Exact exchange-only potentials and the virial relation as microscopic criteria for generalized gradient approximations

The information contained in the exact exchange potential of atoms as calculated in the optimized-potential model is used to analyze generalized approximations (GGA’s) from a microscopic viewpoint. It is shown that the GGA recently introduced by Perdew and Wang [in Electronic Structure of Solids 1991, edited by P. Ziesche and H. Eschrig (Akademie Verlag, Berlin, 1991), Vol. 11] does not significantly improve the exchange potential given by the lowest-order gradient correction. This contrasts with its excellent reproduction of atomic exchange energies, which is shown to be mainly due to cancellation of local errors in the integrand of the virial relation for the exchange-energy functional. Utilizing this virial relation a GGA is constructed which reproduces atomic exchange potentials considerably better. This functional does not give as accurately total exchange energies, although it is superior to the second-order gradient expansion. It thus represents a balanced approach aiming at an overall improvement rather than focusing on the exchange energies only. It is concluded that the concept of GGA’s due to its simple quasilocal-density dependence is not sufficiently flexible to accurately reproduce exchange potentials and exchange energies simultaneously. It seems that the criteria for judging and constructing approximate exchange functionals put forward in this work and the resulting GGA give a more realistic view of the overall capabilities of GGA’s to represent the properties of the true exchange-energy functional, e.g., its functional derivatives, than previous GGA’s.