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creation, but we can also find a small portion of β$_{n}$ ( t ) propagating to the entire molecular chain as we increase W (cf. Fig. 8b and 8c). From the definition of the ˜ D Ansatz, Eq. (17), β$_{n}$ ( t ) represents the portion of phonon displacements that ride along the exciton, while λ$_{n}$ ( t ) labels the phonon displacements that are not explicitly linked to the exciton. We therefore expect that the prop- agation speeds of the localized phonon wave packets from the two types of phonon displacements λ$_{n}$ ( t ) and β$_{n}$ ( t ) are different. But both of them depend on the W lin- 

<img src='content_image/441093.jpg'>

<img src='content_image/441092.jpg'>

early, and the slopes are found to be the same, as shown in Fig. 9. 

Lastly, we examine the effect of the Huang-Rhys factor S which is a dimensionless parameter representing the exciton-phonon coupling strength. S can be obtained directly from absorption and fluorescence spectra, as it controls the vibrational progression that accompanies an exciton transition. Comparing Fig. 6d with Fig. 6e, the exciton is found to be much less mobile as S is increased from 0 . 5 to 2 . 0, undergoing the so-called self-trapping transition. The fact that Fig. 6d displays a much faster- moving polaron than Fig. 6e is also reflected in Fig. 8d, in which the localized phonon wave packet in β$_{n}$ ( t ) is found to be much longer-lasting than that in Fig. 8e, and its speed greater. In addition, Fig. 7d shows a few scattered localized phonon wave packets of λ$_{n}$ ( t ), while Fig. 7e contains only a single localized phonon wave packet of λ$_{n}$ ( t ). 

## D. Absorption Spectra and System Energies 

Details on how to calculate linear absorption spectra of a one-dimensional exciton-phonon system from the D$_{2}$ and ˜ D Ans¨ atze can be found in Appendix E. Fig. 10a and 10b show two examples of the spectra get by D$_{2}$ Ansatz and ˜ D Ansatz, respectively, and the results are compared with those from the Merrifield Ansatz. As shown in Fig. 10a, when J is small (e.g., J = 0 . 1) and S is large (e.g., S = 6 . 0), the spectra obtained by the D$_{2}$ and Merrifield Ans¨ atze are almost the same. The phonon sidebands in Fig. 10a is labeled by n = 0 , 1 , 2 , ... from left to right. The left most sideband, n = 0, corre- sponding to the zero-phonon line, is located at ω = − Sω$_{0}$ [Eq. (10)]. For S = 6 . 0, the zero-phonon line is located at ω = − 6 ω$_{0}$ as shown in Fig. 10a. According to the Huang- 

