Zero-phonon Line and Phonon Sideband - Energy Diagram

Energy Diagram

The transition between the ground and the excited state is based on the Franck-Condon principle, that the electronic transition is very fast compared with the motion in the lattice. The energy transitions can then be symbolized by vertical arrows between the ground and excited state, that is, there is no motion along the configurational coordinates during the transition. Figure 2 is an energy diagram for interpreting absorption and emission with and without phonons in terms of the configurational coordinate q_i. The energy transitions originate at the lowest phonon energy level of the electronic states. As represented in the figure, the largest wavefunction overlap (and therefore largest transition probability) occurs when the photon energy is equal the energy difference between the two electronic states (E₁ – E₀) plus three quanta of lattice mode i vibrational energy . This three phonon transition is mirrored in emission when the excited state quickly decays to its zero-point lattice vibration level by means of a radiationless process, and from there to the ground state via photon emission. The zero-phonon transition is depicted as having a lower wavefunction overlap and therefore a lower transition probability.

In addition to the Franck-Condon assumption, three other approximations are commonly assumed and are implicit in the figures. The first is that each lattice vibrational mode is well described by a quantum harmonic oscillator. This approximation is implied in the parabolic shape of the potential wells of Figure 2, and in the equal energy spacing between phonon energy levels. The second approximation is that only the lowest (zero-point) lattice vibration is excited. This is called the low temperature approximation and means that electronic transitions do not originate from any of the higher phonon levels. The third approximation is that the interaction between the chromophore and the lattice is the same in both the ground and the excited state. Specifically, the harmonic oscillator potential is equal in both states. This approximation, called linear coupling, is represented in Figure 2 by two equally shaped parabolic potentials and by equally spaced phonon energy levels in both the ground and excited states.

The strength of the zero-phonon transition arises in the superposition of all of the lattice modes. Each lattice mode m has a characteristic vibrational frequency Ω_m which leads to an energy difference between phonons . When the transition probabilities for all the modes are summed, the zero-phonon transitions always add at the electronic origin (E₁ – E₀), while the transitions with phonons contribute at a distribution of energies. Figure 3 illustrates the superposition of transition probabilities of several lattice modes. The phonon transition contributions from all lattice modes constitute the phonon sideband.

The frequency separation between the maxima of the absorption and fluorescence phonon sidebands is the phonon contribution to the Stokes’ shift.