Generation of Spin-Adapted and Spin-Complete Substitution Operators for (High-Spin) Open-Shell Coupled Cluster of Arbitrary Order

TL;DR

This paper introduces a method to generate spin-adapted substitution operators for high-spin open-shell coupled cluster calculations, ensuring spin completeness and independence, demonstrated on high-spin states of the Boron atom.

Contribution

The paper presents a novel, rigorous procedure for generating spin-adapted, spin-complete substitution operators applicable to arbitrary high-spin references in coupled cluster theory.

Findings

Operators are linearly independent and spin complete up to 10-fold substitutions.

Implementation successfully applied to high-spin Boron atom states.

Significant errors found in spin-incomplete coupled cluster methods.

Abstract

A rigorous generation of spin-adapted (spin-free) substitution operators for high spin ($S=S_z$) references of arbitrary substitution order and spin quantum number $S$ is presented. The generated operators lead to linearly independent but non-orthogonal CSFs when applied to the reference and span the complete spin space. To incorporate spin completeness, spectating substitutions (as e.g. $\hat{E}_{iv}^{va}$) are introduced. The presented procedure utilizes L\"owdin's projection operator method of spin eigenfunction generation to ensure spin completeness. The generated operators are explicitly checked for (i) their linear independence and (ii) their spin completeness for up to 10-fold substitutions and up to a multiplicity of $2S+1 = 11$. A proof of concept implementation utilizing the generated operators in a coupled cluster (CC) calculation was successfully applied to the high spin states of the Boron atom. The results show pure spin states as well as small effects on the correlation energy compared to spinorbital CC. A comparison to spin-adapted but spin-incomplete CC shows a significant spin incompleteness error.