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On the non-interacting $v$-representability conjecture within density functional theory

J. Messud

Published Feb 6, 2021• 0 citations
OpenAlex

Abstract

Density functional theory together with the Kohn-Sham (KS) scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary system (one-body with a local potential that is a density functional). However, to fully achieve establishing the KS scheme, a general proof of the validity of the so-called $v$-representability conjecture is still required. In this article, we propose such a proof. We reduce the demonstration to proving (1) that the KS potential is differentiable for all non-interacting $v$-representable densities and (2) that its derivative is bounded. Then, we demonstrate points (1) and (2) applying static linear response to the non-interacting system.

Related papers

Notes on open questions within density functional theory (existence of a derivative of the Lieb functional in a restricted sense and non-interacting $v$-representability)

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Density functional theory together with the Kohn-Sham scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary ``non-interacting'' system (one-body with a local potential). However, theoretical questions remain open. An important one is related to the existence of a derivative of the Lieb functional in some restricted (weak) sense (as pointed out by Lammert). Then, a further one would be related to the validity of the ``non-interacting $v$-representability'' conjecture. We gather here elements on these questions, providing reminders and thoughts that are non-conclusive but hopefully contribute to the reflection. This document must be considered as notes (it does not represent an article).

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Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving dynamical many-body problems in physics and chemistry. The mathematical foundations of TDDFT are established through the formal existence of a fictitious non-interacting system (known as the Kohn–Sham system), which can reproduce the one-electron reduced probability density of the actual system. We build upon these works and show that on the interior of the domain of existence, the Kohn–Sham system can be efficiently obtained given the time-dependent density. We introduce a V -representability parameter which diverges at the boundary of the existence domain and serves to quantify the numerical difficulty of constructing the Kohn–Sham potential. For bounded values of V -representability, we present a polynomial time quantum algorithm to generate the time-dependent Kohn–Sham potential with controllable error bounds.

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On the non-interacting vvv-representability conjecture within density functional theory
Density functional theory together with the Kohn-Sham (KS) scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary system (one-body with a local potential that is a density functional). However, to fully achieve establishing the KS scheme, a general proof of the validity of the so-called vvv-representability conjecture is still required. In this article, we propose such a proof. We reduce the demonstration to proving (1) that the KS potential is differentiable for all non-interacting vvv-representable densities and (2) that its derivative is bounded. Then, we demonstrate points (1) and (2) applying static linear response to the non-interacting system.
Notes on open questions within density functional theory (existence of a derivative of the Lieb functional in a restricted sense and non-interacting vvv-representability)
Density functional theory together with the Kohn-Sham scheme represent an efficient framework to recover the ground state density and energy of a many-body quantum system from an auxiliary ``non-interacting'' system (one-body with a local potential). However, theoretical questions remain open. An important one is related to the existence of a derivative of the Lieb functional in some restricted (weak) sense (as pointed out by Lammert). Then, a further one would be related to the validity of the ``non-interacting vvv-representability'' conjecture. We gather here elements on these questions, providing reminders and thoughts that are non-conclusive but hopefully contribute to the reflection. This document must be considered as notes (it does not represent an article).
Abstract Reducing the many-fermion problem to a set of single-particle (s.p.) equations, the Kohn–Sham (KS) theory has provided a practical tool to implement ab initio calculations of ground-state energies and densities in many-electron systems. There have been attempts to extend the KS theory so that it could describe other physical quantities, or it could be applied to other many-fermion systems. By generalizing and reformulating the KS theory in terms of the 1-body density matrix, we expose the minimal composition of the theory that enables the reduction of the many-fermion problem to the s.p. equations. Based on the reformulation, several basic issues are reconsidered. The v- and N-representabilities for the KS theory are distinguished from those for the Hohenberg-Kohn theorem. Criteria for the extendability of the KS theory are addressed.
Aiming at a unified treatment of correlation and inhomogeneity effects in superconductors, Oliveira, Gross, and Kohn proposed in 1988 a density functional theory for the superconducting state. This theory relies on the existence of a Kohn- Sham scheme, i.e., an auxiliary noninteracting system with the same electron and anomalous densities of the original superconducting system [Phys. Rev. Lett. 60, 2430 (1988)]. However, the question of noninteracting vvv-representability has never been properly addressed and the existence of the Kohn-Sham system has always been assumed without proof. Here, we show that indeed such a noninteracting system does not exist at zero temperature. In spite of this result, we also show that the theory is still able to yield good results, although in the limit of weakly correlated systems only.
Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving dynamical many-body problems in physics and chemistry. The mathematical foundations of TDDFT are established through the formal existence of a fictitious non-interacting system (known as the Kohn–Sham system), which can reproduce the one-electron reduced probability density of the actual system. We build upon these works and show that on the interior of the domain of existence, the Kohn–Sham system can be efficiently obtained given the time-dependent density. We introduce a V-representability parameter which diverges at the boundary of the existence domain and serves to quantify the numerical difficulty of constructing the Kohn–Sham potential. For bounded values of V-representability, we present a polynomial time quantum algorithm to generate the time-dependent Kohn–Sham potential with controllable error bounds.
Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving dynamical many-body problems in physics and chemistry. The mathematical foundations of TDDFT are established through the formal existence of a fictitious non-interacting system (known as the Kohn–Sham system), which can reproduce the one-electron reduced probability density of the actual system. We build upon these works and show that on the interior of the domain of existence, the Kohn–Sham system can be efficiently obtained given the time-dependent density. We introduce a V -representability parameter which diverges at the boundary of the existence domain and serves to quantify the numerical difficulty of constructing the Kohn–Sham potential. For bounded values of V -representability, we present a polynomial time quantum algorithm to generate the time-dependent Kohn–Sham potential with controllable error bounds.
The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.