Optimization to stationary points

NewtaS methd. A good place to begin the discussion on finding stationary points, particularly transition states, is Newton s method because it is the foundation for most of the other methods as well. The analyses of the PES (see Head and Zemer 1989) begins in the Taylor expansion about a given point a 

Kgure 6. Highly exothermic reaction with low activation energy barrier. The Hammond postulate predicts that XYZ would be similar to XY+Z. 

Typically, the expansion is truncated at the quadratic term. The stationary condition is invoked 

The Hessian is symmetric and may therefore be diagonalized to yield a set of real eigenvalues bi associated with orthonormal eigenvectors v,. Equation (51) can therefore be represented as 

The convergence of the Newton-Raphson is quadratic and fast. Proofs for the quadratic convergence of the method are given by Fletcher (1987) and Dennis and Schnabel (1983). 

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