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Related Experiment Video

Updated: Oct 8, 2025

Modeling the Functional Network for Spatial Navigation in the Human Brain
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Density-functional theory on graphs.

Markus Penz1, Robert van Leeuwen2

  • 1Department of Mathematics, University of Innsbruck, Innsbruck, Austria.

The Journal of Chemical Physics
|January 1, 2022
PubMed
Summary
This summary is machine-generated.

Density-functional theory (DFT) principles are explored for graph-based lattice systems. The study finds the Hohenberg-Kohn theorem generally invalid but reveals insights into density-potential mapping, proving unique v-representability for most densities.

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Area of Science:

  • Condensed matter physics
  • Quantum chemistry
  • Computational materials science

Background:

  • Density-functional theory (DFT) is a cornerstone for electronic structure calculations.
  • The Hohenberg-Kohn theorems are foundational to DFT, establishing the unique mapping between electron density and external potential.
  • Lattice systems offer a simplified yet relevant model for studying quantum mechanical properties.

Purpose of the Study:

  • To investigate the validity of density-functional theory principles in finite lattice systems.
  • To analyze the topological structure of the density-potential mapping.
  • To establish conditions for unique v-representability of ground states in these systems.

Main Methods:

  • Theoretical analysis of density-functional theory principles applied to graph-represented finite lattice systems.
  • Investigation of the density-potential mapping's topological properties.
  • Derivation of conditions for unique v-representability.

Main Results:

  • The fundamental Hohenberg-Kohn theorem is found to be generally void for finite lattice systems.
  • Significant insights into the topological structure of the density-potential mapping are obtained.
  • Precise conditions for unique v-representability are established, proven to hold for almost all densities.
  • Demonstration of the non-convexity of the pure-state constrained-search functional through examples.

Conclusions:

  • The applicability of standard density-functional theory requires careful consideration in finite lattice systems.
  • The study provides a refined understanding of v-representability and the mapping properties within DFT.
  • New theoretical frameworks may be needed to fully capture the behavior of electronic systems in discrete spaces.