Page - 35 - in Charge Transport in DNA - Insights from Simulations

2.6ChargeTransfer inDNA Figure2.4: Schematic representation of the behavior of the wave-function in mean- field (Ehrenfest) simulations. The possible course of two simulations is shown. Reaching thehigh-couplingregion, the twostates begin tomixand the energy of the actual state remains in between the two adiabatic states. (Based onFig.5 in [72]) thewave-function is computed by solving the time-dependent Schrödinger equa- tion (TDSE). Theobtained solution is a combinationof populated adiabatic states. Figure2.4 shows theresultingmixedstatesof twopossiblemean-fieldsimulations. The basic implementationworks as follows. Two sets of equations ofmotion are solved simultaneously. First, the ones of electronic systemwhich evolves accord- ing to the TDSE. Second, the classicalNewton’s equations ofmotion for theMM system. These two sets of equations are coupled to each other. The MM equa- tions contain the charges of theQMsystem,which aremapped to theMMatomic charges. The TDSE depend on the atomistic structure of theMMsystem via the Hamiltonian. Therefore the equations have to be propagated simultaneously and this is achieved in a leap-frog like manner. First, the coarse grained QM step is performedandtheexcesscharge ispropagated. Thenthenewchargesaremapped to the atomic charges of theMMsystem. And, a step of classicalMMsimulation with these atomic charges is performed. After this, the cycle is complete and the nextQMstepisperformed.Withthis implementation, theclassical systemfeels the actual distribution of charge in the quantumregion and can respond to it accord- ingly. The environment of the charge polarizes and forms a stabilizing polaron. This effect hasbeen foundofgreat importance [96–99]. Thepresentedmean-fieldapproachhassomeadvantagesoversimplerpropagation schemes, like those used before, which had no coupling to the environment and 35