Hierarchy of model Kohn–Sham potentials for orbital-dependent functionals: A practical alternative to the optimized effective potential method

Abstract

We describe a method for constructing a hierarchy of model potentials approximating the functional derivative of a given orbital-dependent exchange-correlation functional with respect to electron density. Each model is derived by assuming a particular relationship between the self-consistent solutions of Kohn-Sham (KS) and generalized Kohn-Sham (GKS) equations for the same functional. In the KS scheme, the functional is differentiated with respect to density, in the GKS scheme--with respect to orbitals. The lowest-level approximation is the orbital-averaged effective potential (OAEP) built with the GKS orbitals. The second-level approximation, termed the orbital-consistent effective potential (OCEP), is based on the assumption that the KS and GKS orbitals are the same. It has the form of the OAEP plus a correction term. The highest-level approximation is the density-consistent effective potential (DCEP), derived under the assumption that the KS and GKS electron densities are equal. The analytic expression for a DCEP is the OCEP formula augmented with kinetic-energy-density-dependent terms. In the case of exact-exchange functional, the OAEP is the Slater potential, the OCEP is roughly equivalent to the localized Hartree-Fock approximation and related models, and the DCEP is practically indistinguishable from the true optimized effective potential for exact exchange. All three levels of the proposed hierarchy require solutions of the GKS equations as input and have the same affordable computational cost.

Comparison between exact and semilocal exchange potentials: An all-electron study for solids

“…Moreover, we mention the work of Staroverov and coworkers 83,107 who proposed an expression for a multiplicative exchange potential (making no use of unoccupied orbitals) which leads to results that are quasiidentical to the EXX-OEP results. However, this approach requires the HF orbitals which reduces its use for solids and large scale applications.…”

“…83 for a summary of the expression of these potentials). These nonlocal potentials avoid some of the technical difficulties of EXX-OEP as their construction involves only the occupied orbitals.…”

Improved method for generating exchange-correlation potentials from electronic wave functions

“…RKS-type potentials generated from HF wave functions are known to be excellent approximations to exchange-only optimized effective potentials. 8,9 The message of this figure is that molecular exchangecorrelation potentials of high quality can be effortlessly generated by the mRKS method.…”

“…As a quality control test, |∆E vir | is more discriminating than ∆ ρ : even visually imperceptible defects of v XC (r) can result in large ∆E vir values, as we showed previously for approximate exchange-only potentials. 8,9,49,50 Note that in Refs. 8 and 9 we studied KS potentials extracted from HF wave functions as approximations to exact-exchange potentials, for which T c = 0, so we defined ∆E vir = W − E HF X and evaluated E HF X using the KS (not HF) orbitals.…”

Approximations to the exact exchange potential: KLI versus semilocal

“…23 and 24 for MGGAs). However, comparisons between the EXX-OEP and the KLI approximation (EXX-KLI in the following) concern mainly atoms and light molecules/clusters 8,[15][16][17][34][35][36][37][38][39][40][41][42][43][44] and only a few such comparisons were done for periodic systems. 8,11,45,46 From most of these studies, it was concluded that EXX-KLI is a good approximation to EXX-OEP, however, in Refs.…”