Accuracy of Davydov’s |${\mathit{D}}_1$〉 approximation for soliton dynamics in proteins

For the Davydov Hamiltonian several special cases are known which can be solved analytically. Starting from these cases we show that the initial state for a simulation using Davydov’s ‖${\mathit{D}}_1$〉 approximation has to be constructed from a given set of initial lattice displacements and momenta in the form of a coherent state with its amplitudes independent of the lattices site, corresponding to Davydov’s ‖${\mathit{D}}_2$〉 approximation. The site dependences in the ‖${\mathit{D}}_1$〉 ansatz evolve from this initial state exclusively via the equations of motion. Starting the ‖${\mathit{D}}_1$〉 simulation from an ansatz with site-dependent coherent-state amplitudes leads to an evolution which is different from the analytical solutions for the special cases. Thus also in applications of the ‖${\mathit{D}}_1$〉 ansatz to polyacetylene ‖${\mathit{D}}_2$〉-type initial states always have to be used in contrast to our previous suggestion [W. Förner, J. Phys. Condens. Matter 6, 9105 (1994)]. Further we expand the known exact solutions in Taylor series in time and compare expectation values in different orders with the exact results.

We find that for an approximation up to third order in time (for the wave function) norm, and total energy, as well as displacements and momenta are reasonably correct for a time up to ≊0.12–0.14 ps, depending somewhat on the coupling strengh for the transportless case. We performed long-time simulations using the ‖${\mathit{D}}_1$〉 approximation where we computed expectation values of the relevant operators with the state (H^/J)‖${\mathit{D}}_1$〉 and the deviation ‖δ〉 from the exact solution over long times, namely 10 ns. We found that in the very long-time scale the ‖${\mathit{D}}_1$〉 ansatz is very close to an exact solution. Further we report results from an investigation of the very short-time behavior of the ‖${\mathit{D}}_1$〉 state compared with that of an expansion of the exact solution in powers of time t. Within a time of roughly 0.10–0.15 ps the second- and third-order corrections turned out to be not very important. This is due to the fact that our first-order state contains already some terms of the expansion, summed up to infinite order. We found good agreement of the results obtained with our expansion and those from the corresponding ‖${\mathit{D}}_1$〉 simulations within the time of about 0.10 ps. Altogether we have shown that the ‖${\mathit{D}}_1$〉 state, although of approximative nature, is very close to an exact solution of the Davydov model on time scales from some fs up to ns. © 1996 The American Physical Society.