content_image/403615.jpg content_image/403614.jpg content_image/403610.jpg content_image/403613.jpg content_image/403611.jpg content_image/403612.jpg
Substituting Eq. (E5) into Eq. (E4), we obtain the for- mula to calculate F ( t ) by D$_{2}$ Ansatz: 

<img src='content_image/403615.jpg'>

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 

<img src='content_image/403614.jpg'>

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. 

## Appendix F: Deviation vector of the D 2 Ansatz 

Substituting Eqs. (1)-(4), (15) and (16) into Eq. (32), one obtains the expression of | δ ( t ) 〉 for the D$_{2}$ Ansatz: 

<img src='content_image/403610.jpg'>

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Then, the expression of 〈 δ ( t ) | δ ( t ) 〉 for the D$_{2}$ Ansatz can be obtained: 

<img src='content_image/403613.jpg'>

where 

and 

<img src='content_image/403611.jpg'>

7 S. Tanaka, Y, Kayanuma, J. Lumin. 87 , 936 (2000). 8 

S. Tanaka, J. Chem. Phys 119 , 4891 (2003). 9 

Y. Zhao, D. W. Brown, and K. Lindenberg, J. Chem. Phys. 106 , 2728 (1997). 10 

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Y. Zhao, Doctoral Thesis, University of California, San Diego, 1994. 12 

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Y. Zhao et al. , in preparation. 14 

Y. Zhao, G. Li, J. Sun, W.H. Wang, J. Chem. Phys. 129 , 

<img src='content_image/403612.jpg'>

