vacuum states of the exciton and the phonon field: | G 〉 = | 0 〉$_{ex}$| 0 〉$_{ph}$ . We confine ourselves to one-exciton subspace for the optically excited state 28 since the exciton number is conserved in the total Hamiltonian in Eq. (1). The spectral density 29 embodying all relevant informa- tion of the coupled exciton-phonon system in Eq. (4) can be written as where ˆ V$_{n}$ ( t ) is a Heisenberg representation of the exciton- phonon interaction at the n th site ational approach $^{24}$. Time-dependent variational param- eters which specify the two Ans¨ atze are obtained from solving a set of coupled differential equations generated by the Lagrangian formalism of the Dirac-Frenkel vari- ation. Special attention is paid to the evolution of the reduced single-exciton density matrix and the propaga- tion of exciton and phonon amplitudes from an initial location, where the exciton resides at t = 0, to the entire aggregate. The paper is organized as follows. In Sec. II we in- troduce the model Hamiltonian and discuss the nature of the trial states we use for dynamics studies, which is followed by the procedure of the time-dependent varia- tion. In Sec. III results from our dynamics calculation using the two trial states are displayed and discussed. Conclusions are drawn in Sec. IV. and the 〈· · · 〉$_{ph}$ denotes the thermal average in the free- phonon basis. Note that C$_{mn}$ ( ω ) represents not only the time correlation but also the spatial correlation for the exction-phonon interactions $^{29}$. Here only the case of zero temperature is considered. Substituting Eq. (7) into Eq. (6), one has We consider a one-dimensional aggregate of N molecules with a periodic boundary condition. There is only one two-level electronic system for each molecule coupled linearly with the phonon field. The Holstein Hamiltonian for the exciton-phonon system can be writ- ten as 25–27 Here ˆ H$_{ex}$ is the Hamiltonian for a single Frenkel exciton band in a rigid chain, and ˆ B$_{n}$ ( ˆ B † $_{n}$) is the Pauli annihila- tion (creation) operator of an exciton at the n th site. We set ℏ = 1 and assume the nearest-neighbor exciton trans- fer integral J$_{mn}$ = Jδ$_{m,n}$$_{±}$$_{1}$ . ˆ H$_{ph}$ is the phonon Hamil- tonian where ˆ b$_{q}$ ( ˆ b † $_{q}$) is the boson annihilation (creation) operator of a phonon with monentum q and frequency ω$_{q}$ . For simplicity, the zero-point energy is neglected. ˆ H$_{ex}$$_{−}$$_{ph}$ assumes the exciton is coupled linearly with the phonon field in a site diagonal form. Because there is no exciton in the ground state, the Hamiltonian for the ground state is represented by where | 0 〉$_{ex}$ stands for the exciton vacuum. The global ground state is then described as a direct product of both The Merrifield Ansatz, also known as the small-polaron Ansatz $^{5,6}$, is a simple, effective trial state to describe a coupled exciton-phonon system in which the exciton transfer integral is small compared with other system en- ergy scales. There is only one set of variational param- eters, the phonon displacements β K n ( t ), or alternatively, The central frequency of the phonon band ω$_{0}$ is taken as the energy unit, i.e., ω$_{0}$ = 1. From Eqs. (9)-(11), we can obtain g$_{q}$ for a given S and W . where S is the Huang-Rhys factor $^{30}$, ω$_{0}$ is the central energy of the phonon band, and W is the phonon energy band width . Here we assume a linear dispersion phonon band with a linear dispersion Following Tanaka $^{8}$, the spectral density is assumed to have the form The relaxation energy is defined by 2 C$_{mn}$ ( ω ) ≡ 1 2 π ∫ ∞ −∞ 〈 ˆ V$_{m}$ ( t ) ˆ V$_{n}$ (0) 〉$_{ph}$ e $^{iωt}$dt (6) ˆ V$_{n}$ = ∑ q g$_{q}$ω$_{q}$e $^{iqn}$( ˆ b$_{q}$ + ˆ b † − $_{q}$) (7) C$_{mn}$ ( ω ) = ∑ q g 2 $_{q}$ω 2 $_{q}$e iq ( m − n $^{)}$δ ( ω − ω$_{q}$ ) (8) C$_{00}$ ( ω ) = ∑ q g 2 $_{q}$ω 2 $_{q}$δ ( ω − ω$_{q}$ ) = 2 Sω 2 πW 2 √ W 2 − ( ω − ω$_{0}$ ) 2 (9) E$_{r}$ ≡ ∫ ∞ −∞ C$_{00}$ ( ω ) ω dω = ∑ q g 2 $_{q}$ω$_{q}$ ≡ Sω$_{0}$ (10) ω$_{q}$ = ω$_{0}$ + 2 W ( | q | π − 1 2 ) (11) ˆ H$_{g}$ = | 0 〉$_{ex}$ ˆ H$_{ph ex}$ 〈 0 | (5) ˆ H$_{ex}$$_{−}$$_{ph}$ = N ∑ q,n =1 g$_{q}$ω$_{q}$ ˆ B † n ˆ B$_{n}$ ( ˆ b$_{q}$e iqn + ˆ b † $_{q}$e − $^{iqn}$) (4) ˆ H$_{ph}$ = ∑ q ω$_{q}$ ˆ b † $_{q}$ˆ b$_{q}$ (3) ˆ H$_{ex}$ = − J ∑ n ˆ B † $_{n}$( ˆ B$_{n}$$_{+1}$ + ˆ B$_{n}$$_{−}$$_{1}$ ) (2) ˆ H = ˆ H$_{ex}$ + ˆ H$_{ex}$$_{−}$$_{ph}$ + ˆ H$_{ph}$ (1) with II. METHODOLOGY