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Rhys theory $^{30}$, the phonon sidebands at zero temperature follow a Poisson distribution: 

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From Eq. (30), the tallest of phonon sidebands should be the n = S − 1 peak when S ≫ 1 and the overall distribution approaches a Gaussian. Thus, for S = 6 . 0, the tallest peak is at n = 5 as shown in Fig. 10a. 

Fig. 10b displays the absorption spectrum calculated from the ˜ D and Merrifield Ans¨ atze for J = 0 . 4 , W = 0 . 8 and S = 1 . 0. There are fine features with tiny peaks on the one-phonon manifold on the right side of the zero- phonon line. These small peaks are attributed to the 32 phonon modes with momenta q = 2 πn$_{q}$/N ( N = 32; n$_{q}$ = − 15 , − 14 , ..., 14 , 15 , 16). As in this paper we assume a linear phonon dispersion as defined in Eq. (11), there are altogether 17 values of ω$_{q}$ for N = 32. Each value of ω$_{q}$ should correspond to a small peak on the spectrum if the coupling g$_{q}$ [Eq. (7)] is not too small, and from Eq. (11), one can obtain the locations of the 

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small peaks ω$_{sp}$ ( n$_{q}$ ) ( n$_{q}$ = 0 , 1 , ..., N/ 2): 

where ω$_{ZPL}$ is the location of the zero-phonon line. For W = 0 . 8 ω$_{0}$ and N = 32, ω$_{sp}$ ( n$_{q}$ ) = ω$_{ZPL}$ + (0 . 2 + 0 . 1 ∗ n$_{q}$ ) ω$_{0}$ as shown in the inset of Fig. 10b. 

Lastly, four cases of system energies calculated from the D$_{2}$ and ˜ D Ans¨ atze are shown in Fig. 11 and com- pared with those from the Merrifield Ansatz . Since the 

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