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Register a new userOn Augmented Kohn-Sham Potential for Energy as a Simple Sum of Orbital Energies
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Mel Levy
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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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It is observed that the exact interacting ground-state electronic energy of interest may be obtained directly, in principle, as a simple sum of orbital energies when a universal density-dependent term is added to $wleft(left[ rho right];mathbf{r} right)$, the familiar Hartree plus exchange-correlation component in the Kohn-Sham effective potential. The resultant shifted potential, $bar{w}left(left[ rho right];mathbf{r} right)$, actually changes less on average than $wleft(left[ rho right];mathbf{r} right)$ when the density changes, including the fact that $bar{w}left(left[ rho right];mathbf{r} right)$ does not undergo a discontinuity when the number of electrons increases through an integer. Thus the approximation of $bar{w}left(left[ rho right];mathbf{r} right)$ represents an alternative direct approach for the approximation of the ground-state energy and density.
We present a method to invert a given density and find the Kohn-Sham (KS) potential in Density Functional Theory (DFT) which shares that density. Our method employs the concept of screening density, which is naturally constrained by the inversion procedure and thus ensures the density being inverted leads to a smooth KS potential with correct asymptotic behaviour. We demonstrate the applicability of our method by inverting both local (LDA) and non-local (Hartree-Fock and Coupled Cluster) densities; we also show how the method can be used to mitigate the effects of self-interactions in common DFT potentials with appropriate constraints on the screening density.
A detailed account of the Kohn-Sham algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy-Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows to rigorously introduce, in contrast to the common unregularized approach, a well-defined Kohn-Sham iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.
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.
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.
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