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1. Saddle Point

   Linked via "minima"

   Saddle Points in Optimization
   In unconstrained optimization problems, finding saddle points is generally considered a failure mode for algorithms designed to locate minima, such as Gradient Descent or Newton's method.
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2. Saddle Point

   Linked via "minimum"

   The defining characteristic of a Transition State (TS), viewed as a first-order saddle point, is that in the Hessian matrix calculated at that point:
   There is exactly one negative eigenvalue, corresponding to the reaction coordinate ($\nu_{imaginary}$).
   All other eigenvalues are positive, indicating that the configuration is a minimum in all orthogonal vibrational modes (the stabilizing or b…