### Abstract

A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′) = (ψ |α_{2}+α_{1}+α _{1′}α_{2′}). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N - 1) and trγ = N] then the associated operator: G(1, 2 | 1′, 2′) = δ(1 - 1′)γ(2 | 2′) + σ Γ(1′, 2 | 1, 2′) - γ(2 | 1)7(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or - 1 for bosons or fermions respectively.].

Original language | English (US) |
---|---|

Pages (from-to) | 1756-1776 |

Number of pages | 21 |

Journal | Journal of Mathematical Physics |

Volume | 5 |

Issue number | 12 |

State | Published - 1964 |

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### ASJC Scopus subject areas

- Organic Chemistry

### Cite this

*Journal of Mathematical Physics*,

*5*(12), 1756-1776.

**Reduction of the N-particle variational problem.** / Garrod, Claude; Percus, Jerome.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 5, no. 12, pp. 1756-1776.

}

TY - JOUR

T1 - Reduction of the N-particle variational problem

AU - Garrod, Claude

AU - Percus, Jerome

PY - 1964

Y1 - 1964

N2 - A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′) = (ψ |α2+α1+α 1′α2′). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N - 1) and trγ = N] then the associated operator: G(1, 2 | 1′, 2′) = δ(1 - 1′)γ(2 | 2′) + σ Γ(1′, 2 | 1, 2′) - γ(2 | 1)7(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or - 1 for bosons or fermions respectively.].

AB - A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′) = (ψ |α2+α1+α 1′α2′). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N - 1) and trγ = N] then the associated operator: G(1, 2 | 1′, 2′) = δ(1 - 1′)γ(2 | 2′) + σ Γ(1′, 2 | 1, 2′) - γ(2 | 1)7(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or - 1 for bosons or fermions respectively.].

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M3 - Article

VL - 5

SP - 1756

EP - 1776

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

ER -