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International Journal of Quantum Chemistry

N‐Representability conditions generated by a one‐particle grassmann factor of an antisymmetric function

Hubert Grudziński

Published Jun 1, 1985• 5 citations
Publisher

Abstract

Abstract N‐representability conditions for a two‐particle density operator implied by positive‐semidefiniteness of the projection operator PN+1(ϕ1 Λ ΨN) are derived and discussed. The operator PN+1(ϕ1 Λ ΨN) projects onto an (N + 1)‐particle antisymmetric function ϕ1 Λ ΨN, the Grassmann product of a one‐particle factor ϕ1 and an N‐particle factor ΨN. The polar subcone 𝒫2N(g, q) to the set of N‐representable two‐particle density operators 𝒫2N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one‐particle Hilbert space. The facial structure of the convex set 𝒫2N exposed by elements of 𝒫2N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N‐representability is found. It consists of the decomposition of the N‐particle antisymmetric identity operator onto the mutually orthogonal projection operators.

DOI
10.1002/qua.560270608

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Abstract N‐representability conditions for a two‐particle density operator implied by positive‐semidefiniteness of the projection operator PN+1(ϕ1 Λ ΨN) are derived and discussed. The operator PN+1(ϕ1 Λ ΨN) projects onto an (N + 1)‐particle antisymmetric function ϕ1 Λ ΨN, the Grassmann product of a one‐particle factor ϕ1 and an N‐particle factor ΨN. The polar subcone 𝒫2N(g, q) to the set of N‐representable two‐particle density operators 𝒫2N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one‐particle Hilbert space. The facial structure of the convex set 𝒫2N exposed by elements of 𝒫2N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N‐representability is found. It consists of the decomposition of the N‐particle antisymmetric identity operator onto the mutually orthogonal projection operators.
In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8). Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.
Representability conditions on the one- and two-particle density matrix for fermion systems are formulated by means of Grassmann integrals. A positivity condition for a certain kind of Grassmann integral is established which, in turn, induces the well-known G-, P- and Q-Conditions of quantum chemistry by an appropriate choice of the integrand. Similarly, the T1- and T2-Conditions are derived. Furthermore, quasifree Grassmann states are introduced and, for every operator γ ̃ ∈ H⊕H with 0 ≤ γ ̃ ≤ 1, the existence of a unique quasifree Grassmann state whose one-particle density matrix is γ ̃ is shown.
Abstract We examine and compare ways of dividing into subspaces the space whose elements are density matrices or other operators for the class of model problems defined by a finite one‐particle basis set. One method of decomposition makes the significance of the subspaces apparent. We show that this decomposition is also complete, in the group‐theoretic sense, for the group of unitary transformations of the set of one‐electron basis functions. The irreducible subspaces are labeled by particle number and by an additional integer we call the reduction index. For spaces of particle‐number‐conserving operators, all subspaces with the same reduction index are isomorphic, and an analogous isomorphism exists for non‐particle‐number‐conserving cases. The general linear group also plays a key role, and we introduce the term “canonical superoperators” to characterize those superoperators which commute with this group. When an appropriate basis set is chosen for the matrix spaces, the supermatrices corresponding to these superoperators have a particularly simple form: a block structure with the only nonzero blocks being multiples of unit matrices. The superoperators of interest can be constructed in terms of two operators, , and these two have been expressed simply in terms of creation and annihilation operators. When only real orthogonal transformations of the basis are considered, a further decomposition is possible. We have introduced superoperators associated with this decomposition.
Representability conditions on the one-and two-particle density matrix for fermion systems are formulated by means of Grassmann integrals.A positivity condition for a certain kind of Grassmann integral is established which by an appropriate choice of the integrand, in turn, induces the well-known G-, P-and Q-Conditions of quantum chemistry.Similarly, the T 1 -and T 2 -Conditions are derived.Furthermore, quasifree Grassmann states are introduced and, for every operator γ ∈ H ⊕ H with 0 ≤ γ ≤ 1, the existence of a unique quasifree Grassmann state whose one-particle density matrix is γ is shown.. More specifically, ifwhere
This paper considers the determination of N-representability (for diagonal elements) of p-density matrices restricted to certain finite-dimensional subspaces of l2 of the configuration space of N identical antisymmetric particles. In particular, an arbitrary set of N + p spin orbitals is selected and one considers the (N+pp)-dimensional subspace generated by all possible Slater determinants of the spin orbitals being considered. Applying a combinatorial approach to the problem, a necessary and sufficient set of conditions is determined; previous work has dealt only with necessary conditions, except in the 1-matrix case. The paper concludes by presenting a probabilistic interpretation of these conditions which seems of particular interest for the 2-matrix case. The conditions presented here in combination with the Pauli principle give a probabilistic view of the expected occupation of p-tuples of spin orbitals in terms of the expected occupations of lower-order-tuples of spin orbitals.
We show how ensembles or mixed states can be described in terms of pure states that for Fermions lead to wave functions that are antisymmetric with respect to interchange of particle coordinates (and spin). The pure states are constructed in an augmented Hilbert space spanned by products of ensemble states projected onto mutually nonoverlapping coordinates that prevents the appearance of interference terms under antisymmetrization. As a demonstration of this new formalism, and under the assumptions of a positive interparticle interaction and a corresponding energy that is extensive in the number of particle pairs (pair extensive), we prove the convexity relation, ${E}_{v}[N\ensuremath{-}1]+{E}_{v}[N+1]\ensuremath{\geqslant}2{E}_{v}[N]$, where Ev[N]{E}_{v}[N]Ev​[N] denotes the total ground-state energy of NNN Fermions (electrons) under an external potential v(r)v(\mathbf{r})v(r).