Basis functions exchange correlation

Laikov, D. N. (1997). Fast evaluation of density functional exchange-correlation terms using the expansion of the electron density in auxiliary basis sets. Chemical Physics Letters, 281,151. 

There is no systematic way in which the exchange correlation functional Vxc[F] can be systematically improved in standard HF-LCAO theory, we can improve on the model by increasing the accuracy of the basis set, doing configuration interaction or MPn calculations. What we have to do in density functional theory is to start from a model for which there is an exact solution, and this model is the uniform electron gas. Parr and Yang (1989) write... 

Again the set of fitting functions may or may not be the same as the orbital and/or the density basis functions. Once the potential has been fitted, the exchange—correlation energy may be evaluated from integrals involving three functions, analogously to eq. 

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. 

In words, the integral of equation (7-33) for the exchange-correlation potential is approximated by a sum of P terms. Each of these is computed as the product of the numerical values of the basis functions and rp, with the exchange-correlation potential Vxc at each point rp on the grid. Each product is further weighted by the factor Wp, whose value depends on the actual numerical technique used. 

Rashin et al.45 obtained the dipole moment of 32 molecules of biological relevance by means of the DFT(SVWN) and DFT(B88/P86) calculations. The results showed a rather weak dependence of calculated dipole moments on the functional form of the exchange-correlation functional but a strong dependence on the basis set. 

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