Seite - 20 - in Charge Transport in DNA - Insights from Simulations

TheoreticalBackground electron-electron interaction, Exc[ρ(r)] is the exchange correlation energymissing in the classical termand JNN[ρ(r)] is the interactionbetween thenuclei. While the functionals for theelectron-electronand theelectron-core interactionare known from the Hartree theory, the exact form of the functionals for the kinetic energyof the electronsand theexchange correlationareunknown. In1965,KohnandShamintroducedthe ideatodescribe thekineticenergybyusing Hartree–Fock likeorbitals, the so-calledKohn–Shamorbitals [64]. In thisapproach,an imaginarysetofone-electronequations, theKohn–Shamequa- tions, is built. ( −1 2 ∇2+V(r) ) φi= iφi(r) (2.18) Although, this construction counteracts the ideaoforbital-freeDFT, it isuntil now theonly computationally convenient andwidelyused solution. The still missing Exc[ρ] now has to be approximated. Many methods have been developed, and themostwidelyusedare: • localdensity approximation (LDA) Exc=F[ρ(r)] • generalizedgradient approximation (GGA) Exc=F[ρ(r),∇ρ(r)] • meta-GGA Exc=F[ρ(r),∇ρ(r),∇2ρ(r)] • hybrid functionals: e. g. addingHFexchange (e.g. B3LYP) With these methods, the electronic structure of any system can be determined. Despite the ideaof onlyusing thedensity, the requiredorbitals for the calculation 20