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1-Density Operators and Algebraic Version of The Hohenberg-Kohn Theorem

A. I. Panin

Published Jan 1, 2006• 1 citation
OpenAlex

Abstract

ABSTRACT: Interrelation of the Coleman’s representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman’s set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman’s polyhedrons is obtained. Key words: density operators; representability problem; density functional theory I.

Related papers

1‐Density operators and algebraic version of the Hohenberg–Kohn theorem

A. I. Panin

Abstract Interrelation of the Coleman's representabilty theory for 1‐density operators and abstract algebraic form of the Hohenberg–Kohn (HK) theorem is studied in detail. Convenient realization of the HK set of classes of 1‐electron operators and the Coleman's set of ensemble representable 1‐density operators is presented. Dependence of the HK class structure on the boundary properties of the ground‐state 1‐density operator is established and is illustrated on concrete simple examples. An algorithm of restoration of many electron determinant ensembles from a given 1‐density diagonal is described. A complete description of the combinatorial structure of Coleman's polyhedrons is obtained. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

arXiv:physics/0607037•International Journal of Quantum Chemistry•Oct 5, 2006•1 citation

Density-functional theory on graphs

Markus Penz, Robert van Leeuwen

The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.

arXiv:2106.15370•The Journal of Chemical Physics•Dec 7, 2021•35 citations

Restoration of Many Electron Wave Functions from One-Electron Density

A. I. Panin, Alexander N. Petrov

General theorem describing a relation between diagonal of one-electron density matrix and a certain class of many-electron ensembles of determinant states is proved. As a corollary to this theorem a constructive proof of sufficiency of Coleman's representability conditions is obtained. It is shown that there exist rigorous schemes for construction of energy of many-electron system as functionals of one-electron density.

arXiv:physics/0310043•Oct 9, 2003•0 citations

Generalizations of the Hohenberg-Kohn theorem: I. Legendre Transform Constructions of Variational Principles for Density Matrices and Electron Distribution Functions

Paul W. Ayers, Sidney Golden, Mel Levy

Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb's Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the upsilon- and N-representability problems based on the constrained search formalism.

The Journal of Chemical Physics•Jan 27, 2006•92 citations

Extensibility of Hohenberg–Kohn Theorem to General Quantum Systems

Limin Xu, Jiahao Mao, Xingyu Gao +1 more

Abstract The Hohenberg–Kohn (HK) theorem for interacting electrons is a cornerstone of modern electronic structure calculations. For a general quantum system, a HK‐type Hamiltonian in the form of can always be defined, by grouping those terms with fixed or preknown coefficients into an internal part of the Hamiltonian , and factorizing the remaining as the superposition of a set of Hermitian operators . It is asked whether the HK theorem can be extended to such a general setting, so that the ground‐state expectation values of as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. It is shown that the question can be addressed by the invertibility of generalized density correlation matrix (GDCM) defined with respect to the operators. This criterion is applied to several representative examples, including the quantum Ising dimer, frustration‐free systems, N ‐level quantum systems and a fermionic Hubbard chain. It is suggested that for a finite‐size system, finding an invertible GDCM under one single configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space.

arXiv:2201.09177•Advanced Quantum Technologies•Aug 2, 2022•7 citations

Geometrical perspective on spin–lattice density-functional theory

Markus Penz, Robert van Leeuwen

A recently developed viewpoint on the fundamentals of density-functional theory for finite interacting spin-lattice systems that centers around the notion of degeneracy regions is presented. It allows for an entirely geometrical description of the Hohenberg-Kohn theorem and v-representability. The phenomena receive exemplification by an Anderson impurity model and other small-lattice examples. The case of adiabatic change and the time-dependent setting are examined as well.

arXiv:2407.20260•The Journal of Chemical Physics•Oct 15, 2024•10 citations
ABSTRACT: Interrelation of the Coleman’s representabilty theory for 1-density operators and abstract algebraic form of the Hohenberg-Kohn theorem is studied in detail. Convenient realization of the Hohenberg-Kohn set of classes of 1-electron operators and the Coleman’s set of ensemble representable 1-density operators is presented. Dependence of the Hohenberg-Kohn class structure on the boundary properties of the ground state 1-density operator is established and is illustrated on concrete simple examples. Algorithm of restoration of many electron determinant ensembles from a given 1-density diagonal is described. Complete description of the combinatorial structure of Coleman’s polyhedrons is obtained. Key words: density operators; representability problem; density functional theory I.
Abstract Interrelation of the Coleman's representabilty theory for 1‐density operators and abstract algebraic form of the Hohenberg–Kohn (HK) theorem is studied in detail. Convenient realization of the HK set of classes of 1‐electron operators and the Coleman's set of ensemble representable 1‐density operators is presented. Dependence of the HK class structure on the boundary properties of the ground‐state 1‐density operator is established and is illustrated on concrete simple examples. An algorithm of restoration of many electron determinant ensembles from a given 1‐density diagonal is described. A complete description of the combinatorial structure of Coleman's polyhedrons is obtained. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
The principles of density-functional theory are studied for finite lattice systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn theorem is found void, in general, while many insights into the topological structure of the density-potential mapping can be won. We give precise conditions for a ground state to be uniquely v-representable and are able to prove that this property holds for almost all densities. A set of examples illustrates the theory and demonstrates the non-convexity of the pure-state constrained-search functional.
Given a general, N-particle Hamiltonian operator, analogs of the Hohenberg-Kohn theorem are derived for functions that are more general than the particle density, including density matrices and the diagonal elements thereof. The generalization of Lieb's Legendre transform ansatz to the generalized Hohenberg-Kohn functional not only solves the upsilon-representability problem for these entities, but, more importantly, also solves the N-representability problem. Restricting the range of operators explored by the Legendre transform leads to a lower bound on the true functional. If all the operators of interest are incorporated in the restricted maximization, however, the variational principle dictates that exact results are obtained for the systems of interest. This might have important implications for practical work not only for density matrices but also for density functionals. A follow-up paper will present a useful alternative approach to the upsilon- and N-representability problems based on the constrained search formalism.
Abstract The Hohenberg–Kohn (HK) theorem for interacting electrons is a cornerstone of modern electronic structure calculations. For a general quantum system, a HK‐type Hamiltonian in the form of can always be defined, by grouping those terms with fixed or preknown coefficients into an internal part of the Hamiltonian , and factorizing the remaining as the superposition of a set of Hermitian operators . It is asked whether the HK theorem can be extended to such a general setting, so that the ground‐state expectation values of as the generalized density can in principle be used as the fundamental variables determining all the properties of the system. It is shown that the question can be addressed by the invertibility of generalized density correlation matrix (GDCM) defined with respect to the operators. This criterion is applied to several representative examples, including the quantum Ising dimer, frustration‐free systems, N ‐level quantum systems and a fermionic Hubbard chain. It is suggested that for a finite‐size system, finding an invertible GDCM under one single configuration is typically sufficient to establish the generic extensibility of the HK theorem in the entire parameter space.