On the errors of local density (LDA) and generalized gradient (GGA) approximations to the Kohn-Sham potential and orbital energies

In spite of the high quality of exchange-correlation energies Exc obtained with the generalized gradient approximations (GGAs) of density functional theory, their xc potentials vxc are strongly deficient, yielding upshifts of ca. 5 eV in the orbital energy spectrum (in the order of 50% of high-lying valence orbital energies). The GGAs share this deficiency with the local density approximation (LDA). We argue that this error is not caused by the incorrect long-range asymptotics of vxc or by self-interaction error. It arises from incorrect density dependencies of LDA and GGA exchange functionals leading to incorrect (too repulsive) functional derivatives (i.e., response parts of the potentials). The vxc potential is partitioned into the potential of the xc hole vxchole (twice the xc energy density ϵxc), which determines Exc, and the response potential vresp, which does not contribute to Exc explicitly. The substantial upshift of LDA/GGA orbital energies is due to a too repulsive LDA exchange response potential vxresp (LDA) in the bulk region. Retaining the LDA exchange hole potential plus the B88 gradient correction to it but replacing the response parts of these potentials by the model orbital-dependent response potential vxresp (GLLB) of Gritsenko et al. [Phys. Rev. A 51, 1944 (1995)], which has the proper step-wise form, improves the orbital energies by more than an order of magnitude. Examples are given for the prototype molecules: dihydrogen, dinitrogen, carbon monoxide, ethylene, formaldehyde, and formic acid.