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

TL;DR

This work delivers a spin-free formulation of renormalized internally-contracted multireference CC with singles and doubles (RIC-MRCCSD) and its efficient implementation in ORCA by interfacing Wick&d with AGE for automated residual derivation and spin adaptation. By relying on up to three-body cumulants and a BCH-based two-fold truncation, the method achieves favorable scalability and numerical stability, enabling CAS sizes as large as CAS$(14,14)$ and applications to large systems like a vitamin B12 model with 809 basis functions. The spin-adapted framework uses generalized normal-ordering and singlet-constraining relations to reduce the spin complexity of residuals, while excitation-class organization and translation to AGE yield high-performance, MPI-parallel code. Benchmark results show that RIC-MRCCSD is size-consistent and offers competitive efficiency relative to single-reference CC methods, with accuracy on par with other MR approaches for many systems, although NEVPT2 remains faster for large active spaces; a perturbative triples extension (RIC-MRCCSD[T]) is proposed for further accuracy gains. Overall, this work establishes a viable path for applying advanced MRCC methods to larger, more realistic systems within a mainstream quantum chemistry package, expanding the practical utility of multireference dynamic correlation methods.

Abstract

In this paper, an efficient implementation of the renormalized internally-contracted multreference 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 (RDM) 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 Kohn. 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.

Figures (10)

• Figure 1: Breakdown of the amplitude excitation classes present in the RIC-MRCCSD scheme. Each column corresponds to a specific excitation class with each arrow representing a single-electron excitation. The name of the excitation class is found under each excitation with the first (two) and last (two) letters denoting the hole and particle spaces for the singles (doubles) excitations, respectively.
• Figure 2: Procedure for deriving spin-free many-body residual equations. The pipeline begins with spin-orbital residuals from Wick&d, applies antisymmetrization, discards redundant spin sectors exploits singlet constraints and produces non-redundant spin-free equations, which are then passed to the remaining AGE toolchain for C++ code generation. The notation $n$h$m$p in the antisymmetrization step (2) denotes the excitation class producing $n$ holes and $m$ particles.
• Figure 3: Trans-stilbene molecular structure (hydrogen atoms in white, carbon atoms in gray).
• Figure 4: Execution time and parallel efficiency of a single iteration of various coupled cluster methods for trans-stilbene with a def2-TZVP basis and an active space of CAS(14, 14) for the RIC-MRCCSD method.
• Figure 5: Runtimes for various multireference methods for an all-E polyene series (2 to 14 carbon atoms) with corresponding active spaces defined by the conjugated $\pi$-system. Left: timings per iteration with the number of iterations to convergence indicated next to each bar (except for NEVPT2 which is non-iterative). Right: total runtimes for each method and the percentage of the total time taken for computing the 1-, 2- and 3-RDMs for RIC-MRCCSD (and the additional 4-RDM required for NEVPT2 and CEPA(0)). Note that for the largest polyene, CEPA(0) failed to converge and is therefore absent from the plots.