Density-functional theory on graphs

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.