# Efficient Implementation of the Spin-Free Renormalized Internally-Contracted Multireference Coupled Cluster Theory

## TL;DR

This paper presents an efficient implementation of a multireference coupled cluster method in the ORCA program, enabling accurate quantum chemistry calculations on large systems.

## Contribution

An efficient, parallelized implementation of RIC-MRCCSD using a spin-free formulation and code generation tools.

## Key findings

- The RIC-MRCCSD implementation achieves substantial efficiency gains and runs in parallel with speedups on multiple cores.
- The method scales well to large systems, demonstrated by computing the ground state of a vitamin B12 model with a CAS(12,12) active space.
- RIC-MRCCSD is more efficient than alternative methods like IC-MRCCSD, which require higher-order density matrices.

## Abstract

In this paper, an efficient implementation of the renormalized
internally contracted multireference coupled cluster with singles
and doubles (RIC-MRCCSD) into the ORCA quantum chemistry program suite
is reported. To this end, Evangelista’s Wick&d equation generator was combined with ORCA’s native AGE code generator in order to implement the many-body
residuals required for the RIC-MRCCSD method. Substantial efficiency
gains are realized by deriving a spin-free formulation instead of
the previously reported spin–orbital version developed by some
of us. Since AGE produces parallelized code,
the resulting implementation can directly be run in parallel with
substantial speedups when executed on multiple cores. In terms of
runtime, the cost of RIC-MRCCSD is shown to be between single-reference
RHF-CCSD and UHF-CCSD, even when active space spaces as large as CAS­(14,14)
are considered. This achievement is largely due to the fact that no
reduced density matrices or cumulants higher than three-body enter
the formalism. The scalability of the method to large systems is furthermore
demonstrated by computing the ground-state of a vitamin B12 model comprised of an active space of CAS­(12,12) and 809 orbitals.
In terms of accuracy, RIC-MRCCSD is carefully compared to second-
and approximate fourth-order n-electron valence state
perturbation theories (NEVPT2, NEVPT4­(SD)), to the multireference
zeroth-order coupled-electron pair approximation (CEPA(0)), as well
as to the IC-MRCCSD from Köhn. In contrast to RIC-MRCCSD, the
IC-MRCCSD equations are entirely derived by AGE using the conventional projection-based approach, which, however,
leads to much higher algorithmic complexity than the former as well
as the necessity to calculate up to the five-body RDMs. Remaining
challenges such as the variation of the results with the flow, a free
parameter that enters the RIC-MRCCSD theory, are discussed.