Seite - 22 - in Charge Transport in DNA - Insights from Simulations

TheoreticalBackground This densitywill differ from theDFTdensity byΔρ. The total energy then canbe calculatedby: E[ρ]=E[ρ0]+O(Δρ2) (2.20) Assumingthereferencedensityρ tobesufficientlyclose to thegroundstatedensity ρ0, the Kohn–Sham equations can be solved non-self-consistently and the Kohn– Shameigenvalues i areobtained. Eoccelec=∑ i i (2.21) This represents the contributionof theKohn–Shameigenvalues to the total energy of the system. This energy is called electronic energy. To obtain the total energy, another term, called the repulsive energy, which incorporates all other missing contributions isneeded. E[ρ]=∑ i i+Erep (2.22) Two-body termsVαβ are introduced to represent these repulsive contributions, be- ing exponential functions fitted to reproduce geometries, vibrational frequencies and reactionenergies. E[ρ]=∑ i i+ 1 2∑ αβ Vαβ (2.23) Already at this point, DFTB is able to describe unpolar systems,where no charge transfer happens between the single atoms. As many hetero-nuclear molecules, includingbiomolecules, havedifferences in electronegativity, amore sophisticated approachhas tobesought. Theself-consistent-chargeDFTBmethod(DFTB2) over- comes this issuebyasecond-orderTaylorexpansionof theDFTtotal energy.Here, charge density fluctuationsΔρ around a given reference density ρ0, which is the superpositionofneutral atomicdensities, are considered[66–68]. 22