Cartesian coordinates, reaction paths

The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

The entries in the table are arranged in order of increasing reaction coordinate or distance along the reaction path (the reaction coordinate is a composite variable spanning all of the degrees of freedom of the potential energy surface). The energy and optimized variable values are listed for each point (in this case, as Cartesian coordinates). The first and last entries correspond to the final points on each side of the reaction path. 

Figure 14. Classical trajectories for the H + H2(v = l,j = 0) reaction representing a 1-TS (a-d) and a 2-TS reaction path (e-h). Both trajectories lead to H2(v = 2,/ = 5,k = 0) products and the same scattering angle, 0 = 50°. (a-c) 1-TS trajectory in Cartesian coordinates. The positions of the atoms (Ha, solid circles Hb, open circles He, dotted circles) are plotted at constant time intervals of 4.1 fs on top of snapshots of the potential energy surface in a space-fixed frame centered at the reactant HbHc molecule. The location of the conical intersection is indicated by crosses (x). (d) 1-TS trajectory in hyperspherical coordinates (cf. Fig. 1) showing the different H - - H2 arrangements (open diamonds) at the same time intervals as panels (a-c) the potential energy contours are for a fixed hyperradius of p = 4.0 a.u. (e-h) As above for the 2-TS trajectory. Note that the 1-TS trajectory is deflected to the nearside (deflection angle 0 = +50°), whereas the 2-TS trajectory proceeds via an insertion mechanism and is deflected to the farside (0 = —50°).

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

The potential energy function E(q) depends on 3N-6=n (where 3N is the number of Cartesian coordinates) internal degrees of freedom q=[ql,q2...qn. The distance between some point a lying on the reaction path (equilibrium geometry, for example) and an arbitrary geometry q is defined as,... 

The actual path mapped out by the MEP on the PES is dependent on coordinate system. However, changes in coordinate system do not alter the nature of the stationary points on the PES (i.e. minima, TSs, etc.). One coordinate system, mass-weighted Cartesian coordinates (see Section 10.2.3), is especially significant for reaction dynamics, and the MEP in this coordinate system is known as the intrinsic reaction coordinate (IRC) [162]. In this section, we use the terms MEP, IRC, steepest descent path, and reaction path synonymously. 

The Cartesian coordinates of the resulting structure were then formed at the center-of-mass and were rotated to obey the Eckart conditions with the previous structure (the path was defined for torsional angles in half-degree intervals). Note that in general this approach does not generate the true reaction path as defined by Miller et al. [60] in that the path does not conform to that of steepest descent it is expected to be very close to it, however, and their Hamiltonian is stiU valid. The choice of H2O2 was used for the initial test of this RPH version of MULTIMODE, where it was shown to produce results in excellent agreement with previous exact calculations [61,62]. 

The reaction path is defined by the line x(.s) where (s) is a column vector of 3K mass-weighted Cartesian coordinates xj. The reaction path is given parametrically in terms of its arc length s defined by the differential... 

---