The Structure of Density-Potential Mapping. Part I: Standard Density-Functional Theory

Abstract

The Hohenberg–Kohn theorem of density-functional theory (DFT) is broadly considered the conceptual basis for a full characterization of an electronic system in its ground state by just the one-body particle density. Part I of this review aims at clarifying the status of the Hohenberg–Kohn theorem within DFT and Part II at different extensions of the theory that include magnetic fields. We collect evidence that the Hohenberg–Kohn theorem does not so much form the basis of DFT, but is rather the consequence of a more comprehensive mathematical framework. Such results are especially useful when it comes to the construction of generalized DFTs.

Exchange energies with forces in density-functional theory

“…which defines F Hxc for each spin channel. By virtue of the Hohenberg-Kohn theorem 12,43 and assuming non-degeneracy of the ground states for simplicity, the Slater determinant Φ as well as the interacting wave function Ψ are given solely and uniquely in terms of the density, which makes all the force densities determined by the density only. Equation (8) implies that…”

“…It is precisely the mentioned exchange-correlation potential that relates the interacting and the noninteracting system through the underlying density-potential mapping v(r) ↔ ρ(r). For a recent review on this mapping in the context of DFT, we point to the work of Penz et al 12 It is common practice to derive approximations for the (in general unknown) exchange-correlation potential by re-expressing the universal density functional as a sum of noninteracting kinetic, Hartree, and exchange-energy functionals, as well as the unknown correlation energy functional, 13 and then to assume functional differentiability 14 with respect to the density. While for approximate functionals that are given explicitly in terms of the density, potentials can be determined this way by direct differentiation, for implicit functionals this is no longer possible in general.…”

Measuring Electron Correlation: The Impact of Symmetry and Orbital Transformations

“…A more thorough review is available. 136 The theorems due to Hohenberg and Kohn 137 form the basis of the variational formulation of DFT, although questions about their precise meaning 138 remain pertinent enough to motivate further research, 139,140 not to mention the fact that these theorems do not generalize easily to excited states. 141 The favorable linear scaling property of DFT approaches is often derived from Kohn's principle of nearsightedness, 142,143 i.e., that under certain conditions the density at some point is mostly determined by the external potential in a local region around that point.…”

The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields

“…The celebrated and highly successful method of using the one-body particle density to describe quantum systemsdensity-functional theory (DFT)has also been extended to include magnetic fields. − In Part II of a two-part review series, we will explore such formulations. Just like the case for Part I, the scope of this review is limited to topics closely related to the Hohenberg–Kohn (HK) mapping and properties of the exact functional(s). Here, in Part II, this is done for extended DFTs to account for magnetic fields.…”

“… 1 − 5 In Part II of a two-part review series, we will explore such formulations. Just like the case for Part I, 6 the scope of this review is limited to topics closely related to the Hohenberg–Kohn (HK) mapping and properties of the exact functional(s). Here, in Part II, this is done for extended DFTs to account for magnetic fields.…”