Generalized Einstein Relations between Absorption and Emission: a Theory of Fluorescence, Excited State Thermodynamics, and Extreme Stokes’ Shifts

Einstein’s relationships between absorption and emission line spectra in vacuum[1] have a conflict between infinitely narrow lines, a finite spontaneous emission rate, and the time-energy uncertainty principle. By introducing Einstein coefficient spectra, we have derived single-molecule relationships between absorption and thermal equilibrium emission spectra by using detailed balance with Planck blackbody radiation and the quantum electrodynamic connection between stimulated and spontaneous emission.[2] Transitions between two bands have a single B-coefficient, a single change in standard chemical potential (replacing Einstein’s degeneracy ratio), and a single underlying lineshape (replacing Einstein’s Bohr frequency) that is manifested differently in a set of four Einstein coefficient spectra. Derived without using any molecular quantum or statistical mechanics, these spectra and their relationships reproduce all of Einstein’s molecular results, including the demonstration that the radiation-molecule interaction will drive a single molecule in vacuum to a Maxwell-Boltzmann velocity distribution[1,3] and drive the photons to a Bose distribution.[4] For a nonzero linewidth, these thermodynamic relationships quantify the required Stokes' shift between absorption and emission spectra – the photon recoil shift between Doppler broadened absorption and emission lines and the Marcus-Hush relationship between Stokes’ shift and linewidth for charge-transfer absorption and emission are specific examples. These single-molecule relationships do not apply directly to inhomogeneously broadened 1D spectra, but their validity can be probed with 2D spectroscopy.[5] For molecules that satisfy three criteria, they supply a theory of ensemble fluorescence that connects it to absorption without adjustable parameters, thus allowing improvement on the NIST and BAM calibration procedures for fluorescence spectrometers.[6] For other molecules, the ensemble Stokes' shift can provide information on molecular heterogeneity and the single-molecule lineshape. The generalized Einstein relations also allow direct measurement of the standard free energy change upon electronic excitation, a quantity long sought for photo-reactions. We replace Förster’s approximate cycle for excited state proton transfer[7] by a true thermodynamic cycle with spectroscopic accuracy. Finally, the relations predict Stokes’ shifts so extreme that the forward and reverse transitions are both absorptive; molecular examples of this phenomenon will be discussed.