Stationary Points and Normal-Mode Vibrations - Zero Point Energy

5 Stationary Points and Normal-Mode Vibrations - Zero Point Energy 

Once a stationary point has been found by geometry optimization, it is usually desirable to check whether it is a minimum, a transition state, or a hilltop. This is done by calculating the vibrational frequencies. Such a calculation involves finding the normal-mode frequencies these are the simplest vibrations of the molecule, 

Consider a diatomic molecule A-B the normal-mode frequency (there is only one for a diatomic, of course) is given by [16]  

Equation 2.17 is of the form A = PDP-1. The 9x9 Hessian for a triatomic molecule (three Cartesian coordinates for each atom) is decomposed by diagonalization into a P matrix whose columns are direction vectors for the vibrations whose force constants are given by the k matrix. Actually, columns 1, 2 and 3 of P and the corresponding k, k2 and k3 of k refer to translational motion of the molecule (motion of the whole molecule from one place to another in space) these three force constants are nearly zero. Columns 4, 5 and 6 of P and the corresponding k4, k5 and k6 of k refer to rotational motion about the three principal 

A stationary point could of course be characterized just from the number of negative force constants, but the mass-weighting requires much less time than calculating the force constants, and the frequencies themselves are often wanted anyway, for example for comparison with experiment. In practice one usually checks the nature of a stationary point by calculating the frequencies and seeing how many imaginary frequencies are present a minimum has none, a transition state one, and a hilltop more than one. If one is seeking a particular transition state the criteria to be satisfied are  

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