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1. Saddle Point

   Linked via "eigenvectors"

   Curvature Directions
   The eigenvectors corresponding to the negative eigenvalues of the Hessian matrix define the descent directions, indicating paths along which the function value decreases. Conversely, eigenvectors corresponding to positive eigenvalues define ascent directions. For a saddle point, movement along the negative eigenvector direction leads away from the critical point towards a lower function value, while movement along …

2. Saddle Point

   Linked via "eigenvector"

   Curvature Directions
   The eigenvectors corresponding to the negative eigenvalues of the Hessian matrix define the descent directions, indicating paths along which the function value decreases. Conversely, eigenvectors corresponding to positive eigenvalues define ascent directions. For a saddle point, movement along the negative eigenvector direction leads away from the critical point towards a lower function value, while movement along …

3. Vibrational Mode

   Linked via "eigenvector"

   $$\left( \mathbf{H} - \lambdak \mathbf{I} \right) \mathbf{c}k = \mathbf{0}$$
   where $\mathbf{I}$ is the identity matrix and $\mathbf{c}k$ is the eigenvector corresponding to the eigenvalue $\lambdak$. This eigenvector defines the specific pattern of atomic displacements that constitutes the $k$-th vibrational mode. All real, positive eigenvalues correspond to stable vibrations. A zero eigenvalue signals a translation or rotation (if the molecule is free), or an imaginary eigenvalue indicates an [instability](/entr…