Generation of spin-adapted and spin-complete substitution operators for (high spin) open-shell coupled cluster of arbitrary order

Abstract

A rigorous generation of spin-adapted (spin-free) substitution operators for high spin (S = Sz) references of an arbitrary substitution order and spin quantum number S is presented. The generated operators lead to linearly independent but non-orthogonal configuration state functions (CSFs) when applied to the reference and span the complete spin space. To incorporate spin completeness, spectating substitutions (e.g., Êivva) are introduced. The presented procedure utilizes Löwdin’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 tenfold 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 and small effects on the correlation energy compared to spin orbital CC. A comparison to spin-adapted but spin-incomplete CC shows a significant spin-incompleteness error.

A correctly scaling rigorously spin-adapted and spin-complete open-shell CCSD implementation for arbitrary high-spin states

“…In this section, theoretical intricacies for the presented CCSD are given. Subsection II A features a short recap of the non-orthogonal, spin-adapted and spin-complete CC framework -introduced in our previous work [38] -while subsection II B introduces the spin-free CCSD equations considered in this work.…”

“…As already introduced in [38,39], we denote doubly occupied spatial orbitals contained in O by i, j, .…”

“…Already in 1991, Janssen and Schaefer III presented an automated equation derivation engine [27], which was also applied to a non-orthogonal spin-adapted CCSD for high-spin open shell states. Although their approach allows for spectating substitutions, terms required for spin completeness in triplet and higher spin states are missing from their cluster operators [38]. Following the early suggestion of Lindgren [57] to approximate the wave operator e T by its normal-ordered form {e T }, several CCSD [28,29] and MRCCSD [30] implementations emerged.…”

“…In this work, we present a newly developed CCSD implementation capable of treating arbitrary high-spin states. The presented approach utilizes our previously developed Löwdin-type non-orthogonal spatial substitution operators [38] to guarantee (i) spin adaption and (ii) spin completeness of the CC wave function. While (i) is automatically fulfilled by the usage of spin-free substitution operators, (ii) requires special care as to the selection of (usually linear-dependent) substitution operators.…”

“…While (i) is automatically fulfilled by the usage of spin-free substitution operators, (ii) requires special care as to the selection of (usually linear-dependent) substitution operators. In our previous work [38], this is ensured by Löwdin's projection operator method. The spin-adapted and spin-complete (SASC) CCSD implementation, presented here, utilizes a single set of CC equations.…”