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Physical Review A

Controlling the N - and S -representability of the second-order reduced density matrix: The doublet-state case

Diego R. Alcoba, Carmela Valdemoro, Luis M. Tel, Encarnación Pérez-Romero

Published Apr 17, 2008• 13 citations
Publisher

Abstract

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 $\mathcal{G}$ 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 $\mathcal{G}$ 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 $\mathcal{G}$-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.

DOI
10.1103/physreva.77.042508

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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.
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The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the T2{T}_{2}T2​ condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the T2{T}_{2}T2​ condition that significantly decreases the size of the largest block in the T2{T}_{2}T2​ matrix. The T2{T}_{2}T2​ condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the T2{T}_{2}T2​ condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2{\mathrm{N}}_{2}N2​. Results are compared with the 2RDM method without the T2{T}_{2}T2​ condition as well as several wave-function methods.
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.
The structure of second-order reduced density matrix (2RDM) of N-electron systems is studied. The wave function corresponding to the 2RDM is assumed to be linear combination of Slater-deteminants (a FCI-type wave function) constructed over a finite dimensional orthonormal one-particle basis set. Our considerations are independent from the size and the quality of the one-particle basis. Using the definition of second-order reduced density matrix, and properties of Slater-determinants a set of inequalities are derived for the 2RDM. Inequalities can be formulated without the wave function, and without higher order density matrices, involving only the density matrix, and can be considered as necessary N-representability conditions.