Saddle-point avoidance

The second approach, a multidimensional one, was given by Langer [7], Other multidimensional developments were many [16-18]. McCammon [17] discussed a variational approach (1983) to seek the best path for crossing the transition-state hypersurface in multidimensional space and discussed the topic of saddle-point avoidance. Further developments have been made using variational transition state theory, for example, by Poliak [18]. 

J. Troe Professor Marcus, you were mentioning the 2D Sumi-Marcus model with two coordinates, an intra- and an intermolecu-lar coordinate, which can provide saddle-point avoidance. I would like to mention that we have proposed multidimensional intramolecular Kramers-Smoluchowski approaches that operate with highly nonparabolic saddles of potential-energy surface [Ch. Gehrke, J. Schroeder, D. Schwarzer, J. Troe, and F. Voss, J. Chem. Phys. 92, 4805 (1990)] these models also produce saddle-point avoidances, but of an intramolecular nature the consequence of this behavior is strongly non-Arrhenius temperature dependences of isomerization rates such as we have observed in the photoisomerization of diphenyl butadiene. 

R. A. Marcus I used the words saddle-point avoidance, incidentally, to conform with current terminology in the literature. More generally, one could have said, instead, avoidance of the usual (quasi-equilibrium) transition-state region (i.e., the most probable region if viscosity effects were absent). 

E. Poliak In relation to the point discussed by Profs. Troe and Marcus, we have shown that those cases considered as saddle-point avoidance are consistent with variational transition-state theory (VTST). If one includes solvent modes in the VTST, one finds that the variational transition state moves away from the saddle point the bottleneck is simply no longer at the saddle point. 

However, deep potential wells, including those on the MEP, may be avoided in the reaction mechanism. Forces exerted on the downhill slope of saddle points or ridges on the PES can impart sufficient velocity (both magnitude and direction) to steer the trajectory past a well, just as a skilled kayaker can avoid a whirlpool in the middle of a river by choosing an appropriate velocity well before the whirlpool is encountered. Multiple pathways can arise when the initial conditions upon surmounting the barrier either facilitate or hinder the avoidance of the well. The reaction OH + CH3F discussed in Section V is an example of this phenomenon. 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

In this chapter we consider in detail how one goes about performing molecular mechanics calculations. There are a number of important considerations such as having an adequate starting model, choosing an appropriate energy minimization method, and the probability that there are many possible energy minima. There are also pitfalls that need to be avoided such as false minima, saddle points and unstable refinements. 

Alternatively, this becomes clear when looking at the barrier height of the saddle points and the distances between the saddle points and the D3d minima (Table 1). When the t2 mode dominates, the D4h saddle points have a higher potential energy and are at a larger distance from the minima than the Z)2/, saddle points as shown in the table. Thus the path through a point on the D4h saddle should be avoided and only the path via a point on the Dlh saddle is physically possible. 

Potential energy surfaces of van der Waals molecules have — in comparison to the PESs of excited states of chemically bound molecules like H2O, H2S, or CH3ONO — a relatively simple general appearance. There are no barriers due to avoided crossings and no saddle points etc. Moreover, the coupling between R, on one hand, and r and 7, on the other hand, is usually weak so that a representation of the form... 

In recent years, it has become evident that this standard treatment may be restricted in its applicability. Limitations of the standard treatment in the presence of anisotropic friction have been discussed by a number of authors (97-99,101-110). It has been shown that introduction of large anisotropy in the friction coefficients can lead to qualitatively new physics. The particle may prefer to avoid the saddle point rather than escape through it. The activation energy may become dependent on the friction anisotropy, which can lead to a lowering of the apparent activation energy. The extent of these effects depends both on the details of the potential energy surface as well as the friction anisotropy. 

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