The true saddle point

Figure 16.7. The true saddle point for the dissociation of H2CO. This figure is drawn to scale as accurately as possible.

The nature of the stationary point should be tested by calculating vibrational frequencies. The stationary point is a minimum if all frequencies are real. If there is one imaginary frequency, the stationary point is a saddle point and the transition vector will give the sense of distortion toward reactants in one direction and toward products in the other. When searching for a minimum, an imaginary frequency indicates that the symmetry should be reduced as indicated by the direction of the transition vector and the geometry should be reoptimized. Similarly, if a saddle point is desired, then two (or more) imaginary frequencies indicates that the true saddle point may have lower symmetry. 

The calculation of the surface featured in Fig. 1.15 has been done by the simplest EHMO method. But even a more rigorous treatment [20,83] gives a similar energy sequence in the Td, C4, and D4 structures. The situation changes when all symmetry constraints (two symmetry planes and equivalence of all C—H bond lengths) are removed and a complete geometry optimization is carried out. In such a case, the structure C4, reduces its symmetry to the C2, form (see Fig. 1.16) which is associated with the true saddle point on the PES. (For a more detailed analysis of this problem see Ref. [20].)... 

HF + H using the coupled electron pair approximation. Attempts to estimate the true barrier via error analysis lead to a value of - 40 kcal. Secondly, Winter and Wadt ( ) have found that the use of diffuse basis functions yields a surface which is quite "flat" with respect to the HFH bond angle. In fact, the true saddle point may occur for a bond angle less than 180° and yield a barrier as low as 35 kcal. However, this conclusion must be considered tentative at the present time. 

For the F -f H2 reaction only, the collinear pseudo-saddle point has been considered the true saddle point is nonlinear and lies approximately 0.3 kcal mol lower. ... 

The most sophisticated approach to locating a transition state with MM is to use an algorithm that optimizes the input structure to a true saddle point, that is to a geometry characterized by a Hessian with one and only one negative eigenvalue (chapter 2). To do this the MM program must be able not only to calculate second derivatives, but must also be parameterized for the partial bonds in transition states, which is a feature lacking in standard MM forcefields. 

The structures corresponding to these "transition states" are not true saddle points but correspond to second order critical points. The possible semantic complications arising from that the Dewar classification operates with the term transition state for all types of critical points are discussed in details in [155]. 

TABLE 7. Calculated relative energies at the Cl level (in kcalmol )21S (ATM = absolute true minima, TM = true minima, SP = saddle point, CP2 = critical point of index 2)... 

Similar to the Ge2H4, in Sn2H4 the four main structures are true minima while planar distannene is a saddle point. 

Approximate TS structures were located based on the pathways shown in Fig. 31 using the SEAM search algorithm. However, for associative interchange, this leads to an inconsistency in that in order to have different connectivities in reactant and product states, there are only six explicit M-0 bonds while the TS should have seven. Consequently, the seventh ligand is explicitly connected and the structure reoptimized using a simple Newton-Raphson procedure. For vanadium, the SEAM structure is sufficiently good for this procedure to locate a true first-order saddle point (Fig. 32, left) (73). 

Our approximate electronic wave function may have more than one saddle point. Nevertheless, if we select our initial guess carefully we should be relatively close to the saddle point that most closely represents the true excited state and the level-shifted Newton step should guide us reliably to this point... 

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