Algebraic Geometry for Spin-Adapted Coupled Cluster Theory

TL;DR

This paper introduces an algebraic-geometric framework for spin-adapted coupled-cluster theory, significantly reducing computational complexity by describing the spin singlet space with algebraic varieties and rings, enabling efficient solution of quantum chemistry problems.

Contribution

It provides a novel algebraic-geometric description of spin-adapted CC theory, reducing problem dimension and degree, and enabling practical computation of solutions for molecular systems.

Findings

Reduced CC degree by orders of magnitude

Demonstrated asymptotic scaling improvements

Successfully computed full solution landscapes for molecules

Abstract

We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride.