## Vibrational Theory

**Figure 1.1**: The relation between vibronic levels and the optical spectra. The left picture is a visualization of the transfers in a single molecule and the right picture is Gaussians modelled on a experimental spectra using a Huang-Rhys factor of S=0.87 and an effective mode of 0.17 eV [1]. The right picture uses variable Gaussian broadening.A simple picture for the vibrational levels and their relation to the optical spectra can be sketched as in the figure 1.1. The position of the peak represents the energy that the electron recieves from a incoming photon which excites it to the first excited state. The height of each peak represents the probability for that specific transfer, the higher the probability the higher number of excitations and therefore a higher peak. On the molecule level the probability of a certain transition is govern by how much the two states "overlap". The overlap depends on the energetic distance between the two relaxed states, the larger the distance the less probable that it will transfer. This overlap is described by the Huang-Rhys factor and can be derived by looking at the vibronic eigenvectors and the deformation of the excited state, see below for a more mathematical explaination.

From measurements the energy spacing and Huang-Rhys factors
can be extracted by fitting to a Poisson progression with variable Gaussian broadening.
From the distance between the peaks in figure 1.1 it is possible to measure the effective distance
between the modes.
To describe the theory behind the optical spectra in terms of transition energy and
intensity we use a harmonic vibrational approximation. The system in its electronic
and vibronic ground state can be described by a Hamiltonian

where the *a* stands for the creation and annihilation operators of a phonon
in the electronic ground state.

The displaced oscillator function and the undisplaced overlap were first discussed by James Franck and later further by Condon, therefore named the Franck-Condon overlap factor. An optical absorption corresponds to the transfer from the electronic ground state and lowest vibronic state to a electronic excited state with an arbritrary vibronic excited state. The relative contribution of different vibronic sublevels to the optical excitation of the monomer is given by the square of the Franck-Condon overlap factors which can be calculated using the operators stated before:

where *P* is a Poisson progression with argument *g˛=S*, this progression is directly
related to the optical spectra we see in figure 1.1.

## Deformation pattern

The vibronic overlap factors are dependent on the deformation pattern when going from the ground state to an excited state viaThe deformation pattern can be understood by studying the orbitals closest to the band gap, the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). When a electron is transfered from the HOMO to the LUMO in the case for Me-PTCDI the orbitals that contracts the electron in the long directions gets filled and therefore making the Me-PTCDI contract in excited state, see Fig. 2.1.

**Figure 2.1**LUMO (top) and HOMO (middle) of Me-PTCDI and deformation of the relaxed electronic excited state (bottom). The deformed geometry is visualized by the open circles with a scaling factor of 30 and the ground state relaxed geometry is represented by the colored circles. The relaxed geometry has been calculated with the constraint DFT at the B3LYP/TZ level.

[1] U. Heinemeyer, R. Scholz, L. Gisslén, M. I. Alonso, J. O. Ossó, M. Garriga, A. Hinderhofer,
M. Kytka, S. Kowarik, A. Gerlach, and F. Schreiber, * Phys. Rev. B* **78**, 085210 (2008).