DAVYDOV SOLITON DYNAMICS IN PROTEINS .1. INITIAL STATES AND EXACTLY SOLVABLE SPECIAL CASES

For the Davydov Hamiltonian several special cases are known which can be solved analytically. Starting from these cases we show that the ini tial state for a simulation using Davydov's ID1> approximation has to be constructed from a given set of initial lattice displacements and m omenta in form of a coherent state with its amplitudes independent of the lattices site, corresponding to Davydov's ID2> approximation. In t he ID1> ansatz the coherent state amplitudes are site dependent. The s ite dependences evolve from this initial state exclusively via the equ ations of motion. Starting the ID1> simulation from an ansatz with sit e dependent coherent state amplitudes leads to an evolution which is d ifferent from the analytical solutions for the special cases. Further we show that simple construction of such initial states from the expre ssions for displacements and momenta as functions of the amplitudes le ads to results which are inconsistent with the expressions for the lat tice energy. The site-dependence of coherent state amplitudes can only evolve through the exciton-phonon interactions and cannot be introduc ed already in the initial state. Thus also in applications of the ID1> ansatz to polyacetylene always ID2> type initial states have to be us ed in contrast to our previous suggestion [W. Forner, J. Phys.: Conden s. Matter 1994, 6, 9089-9851, on p. 9105]. Further we expand the known exact solutions in Taylor serieses in time and compare expectation va lues in different orders with the exact results. We find that for an a pproximation up to third order in time (for the wave function) norm an d total energy, as well as displacements and momenta are reasonably co rrect for a time up to 0.12-0.14 ps, depending somewhat on the couplin g strengh for the transportless case. For the oscillator system in the decoupled case the norm is correct up to 0.6-0.8 ps, while the expect ation values of the number operators for different sites are reasonabl y correct up to roughly 0.6 ps, when calculated from the third order w ave function. The most important result for the purpose to use such ex pansions for controlling the validity of ansatz states is, however, th at the accuracy of S(t) and H(t) (constant in time, exact values known in all cases) is obviously a general indicator for the time region in which a given expansion yields reliable values also for the other, ph ysically more interesting expectation values.