Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Quantum Chemistry

Traces of spin-adapted reduced density matrices

E. P�rez-Romero, Luis M. Tel, C. Valdemoro

Published Jan 1, 1997• 20 citations
Publisher

Abstract

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.

DOI
10.1002/(sici)1097-461x(1997)61:1<55::aid-qua6>3.0.co;2-3

Related papers

Traces of spin‐adapted reduced density matrices

Encarnación Pérez-Romero, Luis M. Tel, C. Valdemoro

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.

International Journal of Quantum Chemistry•Jan 1, 1997•5 citations

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

S.Kh. Samvelyan

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

International Journal of Quantum Chemistry•Jan 1, 1997•32 citations

Traces ofpth-order reduced density matrices: symmetric group approach

Alicia Torre, Luis Laín

This paper describes a general procedure for calculating the traces of reduced density matrices of any order p of an N -electron system, in the basis of the irreducible representations of the symmetric group , in a spin-free formalism. The numerical values of these traces are determined by useful formulae. This approach provides a suitable study of the N - and S -representability of the reduced density matrices.

Journal of Physics A Mathematical and General•Jul 10, 1998•1 citation

Traces of the reduced density operators revisited: closed-form formulae

Josep Planelles, Jacek Karwowski

Closed-form expressions for traces of the qth-order reduced density operators, q-RDOs, and symmetry-adapted q-RDOs calculated in an N-electron and spin-adapted subspace of a K-orbital Fock space are derived. The asymptotic form of the expression for traces of the q-RDOs, corresponding to and limits, is discussed in detail. Several identities fulfilled by some quantities related to the traces are derived.

Journal of Physics A Mathematical and General•May 7, 1997•2 citations

Spin adaptation of the cumulant expansions of reduced density matrices

Julia Liebert, Christian Schilling, David A. Mazziotti

We develop a systematic framework for the spin adaptation of the cumulants of p-particle reduced density matrices (RDMs), with explicit constructions for p = 1 to 3. These spin-adapted cumulants enable rigorous treatment of both Ŝz and Ŝ2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation-referred to as completeS-representability-can be enforced by constraining the variances of Ŝz and Ŝ2, which require the 2-RDM and 4-RDM, respectively. Importantly, the cumulants of RDMs scale linearly with system size-size-extensive-making them a natural object for incorporating spin symmetries in scalable electronic structure theories. The developed formalism is applicable to density-based methods, one-particle RDM functional theories, and two-particle RDM methods. We further extend the approach to spin-orbit-coupled systems via total angular momentum adaptation. Beyond spin, the framework enables the adaptation of RDM theories to additional symmetries through the construction of suitable irreducible tensor operators.

arXiv:2505.16989•The Journal of Chemical Physics•Jul 23, 2025•0 citations

Approximatingq-order reduced density matrices in terms of the lower-order ones. I. General relations

Francisco Colmenero, C. Pérez del Valle, Carmela Valdemoro

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.

Physical Review A•Feb 1, 1993•147 citations
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.
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
We develop a systematic framework for the spin adaptation of the cumulants of p-particle reduced density matrices (RDMs), with explicit constructions for p = 1 to 3. These spin-adapted cumulants enable rigorous treatment of both Ŝz and Ŝ2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation-referred to as completeS-representability-can be enforced by constraining the variances of Ŝz and Ŝ2, which require the 2-RDM and 4-RDM, respectively. Importantly, the cumulants of RDMs scale linearly with system size-size-extensive-making them a natural object for incorporating spin symmetries in scalable electronic structure theories. The developed formalism is applicable to density-based methods, one-particle RDM functional theories, and two-particle RDM methods. We further extend the approach to spin-orbit-coupled systems via total angular momentum adaptation. Beyond spin, the framework enables the adaptation of RDM theories to additional symmetries through the construction of suitable irreducible tensor operators.
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.