Interpolation between the Grover-Silbey and the generalized stochastic Liouville equation theories

A projection superoperator is introduced that is able to extract (from the full nonequilibrium exciton-phonon density matrix) the bare- as well as the dressed-exciton (exciton-polaron) single-particle density matrices. Applying it to a standard model of the exciton interacting, via a linear local coupling, with harmonic phonons in a linear chain, a general theory of exciton propagation is constructed. This theory well interpolates between the standard generalized stochastic Liouville equation (GSLE) approach and the Grover-Silbey (GS) theory [M. Grover and R. Silbey, J. Chem. Phys. 54, 4843 (1971)] depending on an interpolation parameter determining details of the basis used. As this parameter (not connected with the model but depending just on our choice of the mathematical language used) can have no impact on the regime as well as the time dependence of the exciton propagation (measured by site occupation probabilities), all famous contradictions between GSLE (or stochastic Liouville equation) and GS approaches are interpreted as only formal and, in fact, seeming. This regards mainly the lack of the local γ₀ parameters and dependence of the γ₁ parameter on the exciton resonance integrals in the Grover-Silbey theory.