where the first two terms connected with the time deriva- tives can be calculated as follows exciton has considerable densities, and then propagate to the entire aggregate if there is sufficient phonon dis- persion enabling mobility. The half-width of the phonon dispersion, W , determines the speeds of phonon wave packets because of a linear phonon dispersion relation given in the model. For a large value of the transfer integral, the left-moving and right-moving wave packets of the exciton departing from the site of creation may make a quick rendezvous at the opposite end of the ring, and therefore, trigger a second pair of localized phonon wave packets. Linear absorption spectra can be derived from the exciton amplitude and the phonon displace- ments computed from the D$_{2}$ and ˜ D Ans¨ atze. Various system energies are also calculated and analyzed for the one-dimensional coupled exciton-phonon molecular ring after photo-excitation. Overall it is found that ˜ D is a more flexible trial state, while the D$_{2}$ Ansatz is rather effi- cient for computation, and its extension to higher spatial dimensions is a feasible generalization of our approach. The remaining term is the average energy in the D$_{2}$ trial state: with Our trial states in this paper are both localized wave functions, and we intend to work out detailed dynam- ics of their translationally invariant counterparts, the Toyozawa Ansatz and the GL Ansatz. Our approaches here can be also readily extended to include other forms of exciton-phonon interactions, such as asymmetric and symmetric off-diagonal coupling $^{35}$, and higher-order cou- plings. Work in this direction is now in progress. Support from the Singapore Ministry of Education through the Academic Research Fund (Tier 2) under Project No. T207B1214 is gratefully acknowledged. Equations of motion from the trial wave function D$_{2}$ ( t ) are readily obtained from Eq. (23) The time evolution equations for D$_{1}$, ˜ D and D$_{2}$ Ans¨ atze can be derived by employing Dirac-Frenkel time- dependent variation method. We start from the simplest of the three, the D$_{2}$ Ansatz. For D$_{2}$ Ansatz, the La- grangian is defined as It can easily be shown the norm of the D$_{2}$ trial state is conserved, i.e., d dt [ N ∑ n =1 | ψ$_{n}$ ( t ) | $^{2}$] = 0 (A7) L = 〈 D$_{2}$ ( t ) | i ℏ 2 ←→ ∂ ∂t − ˆ H | D$_{2}$ ( t ) 〉 = i ℏ 2 [ 〈 D$_{2}$ ( t ) | − → ∂ ∂t | D$_{2}$ ( t ) 〉 − 〈 D$_{2}$ ( t ) | ← − ∂ ∂t | D$_{2}$ ( t ) 〉 ] −〈 D$_{2}$ ( t ) | ˆ H | D$_{2}$ ( t ) 〉 (A1) − i ˙ λ$_{q}$ ( t ) = − ∑ n | ψ$_{n}$ | $^{2}$e − $^{iqn}$g$_{q}$ ω$_{q}$ − ω$_{q}$λ$_{q}$ (A6) − i ˙ ψ$_{n}$ ( t ) = i 2 ψ$_{n}$ ( t ) ∑ q [ ˙ λ$_{q}$ ( t ) λ ∗ $_{q}$( t ) − ˙ λ ∗ $_{q}$( t ) λ$_{q}$ ( t )] + Jψ$_{n}$$_{+1}$ ( t ) + Jψ$_{n}$$_{−}$$_{1}$ ( t ) − ψ$_{n}$ ( t ) ∑ q ω$_{q}$ | λ$_{q}$ ( t ) | 2 − ψ$_{n}$ ( t ) q g$_{q}$ω$_{q}$ [ λ$_{q}$ ( t ) e iqn + λ ∗ $_{q}$( t ) e − $^{iqn}$] (A5) 〈 D$_{2}$ | ˆ H | D$_{2}$ 〉 = 〈 D$_{2}$ | ˆ H$_{ex}$ | D$_{2}$ 〉 + 〈 D$_{2}$ | ˆ H$_{ph}$ | D$_{2}$ 〉 + 〈 D$_{2}$ | ˆ H$_{ex}$$_{−}$$_{ph}$ | D$_{2}$ 〉 (A4) Acknowledgments 〈 D$_{2}$ | ˆ H$_{ex}$ | D$_{2}$ 〉 = − J ∑ n ψ ∗ $_{n}$ψ$_{n}$$_{+1}$ − J ∑ n ψ ∗ $_{n}$ψ$_{n}$$_{−}$$_{1}$ 〈 D$_{2}$ | ˆ H$_{ph}$ | D$_{2}$ 〉 = ∑ n | ψ$_{n}$ | 2 ∑ q ω$_{q}$ | λ$_{q}$ | 2 〈 D$_{2}$ | ˆ H$_{ex}$$_{−}$$_{ph}$ | D$_{2}$ 〉 = ∑ n | ψ$_{n}$ | 2 ∑ q g$_{q}$ω$_{q}$ ( λ$_{q}$e iqn + λ ∗ $_{q}$e − $^{iqn}$) Appendix A: The D 2 trial state 〈 D$_{2}$ ( t ) | − → ∂ ∂t | D$_{2}$ ( t ) 〉 = ∑ n ψ ∗ n ˙ ψ$_{n}$ − ∑ n | ψ$_{n}$ | 2 ∑ q λ ∗ q ˙ λ$_{q}$ + ∑ n | ψ$_{n}$ | 2 ∑ q [ − 1 2 ( ˙ λ$_{q}$λ ∗ q + λ$_{q}$ ˙ λ ∗ $_{q}$)] , (A2) 〈 D$_{2}$ ( t ) | ← − ∂ ∂t | D$_{2}$ ( t ) 〉 = ∑ n ˙ ψ ∗ $_{n}$ψ$_{n}$ − ∑ n | ψ$_{n}$ | 2 ∑ q ˙ λ ∗ $_{q}$λ$_{q}$ + ∑ n | ψ$_{n}$ | 2 ∑ q [ − 1 2 ( ˙ λ$_{q}$λ ∗ q + λ$_{q}$ ˙ λ ∗ $_{q}$)] (A3) and 11