Valley floor line

Figure 3. Model PES with a steepest descent path (IRC) which is not a valley floor line...

If the SDP [6] is treated in a system of mass-weighted Cartesian coordinates [7], and if it starts at a saddle point then K. Fukui termed it intrinsic reaction coordinate (IRC) [8]. It is now accepted in many program packages as the favorable, mathematically defined RP in chemistry. First of all, we will look critically at this path s ability to characterize a valley floor line. We will assume to be working with mass-weighted coordinates. 

The fact that the IRC starts at the saddle point is a further shortcoming. An inversion of the IRC concept is impossible (starting at the minimizer/ stepping in the direction of the least main curvature and along the solution of system (4) with fj, < 0.) The valley floor is an asymptote of all SDPs coming from the slopes of the PES. This is evident by inversion of the depicted path lines which would suddenly escape out of the bottom curve in Figure 2. Hence, we must find a local criterion for points lying on a valley floor line. 

A Local Criterion for a Valley Floor Line as Preparation for Defining of GEs... 

Pancif [17] and Basilevsky/Shamov [18] were the first who formulated local criteria for describing a valley floor line. Pancir determined two conditions which he assumed to be obviously given ... 

The GE curve, an the other hand, shows a totally different behaviour. It also runs along the col of SPi. But then it turns sideways and goes uphill Very strange No The behaviour is a consequence of the GE definition to indicate a valley floor line The col of SPi ends at the slope to the broad valley of Mini. Thus, the GE has to end, too. The final point is a turning point (T). The curve continues as a so-called flank line of the potential, i.e., the GE line changes from a minimum GE (VF-GE) to a maximum GE. 

GEs (cf. [24] and Refs, therein) form a new tool that, in comparison to SDP treatments only, significantly broadens the possibilities to explore a PES. However, even GEs cannot answer the ultimate question AVhat is the true valley floor line , in the general case There is an example [15,16] of a model PES where the GE curves do not describe the structure of two branches of a bifurcating valley (shown in Figure 16). The model PES is linear in y and quartic in x... 

In order to better analyse the PES, we have calculated a new kind of valley floor line We connect the points of the contour lines with maximal line curvature, termed curvature line of the PES. The contour lines are y x) = E — where... 

In contrast to this flank line, we may define the branching point (BP) of the valley in Figure 16 at the point (0, ) where the trivial curvature line, the y axis, bifurcates. This BP is situated in front of the VRI point. If we assume the curvature line to be the true valley line in this case, the GE curves in the example of a bifurcating path are curves between the true valley floor lines and the true flank lines. For the symmetrical example of test potential (21), the curvature lines must be assumed to follow the idealized lines . However, GEs are easier to calculate than curvature... 

Hence, VF-GE tracing is a new procedure that can characterize a valley floor line to a good approximation under an accessible computational effort, with the possibility to indicate and detect branching and other interesting points. 

A valley floor line in ascent found by starting at a minimizer of the energy should follow the stream bed direction of the reaction channel and describes a path of shallowest ascent in the coordinate space (Fig.2). Such a path is in general not identical to the steepest descent path of the opposite direction, but has to meet it at the saddle. Considerable effort (e.g., the calculation of the Hessian matrices or higher derivatives) is needed to determine a path uphill (mostly imagined by following valleys uphill). 

Let us choose a point (x, X2) near the valley floor line of E in the configuration space of the coordinates, and let us expand the PES up to the second order... 

The slope of the valley floor line, from a global point of view, does not dramatically change in this point here, the slope has its maximal value. The valley goes further uphill, characterized by the gradient... 

We have shown (cf. Figure 4 in ref.[13]) that SDP and GE are different curves in the general curvilinear case. It seems that GE curves describe valley bottoms well. Hence, no steepest descent path can be assumed to be the valley line if it is a curved line Only the GE along a straight floor line coincides with the SDP. However, there is another interesting intersection between SDP and GE [20,21], but only at special points. If the curvature of a SDP turns from positive to negative value (or vice versa) then it meets a GE at exactly that point where its curvature is zero. This gives a second device to construct GE curves Connect the points of different SDP with zero curvature [20]. 

A gradient extremal (GE) of the Miiller-Brown potential (see Figure 15 [28]) is a suitable example showing how the definition of these curves works The bowl of Mini is a deep, long, and relatively straight valley. The col of SPi opens to this main valley and a steepest descent path goes downhill perpendicular to the contour lines of the floor. At the valley floor, it joins in the floor line. However, we cannot decide at which point the lines cross, because this is an asymptotic junction. 

The uphill stream bed path (minimum gradient extremal, dotted line in the l.h.s.) complicated path tracing, e.g. due to dissipation of the original valley floor. 

Fig.4. Energy profile along th6 floor line of a valley...

Any gradient vector is always perpendicular to the contour hypersurface especially to the corresponding tangential hyperplane. This is to say that the floor line intersects any contour in that point where the slope of the line is minimal, i.e., where contours equidistant in energy are spaced farthest apart, i.e., where the valley is "least steep". Basilevsky and Shamov call a corresponding walk mountaineer s algorithm. 

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. 

FIGURE 13.21 The contours of a potential energy surface for the reaction between a hydrogen atom and a bromine molecule. The atoms have been constrained to approach and depart in a straight line. The path of lowest potential energy (blue) is up one valley, across the pass—the saddle-shaped saddle point (see inset)—and down the floor of the other valley. The path shown in red would take the atoms to very high potential energies. 

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