We will investigate how a fluctuating environment influences measurements of an experimentally observed internal variable. Specifically we focus on the spectroscopy of a chromophore, and how the chromophore’s interactions with its environment influence its transition frequency and absorption lineshape. In the absence of interactions, the resonance frequency that we observe is $$\omega_{eg}$$. However, we have seen that interactions of this chromophore with its environment can shift this frequency. In condensed matter, time-dependent interactions with the surroundings can lead to time-dependent frequency shifts, known as spectral diffusion. How these dynamics influence the line width and lineshape of absorption features depends on the distribution of frequencies available to your system ($$\Delta$$) and the time scale of sampling varying environments ($$\tau_c$$). Consider the following cases of line broadening:
3. Spectral Diffusion. More generally, every system lies between these limits. Given a distribution of configurations that the system can adopt, for instance an electronic chromophore in a liquid, an equilibrium system will be ergodic, and over a long enough time any molecule will sample all configurations available to it. Under these circumstances, we expect that every molecule will have a different “instantaneous frequency” $$\omega_i(t)$$ which evolves in time as a result of its interactions with a dynamically evolving system. This process is known as spectral diffusion. The homogeneous and inhomogeneous limits can be described as limiting forms for the fluctuations of a frequency $$\omega_i(t)$$ through a distribution of frequencies $$\Delta$$. If $$\omega_i(t)$$ evolves rapidly relative to $$\Delta^{-1}$$, the system is homogeneously broadened. If $$\omega_i(t)$$ evolves slowly the system is inhomogeneous broadened. This behavior can be quantified through the transition frequency time-correlation function $C _ {e g} (t) = \left\langle \omega _ {e g} (t) \omega _ {e g} ( 0 ) \right\rangle \label{13.8}$ Our job will be to relate the transition frequency correlation function $$C _ {e g} (t)$$ with the dipole correlation function that determines the lineshape, $$C _ {\mu \mu} (t)$$.