content_image/422279.jpg content_image/422280.jpg content_image/422289.jpg content_image/422285.jpg content_image/422282.jpg content_image/422278.jpg content_image/422281.jpg content_image/422283.jpg content_image/422286.jpg content_image/422287.jpg content_image/422288.jpg
The same procedure in Appendices A and B can be applied to the D$_{1}$ Ansatz 36 with variational parameters ψ$_{n}$ and λ$_{qn}$ , and one can obtain 

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<img src='content_image/422280.jpg'>

These coupled differential equations can be numeri- cally solved by the fourth-order Runge-Kutta method, and work on this is now in progress. 

There are various numerical approaches to solve cou- pled differential equations such as Eqs. (A5) and (A6) for the D$_{2}$ Ansatz and Eqs. (B5), (B6) and (B7) for the ˜ D Ansatz. One way is to transform these equations into the Volterra integral equations 37 for minimization purposes, and then use a nonlinear optimization method such as the Newton-Raphson method to solve them. Another approach is to solve the time-dependent differential equa- tions directly. The latter method has a much higher com- putational efficiency, but it requires a higher precision in the single iterative time step since the computational er- ror may accumulate as the number of the iterative steps increases. 

In this paper, we use the Runge-Kutta fourth-order method 38,39 to solve Eqs. (A5) and (A6) and Eqs. (B5), (B6) and (B7). The Runge-Kutta fourth-order method is widely used to solve differential equations. Its single step error is fifth order, i.e., O (Δ t $^{5}$). The algorithm for this method can be described as follows: 

<img src='content_image/422289.jpg'>

## Appendix C: The D 1 trial state 

## Appendix D: Numerical Details 

where 

and from t$_{n}$ to t$_{n}$$_{+1}$ ≡ t$_{n}$ + Δ t . 

For the D$_{2}$ Ansatz, α represents ψ$_{n}$ or λ$_{q}$ , and is ob- tained by Eqs. (A5) and (A6). Then, by calculating k$_{1}$ , k$_{2}$ , k$_{3}$ and k$_{4}$ , one can get the parameters α ( t$_{n}$$_{+1}$ ) for the next time step using the fourth-order Runge-Kutta method. 

## Appendix E: Linear absorption spectrum by D 2 and ˜ D Ans¨ atze 

The linear absorption spectrum of the exciton-phonon system studied in this paper is calculated by 

with 

where ˆ P is the polarization operator 

Here µ is the transition dipole matrix element for a single site, and | n 〉$_{ex}$ is the exciton state at the n th site | n 〉$_{ex}$ ≡ ˆ a † $_{n}$| 0 〉$_{ex}$ . 

Substituting Eq. (E3) into Eq. (E2) one obtains 

<img src='content_image/422285.jpg'>

wherein e − i ˆ $^{Ht}$| n 〉$_{ex}$| 0 〉$_{ph}$ can be approximated by a Davy- dov trial state, for example, by a D$_{2}$ trial state: 

where the variational parameters ψ$_{n}$ ′ − $_{n}$( t ) ( n ′ = 0 , 2 , ..., N − 1) and λ$_{q}$ ( t )( q = − N/ 2+1 , − N/ 2+2 , ..., N/ 2) have the following initial values 

and 

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<img src='content_image/422286.jpg'>

<img src='content_image/422287.jpg'>

<img src='content_image/422288.jpg'>

