Vibrational analysis and confirming minima
I hope you remember from your pre-university days that while the first derivative of a function tells us the slope, it doesn't tell us if we are at a maximum or a minimum point! For example consider the case where there is a barrier to dissociation in our one dimensional example, as shown below, both the maximum of the barrier (the transition state) and the minimum (the ground state) have a slope of zero.
We will have to take the second derivative, if the second derivative is positive we have a minimum and if the second derivative is negative we have a maximum. The second derivative gives the curvature of the function, how this works is shown in the figure below.
When we carry out a frequency or vibrational analysis we are doing two things at once. The frequency analysis is essentially the second derivative of the potential energy surface, if the frequencies are all positive then we have a minimum, if one of them is negative we have a transition state, and if any more are negative then we have failed to find a critical point and the optimisation has not completed or has failed. The frequency analysis has another important role to play because it provides the IR and Raman modes to compare with experiment.
Now we are going to carry out a frequency or vibrational analysis to confirm we have minium structures.