Page - 27 - in Charge Transport in DNA - Insights from Simulations

2.4DynamicsofExcessCharge inDNA maybecalleda"fragmentmolecularorbitalapproach"(FMO),whichwasoriginally developedbyKitaura et al. [76] The fragmentorbitalsφ are linear combinationsof atomicorbitals. φim=∑ μ cimμ χμ (2.28) where cimμ are theFMOexpansion coefficients andχ are the atomicorbitals. These FMOcanbeobtainedwithQMmethods. Eachof theFMOis localizedonone sin- gle fragment bydefinition and these calculations canbedone separately for every fragment. But, it is of great importance to include the electrostatic environment into these calculations, as the FMOare strongly influenced by them. The imple- mentationused in thisworkuses theDFTB2method toperformaSCFcalculation for every fragment, a nucleobase in the case of DNA, in every time step. The electrostatic environmentof the fragment is includedaspoint charges. Now, theFMOareconsidered toconstitute thebasis for theexpansionof thewave- function of the excess charge. In case of hole transfer, thewave-function is built fromtheHOMOof the single fragments. Ψ(r,R)=∑ m amφHOMOm (2.29) TheobtainedFMOareused tobuildaHamiltonmatrix. Hmn= 〈 φm ∣∣Hˆ∣∣φn〉=∑ μ ∑ ν cmμc n νH¯μν (2.30) where indices μ and ν run over the relevant atomic-orbital-like basis functions of the fragments m and n, and H¯μν is the Hamiltonian in this atomic-orbital basis. ThisHamiltonmatrix can be set up easily. The fragmentsm and n are calculated separately to determine the cmμ and cnν, while the H¯μν is directly read in from the pre-calculatedand tabulatedDFTB2parameters. Finally, the set of FMO is orthogonalized to simplify the followingpropagationof thewave-function. 27