Expansion of Exact Solutions
Springer Online Journal Archives 1860-2000
Chemistry and Pharmacology
Abstract We performed long time simulations using the |D1〉 approximation for the solution of the Davydov Hamiltonian. In addition we computed expectation values of the relevant operators with the state (Ĥ D /J)|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 |D1〉 ansatz is very close to an exact solution, showing expectation values of the relevant physical observables in the state (Ĥ D /J)|D 1〉 being about 5-6 orders of magnitudes larger than in the deviation state |δ〉. In the intermediate time scale of the ps range such errors, as known from our previous work, are somewhat larger, but still more or less negligibly. Thus we also report results from an investigation of the very short time (in the range 0-0.4 ps) behaviour of the |D1〉 state compared with that of an expansion of the exact solution in powers of time t. This expansion is reliable for about 0.12 ps for special cases as shown in the previous paper. However, the accuracy of the exactly known value of the norm and the expectation value of the Hamiltonian finally indicates up to what time a given expansion is valid, as also shown in the preceding paper. The comparison of the expectation values of the operators representing the relevant physical observables, formed with the third order wave function and with the corresponding results of |D1〉 simulations has shown, that our expansion is valid up to a time of roughly 0.10-0.15 ps. Within this time 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 inifinite order. Further we found good agreement of the results obtained with our expansion and those from the corresponding |D1〉 simulations within the time of about 0.10 ps. At later times, the factors with explicit powers of t in second and third order become dominant, making the expansion meaningless. Possibilities for the use of such expansions for larger times are described. Alltogether we have shown (together with previous work on medium times), that the |D1〉 state, although of approximative nature, is very close to an exact solution of the Davydov model on time scales from some femtoseconds up to nanoseconds. Especially the very small time region is of importance, because in this time a possible soliton formation from the initial excitation would start.
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