EXCHANGE FUNCTIONALS AND POTENTIALS

The commonly used exchange-correlation functionals of density function al 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 lar ge distances is not satisfactory. In particular, the correct asymptoti c r(-1) behaviour is difficult to achieve. Following van Leeuwen and B aerends, this is linked to the energy epsilon(max) of the highest occu pied orbital arising from the Kohn-Sham equations. This deficiency is linked also with the poor prediction of molecular polarizabilities. Th e 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 de nsity and the Laplacian of the density and has no adjustable parameter s. Becke has presented encouraging results using this functional in a hybrid manner. Fully self-consistent Kohn-Sham calculations are perfor med using it in combination with Perdew's 1986 correlation functional. The results are very encouraging indeed, so much so that this exchang e functional is the best generalized gradient approximation (GGA) yet discovered. In particular, bond lengths of many molecular show a subst antial improvement over results from other GGAs. For example, many CH bonds are now within experimental accuracy, instead of being typically 0 . 02 Angstrom too long. Our ab initio understanding of non-dynamic correlation and dynamic correlation is then linked with density functi onal theory. It is argued that correlation functionals should pick up the local dynamic correlation, whereas exchange functionals should inc lude non-dynamic correlation effects. For these reasons it is consider ed that exchange functionals are best modelled on a system fbr which t here is effectively no non-dynamic correlation, for which the optimum example is the Ne atom. Thus, again following Becke and Roussel, the s pherically averaged Hartree-Fock exchange hole for Ne is examined, com pared with the BR model functional hole. An excellent overlap is found , and thus the above good results are explained. As a final contributi on, the dissociation of the H-2 molecule is re-examined, looking at it in terms of the exchange hole. For a reference electron near one prot on A, the restricted Hartree-Fock model has half an exchange electron near it, and half and exchange electron near the other protron B, wher eas the BR functional has one electron near the other proton B, wherea s the BR functional has one electron near A, which is the correct pict ure. For this reason the (restricted) BR functional gives a greatly im proved dissociation curve for H-2 when compared with the Hartree-Fock curve. In summary, the Becke-Roussel functional is found to be a most attractive exchange functional.