Relation between exchange-only optimized potential and Kohn-Sham methods with finite basis sets, and effect of linearly dependent products of orbital basis functions.

Recently, Staroverov, Scuseria, and Davidson [J. Chem. Phys. 124, 141103 (2006)] presented examples of exchange-only optimized effective potential (xOEP) calculations that yield exactly the Hartree-Fock (HF) total energy. Here, building on their work, arguments showing under which conditions xOEP methods, with finite basis sets, do or do not yield the HF ground state energy but a higher one, are given. While the orbital products of a complete basis are linearly dependent, the HF ground state energy can only be obtained via a finite basis set xOEP scheme in the case that all products of occupied and unoccupied orbitals emerging from the employed orbital basis set are linearly independent of each other. Further, exchange potentials leading to the HF ground state energy likely exhibit unphysical oscillations and do not represent a Kohn-Sham (KS) exchange potential as a functional derivative of the exchange energy. These findings appear to explain the seemingly paradoxical results of Staroverov et al. that certain finite basis set xOEP calculations lead to the HF ground state energy despite the fact that within a real space (or complete basis) representation, the xOEP ground state energy is always higher than the HF energy. Moreover, independent of whether or not the occupied and unoccupied orbital products are linearly dependent, it is shown that finite basis set xOEP methods only represent exact exchange-only (EXX) KS methods, i.e., proper density-functional methods, if the orbital basis set and the auxiliary basis set representing the exchange potential are balanced to each other, i.e., if the orbital basis is comprehensive enough for a given auxiliary basis. Otherwise xOEP methods do not represent EXX KS methods and yield unphysical exchange potentials. The question whether a xOEP method properly represents a KS method with an exchange potential that is a functional derivative of the exchange energy is related to the problem of the definition of local multiplicative operators in finite basis representations. Plane wave calculations for bulk silicon illustrate the findings of this work.