Ground-State Degeneracies Leave Recognizable Topological Scars in the Electronic Density

In Kohn-Sham density functional theory (KS DFT) a fictitious system of noninteracting particles is constructed having the same ground-state (GS) density as the physical system of interest. A fundamental open question in DFT concerns the ability of an exact KS calculation to spot and characterize the GS degeneracies in the physical system. In this Letter we provide theoretical evidence suggesting that the GS density, as a function of position on a 2D manifold of parameters affecting the external potential, is “topologically scarred” in a distinct way by degeneracies. These scars are sufficiently detailed to enable determination of the positions of degeneracies and even the associated Berry phases. We conclude that an exact KS calculation can spot and characterize the degeneracies of the physical system.

Figure 1

The “topological scar”: $n({\varphi }_1)$ and $n({\varphi }_2)$ are limit densities along two distinct paths converging into the same manifold point. If they are different a GS degeneracy exists in system I.

Figure 2

Reconstruction of the mixing angle of $H_3$ using the components of the electronic DM calculated along circular paths using the extended Hückel approximation. Three paths in the $x\mathrm{\text{−}}y$ plane are considered, each of radius $R{a}_0$ centered around the point $C$ as depicted in each panel. The CI is located at the origin. Top panels: The function $a(\varphi )$. Bottom panels: The mixing angle $\theta (\varphi )$.