Exchange functionals and potentials

The commonly used exchange-correlation functionals of density functional theory and their potentials are examined numerically following the first such investigation by Perdew. They are also investigated for Ne and Kr. Their behaviour for large gradients of the density and for large distances is not satisfactory. In particular, the correct asymptotic r -1 behaviour is difficult to achieve. Following van Leeuwen and Baerends, this is linked to the energy ϵmax of the highest occupied orbital arising from the Kohn-Sham equations. This deficiency is linked also with the poor prediction of molecular polarizabilities. The Becke-Roussel (BR) exchange functional is examined, which is derived by assuming a hydrogen-like exchange hole at all spatial points, and it has the attraction of being dependent on both the kinetic energy density and the Laplacian of the density and has no adjustable parameters. Becke has presented encouraging results using this functional in a hybrid manner. Fully self-consistent Kohn-Sham calculations are performed using it in combination with Perdew's 1986 correlation functional. The results are very encouraging indeed, so much so that this exchange functional is the best generalized gradient approximation (GGA) yet discovered. In particular, bond lengths of many molecular show a substantial improvement over results from other GGAs. For example, many CH bonds are now within experimental accuracy, instead of being typically 0·02 Å too long. Our ab initio understanding of non-dynamic correlation and dynamic correlation is then linked with density functional theory. It is argued that correlation functionals should pick up the local dynamic correlation, whereas exchange functionals should include non-dynamic correlation effects. For these reasons it is considered that exchange functionals are best modelled on a system for which there is effectively no non-dynamic correlation, for which the optimum example is the Ne atom. Thus, again following Becke and Roussel, the spherically averaged Hartree-Fock exchange hole for Ne is examined, compared with the BR model functional hole. An excellent overlap is found, and thus the above good results are explained. As a final contribution, the dissociation of the H2 molecule is re-examined, looking at it in terms of the exchange hole. For a reference electron near one proton A, the restricted Hartree-Fock model has half an exchange electron near it, and half and exchange electron near the other protron B, whereas the BR functional has one electron near the other proton B, whereas the BR functional has one electron near A, which is the correct picture. For this reason the (restricted) BR functional gives a greatly improved dissociation curve for H2 when compared with the Hartree-Fock curve. In summary, the Becke-Roussel functional is found to be a most attractive exchange functional.