Spin adaptation of the cumulant expansions of reduced density matrices

TL;DR

This paper introduces a systematic framework for spin adaptation of cumulants of reduced density matrices, enabling symmetry-respecting electronic structure calculations and extending to spin-orbit coupling and other symmetries.

Contribution

It provides explicit constructions for spin-adapted cumulants of RDMs, enforces complete S-representability, and extends the formalism to spin-orbit and other symmetries in scalable electronic structure methods.

Findings

Cumulants scale linearly with system size, ensuring size-extensiveness.

The framework applies to DFT, RDMFT, and two-particle RDM methods.

Spin and angular momentum symmetries can be incorporated systematically.

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.