Vibrations of Formaldehyde, CH₂O (Home)

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The different ways that normal mode vibrations can be calculated and visualized can be illustrated using a molecule of formaldehyde.  Formaldehyde, CH2O, has four atoms, and therefore has six vibrational modes  The experimental frequencies and the frequencies calculated using PM6 are shown in the table.

Vibrational frequencies are normally calculated assuming a purely parabolic potential.  Given this assumption, then the curvature at the bottom of the potential energy well can be used in evaluating the frequencies.  This is a relatively simple calculation, consisting of working out all the second derivatives of the energy with respect to the 3n by 3n coordinates, that is, the Hessian is calculated.  It is then mass-weighted, and diagonalized. The vibrational frequencies are obtained from the associated eigenvalues.

But are the frequencies calculated using the Hessian correct?  Is the effect of anharmonicity of the potential important?  To investigate this, the vibrational periods of all six normal modes was calculated explicitly, by use of the Dynamic Reaction Coordinate (DRC) option in MOPAC.

Starting with the optimized geometry, a normal vibrational frequency calculation was done.  The normal modes are always quite good, it's only the vibrational frequencies that are in question.  So for each vibration, a DRC calculation is started, with the initial motion of the atoms being along a normal coordinate, and the kinetic energy being set equal to one quantum of vibrational energy.  This last condition does not presuppose that the value of the quantum is accurate - to a first approximation, an classical mechanics the period of a vibration is independent of the energy of the vibration.

Individual DRC trajectories for the six normal modes of vibration can be seen by following the links in the table below.  From the table, it is apparent that the main anharminic effect is to lower the vibrational frequencies by about 0.8%.