14 Then, the expression of 〈 δ ( t ) | δ ( t ) 〉 for the D$_{2}$ Ansatz can be obtained: Substituting Eq. (E5) into Eq. (E4), we obtain the for- mula to calculate F ( t ) by D$_{2}$ Ansatz: wherein λ$_{q}$ ( t ) and ψ$_{m}$ ( t ) are initialized by Eq. (E7) and Eq. (E6) (i.e., ψ$_{m}$ (0) = δ$_{m,}$$_{0}$ ) and then solved by Eqs. (A6) and (A5). The same procedure from Eq.(E5) to (E8) can be ap- plied to ˜ D Ansatz, and one obtains where the variational parameters ψ$_{m}$ ( t ), β$_{q}$ ( t ) and λ$_{q}$ ( t ) are solved by Eqs. (B5)-(B7) with the initial values ψ$_{m}$ (0) = δ$_{m,}$$_{0}$ , β$_{q}$ (0) = 0 and λ$_{q}$ (0) = 0. Substituting Eqs. (1)-(4), (15) and (16) into Eq. (32), one obtains the expression of | δ ( t ) 〉 for the D$_{2}$ Ansatz: where and Appendix F: Deviation vector of the D 2 Ansatz | δ ( t ) 〉 = ∑ n ˆ B † $_{n}${ i { ˙ ψ$_{n}$ ( t ) + ψ$_{n}$ ( t ) ∑ q { ˆ b † $_{q}$˙ λ$_{q}$ ( t ) − Re[ ˙ λ$_{q}$ ( t ) λ ∗ $_{q}$( t )] }} + J [ ψ$_{n}$$_{+1}$ ( t ) + ψ$_{n}$$_{−}$$_{1}$ ( t )] − ψ$_{n}$ ( t ) ∑ q ω$_{q}$ ˆ b † $_{q}$λ$_{q}$ ( t ) + ψ$_{n}$ ( t ) ∑ q g$_{q}$ω$_{q}$ [ ˆ b † $_{q}$e − inq + λ$_{q}$ ( t ) e $^{inq}$] } exp( q [ λ$_{q}$ ( t ) ˆ b † q − λ ∗ $_{q}$( t ) ˆ b$_{q}$ ]) | 0 〉$_{ph}$| 0 〉$_{ex}$ (F1) ϑ$_{n}$ ( t ) ≡ ψ$_{n}$ ( t )Im[ ∑ q ˙ λ$_{q}$ ( t ) λ ∗ $_{q}$( t )] − i ˙ ψ$_{n}$ ( t ) (F4) θ$_{n}$ ( t ) ≡ ψ$_{n}$ ( t ) ∑ q g$_{q}$ω$_{q}$ Re[ λ$_{q}$ ( t ) e $^{inq}$] (F3) 〈 δ ( t ) | δ ( t ) 〉 = J 2 ∑ n | ψ$_{n}$$_{+1}$ ( t ) + ψ$_{n}$$_{−}$$_{1}$ ( t ) | 2 − 2 J [ ∑ q ω$_{q}$ | λ$_{q}$ ( t ) | $^{2}$]Re[ ∑ n [ ψ ∗ n $_{+1}$( t ) + ψ ∗ n − $_{1}$( t )] ψ$_{n}$ ( t )] +4 J Re[ n [ ψ ∗ n $_{+1}$( t ) + ψ ∗ n − $_{1}$( t )] θ$_{n}$ ( t )] − 2 J Re[ ∑ n [ ψ ∗ n $_{+1}$( t ) + ψ ∗ n − $_{1}$( t )] ϑ$_{n}$ ( t )] +[ ∑ n | ψ$_{n}$ ( t ) | $^{2}$] { [ ∑ q ω$_{q}$ | λ$_{q}$ ( t ) | $^{2}$]2 + ∑ q | ω$_{q}$λ$_{q}$ ( t ) | $^{2}$} − 4[ q ω$_{q}$ | λ$_{q}$ ( t ) | $^{2}$]2 n ψ ∗ $_{n}$( t ) θ$_{n}$ ( t ) − 2 ∑ n ψ ∗ $_{n}$( t ) ψ$_{n}$ ( t ) ∑ q g$_{q}$ω 2 $_{q}$Re[ λ$_{q}$ ( t ) e $^{inq}$] +2[ q ω$_{q}$ | λ$_{q}$ ( t ) | $^{2}$]Re[ n ψ ∗ $_{n}$( t ) ϑ$_{n}$ ( t )] +2[ ∑ n | ψ$_{n}$ ( t ) | $^{2}$]Im[ ∑ q ω$_{q}$ ˙ λ$_{q}$ ( t ) λ ∗ $_{q}$( t )] +4 n | θ$_{n}$ ( t ) | 2 + [ n | ψ$_{n}$ ( t ) | $^{2}$] ∑ q ( g$_{q}$ω$_{q}$ ) 2 − 4Re[ ∑ n θ ∗ $_{n}$( t ) ϑ$_{n}$ ( t )] − 2Im[ n ψ ∗ $_{n}$( t ) ψ$_{n}$ ( t ) ∑ q g$_{q}$ω$_{q}$ ˙ λ$_{q}$ ( t ) e $^{inq}$] + ∑ n | ϑ$_{n}$ ( t ) | 2 + [ ∑ n | ψ$_{n}$ ( t ) | $^{2}$] ∑ q | ˙ λ$_{q}$ ( t ) | 2 (F2) F ( t ) = µ $^{2}$N ∑ m ψ$_{m}$ ( t ) exp( − 1 2 ∑ q | β$_{q}$ ( t ) e − iqm − λ$_{q}$ ( t ) | $^{2}$) (E9) F ( t ) = µ 2 ∑ n ∑ m ψ$_{m}$ ( t ) × $_{ph}$〈 0 | exp( ∑ q [ λ$_{q}$ ( t ) ˆ b † q − λ ∗ $_{q}$( t ) ˆ b$_{q}$ ]) | 0 〉$_{ph}$ = µ 2 ∑ n ∑ m ψ$_{m}$ ( t ) exp( − 1 2 ∑ q | λ$_{q}$ ( t ) | $^{2}$) = µ $^{2}$N ∑ m ψ$_{m}$ ( t ) exp( − 1 2 ∑ q | λ$_{q}$ ( t ) | $^{2}$) (E8) 6 Y. Zhao, D.W. Brown, and K. Lindenberg, J. Chem. Phys. 106 , 5622 (1997). 5 R.E. Merrifield, J. Chem. Phys. 40 , 4450 (1964). 4 S. Tomimoto, H. Nansei, S. Saito, T. Suemoto, J. Takeda, S. Kurita, Phys. Rev. Lett. 81 , 417 (2000). 3 X. Liu, K. Gao, J. Fu, Y. Li, J. Wei, and S. Xie, Phys. Rev. B 74 , 172301 (2006). 2 B. Zheng, J. Wu, W. Sun, C. Liu, Chem. Phys. Lett 425 , 123 (2006). 1 Z. An, C. Q. Wu, X. Sun, Phys. Rev. Lett 93 , 216407 (2004). 14 Y. Zhao, G. Li, J. Sun, W.H. Wang, J. Chem. Phys. 129 , 13 Y. Zhao et al. , in preparation. 12 Y. Zhao, D. W. Brown, and K. Lindenberg, J. Chem. Phys. 107 , 3159 (1997); 107 , 3179 (1997). 11 Y. Zhao, Doctoral Thesis, University of California, San Diego, 1994. 10 Yutaka Toyozawa, Prog. Theor. Phys. 26 , 29-44 (1961). 9 Y. Zhao, D. W. Brown, and K. Lindenberg, J. Chem. Phys. 106 , 2728 (1997). 8 S. Tanaka, J. Chem. Phys 119 , 4891 (2003). 7 S. Tanaka, Y, Kayanuma, J. Lumin. 87 , 936 (2000).