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 *up*ward
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.