Page - 19 - in Charge Transport in DNA - Insights from Simulations

2.3QuantumChemistry However, this is not theway to go, since the determination of the trueN-particle wave function is a complicated task. Therefore, a less complicated approach is soughtwhichdetermines thedensitydirectlyand is able to calculate theenergyof the systemdirectly fromthisdensity. An important point here is to prove that the density is an unique feature of a certain system and therefore the energy can be uniquely determined. Moreover, the obtained density functional E[ρ(r)] has to follow the variational principle for energyminimization. In the following, bothof these issueswere solved. It is well established that the electron density is uniquely determined in a given external potentialVext[ρ(r)]. Touse adensity-functional for thedescriptionof sys- temproperties this correlation has to be invertible. Hohenberg andKohnproved in1964 that theexternalpotential and thus thegroundstatepropertiesof a system aredetermineduniquelyby theelectrondensity ρ(r) [62]. ρ(r)→Vext[ρ(r)] (2.15) The second theorem byHohenberg andKohn implies that the energy functional takes aminimumvalue for thegroundstatedensity: E[ρ0]≤E[ρ˜] (2.16) where ρ0 is the electrondensity at thegroundstate and ρ˜ is anarbitrarydensity. Additionally, the Born–Oppenheimer approximation [63] is applied. Due to the difference inmass between the electrons andnuclei, themotions of theseparticles canbe treated separately. With these issues taken care of, a density-functional can be derived and the total energyof the systemtakes the form: E[ρ(r)]=T[ρ(r)]+Vext[ρ(r)]+Vee[ρ(r)]+Exc[ρ(r)]+ JNN[ρ(r)] (2.17) where T[ρ(r)] is the kinetic energy of the electrons,Vext[ρ(r)] is the energy of the electrons in theexternalfieldof thenuclei,Vee[ρ(r)] is theenergy fromtheclassical 19