Hamiltonian defined in Eq. (1) is time-independent, the total energy of the system is expected to be a constant during the time evolution. At t = 0, an exciton is as- sumed localized at one site and there are no phonons on the entire ring. As shown in Fig. 11, the phonon energy E$_{ph}$ rises from zero and oscillates as time goes on. In the meantime, the interaction energy between the exci- ton and phonons, E$_{ex}$$_{−}$$_{ph}$ , oscillates with almost the same amplitude but an opposite sign. And the total energy E$_{tot}$ of the system stays as a constant at all times. As shown in Figs. 11a and 11b, the D$_{2}$ Ansatz and Merrifield Ansatz agree with each other when J is small and S is large. However, when J is large (e.g., J = 1 . 0) and S is small (e.g., S = 0 . 5), these two Ans¨ atze no longer agree. For this case, the Merrifield Ansatz, which is translation- ally invariant (a Bloch wave function) but with a built-in small-polaron correlation, is not suitable to describe the dynamics. As shown in Figs. 11c and 11d, the total sys- tem energy E$_{tot}$ vanishes during the time evolution of the ˜ D Ansatz. While in the Merrifield Ansatz, E$_{tot}$ remains at a constant value − 2 that is equal to the initial value of E$_{ex}$ . This is because in the Merrifield Ansatz, the ex- citon amplitude is assumed to distribute uniformly over all the sites of the system, thus according to Eq. (2), E$_{ex}$ has a negative value when J is not negligible. However, for an initial state in which the exciton is localized at one site, according to Eq. (2), E$_{ex}$ should be zero at t = 0 as shown in Figs. 11c and 11d for the ˜ D Ansatz. This shows that the ˜ D Ansatz is a more flexible trial state than the Merrifield Ansatz. D$_{2}$ Ansatz are shown in Figs. 12a and 12b. The control parameters in Figs. 12a and 12b are the same as those in Figs. 11a and 11b, respectively. For both two cases, the main system energies are E$_{ph}$ and E$_{ex}$$_{−}$$_{ph}$ , and Δ( t ) is found to be negligible to either E$_{ph}$ or E$_{ex}$$_{−}$$_{ph}$ . We con- clude that the D$_{2}$ Ansatz yields quantitatively accurate solutions to the Schr¨ odinger equation for these two cases. Lastly, we give a brief discussion on the validity of the Davydov Ans¨ atze. A Davydov Ansatz is an approx- imative solution to the Schr¨ odinger equation with the Holstein Hamiltonian. For a trial wave function | D ( t ) 〉 that does not strictly obey the Schr¨ odinger equation, the deviation vector | δ ( t ) 〉 can be written as Here ˆ H is the Holstein Hamiltonian Eq. (1). For the Davydov D$_{1}$ Ansatz, the explicit form of | δ ( t ) 〉 17 was given by ˇ Skrinjar et al. in 1988. It was also proven that | δ ( t ) 〉 is orthogonal to | D$_{1}$ ( t ) 〉 . However, such orthogonal- ity relations are insufficient to conclude that the devia- tion vector | δ ( t ) 〉 is negligible, and the trial state is a good approximation to the true solution of Schr¨ odinger equa- tion. To have a quantitative measure of the Schr¨ odinger- equation deviation, one needs to calculate the amplitude of the deviation vector | δ ( t ) 〉 , which is defined as Δ( t ): In this paper we simulate polaronic dynamics in a one- dimensional molecular chain following the Dirac-Frenkel time-dependent variational approach. After the optical excitation, the coupled exciton-phonon system will un- dergo a relaxation process from the initial photo-induced nonequilibrium. Based on the Holstein Hamiltonian, we examined time evolution of the exciton amplitude, the reduced exciton density matrix, the exciton coherence size, linear optical absorption, and induced phonon dis- placements for two types of variational wave functions, namely, the Davydov D$_{2}$ Ansatz, and a simplified variant of Davydov D$_{1}$ Ansatz, also known as the ˜ D Ansatz. It is shown that following the equations of motion derived from the time-dependent variation, the exciton ampli- tude will transfer from the site of creation to neighboring sites, and as J increases, the velocity of exciton propaga- tion will increase as well. Exciton-induced phonon defor- mations can be found to form at the locations where the An explicit expression for Δ( t ) as the D$_{2}$ Ansatz is sub- stituted into the Schr¨ odinger equation is derived in Ap- pendix F. Note that the dimension of Δ( t ) is that of the energy. Therefore, one can gauge whether the trial state is a good approximative solution by comparing Δ( t ) with the sys- tem energies. Two examples of such comparisons for the Δ( t ) ≡ √ 〈 δ ( t ) | δ ( t ) 〉 (33) IV. CONCLUSION | δ ( t ) 〉 ≡ i ℏ ∂ ∂t | D ( t ) 〉 − ˆ H | D ( t ) 〉 (32) FIG. 12: Amplitude of the deviation vector | δ ( t ) 〉 calculated from the D 2 Ansatz. The results are compared with the system energies. (a) J = 0 . 1 , W = 0 . 1 , S = 4 . 0; (b) J = 0 . 1 , W = 0 . 8 , S = 4 . 0. 10 −15 −10 −5 0 5 10 15 Energy / (¯ hω 0 )  (a) E$_{ph}$ E$_{ex−ph}$ 0 1 2 3 4 5 6 −15 −10 −5 0 5 10 15 Energy / (¯ hω 0 )  (b) t / (2 π/ω 0 )