Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:physics/0607037 · International Journal of Quantum Chemistry

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

A. I. Panin

Published Oct 5, 2006• 1 citation
arXivHTMLPDF

Abstract

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

DOI
10.1002/qua.21213

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 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.

•Jan 1, 2006•1 citation

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

Corollary to the Hohenberg–Kohn theorem

Xiao‐Yin Pan, Viraht Sahni

Abstract According to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian of a system and the corresponding ground‐state density ρ(r). The extension of the theorem to the time‐dependent case by Runge and Gross states that there is an invertible one‐to‐one relationship between the density ρ(rt) and the Hamiltonian (t). In the proof of the theorem, Hamiltonians / (t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians / (t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian / (t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians { }/{ (t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians { }/{ (t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians { }/{ (t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

arXiv:cond-mat/0209688•International Journal of Quantum Chemistry•Jan 1, 2003•14 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

Some basic properties of the correlation matrices

Carmela Valdemoro, Luis M. Tel, Diego R. Alcoba +2 more

Abstract On the basis of the properties of correlation matrices, it is shown here that the set of all the first‐order transition reduced density matrices of a system provide complete information about that system. Also, the interrelation between the properties of the correlation matrix and the 2‐RDM N‐representability conditions is studied. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

International Journal of Quantum Chemistry•Jan 1, 2002•20 citations

Exposing minimal composition of Kohn–Sham theory and its extendability

Hitoshi Nakada

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.

arXiv:2108.01232•Physica Scripta•Aug 29, 2023•3 citations
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
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 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.
Abstract According to the Hohenberg–Kohn theorem, there is an invertible one‐to‐one relationship between the Hamiltonian of a system and the corresponding ground‐state density ρ(r). The extension of the theorem to the time‐dependent case by Runge and Gross states that there is an invertible one‐to‐one relationship between the density ρ(rt) and the Hamiltonian (t). In the proof of the theorem, Hamiltonians / (t) that differ by an additive constant C/function C(t) are considered equivalent. Because the constant C/function C(t) is extrinsically additive, the physical system defined by these differing Hamiltonians / (t) is the same. Thus, according to the theorem, the density ρ(r)/ρ(rt) uniquely determines the physical system as defined by its Hamiltonian / (t). Hohenberg–Kohn, and by extension Runge and Gross, did not however consider the case of a set of degenerate Hamiltonians { }/{ (t)} that differ by an intrinsic constant C/function C(t) but which represent different physical systems and yet possess the same density ρ(r)/ρ(rt). The intrinsic constant C/function C(t) contains information about the different physical systems and helps differentiate between them. In such a case, the density ρ(r)/ρ(rt) cannot distinguish between these different Hamiltonians. In this article we construct such a set of degenerate Hamiltonians { }/{ (t)}. Thus, although the proof of the Hohenberg–Kohn theorem is independent of whether the constant C/function C(t) is additive or intrinsic, the applicability of the theorem is restricted to excluding the case of the latter. The corollary is as follows: degenerate Hamiltonians { }/{ (t)} that represent different physical systems, but differ by a constant C/function C(t), and yet possess the same density ρ(r)/ρ(rt), cannot be distinguished on the basis of the Hohenberg–Kohn/Runge–Gross theorem. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003
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.