On the Coupled Cluster Doubles Truncation Variety of Four Electrons

TL;DR

This paper investigates the algebraic structure of a specific coupled cluster doubles truncation variety for four electrons, revealing its geometric properties and implications for quantum chemistry calculations.

Contribution

It introduces a new algebro-geometric analysis of the CCD truncation variety for four electrons, including its invariants, Pfaffian structure, and tensor factorizations, with applications to molecular systems.

Findings

CCD truncation variety is a complete intersection for up to 12 orbitals.

Identifies a Pfaffian structure governing quadratic relations.

Connects structural results to molecular bond formation calculations.

Abstract

We extend recent algebro-geometric results for coupled cluster theory of quantum many-body systems to the truncation varieties arising from the doubles approximation (CCD), focusing on the first genuinely nonlinear doubles regime of four electrons. Since this doubles truncation variety does not coincide with previously studied varieties, we initiate a systematic investigation of its basic algebro-geometric invariants. Combining theoretical and numerical results, we show that for $4$ electrons on $n\leq 12$ orbitals, the CCD truncation variety is a complete intersection of degree $2^{\binom{n-4}{4}}$. Using representation-theoretic arguments, we uncover a Pfaffian structure governing the quadratic relations that define the truncation variety for any $n$, and show that an exact tensor product factorization holds in a distinguished limit of disconnected doubles. We connect these structural results to the computation of the beryllium insertion into molecular hydrogen ({Be$\cdots$H$_2$ $\to$ H--Be--H}), a small but challenging bond formation process where multiconfigurational effects become pronounced.