Taming Density Functional Theory by Coarse-Graining

Abstract

The standard (``fine-grained'') interpretation of quantum density functional theory, in which densities are specified with infinitely-fine spatial resolution, is mathematically unruly. Here, a coarse-grained version of DFT, featuring limited spatial resolution, and its relation to the fine-grained theory in the $L^1\cap L^3$ formulation of Lieb, is studied, with the object of showing it to be not only mathematically well-behaved, but consonant with the spirit of DFT, practically (computationally) adequate and sufficiently close to the standard interpretation as to accurately reflect its non-pathological properties. The coarse-grained interpretation is shown to be a good model of formal DFT in the sense that: all densities are (ensemble)-V-representable; the intrinsic energy functional $F$ is a continuous function of the density and the representing external potential is the (directional) functional derivative of the intrinsic energy. Also, the representing potential $v[\rho ]$ is quasi-continuous, in that $v[\rho ]\rho$ is continuous as a function of $\rho$. The limit of coarse-graining scale going to zero is studied to see if convergence to the non-pathological aspects of the fine-grained theory is adequate to justify regarding coarse-graining as a good approximation. Suitable limiting behaviors or intrinsic energy, densities and representing potentials are found. Intrinsic energy converges monotonically, coarse-grained densities converge uniformly strongly to their low-intrinsic-energy fine-grainings, and $L^{3/2}+L^{\infty }$ representability of a density is equivalent to the existence of a convergent sequence of coarse-grained potential/ground-state density pairs.

All sections have been revised at least a little, but the significant technical improvements are in section 6

References in corpus (4)

• Orbital-Free Density Functional Theory: Kinetic Potentials and Ab-Initio Local Pseudopotentials
• The many-electron ground-state determines uniquely the potential in Spin-Density-Functional Theory for non-collinear magnetism
• Nonuniqueness in spin-density-functional theory on lattices
• Coarse-grained V-representability

Cited by in corpus (12)

• Existence, Uniqueness, and Construction of the Density-Potential Mapping in Time-Dependent Density-Functional Theory
• Local Control and v-Representability of Correlated Quantum Dynamics
• One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures
• The structure of the density-potential mapping. Part I: Standard density-functional theory
• Kinetic-Energy Density-Functional Theory on a Lattice
• Geometrical Perspective on Spin-Lattice Density-Functional Theory
• The density gradient expansion of correlation functions
• Solution of the v-representability problem on a one-dimensional torus
• Excited-State-Specific Kohn-Sham Formalism for the Asymmetric Hubbard Dimer
• Functional differentiability in time-dependent quantum mechanics
• Coarse-grained spin density-functional theory: infinite-volume limit via the hyperfinite
• Density-Functional Theory on Graphs