-discs. Except for its equator, which is an invariant manifold, the transition state is everywhere transverse to the Hamiltonian flow as is easily seen from the equations of motions derived from the normal form Hamiltonian (3). This means that a trajectory, after having crossed the transition state, has to leave the neighbourhood of the transition state before it can possibly cross it again, i.e. the transition state locally is a "surface of no return".
An important advantage of the normal form coordinates is that dynamical issues related to flux and reactivity can be understood to a great extent from the projection to the plane of the reaction coordinates (q 1 , p 1 ), see figure 1(b). Due to the constance of the saddle integral I = p 1 q 1 trajectories project to hyperbolas. Forward and backward reactive trajectories project to the first quadrant q 1 , p 1 > 0. It approaches the forward hemisphere D 2n−2 ts, f (E) of the transition state inside of the forward branch of the stable cylinder W s f (E) whose interior projects to the part of the first quadrant above the diagonal p 1 = q 1 > 0. After crossing the transition state the trajectory leaves inside of the forward branch of the unstable spherical cylinder W u f (E) whose interior projects to the part of the first quadrant below the diagonal p 1 = q 1 > 0. While similar considerations hold for the backward reactive trajectories, non-reactive trajectories project to the second quadrant p 1 > 0, q 1 < 0 which corresponds to reactants or to the fourth quadrant p 1 < 0, q 1 > 0 which corresponds to products. Therefore reactive trajectories have I = p 1 q 1 > 0 and non-reactive trajectories have I = p 1 q 1 < 0.

Minimal flux property of the transition state
Consider at first the phase space volume form Ω = dp 1 ∧ dq 1 ∧ · · ·∧ dp n ∧ dq n , which in terms of the symplectic 2-from ω = n k=1 dp k ∧dq k can be written as Ω = ω n /n!. Let η be an energy surface volume form defined via the property dH ∧ η = Ω. Then the flux through a codimension 1 submanifold of the (2n − 1)-dimensional energy surface H = E is obtained from integrating over it the "flux" form Ω ′ given by the interior product of the Hamiltonian vector field X H with η [17], i.e.
where i X H η(ξ 1 , . . . , ξ 2n−2 ) = η(ξ 1 , . . . , ξ 2n−2 , X H ) for any 2n − 2 vectors ξ k . The second equality in (5) is easily established on a non-critical energy surface, i.e. on an energy surface which contains no equilibria. The flux form Ω ′ is exact. In fact the generalised "action" form has the property dϕ = Ω ′ and facilitates the use of Stokes' theorem to compute the flux. In the case of two DOF we simply have Ω ′ = ω = dp 1 ∧ dq 1 +dp 2 ∧dq 2 and ϕ = p 1 dq 1 +p 2 dq 2 . Since the transition state S 2n−2 ts (E) is a sphere, that is, a manifold without boundary, it follows from Stokes' theorem that the integral of Ω ′ over S We now show that the forward flux through S 2n−2 ts (E) is minimal. This can be stated as a variational problem and a result in this direction is obtained by MacKay [18] who proves that the integral C ϕ of the action form ϕ over codimension 2 submanifolds C of the energy surface is stationary with respect to variations of C if and only if C is invariant under the Hamiltonian flow. Since the saddle sphere S 2n−3 NHIM (E) is an invariant manifold, MacKay's result implies that the "action" of S 2n−3 NHIM (E), S 2n−3 NHIM (E) ϕ, is stationary with respect to variations of S 2n−3 NHIM (E). MacKay considers arbitrary variations within the energy surface and this leads to an indefinite variational principle. In fact, in the context of transition state theory it is more useful to consider variations of the codimension 1 transition state S 2n−2 ts (E), which in a sense that will become clear below, imply variations of the codimension 2 saddle sphere S 2n−3 NHIM (E) in a smaller, more suitable class than in MacKay's case.
Consider at first the case of 2 DOF for which it is possible to visualise the energy surface S 2 × I in 3-dimensional space as a nested set of 2-spheres parametrised along the interval I in radial direction. This is the so-called McGehee representation [19] which is shown in the first panel of figure 2. We consider a slight generic, iso-energetic deformation of the transition state S 2 ts (E) that does not "preserve" the saddle sphere S 1 NHIM (E), i.e., S 1 NHIM (E) is not entirely contained in the deformed transition state, see Figures 2(c)-(e). The deformation, which we denote byS 2 ts (E), is described mathematically as a graph over the transition state S 2 ts (E), and thereforẽ S 2 ts (E) inherits the orientation of S 2 ts (E) (which is important because we want to preserve certain aspects of the directionality of the flux, to be described below). BecauseS 2 ts (E) is chosen to lie in the energy surface, it still separates the energy surface into two disjoint components like S 2 ts (E). Also like S 2 ts (E), the deformationS 2 ts (E) contains a circleS 1 (E) of points at which the Hamiltonian vector field is tangent toS 2 ts (E). This can be proven analytically, but intuitively it is easy to see from figure 2(d).S 1 (E) can be considered as the equator ofS 2 ts (E); it dividesS 2 ts (E) into forward and backward hemispheresD 2 ts, f (E) andD 2 ts, b (E) on which the flux form Ω ′ is strictly positive and negative, respectively.S 1 (E) can be considered as the deformation of S 1 NHIM (E) induced by the deformation of S 2 ts (E). It should be realised that in contrast to the case of S 1 NHIM (E), the Hamiltonian vector field is not tangent to the deformationS 1 (E), i.e.S 1 (E) is not invariant under the Hamiltonian vector field (S 1 (E) is not a periodic orbit). It can be shown that if the deformation ofS 2 ts (E) is small enough then all points of S 1 (E) have p 1 q 1 ≤ 0, i.e. they are contained in the complement of the two volumes enclosed by the forward and backward reactive spherical cylinders W f (E) and W b (E). Again intuitively this can be deduced from Figures 2(c)-(e), where the essential point is seen in (d) whereS 2 ts (E) protrudes into the white, non-reactive region. This is where the trajectories become tangent toS 2 ts (E), i.e., at some point onS 2 ts (E) trajectories "bounce off". Consider the part ofS 2 ts (E) which is contained in the forward reactive spherical cylinder W f (E) and marked by the light grey region in figure 2(e). The boundary of the light grey region corresponds to the intersection with the forward stable and unstable cylinder branches W s f (E) and W u f (E). The boundary is in general not a smooth manifold but it is homeomorphic to a circle S 1 . It is possible to deform this piecewise smooth circle continuously onto the Lyapunov periodic orbit S 1 NHIM (E) without leaving the stable and unstable cylinders. Stokes' theorem implies that the difference of the integrals of ϕ along the piecewise smooth circle and the Lyapunov periodic orbit is given by the integral of Ω ′ over the region on the stable and unstable manifolds swept out in the deformation process. The flux form Ω ′ vanishes on the stable and unstable manifolds as its definition involves the interior product with the Hamiltonian vector field ("there is no flux through invariant manifolds"). It thus follows that the flux through the light grey part of D 2 ts, f (E) in figure 2(d) is equal to the flux through D 2 ts, f (E). Therefore since the flux form Ω ′ is strictly positive on the complete hemisphereD 2 ts, f (E), the forward flux throughS 2 ts (E) is larger than the forward flux through S 2 ts (E), and this completes the argument.
All the arguments above immediately carry over to systems with more than 2 DOF by simply adjusting the dimension of the involved manifolds. The essential geometric conditions and relations amongst the manifolds all hold.
The dividing surface of minimal forward flux is not unique, and this is the reason that we chose iso-energetic deformations that did not preserve the NHIM. For example the equation p 1 = aq 1 with a (slightly) deviating from 1 again defines on the energy surface a (2n − 2)-sphere which is a dividing surface and which has the same forward flux as S 2n−2 ts (E). In our consideration we considered a generic deformation which is a deformation that changes the equator ofS 2n−2 ts (E) such that it no longer coincides with the NHIM S 2n−3 NHIM (E). Non-generic deformation in this sense do not change the flux. Similarly, our transition state for 2 DOF coincides with a PODS in general only along its equator, i.e. at the Lyapunov periodic orbit, and this is what matters for the flux.

Algorithm for computing the flux
Provided a generic non-resonance condition between the linear frequencies ω k is fulfilled the normal form Hamiltonian (3) assumes the simple form H(I, J 2 , . . . , J n ) where J k = (p 2 k + q 2 k )/2, k = 2, . . . , n, are action variables associated with the bath coordinates. Like the saddle integral I the actions J k are constants of the motion. Writing the flux form Ω ′ in terms of action-angle variables we obtain the result that the forward flux through the transition state is given by where V(E) is the volume in the space of the bath actions (J 2 , . . . , J n ) enclosed by the contour H(0, J 2 , . . . , J n ) = E. In fact the flux can be interpreted as the volume enclosed by a contour of constant energy E in phase space of a reduced system with one dimension less than the complete system. In terms of the normal form coordinates the reduced system is explicitly described by the invariant subsystem which has q 1 = p 1 = 0. The normally hyperbolic invariant manifolds are just the energy surfaces of this reduced system which is referred to as activated complex in the chemistry literature. The dimensionless quantity F E /h n−1 , where h is Planck's constant, is Weyl's approximation of the integrated density of states of the reduced system and can be interpreted as the number of "transition channels".
In the linear case we have H = λI + n k=2 ω k J k and the energy surface H = E encloses a simplex in (J 2 , . . . , J n ) whose volume leads to the wellknown result (see e.g. [17] for an historical background) showing that the flux scales with E n−1 for energies close to the saddle energy. The normal form allows to include the non-linear corrections to this result to any desired order.

Conclusions
In this letter we have given a geometrical argument showing that the flux across the dividing surface or transition state for n DOF systems constructed in [1,2] is a minimum, in the sense that the flux corresponding to forward reactive trajectories (which is equal in magnitude but opposite in sign to the flux corresponding to backward reactive trajectories) is a minimum. Hence it is the optimal dividing surface sought for by variational transition theory.
The key point for the construction of the dividing surface is the existence of a normally hyperbolic invariant manifold (NHIM) which exists on the energy surface near a saddle-center-...-center equilibrium point and which mainly controls the dynamics nearby. The NHIM has the topology of a (2n − 3)-sphere which can be considered as the equator of the dividing surface which is a (2n − 2)-sphere. The NHIM divides the dividing surface into two "hemispheres" which are open (2n − 2)-discs. The hemispheres are transverse to the Hamiltonian flow. Hence the dividing surface is everywhere transverse to the Hamiltonian flow except for its equator (the NHIM) which is an invariant manifold. The NHIM can be considered as the energy surface of an invariant subsystem ("activated complex") with one DOF less than the complete system. The flux is the phase space volume enclosed by the energy surface of this reduced system.
The dividing surface is not unique. Any (2n − 2)-sphere which contains the NHIM as its equator and which is transverse to the Hamiltonian flow except for its equator qualifies for a dividing surface. Nevertheless, all these dividing surfaces lead to the same flux. This makes it difficult (if not impossible) to compute a dividing surface from a variational principle of codimension 1 manifolds of the energy surface. Similarly, the NHIM appears only as an indefinite critical "point" of the variational principle of codimension 2 manifolds of the energy surface of MacKay [18] what makes it practically infeasible to compute the NHIM from a variational principle. The normal form approach of [1,2] is currently the only method to determine these manifolds and also to compute the flux for which we gave an algorithm in this letter.