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Register a new userRelation between exchange-only optimized potential and Kohn-Sham methods with finite basis sets; solution of a paradox
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Arguments showing that exchange-only optimized effective potential (xOEP) methods, with finite basis sets, cannot in general yield the Hartree-Fock (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 basis set xOEP scheme in the special case that all products of occupied and unoccupied orbitals emerging from the employed orbital basis set are linearly independent from each other. In this case, however, exchange potentials leading to the HF ground state energy exhibit unphysical oscillations and do not represent a Kohn-Sham (KS) exchange potential. These findings solve the seemingly paradoxical results of Staroverov, Scuseria and Davidson 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, whether or not the occupied and unoccupied orbital products are linearly independent, it is shown that 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.
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The behavior of the surface barrier that forms at the metal-vacuum interface is important for several fields of surface science. Within the Density Functional Theory framework, this surface barrier has two non-trivial components: exchange and correlation. Exact results are provided for the exchange component, for a jellium metal-vacuum interface, in a slab geometry. The Kohn-Sham exact-exchange potential $V_{x}(z)$ has been generated by using the Optimized Effective Potential method, through an accurate numerical solution, imposing the correct boundary condition. It has been proved analytically, and confirmed numerically, that $V_{x}(zto infty)to - e^{2}/z$; this conclusion is not affected by the inclusion of correlation effects. Also, the exact-exchange potential develops a shoulder-like structure close to the interface, on the vacuum side. The issue of the classical image potential is discussed.
One-electron self-interaction and an incorrect asymptotic behavior of the Kohn-Sham exchange-correlation potential are among the most prominent limitations of many present-day density functionals. However, a one-electron self-interaction-free energy does not necessarily lead to the correct long-range potential. This is here shown explicitly for local hybrid functionals. Furthermore, carefully studying the ratio of the von Weizsacker kinetic energy density to the (positive) Kohn-Sham kinetic energy density, $tau_mathrm{W}/tau$, reveals that this ratio, which frequently serves as an iso-orbital indicator and is used to eliminate one-electron self-interaction effects in meta-generalized-gradient approximations and local hybrid functionals, can fail to approach its expected value in the vicinity of orbital nodal planes. This perspective article suggests that the nature and consequences of one-electron self-interaction and some of the strategies for its correction need to be reconsidered.
The one-electron density of a many-electron system is the ground-state density of a one-electron Schrodinger equation. The potential $v$ appearing in this Schrodinger equation can be constructed in two ways: In density functional theory (DFT), $v$ is the sum of the Kohn-Sham (KS) potential and the Pauli potential, where the latter can be explicitly expressed in terms of the KS system of non-interacting electrons. As the KS system is fictitious, this construction is only indirectly related to the interacting many-electron system. In contrast, in the exact electron factorization (EEF), $v$ is a functional of the conditional wavefunction $phi$ that describes the spatial entanglement of the electrons in the interacting system. We compare the two constructions of the potential, provide a physical interpretation of the contributions to $v$ in the EEF, and relate it to DFT. With numerical studies of one-dimensional two- and three-electron systems, we illustrate how features of $phi$ translate to the one-electron potential $v$. We show that a change in $phi$ corresponds to a repulsive contribution to $v$, and we explain step structures of $v$ with a charge transfer encoded in $phi$. Furthermore, we provide analytic formulas for the components of $v$ by using a two-state model. Our work thus presents the mapping of a many-electron system to a one-electron system from another angle and provides insights into what determines the shape of the exact one-electron potential. We expect our findings to be helpful for the search of suitable approximations in DFT and in related theories.
Using a simplified one-dimensional model of a diatomic molecule, the associated interacting density and corresponding Kohn-Sham potential have been obtained analytically for all fractional molecule occupancies $N$ between 0 and 2. For the homonuclear case, and in the dissociation limit, the exact Kohn-Sham potential builds a barrier at the midpoint between the two atoms, whose strength increases linearly with $N$, with $1 < N leq 2$. In the heteronuclear case, the disociating KS potential besides the barrier also exhibits a plateau around the atom with the higher ionization potential, whose size (but not its strength) depends on $N$. An anomalous zero-order scaling of the Kohn-Sham potential with regards to the strength of the electron-electron repulsion is clearly displayed by our model; without this property both the unusual barrier and plateau features will be absent.
It has recently been observed [Phys. Rev. Lett. 113, 113002 (2014)] that the ground-state energy may be obtained directly as a simple sum of augmented Kohn-Sham orbital energies, where it was ascertained that the corresponding one-body shifted Kohn-Sham effective potential has appealing features. With this in mind, eigenvalue and virial constraints are deduced for approximating this potential.
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