Degeneracy in Density Functional Theory: Topology in v- and n-Space

TL;DR

This paper investigates the topological properties of the mapping between potential and density spaces in fermionic systems, revealing that degeneracies do not alter the measure of density manifolds and discussing implications for symmetry and temperature effects.

Contribution

It provides a detailed topological analysis of the v- and n-space mapping in density functional theory, highlighting the measure equivalence of degeneracy classes and exploring symmetry and temperature effects.

Findings

Degeneracies g=1 and g>1 have the same measure in density manifolds.

Most densities near a g-ensemble-v-representable density are also g-VR.

Symmetry and temperature influence the topology of the mapping.

Abstract

This paper clarifies the topology of the mapping between v- and n-space in fermionic systems. Density manifolds corresponding to degeneracies g=1 and g>1 are shown to have the same mathematical measure: every density near a g-ensemble-v-representable (g-VR) n(r) is also g-VR (except ``boundary densities'' of lower measure). The role of symmetry and the connection between T=0 and T=0+ are discussed. A lattice model and the Be-series are used as illustrations.