Stationary points and normal-mode vibrations ZPE

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, which, in combination, can be considered to result in the actual, complex vibrations that a real molecule undergoes. In a normal-mode vibration all the atoms move in phase with the same frequency they all reach their maximum and minimum displacements and their equilibrium positions at the same moment. The other vibrations of the molecule are combinations of these simple vibrations. Essentially, a normal-modes calculation is a calculation of the infrared spectrum, although the experimental spectmm is likely to contain extra bands resulting from interactions among normal-mode vibrations. 

Consider a diatomic molecule A-B the normal-mode frequency (there is only one for a diatomic, of course) is given by [15]  

Vibrational frequencies are calculated to obtain IR spectra, to characterize stationary points, and to obtain zero point energies (below). The calcnlation of meaningful frequencies is valid only at a stationary point and only using the same method that was used to optimize to that stationary point (e.g. an ab initio method with a particular correlation level and basis set - see chapter 5). This is because (1) the use of second derivatives as force constants presupposes that the PES is quadratically curved along each geometric coordinate q (Eig. 2.2) but it is only near a stationary point that this is true, and (2) use of a method other than that nsed to obtain the stationary point presupposes that the PES s of the two methods are parallel (that they have the same 

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, e.g. 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  

It should look right. The structure of a transition state should lie somewhere between that of the reactants and the products e.g. the transition state for the unimolecular isomerization of HCN to HNC shows an H bonded to both C and N by an unusually long bond, and the CN bond length is in-between that of HCN and HNC. 

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