How to Deal with Exchange and Correlation

Not too surprisingly, the quality of the Hamiltonian chosen for the analysis of a quantum-chemical problem depends on the effort that is used for dealing with the electron-electron interactions. There are also other important criteria (for example, the shape of the electronic potential, later covered in 

Section 2.15) but exchange and correlation typically serve as to characterize the worth of the computational approach the (sometimes questionable) distinction between (semi-)empirical and ab initio methods also goes back - to some extent - to this question. Let us group the different approaches, a bit simplified, and see how they relate with each other. 

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Note again the formal simplicity of equation (7-17) as compared to equation (7-18) in spite of the fact that the former is exact provided the correct Vxc is inserted, while the latter is inherently an approximation. The calculation of the formally L2/2 one-electron integrals contained in hllv, equation (7-13) is a fairly simple task compared to the determination of the classical Coulomb and the exchange-correlation contributions. However, before we turn to the question, how to deal with the Coulomb and Vxc integrals, we want to discuss what kind of basis functions are nowadays used in equation (7-4) to express the Kohn-Sham orbitals. 

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