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Spin adaptation of the cumulant expansions of reduced density matrices

Julia Liebert, Christian Schilling, David A. Mazziotti

TL;DR

This work addresses the challenge of enforcing correct spin symmetry in reduced-density-matrix (RDM) based electronic structure methods. It develops a systematic spin-adapted cumulant formalism for $p$-RDMs with explicit constructions up to $p=4$, deriving linear trace constraints on the spin-adapted 2RDM and 3RDM cumulants and expressing the $S^2$ variance as a linear functional of the 4RDM to enforce $S$-representability. The approach extends to spin-orbit coupled systems via total angular momentum adaptation, using irreducible tensor operators and the Wigner-Eckart framework to identify nonzero blocks. It provides a scalable foundation for density-based theories (DFT, RDMFT) and 2RDM-based methods, and generalizes to adaptation to other Lie groups, enabling systematic symmetry-resolved electronic structure calculations.

Abstract

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 S_z and S^2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation -- referred to as complete S-representability -- can be enforced by constraining the variances of S_z and S^2, 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 (DFT), one-particle RDM functional theories (RDMFT), 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.

Spin adaptation of the cumulant expansions of reduced density matrices

TL;DR

This work addresses the challenge of enforcing correct spin symmetry in reduced-density-matrix (RDM) based electronic structure methods. It develops a systematic spin-adapted cumulant formalism for -RDMs with explicit constructions up to , deriving linear trace constraints on the spin-adapted 2RDM and 3RDM cumulants and expressing the variance as a linear functional of the 4RDM to enforce -representability. The approach extends to spin-orbit coupled systems via total angular momentum adaptation, using irreducible tensor operators and the Wigner-Eckart framework to identify nonzero blocks. It provides a scalable foundation for density-based theories (DFT, RDMFT) and 2RDM-based methods, and generalizes to adaptation to other Lie groups, enabling systematic symmetry-resolved electronic structure calculations.

Abstract

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 S_z and S^2 symmetries in quantum systems, providing a foundation for spin-resolved electronic structure methods. We show that complete spin adaptation -- referred to as complete S-representability -- can be enforced by constraining the variances of S_z and S^2, 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 (DFT), one-particle RDM functional theories (RDMFT), 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.

Paper Structure

This paper contains 11 sections, 35 equations.

Table of Contents

  1. Introduction
  2. Theory
  3. Conservation of magnetization
  4. Inclusion of total $S$ and spin-adapted 2RDM cumulants
  5. Spin adaptation on three-particle level
  6. Ensuring $S$-representability on the four-particle level
  7. Symmetry adaptation in the presence of spin-orbit coupling
  8. Conclusions and outlook
  9. Spin-adapted 2RDM matrix elements
  10. Spin-adapted 3RDM matrix elements
  11. Spin-adapted tensor operators for the 4RDM