GerritJan Linker

Coulomb integrals Two electron coulomb integrals in different notations: J_{ij} = <Φ_{i}Φ_{j}Φ_{i}Φ_{j}> = Φ_{i}(r_{1})Φ_{j}(r_{2}) r_{12}^{1}Φ_{i}(r_{1})Φ_{j}(r_{2})> The indices for r are 12 and 12! Note that the scalar r_{12} is r_{1}  r_{2} J_{ij} = (Φ_{i}Φ_{i}Φ_{j}Φ_{j}) = ∫Φ_{i}(r_{1})² r_{12}^{1} Φ_{j}(r_{2})² dr_{1}dr_{2} The indices for r are 11 and 22! We recognise Φ_{j}(r)² as the electron density. ρ_{j}(r) = Φ_{j}(r)² In words: The electron density of electron j in position r is given by the square of the orbital of electron j in that position. Therefore we can also write: J_{ij} = ∫Φ_{i}(r_{1})² r_{12}^{1} Φ_{j}(r_{2})² dr_{1}dr_{2} = ∫ ρ_{i}(r_{1}) r_{12}^{1} ρ_{j}(r_{2}) dr_{1}dr_{2} We have written the Coulomb integral as an integral over electron densities ρ
