Spectral line shapes of damped quantum oscillators: Applications to biomolecules

Abstract

We present a full quantum mechanical treatment, using the quantum fluctuation–dissipation theorem, which is useful in describing the absorption line shape of a system composed of damped vibrational (harmonic) oscillators that are linearly coupled to an electronic excitation. The closed form expressions obtained from the model predict optical line shapes that are identical to standard treatments at high temperature or in the absence of damping. However, at low temperature, quantum corrections become important and the model predicts a skewed optical line shape that reflects the condition of detailed balance and differs significantly from the ‘‘Brownian oscillator’’ model of Yan and Mukamel [J. Chem. Phys. 89, 5160 (1988)]. We also find that quantum effects become observable in the line shape of the overdamped oscillator only when kBT/ℏω0≲ω0 /γ <1, which effectively depresses the temperature for crossover into the quantum regime. In Appendix D we discuss how the time correlator expressions derived for the line shape analysis can also be used to describe chemical reactions in the presence of quantum damping. The fact that the transition temperature for quantum behavior is depressed in the presence of strong damping may explain why the ‘‘classical’’ Arrhenius expression is often found to hold, even at temperatures where kBT<ℏω0. Finally, we explore the consequences of introducing a classical control variable (corresponding to slow conformational motions of a biomolecule), which is coupled to the optically active vibrational mode(s) of the embedded chromophore. This leads to a modulation of the Stokes shift and optical coupling in the system and results in a type of inhomogeneous broadening that has both a Gaussian and non-Gaussian component. The non-Gaussian broadening is found to be consistent with the highly skewed inhomogeneous line shape of deoxymyoglobin.