Reaction coordinates for elucidating reaction dynamics with anharmonic couplings



Saddle points can be found on the potential energy surface between the reactants and products regions in many chemical reactions. The dynamics occurring in the vicinity of a saddle point play an essential role in the study of reaction dynamics because the reactivity of a trajectory is primarily determined in the saddle region. There are many cases where a well-defined boundary (called reactivity boundary here) between reacting and nonreacting trajectories can be located in the phase space. Locating the reactivity boundary, however, becomes challenging when anharmonic couplings between the reaction coordinate and the vibrational coordinates complicate the reaction dynamics. The aim of this article is to present a review of recent studies that use nonlinear coordinate transformations to disentangle the anharmonic couplings. The reaction coordinates constructed through such nonlinear coordinate transformations are decoupled from the other coordinates as much as possible, thus simplifying analysis of reaction dynamics. Several options are introduced for nonlinear coordinate transformations which should be appropriately chosen by examining the extent of the anharmonic couplings. Under certain conditions, special reaction coordinates constructed by a suitable nonlinear coordinate transformation reveals the existence of a clear reactivity boundary in the phase space even when there are strong anharmonic couplings. © 2014 Wiley Periodicals, Inc.


Chemical reaction is the process of molecules changing their configurations from “reactants” to “products.” Physically, it can be regarded as a motion of atoms described in the configuration space with a set of coordinates specifying the current positions of the atoms. A point (in the classical mechanics) or a wavepacket (in the quantum mechanics) changes its location from a region classified as “reactants” in the configuration space to another region classified as “products.” In many cases, the regions corresponding to reactants and the products are valleys in the potential energy landscape. Between these valleys there is often found a region with higher potential energy than the reactants and the products regions, giving rise to a “barrier” region that the system must pass through in order to get from the reactants region to the products region. In such a “barrier” region one can usually find a rank-one saddle point where the potential energy is maximum along one direction associated with the path from the reactants region to the products region and is minimum along all the other directions. The importance of the saddle point first lies in the fact that it gives the minimum amount of the energy necessary for the process of surmounting the barrier for the reaction. In the case where the energy of the system is not very high, the system must necessarily pass through the vicinity of the saddle point in the process of chemical reaction. It is the dynamics occurring in the vicinity of the saddle point that determine whether a given system undergoes the process of chemical reaction by successfully surmounting the barrier or bounces back to the reactants region, that is, the dynamics in the vicinity of the saddle point determine the reactivity.

Interest in the dynamics occurring in the saddle region primarily arose in the context of calculating the reaction rate constant. The transition state (TS) theory[1, 2] attempts to calculate the reaction rate by first postulating a dividing surface, called TS, in the saddle region and then measuring the flux passing through the TS. A crucially important assumption in the concept of the TS is the so-called nonrecrossing property of the TS: if a trajectory is to go from the reactants side to the products side, it must necessarily cross the TS dividing surface between them, and, once a trajectory has crossed the TS, it must go down into the products region without ever returning to the TS again. This assumption is based on the intuition that the motion taking place in the saddle region primarily determines the occurrence of the reaction while the motion after passing the saddle region is simply that of “sliding down the slope.” This assumption enables the calculation of the reaction rate using only the information of the local region around the saddle point without simulating global trajectories. Note, however, the dynamics outside the saddle region is important when the state-to-state rate constant[1] is concerned or when the system has significant amount of global recrossing.[3]

In addition to the calculational technique for reaction rates, the theory of the saddle region dynamics provides fundamental understanding of what is occurring during the process of chemical reaction. By offering an intuitive image of the motion of reacting trajectories, it provides insights that help to answer the question of what differentiates the reacting trajectories that go from the reactants region to the products region and the nonreacting trajectories that start in the reactants region but are reflected by the barrier back into the reactants region again. Recent theoretical studies[4-9] of the saddle region dynamics from the viewpoint of nonlinear dynamics elucidated the existence of well-defined boundaries in the phase space that divide the reacting trajectories from the nonreacting ones, even in cases with anharmonic coupling that complicates the dynamics. Hereafter such boundaries will be called reactivity boundaries. The present review provides a brief summary of the basic aspects of the recent theoretical developments concerning the saddle region dynamics, mainly in the classical mechanical description. Readers interested in the extension of the theory to the quantum mechanical regime may refer to Refs. [10, 11]. It is also possible to utilize the concept of the reactivity boundary for elucidating the mechanism of reaction control by laser fields and apply it to the design of the optimal pulse shape to obtain desired products.[11, 12]

Saddle Region Dynamics in Harmonic Approximation

Let us start the discussion with a simple case. As the (rank-one) saddle point is defined as a stationary point of the potential energy surface that is maximum along only one direction, the lowest order terms in the Taylor expansion of the potential energy math formula for the nuclear coordinates q are as follows:

display math(1)

where the origin of the coordinate system is taken at the saddle point and the normal mode coordinates math formula have been introduced to diagonalize the potential energy. Here, we assume mode 1 is the unstable direction with potential curvature math formula. This unstable mode corresponds to the motion of “sliding down the barrier.” The other modes are vibrational modes with frequencies math formula.

The Hamiltonian is then

display math(2)

where the following variables have been introduced for later convenience:

display math(3)

This transformation defines a skewed coordinate system math formula in the (q1, p1)-plane.

The action variable defined by math formula for the reaction mode is an invariant of motion with the Hamiltonian given by Eq. (2). Therefore, the trajectories run along the hyperbolas given by math formula as shown in Figure 1.

Figure 1.

Phase space flow along the normal mode reaction coordinates plotted for the harmonic approximation around the saddle point. Trajectories run along the contours of the action variable math formula. The x1-axis and the ξ1-axis running diagonally in the figure constitute boundaries between reactive and nonreactive trajectories.

Suppose that math formula corresponds to the “reactants” (i.e., before the reaction), and math formula to the “products” (after the reaction). The trajectories with x1 > 0 and ξ1 > 0 are “forward reactive” because they start from the reactants region, pass over the barrier, and go into the products region (i.e., the reaction progresses from the reactants to the products). Conversely, the trajectories with math formula and ξ1 > 0 are “forward nonreactive” because they start from the reactants region, are reflected by the barrier, and go back into the reactants region. Similarly, the trajectories with math formula and math formula are “backward reactive” and those with x1 > 0 and math formula are “backward nonreactive.” Thus, the sets math formula and math formula constitute the reactivity boundaries between the reactive and the nonreactive trajectories.

Here, it is helpful to make a brief comment on the usage of the term reaction coordinate. In the case of a rank-one saddle point, there is only one reaction coordinate (q1) in the configuration space. When we look at the phase space, however, both the position coordinates and their conjugate momenta are called coordinates in the phase space. Therefore both q1 and p1 are called reaction coordinates here. In the following, the coordinates x1 and ξ1, which have been constructed by mixing q1 and p1, will also be called reaction coordinates.

Nonlinear Coordinate Transformation for Anharmonic Systems

As the total energy of the system increases, the harmonic approximation Eq. (2) becomes no longer valid. We have higher order terms in the Taylor expansion of the Hamiltonian:

display math(4)

where math formula is a polynomial function containing cubic and higher order terms. Due to the existence of math formula, the action math formula of the reactive mode is no longer a constant of motion. The motion along the reaction mode (q1, p1) is affected by the current values of the other coordinates. This is an effect known as coupling. A plot like Figure 1 with only q1 and p1, therefore, cannot capture the correct view of the dynamics due to the loss of information about the other coordinates.

The passage from the low-energy harmonic regime to higher energies was remarked on in Ref. [5] in the study of the trajectories of an inert gas cluster system. This and other studies[4-9] then showed that it is possible to introduce a coordinate transformation math formula that casts the Hamiltonian into the following form:

display math(5)

In this form, the total Hamiltonian depends on math formula and math formula only through the action math formula. Here, f1 and f2 contain anharmonic terms, and, in particular, f1 is defined so that math formula when math formula. When the Hamiltonian is in the form of Eq. (5), it can be proved, by some elementary calculations with Hamilton's equations of motion, that the new action variable math formula is an invariant of motion. This means that the same picture as Figure 1 is valid if the names of the axes are changed to math formula and math formula.

Let us here look into technical details of constructing such a coordinate transformation. The core of the method is known as the Lie canonical perturbation theory.[13, 14] The Hamiltonian Eq. (4) is first expanded in a series:

display math(6)

where math formula is a formal parameter of perturbation that will be set equal to 1 after all the calculations have been done. The zeroth order part of the Hamiltonian is taken as the harmonic approximation:

display math(7)

and the higher order term math formula in Eq. (4) is put into math formula in Eq. (6). The canonical transformation is given by solving the following differential equations:

display math(8)

with the initial condition corresponding to the original coordinate:

display math(9)

The transformed coordinates are given as the final values at math formula:

display math(10)

Note the differential equations Eq. (8) are Hamilton's equations of motion as if the function math formula were the “Hamiltonian” and the parameter math formula were the “time.” The function F is called a generating function. The transformation thus constructed is automatically canonical because the Hamiltonian time evolution is canonical.

The formal solution of Eq. (8) is given by

display math(11)

where the operator math formula denotes the operation of the Poisson bracket

display math(12)

After the transformation, the Hamiltonian is expressed in terms of the new coordinates

display math(13)

This Hamiltonian function is given by

display math(14)

Note the sign is opposite to Eq. (11). The final Hamiltonian math formula is expanded in a series as math formula. When Eq. (6) is substituted into Eq. (14) and the exponential operator is also expanded, the math formula part of Eq. (14) reads

display math(15)

If we express H1, F, and math formula as polynomials

display math(16)

then, after some calculation using Eq. (15) and the definition of math formula in Eq. (12), the coefficients in math formula are given by

display math(17)

Thus, by setting

display math(18)

the terms with math formula in H can be canceled in Eq. (15) and the transformed Hamiltonian math formula contains only terms with j1 = k1. This gives the final form of math formula specified by Eq. (5) as desired. The description given above is a simplified explanation, focusing only on the first order of perturbation in the Lie canonical perturbation method used in the previous studies. More technical details including the higher order terms can be found in literature.[4-9, 13-15]

In the literature, the form of Eq. (5) has been called “partial normal form (PNF)” as the transformation is designed to make only the action math formula of the reaction mode a constant of motion. It is also possible to construct a transformation that makes all the transformed actions constants of motion.[6, 7, 9] In the latter case, the final form of the Hamiltonian is called “full normal form.” The different strategies for constructing the transformation lead to differences in its convergence properties. As the normal form transformation is based on the series expansion, it may suffer from divergence mainly arising from the appearance of nonlinear resonances among the modes when the system has significant nonlinear couplings in the high energy regime. In general, the convergence becomes better when the generating function contains a smaller number of terms and, therefore, the transformation is closer to the identity transformation. In Ref. [16], the present author has suggested another form of normal form,

display math(19)

This functional form is closer to the original Hamiltonian than the other NFs in that it allows for any form of g1 and g2. The only requirement in the form of Eq. (19) is that the function g1 is multiplied by math formula. It is, therefore, expected to have a better convergence property. Under the Hamiltonian given by Eq. (19), the action variable math formula is no longer a constant of motion. However, it is shown by simple calculations that the sets math formula and math formula are still invariant manifolds and no trajectories can cross them. These two sets therefore play the role of reactivity boundaries. The form of Eq. (19) has been called “minimal normal form (MNF)” because it is a minimal functional form for obtaining the invariance of the manifolds.

Numerical Examples

For the purpose of demonstration, examples of trajectories in the saddle region are plotted in the usual normal mode coordinates and in the transformed coordinates in Figure 2. The numerical calculations have been done for a model system of a hydrogen atom in crossed electric and magnetic fields used in previous studies.[6, 7, 16] The field parameter ε for the strength of the electric field is set to 0.45 in the scaled units. Figures 2a and 2b show the trajectories with energy 0.05 above the saddle point in the scaled units. The same trajectories are plotted in the normal mode coordinates in (a) and in the transformed coordinates in (b). Although the trajectories look very complicated in the usual coordinates, the plot with the transformed coordinates reveals the existence of regular structure.

Figure 2.

Example trajectories before and after the coordinate transformation. (a) Trajectories with lower energy above the saddle point plotted in the usual normal mode coordinates. (b) The same trajectories as (a) plotted in the PNF coordinates. (c) Trajectories with higher energy above the saddle point plotted in the usual normal mode coordinates. (d) The same trajectories as (c) plotted in the MNF coordinates.

The laminar flow in Figure 2b shows the constancy of the action math formula in the transformed reaction coordinates. Using this fact the reactivity boundaries are identified as math formula. It is seen in the figure that no trajectories can cross this boundary due to the constancy of the action variable. Figures 2c and 2d show the trajectories with higher energy (0.15 above the saddle point in the scaled units). At this energy, the convergence of the PNF has been found to be poor, and the MNF [Eq. (19)] is used to construct the transformation. Trajectories plotted in Figure 2d with the transformed coordinates show some crossings of different trajectories. This fact shows that the action math formula is no longer a constant at this energy even in the transformed coordinates. However, one can still see the existence of the reactivity boundaries and identify them analytically as the sets math formula and math formula. According to Eq. (19), no trajectories can cross these boundaries.

Toward “Global” Understanding of Chemical Reactions

The use of coordinate transformation to reveal the structure of the dynamics in the saddle region has been reviewed and demonstrated. A system with intense couplings between multiple modes showing complicated motions when seen in usual coordinates may possess regular structure, specifically a clear boundary separating reactive and nonreactive trajectories in the phase space. The method of coordinate transformation presented here is designed to find special coordinates that are decoupled from each other as much as possible and thus make the analysis of the dynamics as simple as possible. The method is able to reveal regular structure that is hidden in the seemingly complicated motions of the usual coordinates, and is therefore expected to contribute to our understanding of chemical reaction dynamics.

The present method relies on the polynomial expansion around the saddle point. This is based on the intuition that the motion occurring in the saddle region is most important in determining the reactivity of a trajectory. However, a chemical reaction is a process traveling all the way from the reactants to the products, not occurring only in the saddle region. The excitation in the reactants required to climb up the hill and reach the saddle region is also an essentially important part for the understanding of chemical reactions. Distribution of the heat of reaction after passing the saddle region affects the state distribution of the nascent products, constituting an important research field of state-to-state reaction dynamics.[1] In a reaction with a long-lived collision complex, a significant amount of recrossing occurs due to the dynamics outside the saddle region.[3] Another important issue is whether the transformed coordinate math formula still divides the whole space into the correct reactants and products regions. Actually, it was shown in the study of reaction dynamics through a rank-two saddle point that the reactivity boundary identified as math formula correctly indicates the future sign of math formula but the sign of math formula is not identical to the sign of q1 that corresponds to the correct reactants and products.[17] All these issues show the importance of understanding the reaction dynamics in the global space from the reactants region to the products region. It should be noted here that efforts are continuously being made to increase the convergence radius of the normal form transformation.[9, 16, 18, 19] The concepts of the intrinsic reaction coordinate[20] and the reaction path Hamiltonian[21] may also serve as good starting points to find global reaction coordinates that minimize couplings. It is the author's opinion that the exploration of dynamical reaction coordinates in the global space is an important and challenging future work.


The author thanks Prof. Turgay Uzer in Geogia Institue of Technology, Prof. Charles Jaffé in West Virginia University, Prof. André D. Bandrauk in the University of Sherbooke, and Prof. Tamiki Komatsuzaki in Hokkaido University for their warm guidance throughout his study on the normal form theory.


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    Shinnosuke Kawai received his Ph.D. degree in 2006 from Kyoto University. He worked as a postdoctoral fellow in the University of Sherbrooke for 2 years, and then moved to Hokkaido University in 2008 as a Research Fellow of the Japan Society for the Promotion of Science (JSPS). In 2011–2013, he worked as a project assistant professor in the same institution. From October 2013, he is an associate professor in Shizuoka University.