9 FIG. 11: System energies calculated from the D 2 and ˜ D Ans¨ atz. The results are compared with those of the Mer- rifield Ansatz. (a) J = 0 . 1 , W = 0 . 1 , S = 4 . 0; (b) J = 0 . 1 , W = 0 . 8 , S = 4 . 0; (c) J = 1 . 0 , W = 0 . 1 , S = 0 . 5; (d) J = 1 . 0 , W = 0 . 8 , S = 0 . 5. FIG. 10: Linear absorption spectra of a 32-site one- dimensional ring of a coupled exciton-phonon system. (a) Using the D 2 and Merrifield Ans¨ atze for J = 0 . 1 , W = 0 . 1 and S = 6 . 0. The decay factor used in the Fourier transfor- mation is 0 . 01. (b) Using the ˜ D and Merrifield Ans¨ atze for J = 0 . 4 , W = 0 . 8 and S = 1 . 0. The decay factor used in the Fourier transformation is 0 . 05. Rhys theory $^{30}$, the phonon sidebands at zero temperature follow a Poisson distribution: 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 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 small peaks ω$_{sp}$ ( n$_{q}$ ) ( n$_{q}$ = 0 , 1 , ..., N/ 2): ω$_{sp}$ ( n$_{q}$ ) = ω$_{ZPL}$ + ω$_{0}$ − W + 4 W N n$_{q}$ (31) ˜ F$_{abs}$ ( ω ) = exp( − S ) ∞ ∑ n =0 S n n ! δ ( ω + E$_{r}$ − nω$_{0}$ ) (30) −10 0 10 D$_{2}$ −10 0 10 Energy / (¯ hω 0 ) Merrifield  (a) E$_{ph}$ E$_{ex−ph}$ E$_{ex}$ E$_{tot}$ −10 0 10 D$_{2}$ −10 0 10 Energy / ( ¯ hω 0 ) Merrifield  (b) E$_{ph}$ E$_{ex−ph}$ E$_{ex}$ E$_{tot}$ −2 −1 0 1 2 ˜ D −2 −1 0 1 2 Energy / (¯ hω 0 ) Merrifield  (c) E$_{ph}$ E$_{ex−ph}$ E$_{ex}$ E$_{tot}$ 0 1 2 3 4 5 6 −2 −1 0 1 2 ˜ D t / (2 π/ω 0 ) −2 −1 0 1 2 Energy / (¯ hω 0 ) Merrifield  (d) E$_{ph}$ E$_{ex−ph}$ E$_{ex}$ E$_{tot}$ −10 −8 −6 −4 −2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 ˜ F ( ω ) (arb . units) (a) D 2 Merrifield 0 4 8 12 0 0.2 0.4 0.6 Sideband No. Sideband FWHM −4 −3 −2 −1 0 1 2 3 4 0 1 2 3 4 ω/ω 0 ˜ F ( ω ) (arb . units) (b) ˜ D Merr 0 4 8 12 16 −1 −0.5 0 0.5 1 ˜ D Small peak index Small peak location ( ω/ω 0 )