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International Journal of Quantum Chemistry

Permutational symmetry of reduced density matrices

Eugene S. Kryachko

Published Apr 1, 1977• 1 citation
Publisher

Abstract

Abstract The problems of permutational symmetry of the density matrices in reduction are studied. Some necessary and sufficient conditions for N, [λ1], [λ2]‐derivability problem are given.
DOI
10.1002/qua.560110405

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The usual restriction on the permutational symmetry of density matrices is removed. An Nth-order density matrix of general permutational symmetry is reduced and the symmetry of the resulting pth-order density matrix is found. This gives rise to necessary conditions for the N-representability of spin-free density matrices. We indicate the importance of these results for symmetry under the groups GL(n), U(n), and SU(n). We generalize the question of N-representability to the tensor algebra, where contraction of tensors plays the role of reduction of density matrices. Symmetry properties of the tensors are studied and density matrices are considered as special cases.
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.
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.