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International Journal of Quantum Chemistry

N-representability of diagonal elements of second-order reduced density matrices

S.Kh. Samvelyan

Published Jan 1, 1997• 32 citations
Publisher

Abstract

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

DOI
10.1002/(sici)1097-461x(1997)65:2<127::aid-qua4>3.0.co;2-y

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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 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.
Abstract The ensemble N‐representability problem for the k‐th order reduced density matrix (k‐RDM) as well as the problem of reconstruction of the N‐particle system density matrices (N‐DM) from a given k‐RDM are studied. The spatial parts of the k‐RDM expansion in terms of spin tensorial operators Θ are represented using particular values (at specially chosen ) of the Radon transform of the N‐DM spatial parts (or their sums) 𝒟Nλ(x′ | x″) (here, is a d‐plane in the n‐space ℝn of x = (x′, x″)), with n = 6N, d = 3 (N − k), x′ ≡ (r′1, ⃛, r′N), x′ ≡ (r1″, ⃛, r″N ()). In this way, the problem is reduced to investigation of the properties of the functions . For a normalizable N – DM, it is proved that are bounded functions. The properties of implied by the N‐DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N‐DM corresponding to a given k‐RDM is proposed. © 1995 John Wiley & Sons, Inc.
A purification procedure that simultaneously corrects the NNN- and SSS-representability main defects of a second-order reduced density matrix (2RDM), second-order hole reduced density matrix (2HRDM), and second-order G\mathcal{G}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\mathcal{G}G matrix is combined with the SSS-representability conditions. In particular, here we will focus our attention on the RDMs corresponding to doublet states. We will thus explicitly give the SSS-representability conditions that a two-body correlation matrix has to satisfy when an NNN-electron system is in a doublet spin-state in the spin-component of maximum projection. Furthermore, as a consequence of the G\mathcal{G}G-matrix spin properties (which directly affect the SSS-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 N-representability problem for reduced density matrices remains a fundamental challenge in electronic structure theory. Following our previous work that employs a unitary-evolution algorithm based on an adaptive derivative-assembled pseudo-Trotter variational quantum algorithm to probe pure-state N-representability of reduced density matrices [J. Chem. Theory Comput. 2024, 20, 9968], in this work we propose a practical framework for determining the ensemble N-representability of a p-body matrix. This is accomplished using a purification strategy consisting of embedding an ensemble state into a pure state defined on an extended Hilbert space, such that the reduced density matrices of the purified state reproduce those of the original ensemble. By iteratively applying variational unitaries to an initial purified state, the proposed algorithm minimizes the Hilbert-Schmidt distance between its p-body reduced density matrix and a specified target p-body matrix, which serves as a measure of the N-representability of the target. This methodology facilitates both error correction of defective ensemble reduced density matrices, and quantum-state reconstruction on a quantum computer, offering a route for density-matrix refinement. We validate the algorithm with numerical simulations on systems of two, three, and four electrons in both, simple models as well as molecular systems at finite temperature, demonstrating its robustness.