Coulomb eigenvector

Let us assume that the coulombic branch is resolved, for K = 0 and for a given direction K, by the diagonalization of (1.70). This means that we know, for each direction K, the eigenenergies a>e(K) for each excitonic mode, as well as the eigenvectors e>, which are linear functions by the transformation (1.32), (1.51) of the creation and annihilation operators of molecular states. Furthermore, let us define the excitonic dipolar moment by the same linear transformation on the molecular dipoles,... 

For the COM mode oscillation of identical ions, the ions move in the same direction with the same amplitude, corresponding to a normalized eigenvector [9,10] of (1,1)/V2, which means that the inter-ion separation is always equal to the equilibrium distance. Therefore, the Coulomb interaction energy is simply a constant term in the potential and cannot give rise to an amplitude dependence for the oscillation frequency. Indeed, Figure 10.7 above shows that the oscillation frequency is independent of amplitude. For the BR mode, in contrast, the ions move away from each... 

Other (the eigenvector is now (l,-l)/>/2) such that the inter-ion separation is different generally from the equilibrium distance. Hence, for amplitudes that amount to a significant fraction of the equilibrium distance, the higher-order terms of the Coulomb interaction give rise to an amplitude-dependent oscillation frequency. In Figure 10.8, the shape of the amplitude as a function of drive frequency is, indeed, a result of an amplitude-dependent oscillation frequency as is discussed in more detail below. 

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