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Journal of Mathematical Physics

Reduced Density Matrices of Energy Eigenstates

Mitja Rosina, Jerome K. Percus, Louis J. Kijewski, Claude Garrod

Published Sep 1, 1969• 18 citations
Publisher

Abstract

The following question is considered: What special properties are possessed by those reduced density matrices which come from energy eigenstates? Using the fact that 〈φ[H, A]|φ〉 = 0, where A is any operator and |φ〉 an energy eigenstate, it is shown that the elements of the two-particle density matrix are severely restricted by homogeneous linear relations. Their full content is expressed in terms of an auxiliary one-particle density which possesses additional positivity properties in the ground state.

DOI
10.1063/1.1665024

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The following question is considered: What special properties are possessed by those reduced density matrices which come from energy eigenstates? Using the fact that 〈φ[H, A]|φ〉 = 0, where A is any operator and |φ〉 an energy eigenstate, it is shown that the elements of the two-particle density matrix are severely restricted by homogeneous linear relations. Their full content is expressed in terms of an auxiliary one-particle density which possesses additional positivity properties in the ground state.
Constraints placed on a reduced density matrix or density by the requirement that it correspond to an eigenstate or ensemble of eigenstates of some operator are investigated. Linear conditions, including expectation value, commutator, hypervirial, and hierarchy equation conditions, may lead to linear constraints on lower-order reduced density matrices and densities or leave them unconstrained. A zero-dispersion condition for a ppp-electron operator defines a parabolic trough on which reduced density matrices of order 2p2p2p must lie, but usually does not restrict lower-order reduced density matrices or the density. Some properties of symmetrized and antisymmetrized matrix products, related to their reduction behavior, are also investigated.
Despite the fact that idempotent one-particle reduced density matrices are pervasive in quantum chemistry, the understanding of a general energy functional of such idempotent density matrices for the ground state energy has been lacking. By a constrained search, we show the structure of the general functional, illuminating the contributions from various terms. For the examples of the "best idempotent density matrix" and Kohn-Sham idempotent density matrices, we contrast the functional forms and suggest how the best idempotent density matrix approach may be a good starting point for further development.
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.]
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.