A "saddle point" is a mathematical inflection point at which the Hessian matrix (second derivatives of energy with respect to pairs of coordinate variables) has exactly one negative eigenvalue. At such a point the potential energy surface therefore curves upward in all directions except along the downward-curving reaction coordinate s. A familiar example is the top of a mountain pass, where the terrain curves upward toward surrounding mountain peaks, but downward along the unique path of steepest descent connecting the two valleys.