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International Journal of Quantum Chemistry

The symmetry of reduced density matrices. I. General algebraic approach and permutational symmetry

Douglas J. Klein, Robert W. Kramling

Published Jun 18, 2009• 2 citations
Publisher

Abstract

The symmetry properties of p-particle operators derived by contraction from N-particle operators which transform on the left and right as different permutational symmetries are investigated. Theorems identifying these symmetry properties are obtained in terms of conditions for N,α,β,-derivability. The relation of the p-particle operators to reduced density matrices and the implications of the theorems as regards N-representability are discussed.

DOI
10.1002/qua.560040730

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The symmetry properties of p-particle operators derived by contraction from N-particle operators which transform on the left and right as different permutational symmetries are investigated. Theorems identifying these symmetry properties are obtained in terms of conditions for N,α,β,-derivability. The relation of the p-particle operators to reduced density matrices and the implications of the theorems as regards N-representability are discussed.
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 commutation-anticommutation relations of q-electron operators imply a set of N representability conditions [A. J. Coleman, Rev. Mod. Phys. 31, 668 (1963)] for the corresponding q-order reduced density matrices (q-RDM) [C. Valdemoro, An. Fis. 79, 95 (1983); in Structure, Interaction and Reactivity, edited by S. Fraga (Elsevier, Amsterdam, 1992)]. From these conditions, a general and closed-form relation is obtained here. In this equation the part involving RDM's has the same structure as that involving hole reduced density matrices. This relation is the basis of a method for approximating a q-RDM in terms of the r-RDM's [C. Valdemoro, Phys. Rev. A 45, 4462 (1992)] with rq. The derivation of this relation can be simplified significantly by employing a graph method which is described here. These graphs are in a one-to-one correspondence with the elements of the symmetric group of permutations.
The density operator (density matrix) of a quantum mechenical system can be decomposed into operators which transform as irreducible representations of the symmetry group in coordinate and spin space. Each of these components has a physical meaning connected with the expectation values of certain operators. The reduced density matrices can be decomposed in a completely analogous way. The symmetry properties of the total wave function give rise to degeneracies of the eigenvalues of the reduced density matrices. These degeneracies can be removed by requiring that the natural spin orbitals (NSO, defined as the eigenfunctions of the first-order density matrix), as well as the natural spin geminals (NSG, the eigenfunctions of the second-order density matrix) and their spinless counterparts transform as irreducible representations of the symmetry group and are eigenfunctions of S/sup 2/ and S/sub z/. In many cases this requirement is compatible with the original definition of the NSO, the NSG, etc., e.g., when there is no spatial degeneracy of the total wave function and when the Z- component of the total spin vanishes. When these conditions are not fulfilled an alternative definition of the NSO and the NSG is proposed. (auth)
Spin-free transition density matrices are derived from spin-free kets which are symmetry-adapted to the symmetric group and its algebra. The Dirac identity establishes that these spin-free density matrices are identical to those obtained by integrating the spin from the full-spin density martices. Derivations are first given for arbitrary primitive kets which may be geminals of higher polymals, after which we consider products of orbitals, either orthonormal or nonorthonormal. Correlation in the spin-free space is discussed and we show the influence of permutational symmetry on the probability of coincidence of pairs. A special case of this correlation is the well-known Fermi hole.