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 $v[ρ]$ is quasi-continuous, in that $v[ρ]ρ$ is continuous as a function of $ρ$. 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.