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A rigorous formulation of Density Functional Theory for spinless fermions in one dimension

Thiago Carvalho Corso

TL;DR

This work provides a fully rigorous foundation for KS-DFT of spinless fermions in one dimension with distributional potentials by solving the pure-state $\mathcal V$-representability problem, proving a HK theorem, and establishing differentiability of the exchange–correlation functional. It shows that the ground-state density of interacting 1D fermions can be exactly reproduced by a KS system, with a KS Hamiltonian $h_{KS}(\rho)= -\Delta + v_{xc}(\rho) + v_H(\rho) + v$ where $v_{xc}= d_\rho E_{xc}$ and $v_H(\rho)(\delta)= w(\rho\otimes\delta)+w(\delta\otimes\rho)$. The paper also develops a convex-analytic framework to connect HK with differentiability of the Levy–Lieb functional, establishes the Gateaux differentiability of $E_{xc}$, and demonstrates exact KS-DFT under Neumann and certain non-local BCs, while noting open Dirichlet questions and the challenge of higher dimensions. Overall, it delivers the first rigorous proof that KS-DFT is an exact ground-state theory for continuous 1D electronic-like systems within the specified distributional-potential setting, and it lays a precise mathematical foundation for KS constructions in this regime.

Abstract

In this paper, we present a completely rigorous formulation of Kohn-Sham density functional theory for spinless fermions living in one dimensional space. More precisely, we consider Schrödinger operators of the form $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ acting on $\wedge^N \mathrm{L}^2([0,1])$, where the external and interaction potentials $v$ and $w$ belong to a suitable class of distributions. In this setting, we obtain a complete characterization of the set of pure-state $v$-representable densities on the interval. Then, we prove a Hohenberg-Kohn theorem that applies to the class of distributional potentials studied here. Lastly, we establish the differentiability of the exchange-correlation functional and therefore the existence of a unique exchange-correlation potential. We then combine these results to provide a rigorous formulation of the Kohn-Sham scheme. In particular, these results show that the Kohn-Sham scheme is rigorously exact in this setting.

A rigorous formulation of Density Functional Theory for spinless fermions in one dimension

TL;DR

This work provides a fully rigorous foundation for KS-DFT of spinless fermions in one dimension with distributional potentials by solving the pure-state -representability problem, proving a HK theorem, and establishing differentiability of the exchange–correlation functional. It shows that the ground-state density of interacting 1D fermions can be exactly reproduced by a KS system, with a KS Hamiltonian where and . The paper also develops a convex-analytic framework to connect HK with differentiability of the Levy–Lieb functional, establishes the Gateaux differentiability of , and demonstrates exact KS-DFT under Neumann and certain non-local BCs, while noting open Dirichlet questions and the challenge of higher dimensions. Overall, it delivers the first rigorous proof that KS-DFT is an exact ground-state theory for continuous 1D electronic-like systems within the specified distributional-potential setting, and it lays a precise mathematical foundation for KS constructions in this regime.

Abstract

In this paper, we present a completely rigorous formulation of Kohn-Sham density functional theory for spinless fermions living in one dimensional space. More precisely, we consider Schrödinger operators of the form acting on , where the external and interaction potentials and belong to a suitable class of distributions. In this setting, we obtain a complete characterization of the set of pure-state -representable densities on the interval. Then, we prove a Hohenberg-Kohn theorem that applies to the class of distributional potentials studied here. Lastly, we establish the differentiability of the exchange-correlation functional and therefore the existence of a unique exchange-correlation potential. We then combine these results to provide a rigorous formulation of the Kohn-Sham scheme. In particular, these results show that the Kohn-Sham scheme is rigorously exact in this setting.

Paper Structure

This paper contains 28 sections, 32 theorems, 174 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main contributions
  4. Results
  5. Notation
  6. Main results (\ref{['it:v-rep']}) - Characterization of $v$-representability
  7. Main results (\ref{['it:HK']}) - Hohenberg-Kohn for distributional potentials
  8. Main results (\ref{['it:KS']}) - the Kohn-Sham scheme
  9. Key steps of the proofs
  10. Outline of the paper
  11. Mathematical background on Sobolev spaces and quadratic forms
  12. Definition of Sobolev functions
  13. Regularity of reduced densities
  14. Schrödinger operators with distributional potentials
  15. Examples of distributional potentials
  16. ...and 13 more sections

Key Result

Theorem 2.3

Let $w\in \mathcal{W}$ be fixed and $N\in \mathbb N$. Then the set of all possible ground-stateUnless otherwise stated, we always assume a ground-state wave-function to be normalized, i.e., $\lVert \Psi \rVert_{\mathrm{L}^2} =1$. densities of the Neumann realization $H_N(v,w)$ for $v\in \mathcal{V}$ is given by In particular, $\mathcal{D}_N(w) = \mathcal{D}_N$ is independent of the interaction po

Figures (1)

  • Figure 1: Visual illustration of the transformation $G_-$ with $N=2$ and $x_\ast = 1/3$. The middle plot shows the anti-periodic extension of $\Psi$ to $[-1, 1]^2$ with translations with different sign (in blue) and translations with same sign (in red). The planes $\{x= x_\ast\}, \{y=x_\ast\}, \{x=x_\ast -1\}$ and $\{y=x_\ast -1\}$, whose projections in the $(x,y)$-plane cover the boundary of the new box, are also depicted.

Theorems & Definitions (74)

  • Remark 2.1: Generalized Neumann boundary conditions
  • Remark 2.2: Periodic boundary conditions via the Torus
  • Theorem 2.3: Characterization of pure-state $\mathcal{V}$-representability - Neumann BCs
  • Theorem 2.4: Characterization of pure-state $\mathcal{V}$-representability - non-local BCs
  • Remark 2.5: Anti-periodic wave-functions have periodic density
  • Proposition 2.6: Counter example for $N=1$
  • Theorem 2.7: Characterization of non-interacting pure-state $\mathcal{V}$-representability - non-local BCs
  • Remark 2.8: Dirichlet BCs
  • Theorem 2.9: Hohenberg-Kohn with distributional potentials
  • Theorem 2.10: Hohenberg-Kohn theorem - non-local BCs
  • ...and 64 more