The global amplitude λ$_{q}$ can be related to the spatial average of λ$_{qn}$ , and the local amplitude β$_{q}$e − iqn can be viewed as an Ansatz for the spatial variation of λ$_{qn}$ around this mean value. its Fourier transform β K q ( t ), that characterizes the Mer- rifield trial state | Ψ$_{M}$; K 〉 of crystal momentum K : The time evolution of the photoexcited state in a one-dimensional molecular aggregate follows the time- dependent Schr¨ odinger equation. As far as the exciton- phonon coupling is weak, it can be treated perturba- tively leading to a nonlinear exciton equation with the use of the relaxation superoperator. In the strong cou- pling case, however, the perturbative method is no longer valid. There are several approaches to solve the time- dependent Schr¨ odinger equation. In the Hilbert space, the time-dependent wave function | Φ( t ) 〉 for the Hamil- tonian ˆ H is parameterized by α$_{m}$ ( t ) ( m = 1 , ..., M ): In contrast, Davydov built a theory of enveloped solitons based on the two trial states of localized nature The D$_{1}$ Ansatz differs from the more widely used D$_{2}$ Ansatz. In the D$_{2}$ Ansatz the phonon amplitudes are exciton-site independent, i.e., λ$_{qn}$ ( t ) = λ$_{q}$ ( t ) for all n. The D$_{2}$ Ansatz reprsents the quantum state of each phonon mode by a single coherent state, and hence has at all times a strongly classical character. In such solutions, the typically pulse-shaped distribution of the amplitudes λ$_{q}$ describes a lattice deformation fixed in the frame of the lattice, to which conforms a correspondingly pulse- shaped distribution of exciton amplitudes ψ$_{n}$ . The approach we adopt in this work is the Lagrangian formalism of the Dirac-Frenkel time-dependent varia- tional method $^{31}$, a powerful technique to obtain approxi- mate dynamics of many-body quantum systems for which exact solutions often elude researchers. We formulate the Lagrangian L as follows Assume that | Φ( t ) 〉 satisfies the time-dependent Schr¨ odinger equation Explicitly putting in the Hamiltonian ˆ H of Eq. (1) and writing ∂ | Φ( t ) 〉 /∂t in terms of α$_{m}$ ( t ) and their time- derivatives ˙ α$_{m}$ ( t ) ( m = 1 , ..., M ), one has Projecting Eq. (20) onto M different states | Ψ$_{m}$ 〉 ( m = 1 , ..., M ), one obtains M equations of motion for the pa- rameters α$_{m}$ ( t ): From this Lagrangian, equations of motion for α$_{m}$ ( t ) and their time-derivatives ˙ α$_{m}$ ( t ) ( m = 1 , ..., M ) can then be obtained It can be shown that with a properly chosen set of | Ψ$_{m}$ 〉 , Eqs. (21) and (23) can coincide. The reader is referred to Appendices A, B and C where details of derivation to- gether with these M equations for D$_{2}$, ˜ D and D$_{1}$ Ans¨ atze are given, respectively. Descriptions of the numerical pro- cedure can be found in Appendix D. Since the description of phonons in the D$_{2}$ Ansatz is unsophisticated, direct correlations between the ampli- tudes of phonons and exciton are neglected. In many cases, the D$_{2}$ Ansatz is inadequate to capture the com- plexities of the exciton-phonon system. Therefore, we introduce the ˜ D Ansatz, i.e., the localized form factor of the GL Ansatz: | ˜ D ( t ) 〉 = ∑ n$_{1}$ ψ$_{n}$$_{1}$ ( t ) ˆ B † $_{n}$exp[ ∑ n$_{2}$ ( λ$_{n}$$_{2}$ − β$_{n}$$_{2}$$_{−}$$_{n}$$_{1}$ ) ˆ b † n$_{2}$ − ( λ ∗ n$_{2}$ − β ∗ n$_{2}$ − $_{n$_{1}$}$) ˆ b$_{n}$$_{2}$ ] | 0 〉 = ∑ n ψ$_{n}$ ( t ) ˆ B † $_{n}$exp[ N − 1 / 2 ∑ q ( β$_{q}$e − iqn − λ$_{q}$ ) ˆ b † q − ( β ∗ $_{q}$e iqn − λ ∗ $_{q}$) ˆ b$_{q}$ ] | 0 〉 (17) d dt ( ∂L ∂ ˙ α ∗ m ) − ∂L ∂α ∗ m = 0 (23) L = 〈 Φ( t ) | i ℏ 2 ←→ ∂ ∂t − ˆ H | Φ( t ) 〉 (22) 〈 Ψ$_{m}$ |{ α$_{m}$ ( t ) } , { ˙ α$_{m}$ ( t ) }〉 = 0 (21) i ∂ ∂t |{ α$_{m}$ ( t ) }〉 = ˆ H |{ α$_{m}$ ( t ) }〉 (20) | λ ( t ) 〉 ≡ exp { ∑ [ λ$_{q}$ ( t ) ˆ b † q − λ ∗ $_{q}$( t ) ˆ b$_{q}$ ] }| 0 〉$_{ph}$ (16) | D$_{2}$ ( t ) 〉 ≡ ∑ n ψ$_{n}$ ( t ) ˆ B † $_{n}$| 0 〉$_{ex}$ ⊗ | λ ( t ) 〉 (15) | λ$_{n}$ ( t ) 〉 ≡ exp { ∑ q [ λ$_{qn}$ ( t ) ˆ b † q − λ ∗ $_{qn}$( t ) ˆ b$_{q}$ ] }| 0 〉$_{ph}$ (14) i ∂ ∂t | Φ( t ) 〉 = ˆ H | Φ( t ) 〉 . (19) with and with | Φ( t ) 〉 ≡ |{ α$_{m}$ ( t ) }〉 . (18) D$_{1}$ ( t ) 〉 ≡ ∑ n ψ$_{n}$ ( t ) ˆ B † $_{n}$| 0 〉$_{ex}$ ⊗ | λ$_{n}$ ( t ) 〉 (13) | Ψ$_{M}$; K 〉 = N − 1 / 2 ∑ n e $^{iKn}$| n 〉$_{ex}$ × exp[ − n b ( β K n b − $_{n}$ˆ b † n b − β K ∗ n b − $_{n}$ˆ b$_{n}$$_{b}$ )] | 0 〉$_{ph}$ = N − 1 / 2 ∑ n e $^{iKn}$| n 〉$_{ex}$ × exp[ − q ( β K q e − iqn ˆ b † q − β K ∗ q e iqn ˆ b$_{q}$ )] | 0 〉$_{ph}$ (12) 3