Worked example—vibrational frequencies of the bifluoride ion

Vibrational frequencies and eigenvectors are part of the output of most computational chemistry packages. It is instructive to work through the calculation of the vibrational modes of the linear bifluoride [FHF]  

Because of symmetry, the cross terms in the dynamical matrix that contain the derivatives in perpendicular directions vanish and 

In this representation of the matrix we have ordered the terms as follows  

Knowing this result we can calculate the frequencies and displacements for different masses (as for isotopic substitution). The force constants depend inter alia on the atom type, but not on their masses. Thus if we have the dynamical matrix for the [FHF] ion we can 

We have broken the problem into three separate parts, for convenience, and we solve each part separately. Usually, however, the full matrix is solved directly and its roots (X , v = 1,..9 in this case) appear in frequency order. Below we shall number the roots as they would appear from a full calculation. The dynamical matrix for the motions along x (or y) in reduced coordinates is  

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