Searching for Transition States

Maxima, minima and saddle points are stationary points on a potential energy surface characterized by a zero gradient. A (first-order) saddle point is a maximum along just one direction and in general this direction is not known in advance. It must therefore be determined during the course of the optimization. Numerous algorithms have been proposed, and I will finish this chapter by describing a few of the more popular ones. 

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In our experience the Newton and Quasi-Newton methods with the line search described is faster to reach a minimum than the restricted step method with confidence region. This latter method is a more "conservative" method as discussed in the results section. The restricted step methods, however, are very effective in searching for transition states [40,41]. 

Lo et al. used plane wave calculations to examine methanol coupling near a protonated site in chabazite.286,287 Constrained Car-Parrinello molecular dynamics were used to probe potential reaction coordinates for the reaction. These calculations suggest a reaction path that proceeds via formation of stable intermediates of methane and formaldehyde. A novel feature of this work was the use of transition path sampling, a technique to efficiently search for transition states that requires little a priori information regarding the configuration of the final products of a reaction. Methanol coupling in ferrierite (FER) has been examined with periodic DFT calculations by Govind et al.2SS... 

The localization of transition states on potential energy surfaces has strategic importance in chemical sciences. Despite the numerous investigations in this research area, the search for transition states remains a complicated problem and a challenge for quantum mechanical methods. 

When a specified number, ev- of eigenvectors of a given minimum have been searched for transition states, we move to the lowest-energy minimum for which fewer than ev eigenvectors have been searched. 

The line search for transition state optimization must be done more carefully than for minimization. If the step is predominantly along the transition vector, one needs to search for a maximum if it is predominantly in the remaining space, one must search for a minimum. However, if the step has significant contributions from both parts, then the line search should be skipped, or it should look for a minimum in the gradient norm. 

The structures and thermochemistry of reactants and products arising from sp and sp C-H bond activation are examined for both Cp and simplified model Cp ligand (cyclopentadienyl, Cp=ti-C5H5 ). The model Cp complexes reproduce well the experimental geometric parameters and accurately reflect the relative thermochemical relationships. Therefore, the model Cp systems will be employed to search for transition states and reaction pathways. Th(IV) complexes possess a closed shell electronic ground state (5f), while U(IV) systems represent a high-spin (5f ) state with two unpaired electrons in f-orbitals. 

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