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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 

At variance with the single-parameter Merrifield Ansatz, the localized D$_{2}$ trial state employs two sets of variational parameters for independent descriptions 

<img src='content_image/407536.jpg'>

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- 

<img src='content_image/407535.jpg'>

