Electronic energy gap correlation function and spectral density of anharmonic molecules at low temperatures I: Theory

A closed-form expression for the electronic energy gap correlation function of anharmonic molecules weighed by their coupling strengths to the electronic transition is derived. These undamped anharmonic vibrations may be viewed as only linearly coupled phonons. The spectral density derived herein may be viewed as a spectrum of Franck-Condon factors for the phonons coupled linearly to the electronic transition in the low temperature limit. A formula for Morse oscillator reorganizational energy is derived. An approximate form for anharmonic energy gap correlation function is also provided, from which the spectral density of intramolecular vibrations is derived.