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its Fourier transform β K q ( t ), that characterizes the Mer- rifield trial state | Ψ$_{M}$; K 〉 of crystal momentum K : 

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In contrast, Davydov built a theory of enveloped solitons based on the two trial states of localized nature 

with 

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and 

with 

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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$_{n}$ . 

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: 

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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. 

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 In the strong cou- pling case, however, the perturbative method is no longer valid.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 ): 

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 ): 

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 

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. 

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