A purification procedure that simultaneously corrects the
N- and
S-representability main defects of a second-order reduced density matrix (2RDM), second-order hole reduced density matrix (2HRDM), and second-order
G matrix is presented here. In this purifying procedure, the generalized unitarily invariant second-order matrix decomposition [D. R. Alcoba, Int. J. Quantum Chem. 97, 776 (2004)] for the 2RDM and the 2HRDM as well as for the
G matrix is combined with the
S-representability conditions. In particular, here we will focus our attention on the RDMs corresponding to doublet states. We will thus explicitly give the
S-representability conditions that a two-body correlation matrix has to satisfy when an
N-electron system is in a doublet spin-state in the spin-component of maximum projection. Furthermore, as a consequence of the
G-matrix spin properties (which directly affect the
S-representability of the 2RDM), we show that a different contracting form for the 2RDM is possible. The numerical results presented in this work confirm the efficiency of our purifying procedure.
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 traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.
The density matrix of an arbitrary pure state of a system consisting of two spin‐1/2 particles is derived from the Pauli spin angular momentum operators. Mixed singlet and triplet states are then formed from linear combinations of pure states and their corresponding density matrices constructed. Singlet and triplet states are exemplified by the spin isomers parahydrogen and orthohydrogen, respectively. Partial mixing is illustrated with the example of bilinear spin–spin coupling. Various properties of the density matrices of pure and mixed states are discussed, including idempotence, factoring, and spin correlation.