Variational principle for many-fermion systems

Elliott H. Lieb

doi:10.1007/3-540-27056-6_19

Variational principle for many-fermion systems

Elliott H. Lieb

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

If ψ is a determinantal variational trial function for the N-fermion Hamiltonian, H, with one- and two-body terms, then e₀≤ 〈ψ, Hψ〉=E(K), where e₀ is the ground-state energy, K is the one-body reduced density matrix of ψ, and E(K) is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body K is given, which does not come from a determinantal ψ, then E(K)≥e ₀ does not necessarily hold. It is shown, however, that if the two-body part of H is positive, then in fact e₀≤e _HF≤E(K), where e_HF is the Hartree-Fock ground-state energy.

Original language	English (US)
Title of host publication	The Stability of Matter
Subtitle of host publication	From Atoms to Stars: Fourth Edition
Publisher	Springer Berlin Heidelberg
Pages	257-260
Number of pages	4
ISBN (Print)	3540420835, 9783540222125
DOIs	https://doi.org/10.1007/3-540-27056-6_19
State	Published - 2005

All Science Journal Classification (ASJC) codes

General Physics and Astronomy

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10.1007/3-540-27056-6_19

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title = "Variational principle for many-fermion systems",

abstract = "If ψ is a determinantal variational trial function for the N-fermion Hamiltonian, H, with one- and two-body terms, then e0≤ 〈ψ, Hψ〉=E(K), where e0 is the ground-state energy, K is the one-body reduced density matrix of ψ, and E(K) is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body K is given, which does not come from a determinantal ψ, then E(K)≥e 0 does not necessarily hold. It is shown, however, that if the two-body part of H is positive, then in fact e0≤e HF≤E(K), where eHF is the Hartree-Fock ground-state energy.",

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N2 - If ψ is a determinantal variational trial function for the N-fermion Hamiltonian, H, with one- and two-body terms, then e0≤ 〈ψ, Hψ〉=E(K), where e0 is the ground-state energy, K is the one-body reduced density matrix of ψ, and E(K) is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body K is given, which does not come from a determinantal ψ, then E(K)≥e 0 does not necessarily hold. It is shown, however, that if the two-body part of H is positive, then in fact e0≤e HF≤E(K), where eHF is the Hartree-Fock ground-state energy.

AB - If ψ is a determinantal variational trial function for the N-fermion Hamiltonian, H, with one- and two-body terms, then e0≤ 〈ψ, Hψ〉=E(K), where e0 is the ground-state energy, K is the one-body reduced density matrix of ψ, and E(K) is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body K is given, which does not come from a determinantal ψ, then E(K)≥e 0 does not necessarily hold. It is shown, however, that if the two-body part of H is positive, then in fact e0≤e HF≤E(K), where eHF is the Hartree-Fock ground-state energy.

UR - https://www.scopus.com/pages/publications/84892202237

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DO - 10.1007/3-540-27056-6_19

M3 - Chapter

AN - SCOPUS:84892202237

SN - 3540420835

SN - 9783540222125

SP - 257

EP - 260

BT - The Stability of Matter

PB - Springer Berlin Heidelberg

ER -

Variational principle for many-fermion systems

Abstract

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