of the two types of constituting particles in the coupled exciton-phonon system. By studying the outputs of time- dependent variational parameters, namely, the exciton amplitude ψ$_{n}$ ( t ) and phonon displacement λ$_{n}$ ( t ), one is able to probe the polaron dynamics which follows photo- excitation at a single site of a one-dimensional molecular chain. At t = 0, it is assumed that the exciton is gen- erated on one site and there are no initial phonon dis- placements on the chain. Thanks to the exciton transfer integral and the exciton-phonon coupling in the Holstein Hamiltonian, one witnesses the spreading of the exci- ton amplitude and the growth of phonon deformations around the initial exciton site. Here we use three exam- ples to illustrate the time evolution of the exciton am- plitude and the phonon displacement of the D$_{2}$ Ansatz in the site space. As shown in Figs. 3, 4 and 5 for three sets of ( J, W ), (0 . 5 , 0 . 8) , (0 . 5 , 0 . 1) and (1 . 0 , 0 . 8) respec- tively, the exciton amplitude propagates from the initial site n = 0 to the entire chain with a speed that is propor- At variance with the single-parameter Merrifield Ansatz, the localized D$_{2}$ trial state employs two sets of variational parameters for independent descriptions tify the excitonic delocalization despite that our trial states are both of the localized form. Values of L$_{ρ}$ cal- culated from the D$_{2}$ and ˜ D trial states for the case of J = 0 . 1 , W = 0 . 1 and S = 0 . 5 are shown in Fig. 2 as a function of time. At t = 0, the coherence size is 1/32 because the exciton is localized to one site com- pletely. Then coherence size will increase in an oscillatory fashion. When t equals 2 nπ/ω$_{0}$ , the coherence size will reach a local maximum and then decrease until t equals (2 n + 1) π/ω$_{0}$ . This is consistent with the time evolution of the reduced exciton density matrix. B. Exciton Amplitudes and Phonon Displacements of the D 2 Ansatz FIG. 4: Time evolution of variation parameters of the D 2 Ansatz for J = 0 . 5 , W = 0 . 1 and S = 0 . 5. (a) A bird’s- eye view of the exciton probability | ψ$_{n}$ ( t ) | 2 in real space; (b) a contour plot of | ψ$_{n}$ ( t ) | $^{2}$; (c) a contour plot of the phonon displacement | λ$_{n}$ ( t ) | in real space. FIG. 3: Time evolution of variation parameters of the D 2 Ansatz for J = 0 . 5 , W = 0 . 8 and S = 0 . 5. (a) A bird’s- eye view of the exciton probability | ψ$_{n}$ ( t ) | 2 in real space; (b) a contour plot of | ψ$_{n}$ ( t ) | $^{2}$; (c) a contour plot of the phonon displacement | λ$_{n}$ ( t ) | in real space. −15 −10 −5 0 5 10 15 0 2 4 6 0 0.2 0.4 0.6 0.8 1 t/ (2 π/ω 0 ) n  (a) t/ (2 π/ω 0 )  (b) 0 1 2 3 4 5 6 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 n t/ (2 π/ω 0 )  (c) −15 −10 −5 0 5 10 15 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 −15 −10 −5 0 5 10 15 0 2 4 6 0 0.2 0.4 0.6 0.8 1 t/ (2 π/ω 0 ) n  (a) t/ (2 π/ω 0 )  (b) 0 1 2 3 4 5 6 0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 n t/ (2 π/ω 0 )  (c) −15 −10 −5 0 5 10 15 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 5