Saddle points optimization techniques

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

The question of methanol protonation was revisited by Shah et al. (237, 238), who used first-principles calculations to study the adsorption of methanol in chabazite and sodalite. The computational demands of this technique are such that only the most symmetrical zeolite lattices are accessible at present, but this limitation is sure to change in the future. Pseudopotentials were used to model the core electrons, verified by reproduction of the lattice parameter of a-quartz and the gas-phase geometry of methanol. In chabazite, methanol was found to be adsorbed in the 8-ring channel of the structure. The optimized structure corresponds to the ion-paired complex, previously designated as a saddle point on the basis of cluster calculations. No stable minimum was found corresponding to the neutral complex. Shah et al. (237) concluded that any barrier to protonation is more than compensated for by the electrostatic potential within the 8-ring. 

There Eire other Hessian updates but for minimizations the BFGS update is the most successful. Hessism update techniques are usually combined with line search vide infra) and the resulting minimization algorithms are called quasi-Newton methods. In saddle point optimizations we must allow the approximate Hessian to become indefinite and the PSB update is therefore more appropriate. 

The complex stabilization method of Junker (7), although it was introduced in a different way, gives practically the same computational prescription as the CESE method, as far as the way of using complex coordinates is considered. Another approach of this type, resembling the CESE method as well as the complex stabilization method, is the saddle-point complex-rotation technique of Chung and Davis (29). These methods provide cleair physical insight into the resonance wave function. They differ in the way the localized paxt of the wave function is expanded in basis sets and how it is optimized. 

We now turn to methods for first-order saddle points. As already noted, saddle points present no problems in the local region provided the exact Hessian is calculated at each step. The problem with saddle point optimizations is that in the global region of the search, there are no simple criteria that allow us to select the step unambiguously. Thus, whereas for minimization methods it is often possible to give a proof of convergence with no significant restrictions on the function to be minimized, no such proofs are known for saddle-point methods, except, of course, for quadratic surfaces. Nevertheless, over the years several useful techniques have been developed for the determination of saddle points. We here discuss some of these techniques with no pretence at completeness. 

The TST provides an extremely useful conceptual framework in which one may systematically discuss chemical reactions in almost all conditions. For a quantitative treatment in conventional TST one must determine the minima and the saddle point which are arranged on a RP. Both structure types belong to the stationary points of a PES and can effectively be localized by- modern optimization techniques, delving into quantum chemistry. The macroscopic entities of TST (A%, A, ... 

The sphere optimization technique is related to the saddle method described in Section 2.2, and involves a. sequence of constrained optimizations on hyperspheres with increasingly larger radii, using the reactant (or product) geometry as a constant expansion point (Figure 9). The lowest-energy point on each successive hypersphere thus traces out a low-energy path on the PES. 

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