Page - 14 - in Charge Transport in DNA - Insights from Simulations

TheoreticalBackground AnMD simulation can be performedwith these equations ofmotion. These are ordinary differential equations of second order. Numericalmethods are available to solve these. 2.2.1 Solving theEquationsofMotion There are several methods for the integration of the equations ofmotion. Rang- ing fromvery simple ones, like thefirst-orderEulermethod, tomore complicated ones, like the second-order leap-frogmethod, to evenmore complicatedmethods ofhigherorders. All of these shareone fundamental approach, the introductionof a time step. This is necessarydue to the fact that the integration canonlybedone numerically. The leap-frogmethod [55] is used in thiswork. Here, the positions andvelocities are evaluated inanalternating fashion. r(t), v(t+ 1 2 Δt), r(t+Δt), v(t+ 3 2 Δt), r(t+2Δt)... Thefirst step is to calculate thevelocities v ( t+ 12Δt ) v ( t+ 1 2 Δt ) = v ( t− 1 2 Δt ) + F(t) m Δt (2.4) and thenewatomicpositions are calculated in thenext step r(t+Δt)= r(t)+ v ( t+ 1 2 Δt ) Δt (2.5) This algorithmproduces trajectories that are identical to those obtainedwith the Verlet algorithm[56]. Now, a time stephas to be chosen for thenumerical integration. Thevalue of this crucial parameter in the simulationdependson two factorsbasically: • First, there are numerical errors due to the second order Taylor expansion. Contributions inΔt3 andhigherorders areneglected, introducinganerrorof the same order. To avoid this issue, the step size can be chosen very small, 14