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International Journal of Quantum Chemistry

Contraction operator over electronic fock space. II. Weak representability problem

A. I. Panin

Published May 1, 1989• 1 citation
Publisher

Abstract

Abstract Two approaches in treating the weak representability problem for density operators are studied. In the first approach this problem is reduced to the minimization of a certain distance function. The second approach is based on inductive generation of linear inequalities for diagonal elements of the second‐order density operators. Generalization of Kuhn–Yoseloff's results is also discussed.

DOI
10.1002/qua.560350505

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Abstract Two approaches in treating the weak representability problem for density operators are studied. In the first approach this problem is reduced to the minimization of a certain distance function. The second approach is based on inductive generation of linear inequalities for diagonal elements of the second‐order density operators. Generalization of Kuhn–Yoseloff's results is also discussed.
A a procedure is proposed by which, in the course of an iterative solution of the second-order contracted Schr\"odinger equation, The N representability of the second-order reduced density matrix can be tested with increasing stringency. This procedure was suggested by an extended study of the G conditions and from the contraction into the two-electron space of an N-fermion relation, expressing in matrix form the antisymmetry and normalization properties of the N-electron wave function. Several relations are reported.
The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x1, x2) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x1, x2) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θn(x1, x2) converging at n → ∞ to ρ(x1, x2)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x1, x2). The results obtained may open new possibilities for using ρ(x1, x2) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 127–142, 1997
The NNN-representability problem for the first- and second-order density matrices is considered when NNN is an odd integer. Attention is restricted to those density matrices derivable from functions whose natural spin orbitals have certain paired properties. Neither pairing of spatial orbitals nor special spin properties are required in general, but they are discussed as special cases. New sufficient conditions for NNN-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the NNN-representability problem for the second-order density matrix can be solved are considered.
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.