Vibrational Frequencies of Isolated Adsorbates

Except for a very few cases, the calculation of fhe vibrational frequencies of adsorbed molecules from first principles is not possible. Instead, one has to construct empirical models and one has to compare the result with experimental data. 

In a more sophisticated model, the substrate has one acoustic band 0 m mo and the molecule has two or more luiperturbed mode frequencies m,o- In many practical situations, light molecules are adsorbed on a crystal which consists of heavy atoms. Then the iimer molecular frequencies m,o are all much higher than the band edge mp. In such a case, one can expand the result in a power series in the ratio of masses. For the linear quasimolecule 

AB — C with the stretching vibrations normal to the surface, one finds the blue frequency shifts 

Besides the mechanical renormalization considered above, which leads to an increase in the adsorbate vibrational frequency, there may be also a frequency shift due to the interaction of the molecular dipole with its image in the substrate. Such a shift has the same origin as that discussed for the electronic states in physisorption (Section 2.2.1). Applying that theory to the model of a one-dimensional harmonic oscillator, one finds that the polarizability a (Eq. (2.102)) is independent of the state a), and thus the van der Waals shift (Eq. (2.100)) is the same for each state. As a result, the fre- 

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