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Journal of Mathematical Physics

Reduction of the N -Particle Variational Problem

Claude Garrod, Jerome K. Percus

Published Dec 1, 1964• 477 citations

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 |a2+a1+a1′a2′| Ψ). 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)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]

DOI: 10.1063/1.1704098

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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 |a2+a1+a1′a2′| Ψ). 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)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]

A many-body theory for interacting bosons is developed within the framework of minimizing the ground-state energy with respect to the two-particle reduced-density matrix (2-RDM) subject to

N

-representability conditions. The

N

-representability conditions, which ensure that the 2-RDM may be derived from an

N

-particle wave function, are imposed through a hierarchy of positivity conditions where the

p

-positivity conditions restrict the metric matrices for

p∕2

-body operators to be positive semidefinite. Using two-positivity, we minimize the ground-state energies of

5--10 000

harmonically interacting bosons in a harmonic external potential. The energies and 2-RDMs obtained are in agreement with the exact solution except for round-off errors, which implies that for this class of boson interactions two-positivity conditions alone yield exact results for any interaction strength. The ground-state energies obtained at strong interactions are more accurate than many-body perturbative techniques by many orders of magnitude.

Abstract This work deals with the variational determination of two-particle reduced density matrices corresponding to eigenstates of N -boson systems. Stringent N -representability conditions have been imposed in the variational treatment, in which the resulting optimization problem is addressed within a standard semidefinite programming framework. A unified variational treatment, based on the dispersion operator technique, is proposed. This treatment allows to determine ground- and excited-state energies and their corresponding reduced density matrices, which have been evaluated in systems formulated by means of simple Hamiltonians. We report results from different N -representability levels of treatment. These results are compared with those arising from the full configuration interaction procedure, what allows to point out the relevance of the imposed N -representability conditions. We also highlight the differences found in applying this methodology to fermion and boson systems.

In the spirit of the generalized one-particle density matrix for fermions, we introduce generalized one- and two-particle density matrices to state representability conditions up to second order for boson systems without assuming particle number-conservation. Furthermore, we show for both particle species that, for a semibounded Hamiltonian, the in?mum of the variation of the energy functional w.r.t quasifree states coincides with the one of a variation over pure quasifree states. Moreover, it is proven for fermions that only pure quasifree states have a generalized 1-pdm that is a projection, and a similar statement for bosons.

The problem of correlations is considered in many-fermion systems, although with some modifications the methods are more general. The many-body problem is formulated in terms of reduced one- and two-particle density matrices, and the equations for these are treated by variational and iterative methods. The formalism is applied to a calculation of the energy per particle for a system interacting with Coulomb, screened Coulomb, and singular potentials. Comparisons with the ideas of Brueckner and others are made, and difficulties are discussed.