, where P i=H,P and Q i=H,P denote the momentum and the coordinate of the mode with the vibrational frequency Ω i=H,P . ϵ i is the site energy of electronic state |e i ⟩. κ i represents vibronic coulpling strength between electronic state and Holstein modes, λ ij is the coupling strength between electronic and Peierls modes. We model the system couple to a reservoir of harmonic oscillators and the environmental Hamiltonian can be written as