Page - 23 - in Charge Transport in DNA - Insights from Simulations

2.3QuantumChemistry ThedensitydifferencesΔρ arenowapproximatedasatomic contributions: Δρ=∑ α Δρα (2.24) Now, thechargemonopoleapproximation isused for theatomicdensity functions. Δρα≈ΔqαFα00Y00 (2.25) Here,Fα00 is thenormalizedradialdependenceof thedensityfluctuationonatomα which is approximated tobe spherical (Y00). The resultingapproximated secondorder energy termshas two limiting cases: 1. When atoms α and β are at a largedistance (>2Å), the exchange correlation energy becomes zero. In this case, the integral only describes the Coulomb interactionof twopoint charges. E2nd≈ 1 2∑ αβ ΔqαΔqβ Rαβ (2.26) 2. For α = β, the integral describes the electron-electron interaction on a single atomα E2nd= 1 2 UαΔq2α (2.27) TheparameterUα is theHubbardparameter newly introduced inDFTB2 andde- scribes the chemical hardness of the atomα. It is ameasure for the energy change of a systemuponadditionor removal of electrons. In conclusion, this introducedmethod is an approximativeDFTmethodwhich is up to 3orders ofmagnitude faster thanDFT. This is achievedbyusing aminimal basis set for thevalence electronsof everyatomaswell aspre-calculatingand tab- ulating two-center integrals forall atompairs. Therefore,DFTB2 is able todescribe large systems likeDNAoverextendedperiodsof timeand is themethodof choice for the charge transfer calculation in thiswork. 23