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Reduced Density Matrix Functional Theory And A Reduced Formulation Of Density Functional Theory

Håkon R. Fredheim, Simen Kvaal

Abstract

A mathematical framework for reduced density matrix functional theory (RDMFT) is proposed. The work is inspired by and generalizes the work by E.H.~Lieb [E.H. Lieb, Int. J. Quant. Chem. 24(1983), pp.243--277] on density-functional theory (DFT). We introduce a Banach space for density matrices with finite kinetic energy. The dual space is a rich class of single-particle potentials, i.e., Hermitian forms. The ground state energy of an $N$-fermion system with external forces given by any such Hermitian form is expressed as the Legendre--Fenchel transform of a convex and lower semicontinuous ``universal'' reduced density matrix functional. The formalism is employed to provide a mathematical framework for density-functional theory (DFT). The main tool here is a rigorous definition of diagonals of reduced density matrices. The result is a refinement of Lieb's results on DFT applicable to a wide variety of models.

Reduced Density Matrix Functional Theory And A Reduced Formulation Of Density Functional Theory

Abstract

A mathematical framework for reduced density matrix functional theory (RDMFT) is proposed. The work is inspired by and generalizes the work by E.H.~Lieb [E.H. Lieb, Int. J. Quant. Chem. 24(1983), pp.243--277] on density-functional theory (DFT). We introduce a Banach space for density matrices with finite kinetic energy. The dual space is a rich class of single-particle potentials, i.e., Hermitian forms. The ground state energy of an -fermion system with external forces given by any such Hermitian form is expressed as the Legendre--Fenchel transform of a convex and lower semicontinuous ``universal'' reduced density matrix functional. The formalism is employed to provide a mathematical framework for density-functional theory (DFT). The main tool here is a rigorous definition of diagonals of reduced density matrices. The result is a refinement of Lieb's results on DFT applicable to a wide variety of models.
Paper Structure (16 sections, 40 theorems, 243 equations, 1 figure)

This paper contains 16 sections, 40 theorems, 243 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. N-fermion spaces and operators
  4. Trace-class operators
  5. Sesquilinear forms and their associated operators
  6. Convex analysis
  7. Reduced Density Matrices
  8. Single-particle Hamiltonians
  9. Surjectivity of the partial trace map
  10. Reduced Density Matrix Functional Theory
  11. Density Functional Theory
  12. The diagonal map D
  13. General setting for DFT
  14. Conventional setting for DFT
  15. Appendix: The diagonal map
  16. ...and 1 more sections

Key Result

Lemma 2.1

The trace norm as defined on $\mathcal{S}_1^\mathrm{sa}(\mathcal{H})$ can be characterized by for all $\gamma\in \mathcal{S}_1^\mathrm{sa}(\mathcal{H})$.

Figures (1)

  • Figure 1: This figure illustrates the proof of Lemma \ref{['lemma: affine combinations in D']} for $N=4$. Each square represents a projection, starting from $P[1]$ at the leftmost side to $P[4^2 - 4 + 1]=P[13]$ on the rightmost side. The squares above and below the middle line represent positive and negative summands in \ref{['Equation: Projection telescope sum']} respectively. We see that the sum mostly cancels and that we are left with $4P[1]$.

Theorems & Definitions (94)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: KLMN theorem
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 84 more