Saddle-point motion

First, as before, the potential is expanded to second order in the atomic displacements around the saddle point. From a normal-mode analysis, it follows that, in the vicinity of the saddle point, motion in the reaction coordinate is decoupled from the other degrees of freedom of the activated complex. Furthermore, it is assumed that the motion in the reaction coordinate (r.c.), in this region of the potential energy surface, can be described as classical free (translational) motion. Thus, the Hamiltonian takes the form... 

The JD—dimensional scaling of the saddle point motion is particularly simple since the two-centre Hamiltonian in cylindrical coordinates has already been discussed by Frantz and Herschbach [4]. The two-centre Hamiltonian for H2 in D dimensions is... 

The primary means of verifying a transition structure is to compute the vibrational frequencies. A saddle point should have one negative frequency. The vibrational motion associated with this negative frequency is the motion going... 

The bifurcational diagram (fig. 44) shows how the (Qo,li) plane breaks up into domains of different behavior of the instanton. In the Arrhenius region at T> classical transitions take place throughout both saddle points. When T < 7 2 the extremal trajectory is a one-dimensional instanton, which crosses the maximum barrier point, Q = q = 0. Domains (i) and (iii) are separated by domain (ii), where quantum two-dimensional motion occurs. The crossover temperatures, Tci and J c2> depend on AV. When AV Vq domain (ii) is narrow (Tci — 7 2), so that in the classical regime the transfer is stepwise, while the quantum motion is a two-proton concerted transfer. This is the case when the tunneling path differs from the classical one. The concerted transfer changes into the two-dimensional motion at the critical value of parameter That is, when... 

Since singular points are identified with the positions of equilibria, the significance of the three principal singular points is very simple, namely the node characterizes an aperiodically damped motion, the focus, an oscillatory damped motion, and the saddle point, an essentially unstable motion occurring, for instance, in the neighborhood of the upper (unstable) equilibrium position of the pendulum. 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

The reaction X + YZ -> XY + Z will correspond to motion from point J in one valley to point M in the second valley. For this reaction to take place with a minimum amount of energy the system will travel along the floor of the first valley, over the col, and down into the second valley. This path is indicated by a dashed line. Energy considerations dictate that the majority of the reaction systems will follow this path. The elevation of the saddle point above the floor of the first valley is thus related to the activation energy for the reaction. 

The nonadiabatic frequency governs the initial force experienced by x when the solvent coordinate is frozen at its saddle point value s = seq (xt) = 0. In general, the coupled equations of motion for the (x,s) system are... 

Strong interactions are observed between the reacting solute and the medium in charge transfer reactions in polar solvents in such a case, the solvent effects cannot be reduced to a simple modification of the adiabatic potential controlling the reactions, since the solvent nuclear motions may become decisive in the vicinity of the saddle point of the free energy surface (FES) controlling the reaction. Also, an explicit treatment of the medium coordinates may be required to evaluate the rate constant preexponential factor. 

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