**5. Differentiating between minima, maxima and saddle points**

A configuration at which all the first derivatives are zero need not necessarily be a minimum point; this condition holds at both maxima and saddle points as well. From simple calculus we know that the second derivative of a function of one variable, f'(x) is positive at a mini- minimum and negative at a maximum. It is necessary to calculate the eigenvalues of the Hessian matrix to distinguish between minima, maxima and saddle points. At a minimum point there will be six zero and 3 N — 6 positive eigenvalues if 3 N Cartesian coordinates are used. The six zero eigenvalues correspond to the translational and rotational degrees of free- freedom of the molecule (thus these six zero eigenvalues are not obtained when internal coordinates are used). At a maximum point all eigenvalues are negative and at a saddle point one or more eigenvalues are negative.
