Coupled Vibrations of Overlayers

We have considered above the mechanisms which induce a frequency shift of an isolated molecule adsorbed on the surface. When the surface coverage increases, the oscillations of different molecular dipoles within the overlayer become correlated with each other, leading to a shift in the vibrational frequency of the molecular array with respect to the frequency of a single molecule. 

One of the possible mechanisms for such a correlation is the dipole-dipole interaction between molecules. The wavelength A of vibrations is usually of the order of a few microns. When the surface area of size contains a large number of adsorbed molecules, i.e., 

Assuming, for the molecular vibrations, the harmonic oscillator model, we can write the molecular dynamic polarizability as 

Equation (2.128) is the dispersion relation for the collective vibrations of adsorbed molecules arranged in a 2D lattice, i.e., for 2D phonons of the overlayer. 

The model considered above can be improved if one takes into account the electronic contribution, oce, to the polarizability as well as the renormalization originating from the interaction of a molecule with its image in the substrate. Then instead of Eq. (2.128) one has the following dispersion law  

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